Engineering-Mathematics

Question 1
A class of 30 students occupy a classroom containing 5 rows of seats, with 8 seats in each row. If the students seat themselves at random, the probability that the sixth seat in the fifth row will be empty is
A
1/5
B
1/3
C
1/4
D
2/5
       Engineering-Mathematics       Probability       ISRO-2018
Question 1 Explanation: 
Step-1: Given, 5 rows with 8 seats in each row.
Total number of seats = 5*40
= 40 seats
Total number of students= 30
Step-2: Given constraint that, 6th seat in the fifth row is empty
When we are deleting 6th seat, 30 students have 39 choices of seats
Step-3: Total number of choices = 39C30
Total ways to choose = 40C30
Step-4: Final probability = 39C30 / 40C30
= 1/4
Question 2
The domain of the function log( log sin(x) ) is
A
0 < x < π
B
2nπ < x < (2n+1)π , for n in N
C
Empty set
D
None of the above
       Engineering-Mathematics       Calculus       ISRO-2018
Question 2 Explanation: 
→ The range of sinx value lies between -1 and 1 and whereas log(sinx) value will be the positive value.
→ So the domain of log(log sin(x)) is undefined which is empty Set.
Question 3
( G, *) is an abelian group. Then
A
x = x-1, for any x belonging to G
B
x = x2, for any x belonging to G
C
(x * y )2 = x2 * y2, for any x, y belonging to G
D
G is of finite order
       Engineering-Mathematics       Set-Theory       ISRO-2018
Question 3 Explanation: 
An abelian group is a commutative group. As per the above options, option C is the correct answer because it follows commutative property.
(x * y )2 = x2 * y2, for any x, y belonging to G
Question 4
A graph with n vertices and n-1 edges that is not a tree, is
A
Connected
B
Disconnected
C
Euler
D
A circuit
       Engineering-Mathematics       Graph-Theory       ISRO-2007
Question 4 Explanation: 
Consider a graph with two nodes(n1&n2) and number of edges are 1, There may be chance self edge with node n1 then graph is disconnected.
Consider the graph with three nodes(n1,n2&n3) and has 2 edges.
n1-->n2 and n2--->n1 then the graph is disconnected.
Question 5
If a graph requires k different colours for its proper colouring, then the chromatic number of the graph is
A
1
B
k
C
k-1
D
k/2
       Engineering-Mathematics       Graphs       ISRO-2007
Question 5 Explanation: 
The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color and if a graph requires k different colours for its proper colouring, then k is the chromatic number of the graph.
Question 6
Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls Note: taller (x, y) is true if x is taller than y.
A
(∃x)(boy(x) → (∀y)(girl(y) ∧ taller(x, y)))
B
(∃x)(boy(x) ∧ (∀y)(girl(y) ∧ taller(x, y)))
C
(∃x)(boy(x) → (∀y)(girl(y) → taller(x, y)))
D
(∃x)(boy(x) ∧ (∀y)(girl(y) → taller(x, y)))
       Engineering-Mathematics       Propositional-Logic       ISRO-2007
Question 6 Explanation: 
Don't confuse with '∧' and '→'
'∧' → predicts statements are always true, no matter the value of x.
'→' → predicts there is no need of left predicate to be true always, but whenever it becomes true, then right predicate must be true.
Option D:
There exists a some boys who are taller than of all girls y.
Question 7
Eigenvectors of

are
A
B
C
D
E
None Of the Above
       Engineering-Mathematics       Linear-algebra       ISRO-2007
Question 7 Explanation: 
Explanation: Excluded for evaluation
Question 8
The set of all Equivalence Classes of a set A of Cardinality C
A
is of cardinality 2c
B
have the same cardinality as A
C
forms a partition of A
D
is of cardinality C2
       Engineering-Mathematics       Set-Theory       ISRO-2007
Question 8 Explanation: 
The set of ll equivalence class of a set A of cardinality C forms a partition of A
Note: Ambiguous between answer with option D
Question 9
Company X shipped 5 computer chips, 1 of which was defective and company Y shipped 4 computer chips, 2 of which were defective. One computer chip is to be chosen uniformly at a random from the 9 chips shipped by the companies. If the chosen chip is found to be defective, what is the probability that the chip came from the company Y?
A
2/9
B
4/9
C
2/3
D
1/2
       Engineering-Mathematics       Probability       ISRO-2007
Question 9 Explanation: 
Probability that chip came chosen from company X = 5/9
Probability that chip came chosen from company Y = 4/9
Number of Defective chips from company X = 1
Number of Defective chips from company Y = 2
Probability that chip is defective from company X = 5/9 * 1/5
Probability that chip is defective from company Y = 4/9 * 2/4
Probability that chip is defective = 5/9 * 1/5 + 4/9 * 2/4
Given chip is defective, probability that it came from the company
Y = P(Defective Company Y)/ P(Defective)
= (4/9 * 2/4) / (5/9 * 1/5 + 4/9 * 2/4)
= 2/3
Question 10
A program consists of two modules executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by
A
f1(t) + f2(t)
B
t0 f1(x), f2(x) dx
C
t0 f1(x), f2(t-x) dx
D
max (f1(t), f2(t))
       Engineering-Mathematics       Probability       ISRO-2007
Question 10 Explanation: 
The two modules executed sequentially. The total runtime of the program is the sum of runtime of the two module. Probability density function of overall time taken
t0 f1(x), f2(t-x) dx
Question 11
Consider the following C code segment   Of the following, which best describes the growth of f(x) as a function of x?
A
Linear
B
Exponential
C
Quadratic
D
Cubic
       Engineering-Mathematics       Calculus       ISRO-2018
Question 11 Explanation: 
Question 12
The number of edges in a regular graph of degree d and n vertices is
A
maximum of n and d
B
n + d
C
nd
D
nd / 2
       Engineering-Mathematics       Graph-Theory       ISRO-2018
Question 12 Explanation: 
Sum of degree of vertices = 2 × no. of edges
d * n = 2 * |E|
∴ |E| = d*n/2
Question 13
If A is a skew-symmetric matrix, then AT
A
diagonal matrix
B
A
C
-A
D
0
       Engineering-Mathematics       Linear-Algebra       ISRO-2017 May
Question 13 Explanation: 
→ In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative, that is, it satisfies the condition.
→ If A is skew symmetric matrix then AT = -A
→ In terms of the entries of the matrix, if aij denotes the entry in the ith row and jth column, then the skew-symmetric condition is equivalent to
→ If A is skew symmetric matrix then aji=-aij
Question 14
If A and B be two arbitrary events, then
A
P(A∩B) = P(A)P(B)
B
P(A∪B) = P(A) + P(B)
C
P(A|B) = P(A ∩ B) + P(B)
D
P(A∪B) <= P(A) + P(B)
       Engineering-Mathematics       Probability       ISRO-2017 May
Question 14 Explanation: 
(A) Happens when A and B are independent.
(B) Happens when A and B are mutually exclusive.
(C) Not happens.
(D) P(A∪B) ≤ P(A) + P(B) is true because P(A∪B) = P(A) + P(B) - P(A∩B).
Question 15
Using Newton Raphson method, a root correct to 3 decimal places of the equation x3 – 3x – 5 = 0
A
2.222
B
2.275
C
2.279
D
None of the above
       Engineering-Mathematics       Newton-Raphson-Method       ISRO-2017 May
Question 15 Explanation: 
Here f(x)= x3-3x-5 and f1(x) = 3x2-3
The Newton-Raphson iterative formula is

Since x3 and x4 are identical upto 3 places of decima,we take x4=2.279 as the required root, correct to three places of the decimal.
Question 16
The symmetric difference of sets A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 6, 7, 8, 9} is
A
{1, 3, 5, 6, 7, 8}
B
{2, 4, 9}
C
{2, 4}
D
{1, 2, 3, 4, 5, 6, 7, 8, 9}
       Engineering-Mathematics       Sets-And Relation       ISRO-2017 May
Question 16 Explanation: 
Symmetrical Difference of A and B=(A–B)∪(B–A)

Question 17
Which one of the following Boolean expressions is NOT a tautology?
A
((a → b) ∧ (b → c)) → (a → c)
B
(a ↔ c) → ( ¬b → (a ∧ c))
C
(a ∧ b ∧ c) → (c ∨ a)
D
a → (b → a)
       Engineering-Mathematics       Propositional-Logic       ISRO-2017 May
Question 17 Explanation: 
Tautology: A universal truth in formal logic.
Note: Above (A↔C) →( ¬B→(A∧C)) is not tautology.
Question 18
If a square matrix A satisfies AAT = I, then the matrix, A is
A
Idempotent
B
Symmetric
C
Orthogonal
D
Hermitian
       Engineering-Mathematics       Linear-algebra       ISRO CS 2008
Question 18 Explanation: 

A square matrix A is an orthogonal matrix if its transpose is equal to its inverse. An orthogonal matrix A is necessarily invertible and unitary.

Conditions for an orthogonal matrix:

AT= A-1 and

A AT = AT A = I,

where I is an Identity matrix
Question 19
Consider the graph shown in the figure below: Which of the following is a valid strong component?
A
a, c, d
B
a, b, d
C
b, c, d
D
a, b, c
       Engineering-Mathematics       Graph-Theory       ISRO CS 2008
Question 19 Explanation: 
A directed graph is strongly connected if there is a path between all pairs of vertices. A strongly connected component (SCC) of a directed graph is a maximal strongly connected subgraph.
The graph has (a, b, c) as a strongly connected component.
Question 20
Suppose A is a finite set with n elements. The number of elements and the rank of the largest equivalence relation on A is
A
{n,1}
B
{n, n}
C
{n2, 1}
D
{1, n2}
       Engineering-Mathematics       Sets-And Relation       ISRO-2017 December
Question 20 Explanation: 
Let us assume a set with 4 elements
S={1,2,3,4}
→ If a set is said to be equivalence, then the set must be
i) Reflexive
ii) Symmetric
iii) Transitive
i) Reflexive Relation: A relation ‘R’ on a set ‘A’ is said to be reflexive if (xRx)∀x∈A.
A = {1,2,3}

R = {(1,1), (2,2), (3,3)}
R = {(1,1), (2,2)} It is false.
R = {(1,1), (2,2), (3,3), (1,2)}

ii) Symmetric Relation: A relation on a set A is said to be symmetric if (xRy). Then (yRx)∀x,y∈A i.e., if ordered pair (x,y)∈R. Then (y,x)∈R ∀x,y∈A.
A={1,2,3}
R1={(1,2), (2,1)}
R2={(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)}

Transitive Relation:
A relation ‘R’ on a set ‘A’ is said to be transitive if (xRy) and (yRz), then (xRz)∀(x,y,z∈A).
A={1,2,3}
R1={ }
R2={(1,1)}
R3={(1,2), (3,1)}
R4={(1,2), (2,1), (1,1)}
⇾ A={1,2,3,4}

Largest ordered set is
S×S={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}
⇒ Total = 16 = 42 = n2
Smallest ordered set = {(1,1),(2,2),(3,3),(4,4)}
⇒ Total=4=n
Note: In question, they are clearly mentioned that Rank of an Equivalence relation is equal to the number of induced Equivalence classes. Since we have maximum number of ordered pairs(which are reflexive, symmetric and transitive ) in largest Equivalence relation, its rank is always 1.
Question 21
Consider the set of integers I. Let D denote “divides with an integer quotient” (e.g. 4D8 but 4D7). Then D is
A
Reflexive, not symmetric, transitive
B
Not reflexive, not antisymmetric, transitive
C
Reflexive, antisymmetric, transitive
D
Not reflexive, not antisymmetric, not transitive
       Engineering-Mathematics       Set-Theory       ISRO-2017 December
Question 21 Explanation: 
Reflexive Relation:
A relation ‘R’ on a set ‘A’ is said to be reflexive if (xRx)∀x∈A.
Example: 4 D 8, 4 D 12, 4 D16, 4 D20…….(Here, D means divide)
8 D 16, 8 D 24……….
In this example, we didn’t get 4D4. So, it is not reflexive.
AntiSymmetric Relation:
For all x ∈ I, R(x,y) and R(y,x) then x=y is antisymmetric. We can easily make a violation as R(-2,2) and R(2,-2) are not antisymmetric.
It is violating. So, not antisymmetric relation.
Transitive relation:
A relation ‘R’ on a set ‘A’ is said to be transitive if (xRy) and (yRz), then (xRz)∀(x,y,z∈A).
Example: 4D8, 4D12, 4D16, 4D20…….(Here, D means divide)
8D16, 8D24……….
{ 4D8, 8D16, 1D16}. So, it is satisfied.
Question 22
A bag contains 19 red balls and 19 black balls. Two balls are removed at a time repeatedly and discarded if they are of the same colour, but if they are different, black ball is discarded and red ball is returned to the bag. The probability that this process will terminate with one red ball is
A
1
B
1/21
C
0
D
0.5
       Engineering-Mathematics       Probability       ISRO-2017 December
Question 22 Explanation: 
Given data,
Step-1: Bag contains 19 Red(R) and 19 Blue(B) balls.
BB (or) RR happen we are discarded.
If we get BR (or) RB then B is discarded and R is returned.
Step-2: There are some conditions that,
→ If black balls will either come with black then both black balls are discarder.
→ If it will come with red then only black balls will be discarded.
→ Suppose 2 red balls will come together means we are discarding both red balls.
Step-3: As per the above constraints, total 19 Red balls it means odd number.
→ Among 19 only 18 will be discarded.
Step-3: Final content of bag at second last trail will be either R,B (or) R,R,R and finally in last
trail bag will left with one red ball in both the cases.
Question 23
If x = -1 and x = 2 are extreme points of f(x) = α log |x| + βx2 + x then
A
α = -6, β = -1/2
B
α = 2, β = -1/2
C
α = 2, β = 1/2
D
α = -6, β =1/2
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 23 Explanation: 
Given data,
Step-1: x= -1 and x=2
f(x) = α log |x| + β x2 + x
f'(x)= α/x + 2βx + 1 = 0
Step-2: for extreme points f'(x)=0
α/x + 2βx + 1=0
Step-3: For x= -1 then we will get α+2β= 1 → (i)
For x= 2: then we will get α+8β= 2 → (ii)
from (i) and (ii) we can get the value of α=2 and β= -1/2
Question 24
Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the range of g(f(x)) then A ∩ B is
A
[-1, 0]
B
[-1, 0)
C
[-∞, 0]
D
[-∞,1]
       Engineering-Mathematics       Set-Theory       ISRO-2017 December
Question 24 Explanation: 
Given data,
Step-1: f(x) = log|x| and given range is [-∞ to +∞]
g(x) = sin(x) and given range is [-1,1]
Step-2: Given 2 variables are A and B
A= f(g(x))
= log|g(x)|
= log|sin(x)|
So, we will get A range is [-∞ ,0]
Step-3: B= g(f(x))
= sin(f(x))
= sin(log|x|)
So, we will get B range is [-1, 1]
Step-4: Common in both A and B is A∩B
A∩B = [-1, 0]
Key point: Ranges [ -1 ≤ sin(x) ≤ 1 and -∞ ≤ log|x| ≤ ∞ ]
Question 25
The proposition (P⇒Q)⋀(Q⇒P) is a
A
tautology
B
contradiction
C
contingency
D
absurdity
       Engineering-Mathematics       Propositional-Logic       ISRO-2017 December
Question 25 Explanation: 
A proposition that is neither a tautology nor a contradiction is called a contingency.
Question 26
If T(x) denotes x is a trigonometric function, P(x) denotes x is a periodic function and C(x) denotes x is a continuous function then the statement “It is not the case that some trigonometric functions are not periodic” can be logically represented as
A
¬∃(x) [ T(x) ⋀ ¬P(x) ]
B
¬∃(x) [ T(x) ⋁ ¬P(x) ]
C
¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]
D
¬∃(x) [ T(x) ⋀ P(x) ]
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 26 Explanation: 
Option A implies "It is not the case that some trigonometric functions are not periodic”.
Hence it is correct .
Option B implies "It is not the case that some are trigonometric functions or they are not periodic”.
Option C implies "It is not the case that some of not trigonometric functions are not periodic”.
Option D implies "It is not the case that some trigonometric functions are periodic”.
Question 27
The number of elements in the power set of { {1,2}, {2,1,1}, {2,1,1,2} } is
A
3
B
8
C
4
D
2
       Engineering-Mathematics       Set-Theory       ISRO-2017 December
Question 27 Explanation: 
Given data,
Step-1: Total number of elements in power set of given set with ‘n’ elements = 2n
Example: {1,2}
{{∅},{1},{2},{1,2}} → Total 4 possibilities.
Step-2: The given set in question contains 3 elements({1,2},{2,1,1}, {2,1,1,2} }. So, number of elements in power set of given set is 23 =8.
Step-3: The power set is {{∅},{X}} or {{∅},{1,2}}.
Question 28
The function f: [0,3]→[1,29] defined by f(x) = 2x3 – 15x2 + 36x + 1 is
A
injective and surjective
B
surjective but not injective
C
injective but not surjective
D
neither injective nor surjective
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 28 Explanation: 
Question 29
A
2/5
B
2
C
3
D
5/2
       Engineering-Mathematics       Vectors       ISRO-2017 December
Question 29 Explanation: 
Given data,
Step-1: It is clearly showing that two vectors are perpendicular. If two vectors are perpendicular
we are using dot product is zero. a.b =0
Step-2: Calculating dot product is “i is multiplied with i” and “j is multiplied with coefficient of j”
Step-3: we can write like this,
= (2i+λj+k).(i-2j+3k)=0
= 2-2λ+3 =0
-2λ = -5
λ=5/2
Question 30
If the two matrices have the same determinant, then the value of x is
A
1/2
B
√2
C
± 1/2
D
± 1/√2
       Engineering-Mathematics       Linear-algebra       ISRO CS 2008
Question 30 Explanation: 
Determinant of 1st Matrix = X2 – 1
Determinant of 2nd Matrix = X2– X
Since the determinants are equal:
X2 - 1 = X2 - 1.
X2 + X - 1 = 0
X = 1/2
Option (A) is correct
Question 31
If G is a graph with e edges and n vertices, the sum of the degrees of all vertices in G is
A
e
B
e/2
C
e2
D
2 e
       Engineering-Mathematics       Graph-Theory       ISRO CS 2009
Question 31 Explanation: 

Handshaking theorem states that the sum of degrees of the vertices of a graph is twice the number of edges.

If G=(V,E) be a graph with E edges,then-

Σ degG(V) = 2E

This theorem applies even if multiple edges and loops are present.

Since the given graph is undirected, every edge contributes as twice in sum of degrees. So the sum of degrees is 2e.
Question 32
Which of the following is true?
A
√3 + √7 = √10
B
√3 + √7 ≤ √10
C
√3 + √7 < √10
D
√3 + √7 > √10
       Engineering-Mathematics       Linear-Equation       ISRO-2016
Question 32 Explanation: 
√3 + √7=1.7320508075688772935274463415059 + 2.6457513110645905905016157536393
=4.3778021186334678840290620951451
√10=3.1622776601683793319988935444327
So, √3 + √7 > √10 is true.
Question 33
A given connected graph is a Euler Graph if and only if all vertices of are of
A
same degree
B
even degree
C
odd degree
D
different degree
       Engineering-Mathematics       Graph-Theory       ISRO-2016
Question 33 Explanation: 
A given connected graph is a Euler Graph if and only if all vertices of are of even degree.
Proof:
→ Let G(V, E) be an Euler graph. Thus G contains an Euler line Z, which is a closed walk.
→ Let this walk start and end at the vertex u ∈ V. Since each visit of Z to an intermediate vertex v of Z contributes two to the degree of v and since Z traverses each edge exactly once, d(v) is even for every such vertex.
→ Each intermediate visit to u contributes two to the degree of u, and also the initial and final edges of Z contribute one each to the degree of u. So the degree d(u) of u is also even.
Question 34
The maximum number of edges in a n-node undirected graph without self loops is
A
n2
B
n(n-1)/2
C
n-1
D
n(n+1)/2
       Engineering-Mathematics       Graph-Theory       ISRO-2016
Question 34 Explanation: 
The set of vertices has size n, the number of such subsets is given by the binomial coefficient C(n,2)⋅C(n,2)=n(n-1)/2.
Question 35
If the pdf of a Poisson distribution is given by f(x) = (e-22x)/x!, then its mean is
A
2x
B
2
C
-2
D
1
       Engineering-Mathematics       Poisson-Distribution       ISRO CS 2009
Question 35 Explanation: 
Given the function f(x) = (e-22x)/x!

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ.

Then, the Poisson probability is: P(x; μ) = (e) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

From the given function the “μ” value is 2 which is the mean.
Question 36
If the mean of a normal frequency distribution of 1000 items is 25 and its standard deviation is 2.5, then its maximum ordinate is
A
(1000/√2π).e-25
B
1000/√2π
C
(1000/√2π ).e-2.5
D
400/√2π
       Engineering-Mathematics       Normal-Distribution       ISRO CS 2009
Question 36 Explanation: 

Question 37
The cubic polynomial y(x) which takes the following values:
y(0) = 1,
y(1) = 0,
y(2) = 1,
y(3) = 10 is
A
x3 +2 x2 + 1
B
x3 + 3x2 + 1
C
x3 + 1
D
x3 – 2 x2 + 1
       Engineering-Mathematics       Functions       ISRO CS 2009
Question 37 Explanation: 
Substitute the values of x (0,1,2 and 3) in the options and check which option gives the corresponding y(x) values. The function x3 – 2 x2 + 1 gives the required values.
x = 0: y(x) = 1
x = 1: y(x) = 0
x = 2: y(x) = 8 - 8 + 1 = 1
x = 3: y(x) = 27 - 18 + 1 = 10
Question 38
x = a cos(t), y = b sin(t) is the parametric form of
A
Ellipse
B
Hyperbola
C
Circle
D
Parabola
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 38 Explanation: 
An ellipse can be defined as the locus of all points that satisfy the equations
x = a cos t
y = b sin t
where:
x,y are the coordinates of any point on the ellipse,
a, b are the radius on the x and y axes respectively,
t is the parameter, which ranges from 0 to 2π radians
Question 39
The value of x at which y is minimum for y = x2 − 3x + 1 is
A
-3/2
B
3/2
C
0
D
-5/4
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 39 Explanation: 
Given function is y = x2 – 3x + 1
To find the minimum value, calculate the derivative until at what point given function is minimum value.
Applying First order derivative to the function y is y’ = 2x – 3
Again applying derivative of y ’is y” = 2 (Since y” > 0, it has a minimum value)
So the minima value at that point is (2x – 3) = 0 and x = 3/2
Question 40
A graph in which all nodes are of equal degree, is known as
A
Multigraph
B
Non regular graph
C
Regular graph
D
Complete graph
       Engineering-Mathematics       Graph-Theory       ISRO CS 2009
Question 40 Explanation: 

Question 41
In a graph G there is one and only one path between every pair of vertices then G is a
A
Path
B
Walk
C
Tree
D
Circuit
       Engineering-Mathematics       Graph-Theory       ISRO CS 2009
Question 41 Explanation: 
A graph is a tree if and only if there is exactly one path between every pair of its vertices.

Let G be a graph and let there be exactly one path between every pair of vertices in G. So G is connected. Now G has no cycles, because if G contains a cycle, say between vertices u and v, then there are two distinct paths between u and v, which is a contradiction. Thus G is connected and is without cycles, therefore it is a tree.

A tree is a minimally connected graph i.e. removing a single edge will disconnect the graph. A tree with n vertices has n−1 edges and only one path exists between every pair of vertices.
Question 42
A simple graph ( a graph without parallel edge or loops) with n vertices and k components can have at most
A
n edges
B
n-k edges
C
(n − k)(n − k + 1) edges
D
(n − k)(n − k + 1)/2 edges
       Engineering-Mathematics       Graph-Theory       ISRO CS 2009
Question 42 Explanation: 
Let G be a graph with k components. Let ni be the number of vertices in the ith component, where 1 ≤ i ≤ k. Then, the number of edges in G is equal to sum of the edges in each of its components.

So, G has maximum number of edges if each component is a complete graph.

Hence, the maximum possible number of edges in the graph G is:

And in every case, (n − k)(n − k + 1)/2 will be greater than or equal to the above expression.

So, at maximum, there can be (n − k)(n − k + 1)/2 edges in a simple graph with n vertices and k components.
Question 43
Consider the polynomial, p(x) = a0 + a1X + a2X2 + a3X3 where ai ≠ 0, ∀i . The minimum number of multiplications needed to evaluate p on an input X is:
A
3
B
4
C
6
D
9
       Engineering-Mathematics       Functions       ISRO CS 2009
Question 43 Explanation: 
P(X)=a0+a1+a2X2+a3X3
=a0+X(a1+X(a2+a3X))
↓ ↓ ↓
③ ② ①
Here, minimum 3 multiplication needed
Question 44
A square matrix A is called orthogonal if A’A =
A
I
B
A
C
-A
D
-I
       Engineering-Mathematics       Linear-Algebra       ISRO CS 2009
Question 44 Explanation: 
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. AAT=ATA =I (Identity matrix)
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: AT=A-1
Question 45
If two adjacent rows of a determinant are interchanged, the value of the determinant
A
becomes zero
B
remains unaltered
C
becomes infinitive
D
becomes negative of its original value
       Engineering-Mathematics       Ls       ISRO CS 2009
Question 45 Explanation: 
If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
The value of the determinant remains unchanged if both rows and columns are interchanged.
Question 46
If A, B, C are any three matrices, then A’+ B’+ C’ is equal to
A
a null matrix
B
A+B+C
C
(A+B+C)’
D
-(A+B+C)
       Engineering-Mathematics       Linear-Algebra       ISRO CS 2009
Question 46 Explanation: 

Question 47
The shift operator E is defined as E[f(xi)] = f(xi + h) and E'[f(xi)] = f(xi – h) then △ (forward difference) in terms of E is
A
E-1
B
E
C
1 – E-1
D
1 – E
       Engineering-Mathematics       Functions       ISRO CS 2009
Question 47 Explanation: 
Question 48
The formula:

is called
A
Simpson rule
B
Trapezoidal rule
C
Romberg’s rule
D
Gregory’s formula
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 48 Explanation: 
The above formula is for trapezoidal rule.
Question 49
The image

A
Newton’s backward formula
B
Gauss forward formula
C
Gauss backward formula
D
Stirling’s formula
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 49 Explanation: 
The above formula is NEWTON’S GREGORY BACKWARD INTERPOLATION FORMULA
Question 50
A
2
B
3
C
4
D
5
       Engineering-Mathematics       ISRO CS 2009
Question 50 Explanation: 
From the given data is determinant = 3
The determinant of 2x2 matrix is (ad-bc), If the elements are a,b,c and d.
From the given matrix, a=3,b=3,c=x and d=5 then ad-bc =15-3x
15-3x=3 ⇒ 3x=12 ⇒ x=4
Question 51
A
779
B
679
C
0
D
256
       Engineering-Mathematics       Linear-Algebra       ISRO CS 2009
Question 51 Explanation: 
Determinant of any symmetric matrix is zero
Question 52
Let f(x) be the continuous probability density function of a random variable x, the probability that a < x ≤ b, is
A
f (b − a)
B
f (b) − f (a)
C
D
       Engineering-Mathematics       Probability       ISRO CS 2009
Question 52 Explanation: 
A non-discrete random variable X is said to be absolutely continuous, or simply continuous, if its distribution function may be represented as
Question 53
Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie down between
A
k and n
B
k-1 and k+1
C
k-1 and n-1
D
k+1 and n-k
       Engineering-Mathematics       Graph-Theory       ISRO CS 2009
Question 53 Explanation: 
→ While a vertex is removed from a graph then that can be itself be forms a new component. The minimum number of components is k-1.
→ If a vertex is removed then it results that all the components are also be disconnected. So removal can create (n-1) components
Question 54
A root of equation f(x) = 0 can be computed to any degree of accuracy if a ‘good’ initial approximation x0 is chosen for which
A
f (x0) > 0
B
f (x0) f (x0)” > 0
C
f (x0) f (x0)” < 0
D
f (x0)” > 0
       Engineering-Mathematics       Functions       ISRO CS 2009
Question 55
Which of the following statement is correct
A
△(UkVk) = Uk△Vk + Vk△Uk
B
△(UkVk) = Uk+1△Vk + Vk+1△Uk
C
△(UkVk) = Vk+1△Uk + Uk△Vk
D
△(UkVk) = Uk+1△Vk + Vk△Uk
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 56
If A and B are square matrices with same order and A is symmetric, then BTAB
A
Skew symmetric
B
Symmetric
C
Orthogonal
D
Idempotent
       Engineering-Mathematics       Linear-Algebra       ISRO CS 2011
Question 56 Explanation: 
For a Symmetric matrix, A’ = A
So, BTAB = B'.A.B
Taking transpose of B’.A.B
(B'.A.B)' = B'.A'.(B')' = B'.A.B // (B')' = B
So, it is a symmetric matrix.
Question 57
n-th derivative of xn is
A
n xn-1
B
nn . n!
C
nxn !
D
n!
       Engineering-Mathematics       Calculus       ISRO CS 2011
Question 57 Explanation: 

Question 58
Three coins are tossed simultaneously. The probability that they will fall two heads and one tail is
A
5/8
B
1/8
C
2/3
D
3/8
       Engineering-Mathematics       Probability       ISRO CS 2011
Question 58 Explanation: 
Three coins tossed means , total number of combinations(possibilities) are 23=8
The combinations are (HHH,HHT,HTH,HTT,TTT,TTH,THT,THH)
The number of combinations with two heads and one tail is HHT,HTH,THH
The the probability is the number of combinations of the event/ total combinations of the event = 3/8
Question 59
How many edges are there in a forest with v vertices and k components?
A
(v+1)−k
B
(v+1)/2 −k
C
v−k
D
v+k
       Engineering-Mathematics       Graph-theory       ISRO CS 2011
Question 59 Explanation: 
Method-1:
→ Suppose, if each vertex is a component, then k=v, then there will not be any edges among them. So, v-k= 0 edges.
Method-2:
→ According to pigeonhole principle, every component have v/k vertices.
→ Every component there will be (v/k)-1 edges.
→ Total k components and edges= k*((v/k)-1)
= v–k
Question 60
Which one of the following is true?
A
R ∩ S = (R ∪ S) − [(R − S) ∪ (S − R)]
B
R ∪ S = (R ∩ S) − [(R − S) ∪ (S − R)]
C
R ∩ S = (R ∪ S) − [(R − S) ∩ (S − R)]
D
R ∩ S = (R ∪ S) ∪ (R − S)
       Engineering-Mathematics       Set-Theory       ISRO CS 2011
Question 60 Explanation: 
This is direct formula for set theory.
R∩S = (R ∪ S) − [(R − S) ∪ (S − R)]
Question 61
What is the matrix that represents the rotation of an object by θ degree about the origin in 2D?
A
B
C
D
       Engineering-Mathematics       Linear-algebra       ISRO CS 2011
Question 61 Explanation: 
→The matrix representation of a counter-clockwise rotation by θ degrees about the origin.
Question 62
The number of edges in a ‘n’ vertex complete graph is ?
A
n * (n-1) / 2
B
n * (n+1) / 2
C
n2
D
n * (n+1)
       Engineering-Mathematics       Graph-theory       ISRO CS 2013
Question 62 Explanation: 
Complete graph is an undirected graph in which each vertex is connected to every vertex, other than itself. If there are ‘n’ vertices then total edges in a complete graph is n(n-1)/2
Question 63
The number of elements in the power set of the set {{A,B},C} is
A
7
B
8
C
3
D
4
       Engineering-Mathematics       Set-theory       ISRO CS 2013
Question 63 Explanation: 
For ‘n’ elements in a set, there are 2n elements in the corresponding power set.
Given set = {{A, B},C}
Power set = { {{A, B}}, {C}, {{A, B}, C}}, ϕ}
So, total 4 elements are present in the power set.
Question 64
Let P(E) denote the probability of the occurrence of event E. If P(A) = 0.5 and P(B) = 1, then the values of P(A/B) and P(B/A) respectively are
A
0.5, 0.25
B
0.25, 0.5
C
0.5, 1
D
1, 0.5
       Engineering-Mathematics       Probability       ISRO CS 2013
Question 64 Explanation: 
Given data is
→P(E) denote the probability of the occurrence of event E
→P(A) = 0.5 and P(B) = 1
→Conditional probability is a measure of the probability of an event (some particular situation occurring) given that another event has occurred.
→If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A | B), or PB(A)
→P(A/B) = P(A ∩ B)/P(B)
→If two events A and B are independent, then the probability of both events is the product of the probabilities for each event: P(A ∩ B) = P(A)P(B)
→P(A/B) = P(A) * P(B) / P(B)
→P(A/B) = 0.5
→Similarly, P(B/A) = P(A) * P(B) / P(A) and P(B/A) = 1
Question 65
What is the least value of the function f(x) = 2x2– 8x – 3 in the interval [ 0 , 5] ?
A
-15
B
7
C
-11
D
-3
       Engineering-Mathematics       Functions       ISRO CS 2013
Question 65 Explanation: 
One method is trial and error method
Substitute all the value of x from 0 to 5

The value of f(x) is -3 when x=0
The value of f(x) is -9 when x=1
The value of f(x) is -11 when x=2
The value of f(x) is -9 when x=3
The value of f(x) is -3 when x=4

The value of f(x) is -7 when x=5

so the least value is -11
Second method:
→We can solve this one by using derivatives.
→Given function is f(x) = 2x2 -8x -3
→ f'(x)=4x-8 (first derivative)
a f''(x)=4 (Second derivative)
Here we got constant value which means that it has minimal value at the point x=2
So we can find the minimum value by substituting value 2 in place of “x” in the given function.
Minimum value is 2⨉(2)2 -8⨉(2) -3= -11
Question 66
Let R be the radius of the circle. What is the angle subtended by an arc of length at the center of the circle?
A
1 degree
B
1 radian
C
90 degrees
D
π radians
       Engineering-Mathematics       Co-ordinate-Geometry       ISRO CS 2014
Question 66 Explanation: 
A degree (in full, a degree of arc, arc degree, or arc degree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.
It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.Because a full rotation equals 2π radians.
The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends;
Question 67
The number of bit strings of length 8 that will either start with 1 or end with 00 is?
A
32
B
128
C
160
D
192
       Engineering-Mathematics       Combinatorics       ISRO CS 2014
Question 67 Explanation: 
→ Number of bit strings of length 8 that start with 1: 27 = 128.
→ Number of bit strings of length 8 that end with 00: 26 = 64.
→ Number of bit strings of length 8 that start with 1 and end with 00: 25 = 32.
→ Applying the subtraction rule, the number is 128+64−32 = 160
Question 68
The conic section that is obtained when a right circular cone is cut through a plane that is parallel to the side of the cone is called
A
parabola
B
hyperbola
C
circle
D
ellipse
       Engineering-Mathematics       Geometry       ISRO CS 2014
Question 68 Explanation: 
Question 69
The rank of a matrix A =
A
0
B
1
C
2
D
3
       Engineering-Mathematics       Linear-Algebra       ISRO CS 2014
Question 69 Explanation: 
Given matrix
Question 70
What is the median of data if its mode is 15 and the mean is 30?
A
30
B
25
C
22.5
D
27.5
       Engineering-Mathematics       Probability       ISRO CS 2014
Question 70 Explanation: 
In a moderately symmetric distribution, the mean, median and mode are connected by the formula:
Mode = 3 Median – 2 Mean
15 = 3 Median - 2(30)
Median = 25
Question 71
Let A be a finite set having x elements and let B be a finite set having y elements. What is the number of distinct functions mapping B into A.
A
xy
B
2 (x+y)
C
yx
D
y! / (y-x)!
       Engineering-Mathematics       Sets-And Relation       ISRO CS 2014
Question 71 Explanation: 
A function on a set involves running the function on every element of the set A, each one producing some result in the set B. So, for the first run, every element of A gets mapped to an element in B. Take this example, mapping a 2 element set A, to a 3 element set B. There are 9 different ways, all beginning with both 1 and 2, that result in some different combination of mappings over to B.
Question 72
The probability that two friends are born in the same month is?
A
1/6
B
1/12
C
1/144
D
1/24
       Engineering-Mathematics       Probability       ISRO CS 2014
Question 72 Explanation: 
Probability of a person to be born in a one month out of 12 months is 1/12
Probability of both friends born in a same month is (1/12) * (1/12)
Suppose, they born in january = 1/12 * 1/12
Suppose, they born in february = 1/12 * 1/12
; ; Suppose, they born in December = 1/12 * 1/12
Probability that two friends are born in the same month is= 12*(1/12)*(1/12)
= 1/12
Question 73
A cube of side 1 unit is placed in such a way that the origin coincides with one of its top vertices and the three axes along three of its edges. What are the coordinates of the vertex which is diagonally opposite to the vertex whose coordinates are (1,0,1)?
A
(0, 0, 0)
B
(0, -1, 0)
C
(0, 1, 0)
D
(1, 1, 1)
       Engineering-Mathematics       Geometry       ISRO CS 2014
Question 73 Explanation: 
Question 74

To guarantee correction of upto t errors, the minimum Hamming distance dmin in a block code must be

A
t+1
B
t−2
C
2t−1
D
2t+1
       Engineering-Mathematics       Graph-Theory       UGC-NET JUNE Paper-2
Question 74 Explanation: 
For detect t-bit errors a hamming distance of t+1 bit is needed but to correct the t-bit error the hamming distance of 2t+1 bit is needed.
Question 75

Consider the following statements :

    (a) False╞ True
    (b) If α╞ (β ∧ γ) then α╞ β and α╞ γ.

Which of the following is correct with respect to the above statements ?

 
A
Both statement (a) and statement (b) are false.
B
Statement (a) is true but statement (b) is false.
C
Statement (a) is false but statement (b) is true.
D
Both statement (a) and statement (b) are true.
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 75 Explanation: 
(a) False╞ True is nothing but False=False|True (or) False=False V True

(b) α╞ (β ∧ γ) also write αV(β ∧ γ)
α╞ β and α╞ γ also write (αVβ)∧(αVγ)

Finally, we proved as LHS=RHS. αV(β ∧ γ) = (αVβ)∧(αVγ)
So, both the statements are correct.
Question 76

Consider the following English sentence :

“Agra and Gwalior are both in India”.

A student has written a logical sentence for the above English sentence in First-Order Logic using predicate In(x, y), which means x is in y, as follows :

 In(Agra, India) ⋁ In(Gwalior, India) 

Which one of the following is correct with respect to the above logical sentence ?

A
It is syntactically valid but does not express the meaning of the English sentence.
B
It is syntactically valid and expresses the meaning of the English sentence also.
C
It is syntactically invalid but expresses the meaning of the English sentence.
D
It is syntactically invalid and does not express the meaning of the English sentence.
       Engineering-Mathematics       First-order-logic       UGC-NET JUNE Paper-2
Question 76 Explanation: 
• In(Agra, India) means Agra is in india.
• In(Gwalior, India) means Gwalior is in india.
• In(Agra, India) ⋁ In(Gwalior, India), in this statement “” means “or” So the entire gives the meaning of Either Agra is in india or Gwalior is in india.
• The statement is not expressing the meaning of English sentence.
Question 77

Consider the following two sentences :

    (a) The planning graph data structure can be used to give a better heuristic for a planning problem.
    (b) Dropping negative effects from every action schema in a planning problem results in a relaxed problem.

Which of the following is correct with respect to the above sentences ?

A
Both sentence (a) and sentence (b) are false.
B
Both sentence (a) and sentence (b) are true.
C
Sentence (a) is true but sentence (b) is false.
D
Sentence (a) is false but sentence (b) is true.
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 77 Explanation: 
• Negative effects put restrictions on the action schema. When these restrictions are put into place, then the number of actions that can be taken to get to the next time step decreases because with each addition of a restriction, the actions that do not meet the restriction are filtered out.
• When these negative effects are dropped, then the number of actions increase and dropping all of the negative effects from the action schema results in a relaxed problem.
• A planning graph is a directed graph organized into levels: first a level S0 for the initial state, consisting of nodes representing each fluent that holds in S0; then a level A0 consisting of nodes for each ground action that might be applicable in S0; then alternating levels Si followed by Ai ; until we reach a termination condition.
• As a tool for generating accurate heuristics, we can view the planning graph as a relaxed problem that is efficiently solvable.
Question 78

A knowledge base contains just one sentence, ∃x AsHighAs (x, Everest). Consider the following two sentences obtained after applying existential instantiation.

    (a) AsHighAs (Everest, Everest)
    (b) AsHighAs (Kilimanjaro, Everest)

Which of the following is correct with respect to the above sentences ?

A
Both sentence (a) and sentence (b) are sound conclusions.
B
Both sentence (a) and sentence (b) are unsound conclusions.
C
Sentence (a) is sound but sentence (b) is unsound.
D
Sentence (a) is unsound but sentence (b) is sound.
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 78 Explanation: 
• The ∃x AsHighAs (x, Everest) means there is one element which as highest as Everest.
• In the statement (a) AsHighAs (Everest, Everest), both are Everest then we can’t compare.
• The statement (b) AsHighAs (Kilimanjaro, Everest) means there kilimanjaro which is as highest as Everest So this valid statement.Because we are comparing Kilimanjaro with Everest.
Question 79

Consider the set of all possible five-card poker hands dealt fairly from a standard deck of fifty-two cards. How many atomic events are there in the joint probability distribution ?

A
2,598,960
B
3,468,960
C
3,958,590
D
2,645,590
       Engineering-Mathematics       Probability       UGC-NET JUNE Paper-2
Question 79 Explanation: 
Joint probability distribution:
Given random variables X, Y, …, that are defined on a probability space, the joint probability distribution for X, Y, … is a probability distribution that gives the probability that each of X, Y, … falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution.
Given data,
- Total cards = 52
- poker hand = 5 cards of all possibilities
Step-1: This problem, we are simply finding combinations of 52C5
= C(n,r) = C(52,5)
= 52! / (5!(52-5)!)
= 2598960
Question 80

Which of the following statements is false about convex minimization problem ?

A
If a local minimum exists, then it is a global minimum
B
The set of all global minima is convex set
C
The set of all global minima is concave set
D
For each strictly convex function, if the function has a minimum, then the minimum is unique
       Engineering-Mathematics       Calculus       UGC-NET JUNE Paper-2
Question 80 Explanation: 
Properties of convex optimization problems:
1. Every local minimum is a global minimum
2. The optimal set is convex
3. If the objective function is strictly convex, then the problem has at most one optimal point.
These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.
Question 81

The following LPP

    Maximize z = 100x1 + 2x2 + 5x3
    Subject to 14x1 + x2 − 6x3 + 3x4 = 7
32x1 + x2 − 12x3 ≤ 10 
3x1 − x2 − x3 ≤ 0 
x, x2 , x3 , x4 ≥ 0 has 
A
Solution : x1 = 100, x2 = 0, x3 = 0
B
Unbounded solution
C
No solution
D
Solution : x1 = 50, x2 = 70, x3 = 60
       Engineering-Mathematics       Calculus       UGC-NET JUNE Paper-2
Question 81 Explanation: 
An unbounded solution of a linear programming problem is a situation where objective function is infinite.
A linear programming problem is said to have unbounded solution if its solution can be made infinitely large without violating any of its constraints in the problem.
Question 82

Digital data received from a sensor can fill up 0 to 32 buffers. Let the sample space be S = {0, 1, 2, .........., 32} where the sample j denote that j of the buffers are full and P(i) = 1/561 (33-i).

Let A denote the event that the even number of buffers are full. Then p(A) is:
A
0.515
B
0.785
C
0.758
D
0.485
       Engineering-Mathematics       Probability       UGC-NET JUNE Paper-2
Question 82 Explanation: 
• Probability of ith buffer getting full = p(i) = 1/562(33−i)
• Probability of all even number of buffers are full is P(A).

• We are going find the values of P(0),P(2),P(4),P(6) …. P(16).
• P(0) is 1/562 (33-0)
• P(2) is 1/562 (33-2)
• …
• P(16) is 1/562 (33-32)
• The probability of all even number of buffers are full is P(A) which is equal to P(0)+P(2)+P(4)+P(6)+ …. P(16).
• P(A) = 1/562(33+31+29+27+....+1)
• P(A) = 1562×289 = 0.51423
Question 83

The equivalence of ¬∃ x Q(x) is :

A
∃x ¬Q(x)
B
∀x ¬Q(x)
C
¬∃ x ¬Q(x)
D
∀x Q(x)
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 83 Explanation: 
We can write it as ¬∃x Q(x) = ∀x ¬Q(x).
Question 84
If Ai = {−i, ... −2,−1, 0, 1, 2, . . . . . i} then  is:
A
Z
B
Q
C
R
D
C
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 84 Explanation: 
• In mathematics, there are multiple number sets: the natural numbers N, the set of integers Z, all decimal numbers D, the set of rational numbers Q, the set of real numbers R and the set of complex numbers C.
Z number set:
• Z is the set of integers, ie. positive, negative or zero.
• Example: ..., -100, ..., -12, -11, -10, ..., -5, -4, -3, -2, - 1, 0, 1, 2, 3, 4, 5, ... 10, 11, 12, ..., 100, ...
• The set N is included in the set Z (because all natural numbers are part of the relative integers).
• The set A consists of Positive, negative and zero numbers.
Question 85

Match the following in List-I and List-II, for a function f :

       List-I                   List-II
(a) ∀x∀y(f(x)=f(y)⟶x=y)      (i) Constant
(b) ∀y∃x(f(x)=y)             (ii) Injective
(c) ∀xf(x)=k                (iii) Surjective 
   

                      

A
(a)-(i), (b)-(ii), (c)-(iii)
B
(a)-(iii), (b)-(ii), (c)-(i)
C
(a)-(ii), (b)-(i), (c)-(iii)
D
(a)- (ii), (b)-(iii), (c)-(i)
       Engineering-Mathematics       Propositional-Logic       UGC-NET JUNE Paper-2
Question 85 Explanation: 
• The function is injective (one-to-one) if each element of the codomain is mapped to by at most one element of the domain. An injective function is an injection. Notationally:
x, x' ∈ X, f(x) = f(x') ⇒ x = x'
Or, equivalently (using logical transposition),
x, x' ∈ X, x ≠ x' f(x) ≠ f(x')
• The function is surjective (onto) if each element of the codomain is mapped to by at least one element of the domain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection. Notationally:
∀y ∈ Y, ∃x ∈ X such that y = f(x)
A constant function is a function whose (output) value is the same for every input value.
For example, the function y(x) = 4 is a constant function because the value y(x) is 4 regardless of the input value x.
Question 86

Which of the relations on {0, 1, 2, 3} is an equivalence relation ?

A
{ (0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3) }
B
{ (0, 0) (1, 1) (2, 2) (3, 3) }
C
{ (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) }
D
{ (0, 0) (0, 2) (2, 3) (1, 1) (2, 2) }
       Engineering-Mathematics       Relations-and-Functions       UGC-NET JUNE Paper-2
Question 86 Explanation: 
→ A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.
1. Reflexivity: f(x) = f(x)
True, as given the same input, a function always produces the same output.
2. Symmetry: if f(x) = f(y) then f(y) = f(x)
True, by the definition of equality.
3. Transitivity: if f(x) = f(y) and f(y) = f(z) then f(x) = f(z)
True, by the definition of equality.
Option-2: { (0,0), (1,1), (2,2), (3,3) }
Has all the properties, thus, is an equivalence relation.
Option-1: { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }
Not reflexive: (1,1) is missing
Not transitive: (0,2) and (2,3) are in the relation, but not (0,3)
Option-3: { (0,0), (0,1) (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3) }
Not symmetric: (1,2) is present, but not (2,1).
Not transitive: (2,0) and (0,1) are in the relation, but not (2,1).
Option-4: Similarly, option-4 also not TRUE
Question 87

Which of the following is an equivalence relation on the set of all functions from Z to Z ?

A
{(f, g) | f(x) - g(x) = 1 ∀ x∈Z}
B
{(f, g) | f(0) = g(0) or f(1) = g(1)}
C
{(f, g) | f(0) = g(1) and f(1) = g(0)}
D
{(f, g) | f(x) - g(x) = k for some k∈Z}
       Engineering-Mathematics       Relations-and-Functions       UGC-NET JUNE Paper-2
Question 87 Explanation: 
An equivalence relation is a binary relation that is reflexive, symmetric and transitive.
The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:
a = a (reflexive property),
if a = b then b = a (symmetric property), and
if a = b and b = c then a = c (transitive property)
Question 88
____ number of undirected graphs can be constructed using V=(v1,v2,..vn).
A
n3
B
2n(n-1)/2
C
n-½
D
2(n-1)/2
       Engineering-Mathematics       Nielit Scentist-B [02-12-2018]
Question 88 Explanation: 
→ With n vertices no. of possible edges = nC2
→ Each subset of these edges will be form a graph.
→ Number of possible undirected graphs is 2(nC2) 2(n(n-1)/2)
Question 89

Which of the following statements is true ?

A
(Z, ≤) is not totally ordered
B
The set inclusion relation ⊆ is a partial ordering on the power set of a set S
C
(Z, ≠) is a poset
D
       Engineering-Mathematics       Set-Theory       UGC-NET JUNE Paper-2
Question 89 Explanation: 
Option 1: (Z, ≤)is false because

Option 2: This is TRUE, because the set inclusion relation ⊆ is a partial ordering on the power set of S.
Example: Suppose the set S = {a,b}, the maximum possibilities are {φ, a, b, ab}

Option 3: (z,≠)is false, because if violates reflexive relation.
Option 4: FALSE.
Question 90
Consider the set S={1,w,w​ 2​ }, where w and w​ 2​ are cube roots of unity. If * denotes the multiplication operation, the structure (S,*) forms:
A
A group
B
A ring
C
An integral domain
D
A field
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 4-12-2016
Question 90 Explanation: 
A Group is an algebraic structure which satisfies
1) closure
2) Associativity
3) Have Identity element
4) Invertible
Over ‘*’ operation the S = {1, ω, ω​ 2​ } satisfies the above properties.
The identity element is ‘1’ and inverse of 1 is 1, inverse of ‘w’ is 'w​ 2​ ' and inverse of 'w​ 2​ ' is 'w'.
Question 91
Two eigenvalues of a 3X3 real matrix P are (2+​ √ ​ -1) and 3. The determinant of P is___
A
0
B
1
C
15
D
-1
       Engineering-Mathematics       Linear-Algebra       Nielit Scientist-B IT 4-12-2016
Question 91 Explanation: 
If an eigenvalue of a matrix is a complex number, then there will be other eigenvalue, which is conjugate of the complex eigenvalue.
So, For the given 3×3 matrix there would be 3 eigenvalues.
Given eigenvalues are : 2+i and 3.
So the third eigenvalue should be 2-i.
As per the theorems, the determinant of the matrix is the product of the eigenvalues. So the determinant is (2+i)*(2-i)*3 = 15.
Question 92
What is the possible number of reflexive relation on a set of 5 elements?
A
210
B
215
C
220
D
225
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 4-12-2016
Question 92 Explanation: 
Let set = ‘A’ with ‘n’ elements,
Definition of Reflexive relation:
A relation ‘R’ is reflexive if it contains xRx ∀ x∈A
A relation with all diagonal elements, it can contain any combination of non-diagonal elements.
Eg:
A={1, 2, 3}

So for a relation to be reflexive, it should contain all diagonal elements. In addition to them, we can have possible combination of (n​ 2​ -n)non-diagonal elements (i.e., 2​ n2-n​ )
Ex:
{(1,1)(2,2)(3,3)} ----- ‘0’ non-diagonal element
{(1,1)(2,2)(3,3)(1,2)} ----- ‘1’ non-diagonal element
{(1,1)(2,2)(3,3)(1,2)(1,3)} “
___________ “
___________ “
{(1,1)(2,2)(3,3)(1,2)(1,3)(2,1)(2,3)(3,1)(3,2)} (n​ 2​ -n) diagonal elements
____________________
Total: 2​ (n2 -n)
For the given question n = 5.
The number of reflexive relations =2​ (25-5)​ =2​ 20
Question 93
Mala has a colouring book in which each english letter is drawn two times. She wants to point each of these 52 prints with one of k colours, such that the colour-pairs used to colour ay two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement?
A
9
B
8
C
7
D
6
       Engineering-Mathematics       Combinatorics       Nielit Scientist-B IT 4-12-2016
Question 93 Explanation: 
No. of letters from A-Z is = 26
Each is printed twice the no. of letters = 26×2 = 52
If Mala has k colours, she can have k pairs of same colours.
She also can have k​ C​ 2 different pairs in which each pair is having different colours.
So total no. of pairs that can be coloured = k+​ k ​ C​ 2
k+​ k​ C​ 2​ ≥ 26
k+k(k-1)/2 ≥ 26
k(k+1)/2 ≥ 26
k(k+1) ≥ 52
k(k+1) ≥ 7*8
k≥7
Question 94
Which of the following statements is/are TRUE for an undirected graph?
P: Number of odd degree vertices is even
Q: Sum of degrees of all vertices is even
A
P only
B
Q only
C
Both P and Q
D
Neither P nor Q
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 94 Explanation: 
Euler’s Theorem 3:
The sum of the degrees of all the vertices of a graph is an even number (exactly twice the number of edges).
In every graph, the number of vertices of odd degree must be even.
Question 95
​ In propositional logic, which of the following is equivalent to p → q?
A
~p → q
B
~p V q
C
~p V ~q
D
p → ~q
       Engineering-Mathematics       Propositional-Logic       Nielit Scientist-C 2016 march
Question 95 Explanation: 
We can use a truth table to determine if two compound propositions are logically equivalent, i.e if they always have the same truth values.
Question 96

Consider the vocabulary with only four propositions A, B, C and D. How many models are there for the following sentence?

(A∨B∨C∨D)
A
8
B
7
C
15
D
16
       Engineering-Mathematics       Propositional-Logic       UGC-NET DEC Paper-2
Question 96 Explanation: 
Here, number of models is nothing but number of TRUEs in final statement. In this proposition logic we got total 15 number of models.
Question 97
Consider the following graph L and find the bridges, if any.
A
No bridges
B
{d,e}
C
{c,d}
D
{c,d} and {c,f}
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 97 Explanation: 
A bridge, ut-edge, or cut arc is an edge of a graph whose deletion increases its number of connected components.
Equivalently, an edge is a bridge if and only if it is not contained in any cycle.
A graph is said to be bridgeless or isthmus-free if it contains no bridges.
If we remove {d,e} edge then there is no way to reach e and the graph is disconnected.
The removal of edges {c,d} and {c,f} makes graph disconnect but this forms a cycle.
Question 98
The following graph nas no Euler circuit because
A
It has 7 vertices
B
It is even-valent(all vertices have even valence)
C
It is not connected
D
It does not have a Euler circuit
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 98 Explanation: 
An Eulerian trail or Euler walk in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian.
Important Properties:
→ An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component.
→ An undirected graph can be decomposed into edge-disjoint cycles if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component.
→ An undirected graph has an Eulerian trail if and only if exactly zero or two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component.
→ A directed graph has an Eulerian cycle if and only if every vertex has equal in degree and out degree, and all of its vertices with nonzero degree belong to a single strongly connected component. Equivalently, a directed graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint directed cycles and all of its vertices with nonzero degree belong to a single strongly connected component.
→ A directed graph has an Eulerian trail if and only if at most one vertex has (out-degree) − (in-degree) = 1, at most one vertex has (in-degree) − (out-degree) = 1, every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph
Question 99
For the graph shown, which of the following paths is a Hamilton circuit?
A
ABCDCFDEFAEA
B
AEDCBAF
C
AEFDCBA
D
AFCDEBA
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 99 Explanation: 
A Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once. A graph possessing a Hamiltonian cycle is said to be a Hamiltonian graph.
Here, Option A: A,F and E are repeated several times.
Option B: It is not a cycle. It means, not closed walk
Option C: It is closed walk and all vertex traversed. So this is final answer.
Option D: It’s not a closed walk.
Question 100
If G is an undirected planar graph on n vertices with e edges then
A
e<=n
B
e<=2n
C
e<=en
D
None of the option
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 100 Explanation: 
Euler's formula: states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then
v − e + f = 2.
→ For a simple, connected, planar graph with v vertices and e edges and faces, the following simple conditions hold for v ≥ 3:
Theorem 1. e ≤ 3v − 6;
Theorem 2. If there are no cycles of length 3, then e ≤ 2v − 4.
Theorem 3. f ≤ 2v − 4.
Theorem (The handshaking theorem):
Let G be an undirected graph (or multigraph) with V vertices and N edges. Then

Exercise:
Suppose a simple graph has 15 edges, 3 vertices of degree 4, and all others of degree 3. How many vertices does the graph have?
3*4+(x-3)*3=30
Question 101
Choose the most appropriate definition of plane graph.
A
A simple graph which is isomorphic to hamiltonian graph
B
A graph drawn in a plane in such a way that if the vertex set of graph can be partitioned into two non-empty disjoint subset X and Y in such a way that each edge of G has one end in X and one end in Y
C
A graph drawn in a plane in such a way that any pair of edges meet only at their end vertices
D
None of the option
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 22-07-2017
Question 101 Explanation: 
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.
In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph.
Question 102
Which of the following propositions is tautology.
A
(p Vq) → q
B
p V(q → p)
C
pV(p → q)
D
Both B) and C)
       Engineering-Mathematics       Propositional-Logic       Nielit Scientist-B CS 22-07-2017
Question 102 Explanation: 
Question 103

The number of substrings that can be formed from string given by

“a d e f b g h n m p” is

A
10
B
45
C
56
D
55
       Engineering-Mathematics       Probability       UGC-NET DEC Paper-2
Question 103 Explanation: 
If we have no repetition in a string then the number of substrings can be found using the formula :
n*(n+1)/2 + 1
We have added 1 because it may include a NULL string also.
The number of substrings = 10*(11)/2 + 1
The number of substrings = 56
Question 104

Match the List 1 and List 2 and choose the correct answer from the code given below

 LIST 1			 LIST 2
(a) Equivalence		 (i) p ⇒ q
(b) Contrapositive     	(ii) p ⇒ q : q ⇒ p
(c) Converse	       (iii) p ⇒ q : ∼q ⇒ ∼p
(d) Implication		(iv) p ⇔ q
Code:
A
(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
B
(a)-(ii), (b)-(i),(c)-(iii), (d)-(iv)
C
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
D
(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
       Engineering-Mathematics       Propositional-Logic       UGC-NET DEC Paper-2
Question 104 Explanation: 
Propositions r and s are logically equivalent if the statement r ↔ s is a tautology.
∼q⇒ ∼p

According to above table,
equivalence means p ⇔ q
Contrapositive means p⇒ q : ∼q⇒ ∼p
Converse means p ⇒ q : q ⇒ p
Implication means p ⇔ q
Question 105
​ Which of the following is FALSE?
Read ⋀ as AND, ⋁ as OR, ~ as NOT, → as one way implication and ⬌ as two way implication?
A
((x→ y)⋀ x)->y
B
((~x→ y)⋀(~x⋀~y)) → x
C
(x→ (x ⋁ y))
D
((x ⋁ y) ⬌ (~x ⋁ ~y))
       Engineering-Mathematics       Propositional-Logic       Nielit Scientist-C 2016 march
Question 105 Explanation: 
Question 106
If f:{a,b}* → {a,b}* be given by f(n)=ax for every value of n∈{a,b}, then f is
A
{a,b,ab,aa}
B
{a,b,ba,bb}
C
{a,b}
D
{aa,ab,ba,bb}
       Engineering-Mathematics       Sets-And Relation       Nielit Scientist-C 2016 march
Question 106 Explanation: 
f:{a,b}* → {a,b}* be given f(n)=ax
→ In option B and D, "ba" is presented which is generated by f. So, we can conclude based on this, option B and D are false.
→ "ab" can be generated by f but which is not present in the option C. So, we can conclude option C is wrong.
→ Option A satisfying all the favorable cases which is generated by f.
Question 107

The K-coloring of an undirected graph G = (V, E) is a function

C: V ➝ {0, 1, ......, K-1} such that c(u)≠c(v) for every edge (u,v) ∈ E

Which of the following is not correct?

A
G has no cycles of odd length
B
G has cycle of odd length
C
G is 2-colorable
D
G is bipartite
       Engineering-Mathematics       Graph-Theory       UGC-NET DEC Paper-2
Question 107 Explanation: 
• A k-colouring of a graph G consists of k different colours and G is then called k-colourable.
• A cycle of length n ≥ 3 is 2-chromatic if n is even and 3-chromatic if n is odd.
• A graph is bi-colourable (2-chromatic) if and only if it has no odd cycles.
• A nonempty graph G is bicolourable if and only if G is bipartite
Question 108
Using Mid Point algorithm, the new coordinates of the point P(x,y) having slope ‘m’ and constant and ‘b’ is calculated using:
A
F(x,y)=mx+b-y
B
F(x,y)=mx-y+b
C
F(x,y)=mx+b
D
F(x,y)=mx-y+b-y
       Engineering-Mathematics       Nielit STA [02-12-2018]
Question 108 Explanation: 
In order to check this, we need to consider the implicit equation:
F(x,y) = mx + b - y
For positive m at any given X,
If y is on the line, then F(x, y) = 0
If y is above the line, then F(x, y) < 0
If y is below the line, then F(x, y) > 0
Question 109

If a graph (G) has no loops or parallel edges and if the number of vertices(n) in the graph is n≥3, then the graph G is Hamiltonian if

    (i) deg(v) ≥ n/3 for each vertex v
    (ii)
    deg(v) + deg(w) ≥ n whenever v and w are not connected by an edge.
    (iii) E(G) ≥ 1/3(n-1)(n-2) + 2

Choose the correct answer for the code given below:

Code:
A
(i) and (iii) only
B
(ii) and (iii) only
C
(iii) only
D
(ii) only
       Engineering-Mathematics       Graph-Theory       UGC-NET DEC Paper-2
Question 109 Explanation: 
Option-2 is correct.
→ Dirac's theorem on Hamiltonian cycles, the statement that an n-vertex graph in which each vertex has degree at least n/2 must have a Hamiltonian cycle.
→ Dirac's theorem on chordal graphs, the characterization of chordal graphs as graphs in which all minimal separators are cliques.
→ Dirac's theorem on cycles in k-connected graphs, the result that for every set of k vertices in a k-vertex-connected graph there exists a cycle that passes through all the vertices in the set.
Question 110
The correct order of the degrees of vertices that would form the connected graph as eulerian is:
A
1,2,3
B
2,3,4
C
2,4,5
D
1,3,5
       Engineering-Mathematics       Nielit STA [02-12-2018]
Question 110 Explanation: 
→ Suppose a graph has n vertices with degrees d1,d2,d3,...,dn. Add together all degrees to get a new number d1 + d2 + d3 + ...+ dn = Dv. Then Dv =2e.
→ In words, for any graph the sum of the degrees of the vertices equals twice the number of edges. Stated in a slightly different way, Dv =2e says that Dv is ALWAYS an even number
→ 1+2+3=6
Question 111
A planar graph is defined as:
A
A graph drawn in a plane in such a way that any pair of edges meet only at their end vertices
B
A graph drawn in a plane in such a way that if the vertex set of graph can be partitioned into two non-empty disjoint subset X and Y in such a way that each edge of G has one end in X and one end in Y.
C
A simple graph which is Isomorphism to Hamiltonian graph.
D
A function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices
       Engineering-Mathematics       Nielit STA [02-12-2018]
Question 111 Explanation: 
→ In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.
→ In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph.
→ A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Question 112
A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20 The minimum degree it can have is
A
1
B
2
C
3
D
4
       Engineering-Mathematics       Calculus       Nielit Scientist-B CS 2016 march
Question 112 Explanation: 
Let's take p(x) = ax + b
p(0) = 5 ⇒ b = 5
p(1) = 4 ⇒ a+b = 4 ⇒ a = -1
p(2) = 9 ⇒ 40+b = 9 ⇒ -4+5 = 9, which is false.
So degree 1 is not possible.
Let's take p(x) = ax​ 2​ + bx +c
p(0) = 5 ⇒ c = 5
p(1) = 4 ⇒ a+b+c = 4 ⇒ a+b = -1 -----(1)
p(2) = 9 ⇒ 4a+2b+c = 9 ⇒ 2a+b = 2 -----(2)
(2) - (1)
⇒ a = 3, b = -1-1 = -4
p(3) = 20 ⇒ 9a+3b+c = 20
⇒ 27-12+5 = 20
⇒ 20 = 20, True
Hence, minimum degree it can have.
Question 113
Consider an undirected random graph of 16 vertices. The probability that there is an edge between a pair of vertices is ½. What is the expected number of unordered cycles of length three?
A
70
B
280
C
120
D
45
       Engineering-Mathematics       Nielit STA [02-12-2018]
Question 113 Explanation: 
Among available ‘16’ vertices, we need to identify the cycles of length ‘3’.

So, the total probability that all three edges of the above exists
= 1/2 × 1/2 × 1/2 (as they are independent events)
= 1/8
Total number of ways in which we can select ‘3’ such vertices among ‘16’ vertices = 16C3 = 560
Total number of cycles of length ‘3’ out of 16 vertices = 560 × 1/8 = 70
Question 114
If S be an infinite set and S​ 1​ ..., S​ n​ be sets such that S​ 1​ U S​ 2​ U...U S​ n​ =S, then
A
At least one of the set S​ i​ is a finite set
B
not more than one of the sets S​ i​ can be finite
C
At least one of the sets S​ i​ is an infinite set
D
not more than one of the sets S​ i​ can be infinite
       Engineering-Mathematics       Sets-And Relation       Nielit Scientist-B CS 2016 march
Question 114 Explanation: 
Given sets are finite union of sets. One set must be infinite to make whole thing to be infinite.
Question 115

A survey has been conducted on methods of commuter travel. Each respondent was asked to check Bus, Train or Automobile as a major methods of travelling to work. More than one answer was permitted. The results reported were as follows :

Bus 30 people; Train 35 people; Automobile 100 people; Bus and Train 15 people; Bus and Automobile 15 people; Train and Automobile 20 people; and all the three methods 5 people. How many people complete the survey form ?

A
160
B
120
C
165
D
115
       Engineering-Mathematics       Probability       UGC-NET DEC Paper-2
Question 116

A full joint distribution for the Toothache, Cavity and Catch is given in the table below:

Which is the probability of Cavity, given evidence of Toothache ?

A
< 0.2, 0.8 >
B
< 0.6, 0.8 >
C
< 0.4, 0.8 >
D
< 0.6, 0.4 >
       Engineering-Mathematics       Probability       UGC-NET DEC Paper-2
Question 117
The number of independent loops for a network with n nodes and b branch is
A
n-1
B
b-n
C
b-n+1
D
independent of "the number of nodes
       Engineering-Mathematics       Graph-Theory       NieLit STA 2016 March 2016
Question 117 Explanation: 
● A loop is a closed path in a circuit. Loop counts starting at a node passing through a set of nodes and returning to the starting node without passing through any node more than once.
● A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop.
● A network with ‘b’ branches, ‘n’ nodes and ‘L’ independent loops will satisfy the fundamental theorem of network topology.
b=L+n−1
L=b−n+1
Question 118
Which of the following is/are tautology?
A
a V b → b ⋀ c
B
a ⋀ b → b V c
C
a V b → (b → c)
D
a → b → (b → c)
       Engineering-Mathematics       Propositional-Logic       NieLit STA 2016 March 2016
Question 118 Explanation: 

Question 119
The number of circuits in a tree with 'n' nodes is
A
Zero
B
One
C
n-1
D
n/2
       Engineering-Mathematics       Graph-Theory       NieLit STA 2016 March 2016
Question 119 Explanation: 
● A tree (e.g. a binary tree) has a root, branches (nodes), and leaf nodes.
● A circuit in such a tree is impossible (unless you have some other data structure as well such as a linked list or you just program the tree so badly it ends up with circuits).
Question 120
A complete graph with "n" vertices is
A
2-chromatic
B
(n/2) chromatic
C
(n-1) chromatic
D
n-Chromatic
       Engineering-Mathematics       Graph-Theory       NieLit STA 2016 March 2016
Question 120 Explanation: 

Question 121
Minimum number of colours required to colour the vertices of a cycle with n nodes in such a way that no two adjacent nodes have the same colour is
A
2
B
3
C
4
D
n-2⌈(n/2)⌉+2
       Engineering-Mathematics       Graph-Theory       NieLit STA 2016 March 2016
Question 121 Explanation: 
We need 3 colors to color a odd cycle and 2 colors to color an even cycle.
Question 122

In mathematical logic, which of the following are statements ?

    (i)  There will be snow in January
    (ii)  What is the time now ?
    (iii)  Today is Sunday
    (iv)  You must study Discrete Mathematics.

Choose the correct answer from the code given below :

Code :
A
(iii) and (iv)
B
(i) and (ii)
C
(i) and (iii)
D
(ii) and (iv)
       Engineering-Mathematics       Propositional-Logic       UGC-NET DEC Paper-2
Question 122 Explanation: 
In mathematical logic, the term statement is variously understood to mean either:
(a) a meaningful declarative sentence that is true or false, or
(b) the assertion that is made by a true or false declarative sentence.
From the above four statements, statement 2 and 4 wont give meaning like true or false answers and statements 1 and 3 will give either true or false answers.
Question 123

Consider the statements below :

“There is a country that borders both India and Nepal.“

Which of the following represents the above sentence correctly ?

A
∃c Border(Country(c), India ∧ Nepal)
B
∃c Country(c) ∧ Border(c, India) ∧ Border(c, Nepal)
C
[∃c Country(c)] ⇒ [Border(c,India) ∧ Border(c, Nepal)]
D
∃c Country(c) ⇒ [ Border(c, India) ∧ Border(c, Nepal)]
       Engineering-Mathematics       Propositional-Logic       UGC-NET DEC Paper-2
Question 123 Explanation: 
→ ∃c Country(c) which represents there is one country “c”
→ Border(c, India) which represents border between c and india
→ Border(c, Nepal) which represents border between c and Nepal
→ “∧” represents both
Option-2 represents the sentence “There is a country that borders both India and Nepal".
Question 124
(PVQ) ​ ⋀ ​ (P → R) ​ ⋀ ​ (Q → R) is equivalent to
A
P
B
Q
C
R
D
True≅T
       Engineering-Mathematics       Propositional-Logic       NieLit STA 2016 March 2016
Question 124 Explanation: 
Question 125

Use Dual Simplex Method to solve the following problem :

    Maximize z = -2x1 - 3x2
    subject to:
    x1 + x2 ≥ 2
    2x1 + x2 ≤ 10
    x1 + x2 ≤ 8
    x1, x2 ≥ 0
A
x1 = 6, x2 = 2 and z = -18
B
x1 = 2, x2 = 6 and z = -22
C
x1 = 2, x2 = 0 and z = -4
D
x1 = 0, x2 = 2 and z = -6
       Engineering-Mathematics       Dual-simplex-method       UGC-NET DEC Paper-2
Question 125 Explanation: 
→ One simple method is substitute the option values in the maximize equation Z to know the maximum value of the problem and remaining equations to satisfies the conditions which are given in the problem statement.
→ Option 3 satisfies all conditions and maximize property in the given problem
Question 126

A box contains six red balls and four green balls. Four balls are selected at random from the box. What is the probability that two of the selected balls will be red and two will be green ?

A
1/35
B
1/14
C
1/9
D
3/7
       Engineering-Mathematics       Probabilty       UGC-NET DEC Paper-2
Question 126 Explanation: 
→ Total red balls are 6
→ Green balls are 4
→ Total 10 balls in the box. We need to select 4 balls from 10 balls in which two red balls out of 6 red balls and two green balls from 4 green balls.
→ Total of number of ways for selecting 4 balls out of 10 balls is C(10,4)
→ Number of ways for selecting two red balls from 6 and two green balls from 4 is C(6,2)*C(4,2)
→ Probability of selecting two balls is number of ways for selecting four balls / total number of ways P = C(6,2)*C(4,2) / C(10,4)
= C(6,2) is 6x5/2 = 15
= C(4,2) is 4x3/2 = 6
= C(10,4) is (10x9x8x7) / (4x3x2x1) = 210
P = C(6,2)*C(4,2) / C(10,4)
= 15x6/210
= 3/7, So option 4 is correct
Question 127
Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to:
A
3
B
4
C
5
D
6
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B CS 4-12-2016
Question 127 Explanation: 
v - e + f = 2
‘v’ is number of vertices and ‘e’ is number of edges
‘f’ is number of faces including bounded and unbounded
10 - 15 + f = 2
f = 7
There is always one unbounded face, so the number of bounded faces = 6
Question 128
Let A,B,C,D be nxn matrices, each with non-zero determinant. If ABCD=1, then B(-1) is:
A
D(-1)C(-1)A(-1)
B
CDA
C
ADC
D
Does not necessarily exist
       Engineering-Mathematics       Linear-Algebra       Nielit Scientist-B CS 4-12-2016
Question 128 Explanation: 
Given
ABCD = I
Multiply LHS, RHS by A −1
A −1 ABCD = A −1 I (position of A −1 on both sides should be left)
⇒ B CD = A −1
⇒ BCDD −1 = A −1 D −1
⇒ B C = A −1 D −1 ⇒BCC −1 = A −1D −1C −1
⇒ B = A −1 D −1 C −1
⇒ B = A −1 D −1 C −1
Now, B −1 = (A −1 D −1 C −1 )−1
B −1 = C DA
Question 129
Consider the function f(x)=sin(x) in the interval [​ π ​ /4, 7​ π ​ /4]. The number and location(s) of the minima of this function are:
A
One, at π/2
B
One, at 3​ π ​ /2
C
Two, at ​ π ​ /2 and 3​ π ​ /2
D
Two, at ​ π ​ /4 and 3​ π ​ /2
       Engineering-Mathematics       Calculus       Nielit Scientist-B CS 4-12-2016
Question 129 Explanation: 
The local minima is at x =3π/2
This is very obvious from the graph of f (x) = sin x
On a second look at the graph below, I believe x =π/4
is also a local minimum. This is because it is lesser than all other values within its locality.
Thus we have two local minima: x = π /4 , 3π
Question 130
Which one of the following is NOT necessarily a property of a group?
A
Commutativity
B
Associativity
C
Existence of inverse for every element
D
Existence of identity
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B CS 4-12-2016
Question 130 Explanation: 
The axioms (basic rules) for a group are:
1. CLOSURE: If a and b are in the group then a • b is also in the group.
2. ASSOCIATIVITY: If a, b and c are in the group then (a • b) • c = a • (b • c).
3. IDENTITY: There is an element e of the group such that for any element a of the group
a • e = e • a = a.
4. INVERSES: For any element a of the group there is an element a​ -1​ such that ○ a • a​ -1​ = e
and
○ a​ -1​ • a = e
Question 131
How many onto(or surjective) functions are there from an n-element(n>=2) set to a 2-element set?
A
2​ n
B
2​ n -1
C
2​ n​ -2
D
2(2​ n​ -2)
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B CS 4-12-2016
Question 131 Explanation: 
The number of onto functions from set of m elements to set of n elements, if m>n is
n​ m​ – (2​ n​ – 2)
i.e., 2​ n​ – (2​ 2​ – 2) = 2​ n​ – 2
If there are 'm' elements in set A, 'n' elements in set B then
The number of functions are : n​ m The number of injective or one-one functions are n​ ​ P​ m
The number of surjective functions are:
If m If m>n, then n! * m​ ​ C​ n
Given that m=n, n=2
2! * n​ ​ C​ 2
Question 132
Given an undirected graph G with V vertices and E edges, the sum of the degrees of all vertices is
A
E
B
2E
C
V
D
2V
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 132 Explanation: 
Theorem (Sum of Degrees of Vertices Theorem):
Suppose a graph has n vertices with degrees d 1 , d 2 , d 3 , ..., d n.
Add together all degrees to get a new number
d 1 + d 2 + d 3 + . .. + d n = D v . Then D v = 2 e .
In words, for any graph the sum of the degrees of the vertices equals twice the number of edges.
Question 133
A path in graph G, which contains every vertex of G and only Once?
A
Euler circuit
B
Hamiltonian path
C
Euler Path
D
Hamiltonian Circuit
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 133 Explanation: 
● Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. If a Hamiltonian path exists whose endpoints are adjacent, then the resulting graph cycle is called a Hamiltonian cycle (or Hamiltonian cycle)
● An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once.
● An Euler path starts and ends at different vertices.
● An Euler circuit starts and ends at the same vertex.
Question 134
Considering the following graph, which one of the following set of edged represents all the bridges of the given graph?
A
(a,b)(e,f)
B
(a,b),(a,c)
C
(c,d)(d,h)
D
(a,b)
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 134 Explanation: 
● A bridge ,cut-edge, or cut arc is an edge of a graph whose deletion increases its number of connected components.
● Equivalently, an edge is a bridge if and only if it is not contained in any cycle.
● The removal of edges (a,b) and (e,f) makes graph disconnected.
Question 135
Which of the following statements is/are TRUE?
S1: The existence of an Euler circuit implies that an euler path exists.
S2: The existence of an Euler path implies that an Euler circuit exists.
A
S1 is true
B
S2 is true
C
S1 and S2 both are true
D
S1 and S2 both are false
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 135 Explanation: 
An Euler circuit in a graph G is a simple circuit containing every edge of G exactly once
An Euler path in G is a simple path containing every edge of G exactly once.
An Euler path starts and ends at different vertices.
An Euler circuit starts and ends at the same vertex.
Question 136
A connected planar graph divides the plane into a number of regions. If the graph has eight vertices and these are linked by 13 edges, then the number of regions is:
A
5
B
6
C
7
D
8
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 136 Explanation: 
Use Euler's formula ,V−E+R=2
Where V is the number of vertices, E is the number of edges, and R is the number of regions.
R=2-V+E=2-8+13=7
Question 137
Power set of empty set has exactly__subset
A
One
B
Two
C
Zero
D
Three
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 137 Explanation: 
● The collection of all subsets of a set A is called the power set of A.
● The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero
● Common notations for the empty set include "{}", " ", and "∅".
Question 138
What is the cartesian product of A={1,2} and B={a,b}?
A
{(1,a),(1,b),(2,a),(b,b)}
B
{(1,1),(2,2),(a,a),(b,b)}
C
{(1,a),(2,a),(1,b),(2,b)}
D
{(1,1),(a,a),(2,a),(1,b)}
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 138 Explanation: 
● A Cartesian product is a ​ mathematical operation​ that returns a ​ set​ (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ​ ordered pairs​ (a, b) where a ∈ A and b ∈ B.
● In the question, Set A consists of two elements and Set B consists of two elements. So the total ordered pairs are four.
● Each ordered pair consists of one element from Set-A and another element from Set-B.
Question 139
What is the cardinality of the power set of the set {0,1,2}?
A
8
B
6
C
7
D
9
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 139 Explanation: 
The cardinality of a set is defined as the total number of distinct items in that set and power set is defined as the set of all subsets of a set.
Power set consists of ​ {}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}
So power set consists of all unique items then cardinality is 8
Question 140
Let G be a simple connected planar graph with 13 vertices and 19 edges. then the number of faces in the planar embedding of the graph is
A
6
B
8
C
9
D
13
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 140 Explanation: 
● Use Euler's formula ,V−E+R=2
● Where V is the number of vertices, E is the number of edges, and R is the number of regions.
● R=2-V+E=2-13+19=8
● We can also term region as face
Question 141
Which of the following statements is false?
A
(P⋀Q)V(~P⋀Q)V(P⋀~Q) is equal to ~Q⋀~P
B
(P⋀Q)V(~P⋀Q)V(P⋀~Q) is equal to QVP
C
(P⋀Q)V(~P⋀Q)V(P⋀~Q) is equal to QV(P⋀~q)
D
(P⋀Q)V(~P⋀Q)V(P⋀~Q) is equal to PV(Q⋀~p)
       Engineering-Mathematics       Propositional-Logic       Nielit Scientist-B IT 22-07-2017
Question 141 Explanation: 
One simple method is, by using truth table we can find the statement is true or not.

The last two columns of the above table are different. So option A is false.
Question 142
There are four bus lines between A and B; And three bus lines between B and C The number of way a person roundtrip by bus from A to C by way of B will be
A
12
B
7
C
144
D
264
       Engineering-Mathematics       Combinatorics       Nielit Scientist-B IT 22-07-2017
Question 142 Explanation: 
The number of bus lines between a and b =4
The number of bus lines between b and c =3
The number of possible combinations between a and c going through b are 4 * 3 = 12.
if you're talking about round trip, he has 12 possible ways to get there and 12 possible ways to get back, so the total possible ways is 12 * 12 = 144.
Question 143
The number of diagonals that can be drawn by joining the vertices of an octagon is
A
28
B
48
C
20
D
None of the option
       Engineering-Mathematics       Graph-Theory       Nielit Scientist-B IT 22-07-2017
Question 143 Explanation: 
Octagon consists of the 8 vertices and we can draw 5 diagonals

So, we can construct 5*8 = 40 diagonals.
But we have constructed each diagonal twice, once from each of its ends. Thus there are 20 diagonals in a regular octagon.
Question 144
A partial ordered relation is transitive, reflexive and
A
Antisymmetric
B
bisymmetric
C
anti reflexive
D
Asymmetric
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 144 Explanation: 
● Let R be a binary relation on a set A.
● R is antisymmetric if for all x,y A, if xRy and yRx, then x=y.
● R is a partial order relation if R is reflexive, antisymmetric and transitive.
Question 145
If B is a Boolean algebra, then which of the following is true?
A
B is a finite but not complemented lattice
B
B is a finite, Complemented and distributive lattice
C
B is a finite,distributive but not complemented lattice
D
B is not distributive lattice
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 145 Explanation: 
Distributive property of boolean algebra
(i)a.(b + c) =a.b + a.c
(ii) a+(b.c) = (a + b).(a + c)
Boolean algebra properites:

Question 146
If A and B are two sets and A U B = A ​ ∩ ​ B then
A
A=​ ∅
B
B=​ ∅
C
A != B
D
A=B
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 146 Explanation: 
● For example, Set A={1,2,3} and Set B={1,2,3}
● AUB={1,2,3}
● A ∩ B ={1,2,3}
● If two sets consists of same elements then ​ A U B = A ​ ∩ ​ B
Question 147
The relation {(1,2),(1,3)(3,1),(1,1),(3,3),(3,2),(1,4),(4,2),(3,4)} is
A
Reflexive
B
Transitive
C
Symmetric
D
Asymmetric
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 147 Explanation: 
Given set consists of elements a,b, and c.
The following conditions need to satisfy for each property.
a ​ = ​ a ​ (reflexive property),
if ​ a ​ = ​ b ​ then ​ b ​ = a ​ (symmetric property), and
if ​ a ​ = ​ b ​ and ​ b ​ = ​ c ​ then ​ a ​ = c ​ (transitive property).
an asymmetric relation is a binary relation on a set X where:For all a and b in X, if a is related to b, then b is not related to a
The above relation is not reflexive because there is no ordered pairs (2,2) and (4,4)
The above relation is not Symmetric because if (1,2) present means the relation should consists of (2,1) but there is no such ordered pair in the relation.
Asymmetric property also invalid because of (1,3) and (3,1) ordered pairs.
Question 148
The probability that top and bottom cards of a randomly shuffled deck are both aces is:
A
4/52 X 4/52
B
4/52 X 3/52
C
4/52 X 3/51
D
4/52 X 4/51
       Engineering-Mathematics       Probability       Nielit Scientist-B IT 22-07-2017
Question 148 Explanation: 
E​ 1​ : First card being ace
E​ 2​ : Last card being ace
Note that E​ 1​ and E​ 2​ are dependent events, i.e., probability of last card being ace if first is ace will be lesser than the probability of last card being ace if first card is not ace.
So, probability of first card being ace = 4/52
Probability of last card being ace given that first card is ace is,
P(E​ 2​ / E​ 1​ ) = 3/51
∴ P(E​ 1​ and E​ 2​ ) = P(E​ 1​ ) ⋅ P(E​ 2​ / E​ 1​ ) = 4/52 * 3/51
Question 149

Let P, Q and R be three atomic prepositional assertions, and

    X : (P ∨ Q) → R
    Y : (P → R) ∨ (Q → R)

Which one of the following is a tautology?

A
X → Y
B
Y → X
C
X ≣ Y
D
~Y → X
       Engineering-Mathematics       Propositional-Logic       JT(IT) 2018 PART-B Computer Science
Question 149 Explanation: 
Question 150

For what values of k, the points(-k+1, 2k),(k, 2-2K) and (-4-k, 6-2k) are collinear?

A
0, 1
B
-1, 1
C
-1, 1/2
D
1/2, -1/2
       Engineering-Mathematics       Vectors       JT(IT) 2018 PART-B Computer Science
Question 150 Explanation: 
There is a restrictions for 3 points to be colLinear and that is
1/2[X1(Y2 - Y3) + X2(Y3 - Y1) + X3(Y1 - Y2)] = 0
Here, X1 = -k + 1, Y1 = 2k, X2 = k, Y2 = 2 – 2k, X3 = -4 - k, Y3 = 6 – 2k
1/2[-k + 1(2 – 2k - 6 + 2k) + k(6 – 2k - 2k) - 4 -k(2k - 2 + 2k)] = 0
1/2[-k + 1(-4) + k(6 - 4k) -4 - k(4k - 2)] = 0
1/2[4k - 4 + 6k - 4k2 - 16k + 8 - 4k2 + 2k)] = 0
1/2(-8k2 - 4k + 4) = 0
-8k2 - 4k + 4 = 0
-8k2 - 8k + 4k + 4 = 0
-8K(k + 1) + 4(k + 1) = 0
(k + 1) (4 - 8k) = 0
k + 1 = 0
k = -1
4 - 8k = 0
k = 4/8
k = 1/2
So, the value of k is -1 and 1/2 .
Question 151

If a connected graph G has planar embedding with 4 faces and 4 vertices, then what will be the number of edges in G?

A
7
B
6
C
4
D
3
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 151 Explanation: 
Euler's Formula for Planar Graphs:
For any(connected) planar graph with v vertices, e edges and faces, we have
V - E + F = 2
= 4 - E + 4 =2
E = 4 - 2 + 4
E = 6
Question 152

What is the area bounded by the parabola 2y = x2 and the line x = y - 4?

A
18
B
36
C
72
D
6
       Engineering-Mathematics       Co-ordinate-Geometry       JT(IT) 2018 PART-B Computer Science
Question 152 Explanation: 
Given parabola 2y = x2 --- (1)
and the line x = y – 4 ------ (2)
Then y = x+4
Now, Substitute Y value in Equation-(1)
x2 = 2 ( x + 4 ) ------ (3)
Solving equation-3 we get x = 4, - 2.
Place Values of x in equation (1) and (2) we will get y = 8, 2.
Then the points of intersection are (8, 4), (2, –2).

After solving the integration, we will get 18.
Question 153

What is the possible number of reflexive relations on a set of 5 elements?

A
225
B
215
C
210
D
220
       Engineering-Mathematics       Sets-And Relation       JT(IT) 2018 PART-B Computer Science
Question 153 Explanation: 
Step-1: To find number of reflexive relation we have standard formula is 2(n2-n)
Step-2: The possible number of reflexive relations on a set of 5 elements 2(n2-n) which is 220 for n=5.
Question 154

Which one of the following is most affected by the presence of outliers in sample data?

A
Variance
B
Mean
C
Median
D
Mode
       Engineering-Mathematics       Probability       JT(IT) 2018 PART-B Computer Science
Question 154 Explanation: 
Let's examine what can happen to a data set with outliers.
For the sample data set:
1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4
We find the following mean, median, mode, and standard deviation:
Mean = 2.58
Median = 2.5
Mode = 2
Standard Deviation = 1.08
If we add an outlier to the data set:
1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 400
The new values of our statistics are:
Mean = 35.38
Median = 2.5
Mode = 2
Standard Deviation = 114.74
Note: Outliers often has a significant effect on your mean and standard deviation.
Question 155

Consider the matrix A defined as follows:

What is the eigenvalue of 3A+ 5A- 6A + 21, where I is an identity matrix?

A
4, 110, 10
B
1, 27, -8
C
1, 9, 4
D
4, 27, 9
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2018 PART-B Computer Science
Question 155 Explanation: 
The eigenvalues of A are 1, 3, –2 ( ∵ The given matrix is upper triangular)
Eigenvalues of A3 are 1, 27 and –8.
Eigenvalues of A2 are 1, 9 and 4.
Eigenvalues of A are 1, 3 and –2.
Eigenvalues of I are 1, 1 and 1.
∴ The eigenvalues of 3A3 + 5A2 – 6A + 2I.
First eigenvalue = 3(1) + 5(1) – 6(1) + 2(1) = 4
Second eigenvalue = 3(27) + 5(9) – 6(3) + 2(1) = 110
Third eigenvalue = 3(–8) + 5(4) – 6(–2) + 2(1) = 10
∴ The required eigenvalues are 4, 110, and 10.
Question 156

If A and B are sets and AUB = A∩B, then which of the following is correct?

A
A=B
B
A=∅
C
B=∅
D
A⊂B
       Engineering-Mathematics       Sets-And Relation       JT(IT) 2018 PART-B Computer Science
Question 156 Explanation: 
For this let x belongs to A implies x belongs to A U B
= x belongs to A intersection B
= x belongs to A and x Belongs to B
= x belongs to B
so A subset of B --- (2)
Now we will let y belong to B which implies y belongs to A U B
= y belongs to A intersection B
= y belongs to A and y belongs to B
= y belongs to A
Therefore, B subset of A --- (3) from (2) and (3) we get A=B.
Question 157
Let L be a lattice. Then for every a and b in L which one of the following is correct?
A
aVb = a​ ⋀ ​ b
B
aV(bVc)=(aVb)Vc
C
aV(b​ ⋀ ​ c)=a
D
aV(bVc)=b
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 157 Explanation: 
Distributive lattice is satisfies this condition aV(bVc)=(aVb)Vc
A distributive lattice is a lattice in which the operations of join and meet distribute over each other.
Question 158
​ Let N={1,2,3,...} be ordered by divisibility, which of the following subset is totally ordered?
A
(2,6,24)
B
(3,5,15)
C
(2,9,16)
D
(4,15,30)
       Engineering-Mathematics       Set-Theory       Nielit Scientist-B IT 22-07-2017
Question 158 Explanation: 
A binary relation R on a set A is a total order on A iff R is a connected partial order on A.
Question 159

Which of the following statements is/are correct?

    (i) If the rank of the matrix of given vectors is equal to the number of vectors, then the vectors are linearly independent.
    (ii) If the rank of the matrix of given vectors is less than the number of vectors, then the vectors are linearly dependent.
A
Both (i) and (ii)
B
Only (ii)
C
Only (i)
D
Neither (i) nor (ii)
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2018 PART-B Computer Science
Question 159 Explanation: 
The Rank of a Matrix
→ You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r elements.
→ The rank of a matrix is defined as (a) the maximum number of linearly independent column vectors in the matrix or (b) the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent.
For an r x c matrix,
1. If r is less than c, then the maximum rank of the matrix is r.
2. If r is greater than c, then the maximum rank of the matrix is c.
→ The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.
Question 160
M is a square matrix of order 'n' and its determinant value is 5, If all the elements of M are multiplied by 2, its determinant value becomes 40. he value of 'n' is
A
2
B
3
C
5
D
4
       Engineering-Mathematics       Linear-Algebra       Nielit Scientific Assistance IT 15-10-2017
Question 160 Explanation: 
M has n rows. If all the elements of a row are multiplied by , the determinant value becomes 2*5. Multiplying all the n rows by 2, will make the determinant value 2​ n ​ *5=40. Solving n= 3.
Question 161
The convergence of the bisection method is
A
Cubic
B
Quadratic
C
Linear
D
none
       Engineering-Mathematics       Convergence-Bisection       Nielit Scientific Assistance IT 15-10-2017
Question 161 Explanation: 
The convergence of the bisection method is very slow. Although the error, in general, does not decrease monotonically, the average rate of convergence is 1/2 and so, slightly changing the definition of order of convergence, it is possible to say that the method converges linearly with rate 1/2.
Note:For the bisection you simply have that ε​ i+1​ /ε​ i​ =1/2, so, by definition the order of convergence is 1 (linearly).
Question 162
Using bisection method, one root of X4-X-1 lies between 1 and 2. After second iteration the root may lie in interval:
A
(1.25,1.5)
B
(1,1.25)
C
(1,1.5)
D
None of the options
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 162 Explanation: 
Given data.
root= X4-X-1.
Root lies Between 1 and 2,
After second iteration=?
Using bisection method.
f(1)=X4-X-1
=1-1-1
= -1
f(2)=X4-X-1
= 24 -2 -1
=13
Given constraint that “root lies between 1 and 2”
Iteration-1: x1=(a+b)/2
=(1+2)/2
= 1.5
f(1.5) = 2.5625
Iteration-2: x2=(a+b)/2
=(1+1.5)/2
=1.25
f(1.25)=0.19140625 >0
Root may lie in between (1, 1.25)
Algorithm - Bisection Scheme
Given a function f (x) continuous on an interval [a,b] and f (a) * f (b) < 0
Do
c=(a+b)/2
if f(a)*f(c)< 0 then b=c
else a=c
while (none of the convergence criteria C1, C2 or C3 is satisfied)
More info:
Bisection method is the simplest among all the numerical schemes to solve the transcendental equations. This scheme is based on the intermediate value theorem for continuous functions .
Consider a transcendental equation f(x)=0 which has a zero in the interval [a,b] and f(a)*f(b)<0. Bisection scheme computes the zero, say c, by repeatedly halving the interval [a,b]. That is, starting with
c = (a+b) / 2
The interval [a,b] is replaced either with [c,b] or with [a,c] depending on the sign of f (a) * f (c) . This process is continued until the zero is obtained. Since the zero is obtained numerically the value of c may not exactly match with all the decimal places of the analytical solution of f (x) = 0 in the interval [a,b]. Hence any one of the following mechanisms can be used to stop the bisection iterations :
C1. Fixing a priori the total number of bisection iterations N i.e., the length of the interval or the maximum error after N iterations in this case is less than | b-a | / 2N.
C2. By testing the condition | ci - c i-1| (where i are the iteration number) less than some tolerance limit, say epsilon, fixed a priori.
C3. By testing the condition | f (ci ) | less than some tolerance limit alpha again fixed a priori.
http://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/bracketing%20methods/bisection/bisection.html
Question 163
Find the smallest number y such that y*162 (y multiplied by 162) is perfect cube:
A
24
B
27
C
36
D
38
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 163 Explanation: 
Prime factorize: 162
⇒162 =2×3×3×3×3 = 33×(2×3)
For (2×3) to be a perfect cube, it should be multiplied by (22×32)
∴ Required number = y = 22×32 = 36
Question 164
In how many ways 8 girl and 8 boys can sit around a circular table so that no two boys sit together?
A
(7!)2
B
(8!)2
C
7!8!
D
15!
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 164 Explanation: 
→ First fix one boy and place other 7 in alternative seats so total ways is 7! Because they are seated in circular table. (n-1)!.
→ Now place each girl between a pair of boys. So, total ways of seating arrangement of girls is 8!
→ Finally, 7!8! Possible ways are possible.
Question 165
Let u and V be two vectors in R 2 whose euclidean norms satisfy |u|=2|v|. What is the value α such that w=u+αv bisects the angle between u and v?
A
2
B
1
C
1/2
D
-1
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 165 Explanation: 

Let u, v be vectors in R2, increasing at a point, with an angle θ.
A vector bisecting the angle should split θ into θ/2,θ/2
Means ‘w’ should have the same angle with u, v and it should be half of the angle between u and v.
Assume that the angle between u, v be 2θ (thus angle between u,w=θ and v,w=θ) Cosθ=(u∙w)/(∥u∥ ∥w∥) ⇾①
Cosθ=(v∙w)/(∥v∥ ∥w∥) ⇾②
①/②⇒1/1=((u∙w)/(∥u∥ ∥w∥))/((v∙w)/(∥v∥ ∥w∥))⇒1=((u∙w)/(∥u∥))/((v∙w)/(∥v∥))
⇒(u∙w)/(v∙w)=(∥u∥)/(∥v∥) which is given that ∥u∥=2 ∥v∥
⇒(u∙w)/(v∙w)=(2∥v∥)/(∥v∥)=2 ⇾③
Given ∥u∥=2∥v∥
u∙v=∥u∥ ∥v∥Cosθ
=2∙∥v∥2 Cosθ
w=u+αv
(u∙w)/(v∙w)=2
(u∙(u+αv))/(v∙(u+αv))=2
(u∙u+α∙u∙v)/(u∙v+α∙v∙v)=2a∙a=∥a∥2
4∥v∥2+α∙2∙∥v∥2 Cosθ=2(2∥v∥2 Cosθ+α∙∥v∥2/sup>)
4+2αCosθ=2(2Cosθ+α)

4+2αCosθ=4Cosθ+2α⇒Cosθ(u-v)+2α-4=0
4-2α=Cosθ(4-2α)
(4-2α)(Cosθ-1)=0 4-2α=0

Question 166

Every cut set of a connected euler graph has____

A
An odd number of edges
B
At least three edges
C
An even number of edges
D
A single edge
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 166 Explanation: 
If every minimal cut has an even number of edges, then in particular the degree of each vertex is even. Since the graph is connected, that means it is Eulerian.
Question 167

A fair coin is tossed 6 times. What is the probability that exactly two heads will occur?

A
11/32
B
1/64
C
15/64
D
63/64
       Engineering-Mathematics       Probability       JT(IT) 2018 PART-B Computer Science
Question 167 Explanation: 
Step-1: We want to calculate the probability of getting exactly k "success" results using Bernoulli's Scheme.

The probability can be calculated as:
n is a total number of experiments
k is an expected number of successes
p is the probability of a success
Step-2: The first task can be simply solved by calculating the probability of 2 successes:
A "success" is "tossing heads in a single toss".
A "failure" is "tossing tails in a single toss".
Step-3: The probability of a success is ½
The number of all experiments is n=6.
The number of successes is k=2
Question 168
Let G be a complete undirected graph on 8 vertices of G are labelled, then the number of distinct cycles of length 5 in G is equal to:
A
15
B
30
C
56
D
60
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 168 Explanation: 
Given data,
8 vertices and distinct cycles length=5
Step-1: To find number of distinct cycles
C(n,r)=C(8,5)
=8! / (5!(8−5)!)
= 56
Question 169

How many solutions does the equation x + y - z = 11 have, where x, y and z are non negative integers?

A
120
B
78
C
156
D
130
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2018 PART-B Computer Science
Question 169 Explanation: 
→ For any pair of natural numbers n and k, the number of distinct n-tuples of non-negative integers whose sum is k is given by the binomial coefficient.

Here, n=3 and k=11, giving you
Question 170

A graph G is dual if and only if G is a ___

A
Euler graph
B
Regular graph
C
Complete graph
D
Planar graph
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 170 Explanation: 
Definition Given a plane graph G, the dual graph G* is the plane graph whose vtcs are the faces of G.
→ The correspondence between edges of G and those of G* is as follows:
if e∈E(G) lies on the boundaries of faces X and Y, then the endpts of the dual edge e*∈(G*) are the vertices x and y that represent faces X and Y of G.
Question 171

Which of the given options is the logical translation of the following statement, where F(x) and P(x) express the terms friend and perfect, respectively?

“None of my friends are perfect”?

A
∃x(~F(x)∧P(x))
B
∃x(F(x)∧ ~P(x))
C
~∃x(F(x)∧P(x))
D
∃x(~F(x)∧~P(x))
       Engineering-Mathematics       Propositional-Logic       JT(IT) 2018 PART-B Computer Science
Question 171 Explanation: 
F(x) represents x is friend.
P(x) represents x is perfect.
Given statement is “None of my friends are perfect”.
F(x)∧P(x) ---> X is both Friend and perfect.
~∃x ----> There is exist no one.
We can also write the above statement as follows
∀x[F(x) ⟹ ¬P(x)]
≡ ∀x[¬F(x) ∨ ¬P(x)]
≡∀x¬[F(x) ∧ P(x)]
≡¬∃x[F(x) ∧ P(x)]
Question 172
Which one is the correct translation of the following statement into mathematical logic?
"None of my friends are perfect"
A
~∃x(p(x) ⋀ q(x))
B
∃x(~p(x) ⋀ q(x))
C
∃x(~p(x) ⋀ ~q(x))
D
∃x(p(x) ⋀ ~q(x))
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 172 Explanation: 
F(x) ⇒ x is my friend
P(x) ⇒ x is perfect
There doesn't exist any person who is my friend and perfect
Question 173
The number of integers between 1 and 500(both inclusive) that are divisible by 3 or 5 or 7 is__
A
269
B
20
C
271
D
272
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 173 Explanation: 

Let A = number divisible by 3
B = numbers divisible by 5
C = number divisible by 7
We need to find “The number of integers between 1 and 500 that are divisible by 3 or 5 or 7" i.e.,|A∪B∪C|
We know,
|A∪B∪C|=|A|+|B|+C-|A∩B|-|A∩C|-|B∩C|+|A∩B|
|A|=number of integers divisible by 3
[500/3=166.6≈166=166]
|B|=100
[500/5=100]
|C|=71
[500/7=71.42]
|A∩B|=number of integers divisible by both 3 and 5 we need to compute with LCM (15)
i.e.,⌊500/15⌋≈33
|A∩B|=33
|A∩C|=500/LCM(3,7) 500/21=23.8≈28
|B∩C|=500/LCM(5,3) =500/35=14.48≈14
|A∩B∩C|=500/LCM(3,5,7) =500/163=4.76≈4
|A∪B∪C|=|A|+|B|+|C|-|A∩B|-|A∩C|-|B∩C|+|A∩B∩C|
=166+100+71-33-28-14+4
=271
Question 174

The average rate of convergence of the bisection method is ____.

A
1/2
B
3
C
2
D
1
       Engineering-Mathematics       Sets-And Relation       JT(IT) 2018 PART-B Computer Science
Question 174 Explanation: 
The average rate of convergence of the bisection method is 1/2.
Question 175

What is the chromatic number of an n-vertex simple connected graph, which does NOT contain any odd-length cycle?

A
N
B
N-1
C
2
D
3
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 175 Explanation: 
→ If n ≥ 2 and there are no odd length cycles, Then it is bipartite graph.
→ A bipartite graph has the chromatic number 2.
Eg: Consider a square, which has 4 edges. It can be represented as bipartite ,with chromatic number 2.
Question 176
When the sum of all possible two digit numbers formed from three different one digit natural numbers are divided by sum of the original three numbers, the result is:
A
26
B
24
C
20
D
22
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 176 Explanation: 
Let a,b,c be 3 different digits, the sum of the different 2 digit numbers formed with them is
=(a*10+b)+(b*10+a)+(a*10+c)+(c*10+a)+(b*10+c)+(c*10+b)
=2(a+b+c)(10+1)
=22
Question 177

Considering an undirected graph, which of the following statements is/are true?

    (i) Number of vertices of odd degree is always even
    (ii) Sum of degrees of all the vertices is always even
A
Neither (i) nor (ii)
B
Both (i) and (ii)
C
Only (ii)
D
Only (i)
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 177 Explanation: 
True: Number of vertices of odd degree is always even.
True: Sum of degrees of all the vertices is always even.
Question 178
Consider two matrices M1 and M2 with M1*M2 =0 and M1 is non singular. Then which of the following is true?
A
M2 is non singular
B
M2 is null Matrix
C
M2 is the identity matrix
D
M2 is transpose of M1
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 178 Explanation: 
M2 is null Matrix
The easiest way we can prove this is by looking at the determinant, since det(AB)=det(A)det(B)
and a matrix A is singular iff det(A)=0
Question 179
Let G be a simple undirected graph on n=3x vertices (x>=1) with chromatic number 3, then maximum number of edges in G is:
A
n(n-1)/2
B
nn-2
C
nx
D
n
       Engineering-Mathematics       Nielit Scientist-B 17-12-2017
Question 179 Explanation: 
Given undirected graph on n=3x vertices (x>=1) with chromatic number 3.
→ Maximum number of edges are=n.

Question 180
The solution of the recurrence relation a​ r​ =a​ r-1​ +2a​ r-2​ with a​ 0​ =2,a​ 1​ =7
A
a​ r​ =(3)​ r​ +(1)​ r
B
2a​ r​ =(2)​ r​ /3 -(1)​ r
C
a​ r​ =3​ r+1​ -(-1)​ r
D
a​ r​ =3(2)​ r​ -(-1)​ r
       Engineering-Mathematics       Combinatorics       Nielit Scientific Assistance IT 15-10-2017
Question 180 Explanation: 
Given the recurrence relation a​ r​ =a​ r-1​ +2a​ r-2 and a​ 0​ =2,a​ 1​ =7.
For r =2, a​ 2=​ a​ 2-1​ +2a​ 2-2​ =a​ 1​ +2a​ 0​ =7+2*2=7+4=11
For r=3,a​ 3=​ a​ 3-1​ +2a​ 3-2​ =a​ 2​ +2a​ 1​ =11+2*7=11+14=25
From the above options , Substitute the r values 0,1,2,3 then option -D gives the solution to recurrence relation.
Question 181

Select the option that best describes the relationship between the following graphs:

A
G and H are directed
B
G and H are isomorphic
C
G and H are homomorphic
D
G and H are not isomorphic
       Engineering-Mathematics       Graph-Theory       JT(IT) 2018 PART-B Computer Science
Question 181 Explanation: 
If G1 ≡ G2 then
1. |V(G1)| = |V(G2)|
2. |E(G1)| = |E(G2)|
3. Degree sequences of G1 and G2 are same.
4. If the vertices {V1, V2, ... Vk} form a cycle of length K in G1, then the vertices {f(V1), f(V2), … f(Vk)} should form a cycle of length K in G2.
→ Graph G and H are having same number of vertices and edges. So, it satisfied first rule.
→ u1 degree 2, u2 degree 3, u3 degree 3, u4 degree 2, u5 degree 3, u6 degree 3.
→ V1 degree 2, V2 degree 3, V3 degree 3, V4 degree 2, V5 degree 3, V6 degree 3.
→ But it violates the property of number of cycles. The number of cycles are not same.
Question 182

Consider the following statements. Which one is/are correct?

    (i) The LU decomposition method fails if any of the diagonal elements of the matrix is zero.
    (ii) The LU decomposition is guaranteed when the coefficient matrix is positive definite.
A
Both (i) and (ii)
B
Only (i)
C
Neither (i) nor (ii)
D
Only (ii)
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2018 PART-B Computer Science
Question 182 Explanation: 
Lower–Upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
1. The LU decomposition method fails if any of the diagonal elements of the matrix is zero.
2. The LU decomposition is guaranteed when the coefficient matrix is positive definite.
Question 183

Which of the following Hasse diagrams is are lattice(s)?

A
Only (a), (b) and (c)
B
Only (a) and (b)
C
Only (b) and (c)
D
Only (a) and (c)
       Engineering-Mathematics       Sets-And Relation       JT(IT) 2018 PART-B Computer Science
Question 183 Explanation: 
A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
Question 184
Total number of simple graphs that can be drawn using six vertices are:
A
215
B
214
C
213
D
212
       Engineering-Mathematics       Nielit STA 17-12-2017
Question 184 Explanation: 
To get total number of simple graphs, we have a direct formula is 2(n(n-1)/2).
=26(5)/2
=215
Question 185
If a planner graph, having 25 vertices divides plane into 17 different regions. Then how many edges are used to connect the vertices in this graph
A
20
B
30
C
40
D
50
       Engineering-Mathematics       Nielit STA 17-12-2017
Question 185 Explanation: 
Euler’s Formula : For any polyhedron that doesn’t intersect itself (Connected Planar Graph), F(faces) + V(vertices) − E(edges) = 2.
Here as given, F=?,V=25 and E=17
→ F+25-17=2
→ 40
Question 186
If a random coin is tossed 11 times, then what is the probability that for 7th toss head appears exactly 4 times?
A
5/32
B
15/128
C
35/128
D
None of the options
       Engineering-Mathematics       Nielit STA 17-12-2017
Question 186 Explanation: 
→ To find probability that for the 7th toss head appears exactly 4 times. We have find that past 6 run of tosses.
→ Past 6 tosses, What happen we don’t know. But according to 7th toss, we will find that head appeared for 3 times.
→ Any coin probability of a head=½ and probability of a tail=½
→ If we treat tossing of head as success,then this leads to a case of binomial distribution.
→ According to binomial distribution, the probability that in 6 trials we get 3 success is
6C3*(½)3 *(½)3= 5/16 (3 success in 6 trials can happen in 6C3 ways). → 7th toss, The probability of obtaining a head=1/2
→ In given question is not mention that what happens after 7 tosses.
→ Probability for that for 7th toss head appears exactly 4 times is =(5/16)*(1/2)
=5/32.
Question 187
The number of the edges in a regular graph of degree 'd' and 'n' vertices is____
A
Maximum of n,d
B
n+d
C
nd
D
nd/2
       Engineering-Mathematics       Graph-Theory       Nielit Scientific Assistance CS 15-10-2017
Question 187 Explanation: 
Sum of degree of vertices = 2*no. of edges
d*n = 2*|E|
∴ |E| = (d*n)/2
Question 188
The convergence of the bisection method is
A
Cubic
B
Quadratic
C
Linear
D
none
       Engineering-Mathematics       Convergence-Bisection       Nielit Scientific Assistance CS 15-10-2017
Question 188 Explanation: 
The convergence of the bisection method is very slow. Although the error, in general, does not decrease monotonically, the average rate of convergence is 1/2 and so, slightly changing the definition of order of convergence, it is possible to say that the method converges linearly with rate 1/2.
Note:For the bisection you simply have that ε​ i+1​ /ε​ i​ =1/2, so, by definition the order of convergence is 1 (linearly).
Question 189

Choose the option that correctly matches each element of LIST-1 with exactly one element of LIST-2:

        LIST-1				LIST-2
(i)  Newton-Raphson method	(a) Solving nonlinear equation
(ii) Simpson’s rule		(b) Solving ordinary differential equations
(iii)Runge-Kutta method		(c) Numerical integration
(iv) Gauss elimination		(d) Interpolation
A
(i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
B
(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
C
(i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
D
(i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
       Engineering-Mathematics       Newton-Raphson-Method       JT(IT) 2018 PART-B Computer Science
Question 189 Explanation: 
Newton-Raphson method→ Interpolation
Simpson’s rule→ Numerical integration
Runge-Kutta method→ Solving ordinary differential equations
Gauss elimination→ Solving nonlinear equation
Question 190

How many ways are there to arrange the nine letters of the word ALLAHABAD?

A
4560
B
4000
C
7560
D
7500
       Engineering-Mathematics       Combinatorics       JT(IT) 2018 PART-B Computer Science
Question 190 Explanation: 
Step-1: There are 4A, 2L, 1H, 1B and 1D.
Step-2: Total number of alphabets! / alphabets are repeating using formula (n!) / (r1! r2!..).
Step-3: Repeating letters are 4A's and 2L's
= 9! / (4!2!)
= 7560
Question 191

How many bit strings of length eight (either start with a 1 bit or end with the two bits 00) can be formed?

A
64
B
255
C
160
D
128
       Engineering-Mathematics       Combinatorics       JT(IT) 2018 PART-B Computer Science
Question 191 Explanation: 
Use the subtraction rule.
1. Number of bit strings of length eight that start with a 1 bit: 27 = 128
2. Number of bit strings of length eight that end with bits 00: 26 = 64
3. Number of bit strings of length eight that start with a 1 bit and end with bits 00: 25 = 32
The number is 128+64+32 = 160
Question 192

Which of the following statements are logically equivalent?

    (i) ∀x(P(x))
    (ii) ~∃x(P(x))
    (iii) ~∃x(~P(x))
    (iv) ∃x(~P(x))
A
Only (ii) and (iv)
B
Only (ii) and (iii)
C
Only (i) and (iv)
D
Only (i) and (ii)
       Engineering-Mathematics       Propositional-Logic       JT(IT) 2018 PART-B Computer Science
Question 192 Explanation: 
∀x(P(x)) = ~∃x(~P(x))
Question 193

What is the solution of the recurrence relation an = 6an-1 - 9an-2 with initial conditions a0=1 and a1=6?

A
an = 3n + 3n2
B
an = 3n + n3n
C
an = 3n + 3nn
D
an = n3 + n3n
       Engineering-Mathematics       Combinatorics       JT(IT) 2018 PART-B Computer Science
Question 193 Explanation: 
→ Let c1, c2, ..., ck be real numbers.
→ Suppose that the characteristic equation rk −c1rk−1 −···−ck = 0 has k distinct roots r1, r2, ..., rk.
→ Then, a sequence {an} is a solution of the recurrence relation: an = c1an−1 + c2an−2 + ··· + ckan−k
→ if and only if an = α1rn1 + α2rn2 + ··· + αk rnk for n = 0,1,2,..., where α1, α2, ..., αk are constants.
→ r2 − 6r + 9 = 0 has only 3 as a root.
→ So the format of the solution is an = α13n + α2n3n.
→ Need to determine α1 and α2 from initial conditions:
a0 = 1 = α1
a1 = 6 = α13 + α23
Solving these equations we get α1=1 and α2=1
Therefore, an = 3n + n3n.
Question 194

Which of the following statements about the Newton-Raphson method is/are correct?

    (i) It is quadratic convergent
    (ii) If f'(x) is zero, it fails
    (iii) It is also used to obtain complex root
A
(i), (ii) and (iii)
B
only (i) and (iii)
C
only (i) and (ii)
D
only (i)
       Engineering-Mathematics       Calculus       JT(IT) 2018 PART-B Computer Science
Question 194 Explanation: 
Above all statements are true for Newton-Raphson method.
Note: Newton-Raphson method is mainly used for Interpolation.
Question 195

What is the minimum number of students in a class to be sure that three of them are born in the same month?

A
24
B
12
C
25
D
13
       Engineering-Mathematics       Combinatorics       JT(IT) 2018 PART-B Computer Science
Question 195 Explanation: 
With the help of Pigeonhole principle we can get the solution. The idea is that if you n items and m possible containers. If n > m, then at least one container must have two items in it. So, total 13 minimum number of students.
Question 196
What is the maximum value of the function f(x)=2x​ 2​ -2x +6 in the interval [0,2]?
A
6
B
10
C
12
D
5,5
       Engineering-Mathematics       Calculus       Nielit Scientific Assistance CS 15-10-2017
Question 196 Explanation: 
An open interval does not include its endpoints, and is indicated with parentheses. For example, (0,1) means greater than 0 and less than 1.
A closed interval is an interval which includes all its limit points, and is denoted with square brackets. For example, [0,1] means greater than or equal to 0 and less than or equal to 1.
A half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. (0,1] means greater than 0 and less than or equal to 1, while [0,1) means greater than or equal to 0 and less than 1.
Given function f(x)=2x​ 2​ -2x +6 and the given interval is [0,2]
According to the given interval , we need to check the function at the values 0,1,2.
f(0)=2x0-2x0+6=6
f(1)=2x1​ 2​ -2x1+6=2-2+6=6
f(2)=2x2​ 2​ -2x2+6=8-4+6=10
Question 197
The value of the integral
A
(x+2)/2
B
2/(π-2)
C
π-2
D
π+2
       Engineering-Mathematics       Calculus       Nielit Scientific Assistance CS 15-10-2017
Question 197 Explanation: 
The integral value of x​ 2​ sin(x)dx is [−x​ 2​ cos(x)+2xsin(x)+2cos(x)+C]​ with the interval [π/2,0]
=[−(​ π/2)​ 2​ cos(​ π/2​ )+2​ π/2​ sin(​ π/2​ )+2cos(​ π/2​ )+C]​ - ​ [−(​ 0)​ 2​ cos(​ π/2​ )+2(0)sin(​ π/2​ )+2cos(​ 0 ​ )+C]
=[0+​ π+0+C-2-C]
=π-2
Question 198
The solution of the recurrence relation a​ r​ =a​ r-1​ +2a​ r-2​ with a​ 0​ =2,a​ 1​ =7
A
a​ r​ =(3)​ r​ +(1)​ r
B
2a​ r​ =(2)​ r​ /3 -(1)​ r
C
a​ r​ =3​ r+1​ -(-1)​ r
D
a​ r​ =3(2)​ r​ -(-1)​ r
       Engineering-Mathematics       Combinatorics       Nielit Scientific Assistance CS 15-10-2017
Question 198 Explanation: 
● Given the recurrence relation a​ r​ =a​ r-1​ +2a​ r-2 and a​ 0​ =2,a​ 1​ =7.
● For r =2, a​ 2=​ a​ 2-1​ +2a​ 2-2​ =a​ 1​ +2a​ 0​ =7+2*2=7+4=11 ● For r=3,a​ 3=​ a​ 3-1​ +2a​ 3-2​ =a​ 2​ +2a​ 1​ =11+2*7=11+14=25
● From the above options ,Substitute the r values 0,1,2,3 then option -D gives the solution to recurrence relation.
Question 199
M is a square matrix of order 'n' and its determinant value is 5. If all the elements of M are multiplied by 2, Its determinant value becomes 40. The value of 'n' is
A
2
B
3
C
5
D
4
       Engineering-Mathematics       Linear-Algebra       Nielit Scientific Assistance CS 15-10-2017
Question 199 Explanation: 
● From the matrix property : If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
● Consider matrix order is 3, there are 3 rows and each row is multiplied by 2 means we need to multiply 8 to the existing determinant.
● The given existing determinant is 5 and each row multiplied by 2 means 8 *5 =40.
Question 200
The figure below represents a
A
Hilbert curve order 3
B
Hilbert curve order 2
C
Hilbert curve of order 1
D
Hilbert curve of order 4
       Engineering-Mathematics       Hilbert-Curve       KVS DEC-2013
Question 200 Explanation: 
→ The Hilbert Curve is a space filling curve that visits every point in a square grid.
→ The path taken by a Hilbert Curve appears as a sequence - or a certain iteration - of up, down, left, and right.
Question 201

If a random variable takes a finite set of values it is called:

A
Continuous variate
B
Normal variate
C
Discrete variate
D
Exponential variate
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 201 Explanation: 
→ A discrete variable is a variable whose value is obtained by counting.
Examples:
number of students present
number of red marbles in a jar
number of heads when flipping three coins
students’ grade level
→ A discrete random variable X has a countable number of possible values.
Example:
Let X represent the sum of two dice.
→ A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........ Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete.
Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten.
Question 202

If the mean of a poisson distribution is m, then standard deviation of the distribution is:

A
m2
B
m
C
2*m
D
√m
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 202 Explanation: 
Mean and Variance of the Poisson distribution. There is also a formula for the standard deviation, σ, and variance, σ2.
Question 203

The standard deviation of binomial distribution with n observations and probability of success p, probability of failure is q is:

A
√npq
B
Pq
C
Np
D
√pq
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 203 Explanation: 
Mean and Variance of the Binomial Distribution:
The binomial distribution for a random variable X with parameters n and p represents the sum of n independent variables Z which may assume the values 0 or 1. If the probability that each Z variable assumes the value 1 is equal to p, then the mean of each variable is equal to l*p + 0*(l-p) = p, and the variance is equal to p(l-p).
By the addition properties for independent random variables, the mean and variance of the binomial distribution are equal to the sum of the means and variances of the n independent Z variables, so

These definitions are intuitively logical. Imagine, for example 8 flips of a coin. If the coin is fair, then p = 0.5. One would expect the mean number of heads to be half the flips, or np = 8*0.5 = 4.
The variance is equal to np(l-p) = 8*0.5*0.5 = 2.
Question 204

The normal curve is symmetrical about its:

A
Standard deviation
B
Mean
C
Variance
D
Probability
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 204 Explanation: 
Symmetrical distribution occurs when the values of variables occur at regular frequencies and the mean, median and mode occur at the same point. In graph form, symmetrical distribution often appears as a bell curve. If a line were drawn dissecting the middle of the graph, it would show two sides that mirror each other.
The probability density of the normal distribution is

Where
• μ is the mean or expectation of the distribution (and also its median and mode).
• σ is the standard deviation, and
• 2 is the variance
Question 205

The density of uniform distribution over the interval -⍺ < a < b < ⍺ is given by:

A
f(x) = λe-λx , x>=0
B
f(x) = qkp
C
f(x) = 1/(b-a), a
D
f(x) = (⍺/c)x⍺-1
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 205 Explanation: 
The probability density function of the continuous uniform distribution is:

The values of f(x) at the two boundaries a and b are usually unimportant because they do not alter the values of the integrals of f(X) dX over any interval, not of X f(X) dX or any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be 1/(b – a). The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may lake the value of f(a) to be 1|2(b – a), since then the inverse transform of may integral transform of this uniform function will yield back the function itself, rather than a function which equal ‘almost everywhere’, i.e except on a set of points with zero measure. Also, it is consistent with the sign function which has no such ambiguity.
In terms of mean and variance σ2, the probability density may be written as:
Question 206

Exponential distribution is special case of ____ distribution.

A
Theta
B
Alpha
C
Beta
D
Gamma
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 206 Explanation: 
The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.
Question 207
The algebraic sum of the deviations of all the variables from their mean i.e., is:
A
0
B
1
C
-1
D
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 207 Explanation: 
The sum of the deviations from the mean is zero. This will always be the case as it is a property of the sample mean, i.e., the sum of the deviations below the mean will always equal the sum of the deviations above the mean. However, the goal is to capture the magnitude of these deviations in a summary measure.
Question 208

The mean, mode and median are connected by the empirical relationship:

A
Mean-mode = 2(mean-median)
B
Mean-mode = 3(mean-median)
C
Mean-mode = (mean-mode)/2
D
Mean-mode = (mean-mode)/3
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 208 Explanation: 
A distribution in which the values of mean, median and mode coincide (i.e. mean = median = mode) is known as a symmetrical distribution. Conversely, when values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed distribution. In moderately skewed or asymmetrical distribution a very important relationship exists among these three measures of central tendency. In such distributions the distance between the mean and median is about one-third of the distance between the mean and mode, as will be clear from the diagrams 1 and 2. Karl Pearson expressed this relationship as:
Mode = mean - 3 [mean - median]
Mode = 3 median - 2 mean
and Median = mode + ⅔ [mean-mode]
Question 209

The root mean square deviation when measured from the mean is:

A
Greatest
B
Positive
C
Least
D
Negative
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 209 Explanation: 
The root mean square deviation when measured from the mean is least value.
Question 210

The values which divide the frequency into four equal parts are called:

A
Coefficient of variance
B
Range
C
Dispersion
D
Quartiles
       Engineering-Mathematics       Discrete-Distribution       JT(IT) 2016 PART-B Computer Science
Question 210 Explanation: 
The values of a variable that divide a distribution into four equal parts are called quartiles. Since three values are needed to divide a distribution into four parts, there are three quartiles, viz. Q1, Q2 and Q3, known as the first, second and the third quartile respectively. For a discrete distribution, the first quartile (Q1) is defined as that value of the variate such that at least 25% of the observations are less than or equal to it and at least 75% of the observations are greater than or equal to it.
Question 211

How many 3 digit numbers are there with all different odd digits?

A
16
B
48
C
54
D
60
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 211 Explanation: 
• Three digit odd numbers implies the numbers would only be made of digits 1 , 3 , 5 , 7 , 9 With repetition of digits we would have had 5 * 5 * 5 = 125
• But for every hundreds, maximum of 4 tens is possible (avoiding the duplicate of digit used in hundreds)
• And each of these 4 tens, maximum of 3 units is possible (avoiding the duplicate of digits used in tens)
• 5 * 4 * 3 = 60
Question 212

In how many ways can a committee of 4 people be chosen from a group of 12?

A
495
B
595
C
395
D
295
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 212 Explanation: 
As the order of people does not matter, it is C4 12
C(n,r) = C(12,4)
= 12! / [(4!(12−4)!)]
= 495
Hence, a committee of 4 people be selected from a group of 12 people in 495 ways.
Question 213

A straight line which cuts a curve on two points at an infinite distance from the origin and yet is not itself wholly at infinity is called:

A
Spiral
B
Asymptote
C
Parallel
D
Polar
       Engineering-Mathematics       Co-ordinate-Geometry       JT(IT) 2016 PART-B Computer Science
Question 213 Explanation: 
A straight line which cuts a curve in two points at an infinite distance from the origin, but which is not itself wholly at infinity, is called an asymptote to a curve.
Question 214

If k parallel lines of a determinant Δ become identical when x=a, then ____ is a factor of Δ.

A
(x-a)+(k+1)
B
(x-a)/(k-1)
C
(x-a)*(k+1)
D
(x-a)(k-1)
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2016 PART-B Computer Science
Question 214 Explanation: 
• If a determinant D vanishes for x = a, then (x - a) is a factor of D, in other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.
• In general, if k rows (or k columns) become identical (x=a) when a is substituted for x, then (x-a)r-1 is a factor of D.
Question 215

A minimal subgraph G’ of G such that V(G’)=V(G) and G’ is connected is called:

A
A spanning tree
B
A connected graph
C
A directed graph
D
A biconnected component
       Engineering-Mathematics       Graph-Theory       JT(IT) 2016 PART-B Computer Science
Question 215 Explanation: 
Given a connected graph G, a connected subgraph that is both a tree and contains all the vertices of G is called a spanning tree for G.
Question 216

Rank of nonsingular square matrix of order r is:

A
r
B
0
C
r-1
D
1
       Engineering-Mathematics       Linear-Algebra       JT(IT) 2016 PART-B Computer Science
Question 216 Explanation: 
• A square matrix of order ’r’ is nonsingular if its determinant is non zero and therefore its rank is “r”. It's all rows and columns are linearly independent and it is invertible.
• Rank of singular matrix is less than “r”.
Question 217

The newton’s Raphson iterative formula for finding 1/N is:

A
½(xn + N/xn)
B
xn(1 - Nxn)
C
½(xn+1/Nxn)
D
1/k((k-1)xn + N/xnk-1)
       Engineering-Mathematics       Newton-Raphson-Method       JT(IT) 2016 PART-B Computer Science
Question 217 Explanation: 
• Newton's method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is one example of a root-finding algorithm.
• The iterations xk+1 = xk − ( f(xk)/ f ′(xk) ) are called Newton’s iterations.
Question 218

Portability is not a quality factor of:

A
Software coding
B
Software design
C
Software Process
D
Software testing
       Engineering-Mathematics       Probability       JT(IT) 2016 PART-B Computer Science
Question 218 Explanation: 
McCall’s Quality Factors:
This model classifies all software requirements into 11 software quality factors. The 11 factors are grouped into three categories – product operation, product revision, and product transition factors.
1. Product operation factors − Correctness, Reliability, Efficiency, Integrity, Usability.
2. Product revision factors − Maintainability, Flexibility, Testability.
3. Product transition factors − Portability, Reusability, Interoperability.
Question 219

The number of strips required in simpson’s 3/8th rule is a multiple of:

A
1
B
2
C
3
D
6
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 219 Explanation: 
Simpson’s 3/8th rule is also known as Simpson's 2nd rule:
Area = 3h/ 8 [( a + 3 b + 3 c + d )]
Simpson's Second Rule:
Multipliers:
Question 220

A declarative sentence which is either true(1) or false(0) is called:

A
Lattice
B
Tautology
C
Contradiction
D
Proposition
       Engineering-Mathematics       Set-Theory       JT(IT) 2016 PART-B Computer Science
Question 220 Explanation: 
• Declarative sentences are propositions.
• Sentences that assert a fact that could either be true or false.
Question 221

The points at which the function attains extreme values are called:

A
Turning points
B
End points
C
Higher points
D
Extreme points
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 221 Explanation: 
The value of the function, the value of y, at either a maximum or a minimum is called an extreme value. The points at which the function attains extreme values are called Turning points.
Question 222

If f(x) = ax2 + bx + c the f(x-(b/2a)) is:

A
An even function for all a except a=0
B
An even function for all a
C
Neither even nor odd
D
An odd function for all a except a=0
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 222 Explanation: 
• A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
• A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.
Question 223

The sum of n terms of 1/(1*2) + 1/(2*3) + 1/(3*4) + ... is

A
(n+1)/n
B
n/(n+1)
C
n/(2n+1)
D
(2n+1)/n
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 223 Explanation: 
Sum upto n terms = 1/(1*2) + 1/(2*3) + 1/(3*4) + ........ + 1/(n*(n+1))
where
1st term = 1/(1*2)
2nd term = 1/(2*3)
3rd term = 1/(3*4)
.
.
.
.
n-th term = 1/(n*(n+1))
n-th term = 1/(n*(n+1))
i.e. the k-th term is of the form 1/(k*(k+1))
which can further be written as k-th term = 1/k - 1/(k+1)
So, sum upto n terms can be calculated as:
(1/1 - 1/1+1) + (1/2 - 1/2+1) + (1/3 - 1/3+1) + ......... + (1/n-1 - /1n) + (1/n - 1/n+1)
= (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ......... + (1/n-1 - 1/n) + (1/n - 1/n+1)
= 1 - 1/n+1
= ((n+1) - 1)/n+1
= n/n+1
Question 224

How many integers are between 1 and 200 which are divisible by any one of the integers 2,3 and 5(Hint: use set operation)?

A
125
B
145
C
146
D
136
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 224 Explanation: 
A) numbers divisible by 2: 200/2 = 100
B) numbers divisible by 3: 200/3 = 66
C) numbers divisible by 5: 200/5 = 40
Counting twice:
AB) numbers divisible by 6: 200/6 = 33
AC) numbers divisible by 10: 200/10 = 20
BC) numbers divisible by 15: 200/15 = 13
Counting 3 times:
ABC) numbers divisible by 30: 200/30 = 6
Total of numbers = A + B + C - AB - AC - BC + ABC = 100 + 66 + 40 - 33 - 20 -13 + 6 = 146
Question 225

In algebra of logic, the conjunction of two tautologies is:

A
Contradiction
B
Tautology
C
Negation
D
Disjunction
       Engineering-Mathematics       Propositional-Logic       JT(IT) 2016 PART-B Computer Science
Question 225 Explanation: 
Some properties are tautologies:
1. The negation of a contradiction is a tautology.
2. The disjunction of two contingencies can be a tautology.
3. The conjunction of two tautologies is a tautology.
Question 226
The partial differential equation ∂2z/∂t2 = c2(∂2z/∂x2) represents:
A
Harmonic function
B
Laplace equation
C
Wave equation
D
Homogeneous
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 226 Explanation: 
Let us derive the d’Alembert’s formula in an alternate way.
Note that the wave equation can be factored as
Question 227

The number of two digit numbers divisible by the product of digits is:

A
8
B
14
C
13
D
5
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 227 Explanation: 
11=1*1=1 which is a factor of 11
12=1*2=2 which is a factor of 12
15=1*5=5 which is a factor of 15
24=2*4=8 which is a factor of 24
36=3*6=12 which is a factor of 36
Question 228

If a relation < from A={1,2,3,4} to B={1,3,5} i.e., (a,b)∈R if a < b, then R-1 is:

A
{(1,3)(1,5)(2,3)(2,5)(3,5)(4,5)}
B
{(3,1)(5,1)(3,2)(5,2)(5,3)(5,4)}
C
{(3,3)(3,5)(5,3)(5,5)}
D
{(3,3)(3,4)(4,5)}
       Engineering-Mathematics       Sets-And Relation       JT(IT) 2016 PART-B Computer Science
Question 228 Explanation: 
The relation R is {(1,3), (1,5),(2,3),(2,5),(3,5),(4,5) } where (a,b) ∈ R if a The R-1 is { (3,1),(5,1),(3,2),(5,2),(5,3),(5,4) } where (a,b) ∈ R-1 if a>b.
Question 229

If a lattice (L,R) has a greatest and least element then it is said to be:

A
Sub Lattice
B
Complemented Lattice
C
Bounded Lattice
D
Distributive Lattice
       Engineering-Mathematics       Set-Theory       JT(IT) 2016 PART-B Computer Science
Question 229 Explanation: 
→ In a Lattice if Upper Bound and Lower exists then it is called Bounded Lattice.
→ A bounded lattice is an algebraic structure of the form (L, ∨, ∧, 0, 1) such that (L, ∨, ∧) is a lattice, 0 (the lattice's bottom) is the identity element for the join operation ∨, and 1 (the lattice top) is the identity element for the meet operation ∧.
→ Let 'L' be a lattice w.r.t R if there exists an element I∈L such that (aRI)∀x∈L, then I is called Upper Bound of a Lattice L.
Similarly, if there exists an element O∈L such that (ORa)∀a∈L, then O is called Lower Bound of Lattice L.
Question 230

The sum of the series: (1/2) + (1/3) + (1/4) - (1/6) + (1/8) + (1/9) + (1/16) - (1/12) + ... + α is:

A
(1/2) - log(√2)3
B
1 + log(√2)3
C
3/2
D
1/2 + log(√2)3
       Engineering-Mathematics       Combinatorics       JT(IT) 2016 PART-B Computer Science
Question 230 Explanation: 
The sum of the series: (1/2) + (1/3) + (1/4) - (1/6) + (1/8) + (1/9) + (1/16) - (1/12) + ... + α is 1 + log(√2)3.
Question 231

Find the boolean product A⊙B of the two matrices.
A
B
C
D
       Engineering-Mathematics       Linear-Algebra       UGC NET CS 2017 Nov- paper-2
Question 231 Explanation: 
In this problem, they given boolean product A⊙B of two matrices.
The boolean product truth table is

According to the truth table, we have to perform matrix multiplication.
Question 232
How many distinguishable permutations of the letters in the word BANANA are there ?
A
720
B
120
C
60
D
360
       Engineering-Mathematics       Probability       UGC NET CS 2017 Nov- paper-2
Question 232 Explanation: 
Total distinguishable permutations means no repetition.
BANANA= 6 letters
B is appearing 1 time
A is appearing 3 times
N is appearing 2 times
So, = 6! / (3! * 2!)
= 60
Question 233
Let P and Q be two propositions, ¬ (P ↔ Q) is equivalent to:
(I) P ↔ ¬ Q
(II) ¬ P ↔ Q
(III) ¬ P ↔ ¬ Q
(IV) Q → P
A
Only (I) and (II)
B
Only (II) and (III)
C
Only (III) and (IV)
D
None of the above
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2017 Nov- paper-2
Question 233 Explanation: 

So, (I) and (Ii) are TRUE.
Question 234
Negation of the proposition ∃x H(x) is:
A
∃x ⌐H(x)
B
∀x ⌐H(x)
C
∀x H(x)
D
⌐x H(x)
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2017 Nov- paper-2
Question 234 Explanation: 
∃x H(x) = ∀x ⌐H(x)
Question 235
Consider a sequence F​ 00​ defined as : F​ 00​ (0) = 1, F​ 00​ (1) = 1
F​ 00​ (n) = ((10 ∗ F​ 00​ (n – 1) + 100)/ F​ 00​ (n – 2)) for n ≥ 2 Then what shall be the set of values of the sequence F​ 00​ ?
A
(1, 110, 1200)
B
(1, 110, 600, 1200)
C
(1, 2, 55, 110, 600, 1200)
D
(1, 55, 110, 600, 1200)
       Engineering-Mathematics       Combinatorics       UGC NET CS 2017 Jan -paper-2
Question 235 Explanation: 
Given data,
Sequence F​ 00​ defined as
F​ 00​ (0) = 1,
F​ 00​ (1) = 1,
F​ 00​ (n) = ((10 ∗ F​ 00​ (n – 1) + 100)/ F​ 00​ (n – 2)) for n ≥ 2
Let n=2
F​ 00​ (2) = (10 * F​ 00​ (1) + 100) / F​ 00​ (2 – 2)
= (10 * 1 + 100) / 1
= (10 + 100) / 1
= 110
Let n=3
F​ 00​ (3) = (10 * F​ 00​ (2) + 100) / F​ 00​ (3 – 2)
= (10 * 110 + 100) / 1
= (1100 + 100) / 1
= 1200
Similarly, n=4
F​ 00​ (4) = (10 * F​ 00​ (3) + 100) / F​ 00​ (4 – 2)
= (12100) / 110
= 110
F​ 00​ (5) = (10 * F​ 00​ (4) + 100) / F​ 00​ (5 – 2)
= (10*110 + 100) / 1200
= 1
The sequence will be (1, 110, 1200,110, 1).
Question 236
The functions mapping R into R are defined as : f(x) = x​ 3​ – 4x, g(x) = 1/(x​ 2​ + 1) and h(x) = x​ 4​ . Then find the value of the following composite functions : hog(x) and hogof(x)
A
(x​ 2​ + 1)4 and [(x​ 3​ – 4x)​ 2​ + 1]​ 4
B
(x​ 2​ + 1)4 and [(x​ 3​ – 4x)​ 2​ + 1]​ -4
C
(x​ 2​ + 1)-4 and [(x​ 3​ – 4x)​ 2​ + 1]​ 4
D
(x​ 2​ + 1)-4 and [(x​ 3​ – 4x)​ 2​ + 1]​ -4
       Engineering-Mathematics       Calculus       UGC NET CS 2017 Jan -paper-2
Question 236 Explanation: 
Step-1: Given data,
f(x) = x​ 3​ – 4x, g(x) = 1/(x​ 2​ + 1) and h(x) = x​ 4
hog(x)=h(1/(x​ 2​ + 1))
=h(1/(x​ 2​ )+1)​ 4
= 1/(x​ 2​ +1)​ 4
= (x​ 2​ +1)​ -4
hogof(x)= hog(x​ 3​ -4x)
= h(1/(x​ 3​ -4x)​ 2​ +1)
= h(1/(x​ 3​ -4x)​ 2​ +1)​ 4
= h((x​ 3​ -4x)​ 2​ +1)​ -4
So, option D id is correct answer.
Question 237
How many multiples of 6 are there between the following pairs of numbers ?
0 and 100 and –6 and 34
A
16 and 6
B
17 and 6
C
17 and 7
D
16 and 7
       Engineering-Mathematics       Combinatorics       UGC NET CS 2017 Jan -paper-2
Question 237 Explanation: 
Method-1:
0 and 100 → Counting sequentially:
0,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96 Total=17
–6 and 34 → Counting sequentially: -6,0,6,12,18,24,30
Total=7
Method-2: 0 and 100 → Maximum number is 100. Divide ⌊100/6⌋ = 16+1 =17
Here, +1 because of 0.
–6 and 34 → Maximum number is 34. Divide ⌊34/6⌋ = 5+1+1 =7
Here, +1 because of 0 and +1 for -6
Question 238
Consider a Hamiltonian Graph G with no loops or parallel edges and with |V(G)| = n ≥ 3. Then which of the following is true ?
A
deg(v) ≥n/2 for each vertex v.
B
|E(G)| ≥1/2(n – 1) (n – 2) + 2
C
deg (v) + deg(w) ≥ n whenever v and w are not connected by an edge
D
All of the above
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2017 Jan -paper-2
Question 238 Explanation: 
With the help of dirac’s theorem, we can prove above three statements.
Question 239
In propositional logic if (P → Q) ∧ (R → S) and (P ∨ R) are two premises such that
A
P ∨ R
B
P ∨ S
C
Q ∨ R
D
Q ∨ S
E
None of These
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2017 Jan -paper-2
Question 239 Explanation: 
Option-A: Let P be TRUE and R be false, then the conclusion PVR will be TRUE. Now if we make Q as false then premises (P→Q)∧(R→ S)will be false because P→ Q is false. Hence this option is not correct.
Option-B: Let P be TRUE and S be false then the conclusion PVS is TRUE. Now, if we make R as TRUE then the premises (P→ Q)∧(R→ S) will be false because (R→ S) will be false. Hence this option is not correct.
Option-C: Let Q be false, R be TRUE then conclusion QVR will be TRUE. Now if we make S as FALSE then Premises (P→ Q)∧(R→ S) will be FALSE because (R→ S) will be false. Hence, this option is not correct.
Option-D: Let Q be TRUE and S be FALSE then conclusion QVS will be TRUE. Now if we make R as TRUE then premises (P→ Q)∧(R→ S) will be FALSE because (R→ S) will be false.
Therefore None of the given options are correct.
Note: As per UGC NET key, given option D as correct answer.
Question 240
Let A and B be sets in a finite universal set U. Given the following: |A – B|, |A ⊕ B|, |A| + |B| and |A ∪ B| Which of the following is in order of increasing size ?
A
|A – B| ≤ |A ⊕ B| ≤ |A| + |B| ≤ |A ∪ B|
B
|A ⊕ B| ≤ |A – B| ≤ |A ∪ B| ≤ |A| + |B|
C
|A ⊕ B| ≤ |A| + |B| ≤ |A – B| ≤ |A ∪ B|
D
|A – B| ≤ |A ⊕ B| ≤ |A ∪ B| ≤ |A| + |B|
       Engineering-Mathematics       Set-Theory       UGC NET CS 2016 Aug- paper-2
Question 240 Explanation: 
Step-1: Let A and B be sets in a finite universal set U.
Step-2: |A–B| we can also write into |A|-|A ∩ B| and equivalent Venn diagram is

Step-3: |A⊕B| We can also write into |A|+|B|- 2|A∩B| and equivalent Venn diagram is

Step-4: |A| + |B| We can represented into |A| + |B| and equivalent Venn diagram is

Step-5: |A∪B| We can also write into |A|+|B|- |A∩B| and equivalent Venn diagram is

Step-6: |A – B| ≤ |A ⊕ B| ≤ |A ∪ B| ≤ |A| + |B| is correct order.
Question 241
What is the probability that a randomly selected bit string of length 10 is a palindrome?
A
1/64
B
1/32
C
1/8
D
1⁄4
       Engineering-Mathematics       Probability       UGC NET CS 2016 Aug- paper-2
Question 241 Explanation: 
Palindrome is a number that remains the same when its digits are reversed.
Ex: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121...,
Step-1:​ Here, string length is 10. If we consider first 5 numbers as random choices like 0 or 1.
Remaining 5 numbers are fixed. Total number of possibilities are 2​ 10​ . But we are only considering first 5 choices. The probability is 2​ 5​ .
Step-2:​ The probability 2​ 5​ /2​ 10
= 1/2​ 5
= 1/32
Question 242
Given the following graphs:

Which of the following is correct?
A
G​ 1​ contains Euler circuit and G​ 2​ does not contain Euler circuit.
B
G​ 1​ does not contain Euler circuit and G​ 2​ contains Euler circuit.
C
Both G​ 1​ and G​ 2​ do not contain Euler circuit.
D
Both G​ 1​ and G​ 2​ contain Euler circuit.
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2016 Aug- paper-2
Question 242 Explanation: 

Step-1: G1 have odd number of vertices. So, it is not euler circuit.
Step-2: G2 also have odd number of vertices. So, it not euler circuit.
Question 243
How many different equivalence relations with exactly three different equivalence classes are there on a set with five elements?
A
10
B
15
C
25
D
30
       Engineering-Mathematics       Sets-And Relation       UGC NET CS 2016 July- paper-2
Question 243 Explanation: 
Step-1: Given number of equivalence classes with 5 elements with three elements in each class will be 1,2,2 (or) 2,1,2 (or) 2,2,1 and 3,1,1.
Step-2: The number of combinations for three equivalence classes are
2,2,1 chosen in (​ 5​ C​ 2​ *​ 3​ C​ 2​ *​ 1​ C​ 1​ )/2! = 15
3,1,1 chosen in(​ 5​ C​ 2​ *​ 3​ C​ 2​ *​ 1​ C​ 1​ )/2! = 10
Step-3: Total differential classes are 15+10
=25.
Question 244
The number of different spanning trees in complete graph, K​ 4​ and bipartite graph, K​ 2,2​ have ______ and _______ respectively.
A
14, 14
B
16, 14
C
16, 4
D
14, 4
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2016 July- paper-2
Question 244 Explanation: 
Step-1: Given complete graph K​ 4​ .To find total number of spanning tree in complete graph using standard formula is n​ (n-2) Here, n=4
=n​ (n-2)
= 4​ 2
=16
Step-2: Given Bipartite graph K​ 2,2​ . To find number of spanning tree in a bipartite graph K​ m,n​ having standard formula is m​ (n-1)​ * n​ (m-1)​ .
m=2 and n=2
= 2​ (2-1)​ * 2​ (2-1)
= 2 * 2
= 4
Question 245
Suppose that R​ 1​ and R​ 2​ are reflexive relations on a set A. Which of the following statements is correct ?
A
R​ 1​ ∩ R​ 2​ is reflexive and R​ 1​ ∪ R​ 2​ is irreflexive.
B
R​ 1​ ∩ R​ 2​ is irreflexive and R​ 1​ ∪ R​ 2​ is reflexive.
C
Both R​ 1​ ∩ R​ 2​ and R​ 1​ ∪ R​ 2​ are reflexive.
D
Both R​ 1​ ∩ R​ 2​ and R​ 1​ ∪ R​ 2​ are irreflexive.
       Engineering-Mathematics       Sets-And Relation       UGC NET CS 2016 July- paper-2
Question 245 Explanation: 
A binary relation R over a set X is reflexive if every element of X is related to itself. Formally, this may be written ​ ∀ ​ x ∈X : xRx.
Ex: Let set A={0,1}
R​ 1​ ={(0,0),(1,1)} all diagonal elements we are considering for reflexive relation.
R​ 2​ ={(0,0),(1,1)} all diagonal elements we are considering for reflexive relation.
R​ 1​ ∩ R​ 2​ must have {(0,0),(1,1)} is reflexive.
R​ 1​ ∪ R​ 2​ must have {(0,0),(1,1)} is reflexive.
Question 246
There are three cards in a box. Both sides of one card are black, both sides of one card are red, and the third card has one black side and one red side. We pick a card at random and observe only one side. What is the probability that the opposite side is the same colour as the one side we observed?
A
3/4
B
2/3
C
1/2
D
1⁄3
       Engineering-Mathematics       Probability       UGC NET CS 2016 July- paper-2
Question 246 Explanation: 
Given data,
-- 3 cards in a box
-- 1​ st​ card: Both sides of one card is black. The card having 2 sides. We can write it as BB.
-- 2​ nd​ card: Both sides of one card is red. The card having 2 sides. We can write it as RR.
-- 3rd card: one black side and one red side.​ ​ We can write it as BR.
Step-1: The probability that the opposite side is the same colour as the one side we observed is 2⁄3 because total number of cards are 3
Question 247
A clique in a simple undirected graph is a complete subgraph that is not contained in any larger complete subgraph. How many cliques are there in the graph shown below?

A
2
B
4
C
5
D
6
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2016 July- paper-2
Question 247 Explanation: 
Definition of clique is already given in question.
Definition: A clique in a simple undirected graph is a complete subgraph that is not contained in any larger complete subgraph.
Step-1: b,c,e,f is complete graph.

Step-2: ‘a’ is not connected to ‘e’ and ‘b’ is not connected to ‘d’. So, it is not complete graph.
Question 248
Which of the following statement(s) is/are false ?
(a) A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.
(b) A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree.
(c) A complete graph (K​ n​ ) has a Hamilton Circuit whenever n ≥ 3.
(d) A cycle over six vertices (C​ 6​ ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
A
(a) only
B
(b) and (c)
C
(c) only
D
(d) only
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2015 Dec- paper-2
Question 248 Explanation: 
(a)TRUE: A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.
(b)TRUE: A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree.
(c)TRUE: A complete graph (K​ n​ ) has a Hamilton Circuit whenever n ≥ 3. (d) FALSE: A cycle over six vertices (C​ 6​ ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
Question 249
Which of the following is/are not true?
(a) The set of negative integers is countable.
(b) The set of integers that are multiples of 7 is countable.
(c)The set of even integers is countable.
(d)The set of real numbers between 0 and 1⁄2 is countable.
A
(a) and (c)
B
(b) and (d)
C
(b) only
D
(d) only
       Engineering-Mathematics       Set-Theory       UGC NET CS 2015 Dec- paper-2
Question 249 Explanation: 
(a)TRUE: The set of negative integers is countable.
Suppose negative integers set size is 10.
Ex: -1, -2, -3,.....,-10 is countable
(b)TRUE: The set of integers that are multiples of 7 is countable.
Suppose set of integers size is 10.
Ex: 1*7, 2*7, 3*7, .....,10*7 is countable
(c)TRUE: The set of even integers is countable.
Suppose set of even integers size is 10.
Ex: 2,4,6,8,10,....,20
(d) FALSE: The set of real numbers between 0 and 1⁄2 is countable. We can’t count real numbers.
Ex: 0.1, 0.2, 0.3, .....,0.∞
Question 250

Consider the graph given below: The two distinct sets of vertices, which make the graph bipartite are:
A
(v​ 1​ , v​ 4​ , v​ 6​ ); (v​ 2​ , v3​ , v​ 5​ , v​ 7​ , v​ 8​ )
B
(v​ 1​ , v​ 7​ , v​ 8​ ); (v​ 2​ , v​ 3​ , v​ 5​ , v​ 6​ )
C
(v​ 1​ , v​ 4​ , v​ 6​ , v​ 7​ ); (v​ 2​ , v​ 3​ , v​ 5​ , v​ 8​ )
D
(v​ 1​ , v​ 4​ , v​ 6​ , v​ 7​ , v​ 8​ ); (v​ 2​ , v​ 3​ , v​ 5​ )
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2015 Dec- paper-2
Question 250 Explanation: 
A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
→ The two sets U and V may be thought of as a coloring of the graph with two colors.
Option A: FALSE because V​ 5​ , V​ 7​ and V​ 3​ are adjacent. So, it not not bipartite graph.

Option-B FALSE because V​ 5​ , V​ 6​ and V​ 2​ are adjacent. So, it not not bipartite graph.

Option-C TRUE because it follows properties of bipartied and no two colours are adjacent.

Option-D FALSE because because V​ 4​ , V​ 6​ and V​ 8​ are adjacent. So, it not not bipartite graph.
Question 251
A tree with n vertices is called graceful, if its vertices can be labelled with integers 1, 2,....n such that the absolute value of the difference of the labels of adjacent vertices are all different. Which of the following trees are graceful?
(a)
(b)
(c)
A
(a) and (b)
B
(b) and (c)
C
(a) and (c)
D
(a), (b) and (c)
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2015 Dec- paper-2
Question 251 Explanation: 
Above all graphs are graceful.

Question 252
Which of the following arguments are not valid ?
(a) “If Gora gets the job and works hard, then he will be promoted. If Gora gets promotion, then he will be happy. He will not be happy, therefore, either he will not get the job or he will not work hard”.
(b) “Either Puneet is not guilty or Pankaj is telling the truth. Pankaj is not telling the truth, therefore, Puneet is not guilty”.
(c) If n is a real number such that n >1, then n​ 2 ​ >1. Suppose that n​ 2 ​ >1, then n >1.
A
(a) and (c)
B
(b) and (c)
C
(a), (b) and (c)
D
(a) and (b)
       Engineering-Mathematics       Combinatorics       UGC NET CS 2015 Dec- paper-2
Question 253
Match the following terms:

A
(i)(ii)(iii)(iv)
B
(ii)(iii)(i)(iv)
C
(iii)(ii)(iv)(i)
D
(iv)(iii)(ii)(i)
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2015 Dec- paper-2
Question 253 Explanation: 
→ Vacuous proof is a proof that the implication p → q is true based on the fact that p is false.
→ Trivial proof is a proof that the implication p → q is true based on the fact that q is true.
→ Direct proof is a proof that the implication p → q is true that proceeds by showing that q must be true when p is true.
→ Indirect proof is a proof that the implication p → q is true that proceeds by showing that p must be false when q is false.
Question 254
Consider the compound propositions given below as:
(a)p ∨ ~(p ∧ q)
(b)(p ∧ ~q) ∨ ~(p ∧ q)
(c)p ∧ (q ∨ r)
Which of the above propositions are tautologies?
A
(a) and (c)
B
(b) and (c)
C
(a) and (b)
D
only (a)
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2015 Dec- paper-2
Question 254 Explanation: 
Question 255
Which of the following property/ies a Group G must hold, in order to be an Abelian group?
(a)The distributive property
(b)The commutative property
(c)The symmetric property
A
(a) and (b)
B
(b) and (c)
C
(a) only
D
(b) only
       Engineering-Mathematics       Set-Theory       UGC NET CS 2015 Dec- paper-2
Question 255 Explanation: 
An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b. The symbol • is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the abelian group axioms:
Closure: For all a, b in A, the result of the operation a • b is also in A.
Associativity: For all a, b and c in A, the equation (a • b) • c = a • (b • c) holds.
Identity element: There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds.
Inverse element: For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element.
Commutativity: For all a, b in A, a • b = b • a.
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".
Question 256
How many solutions are there for the equation x + y + z + u = 29 subject to the constraints that x ≥ 1, y ≥ 2, z ≥ 3 and u ≥ 0?
A
4960
B
2600
C
23751
D
8855
       Engineering-Mathematics       Permutations-combinations       UGC NET CS 2015 Dec- paper-2
Question 256 Explanation: 
Given data,
-- The equation x+y+z+u = 29
-- Constraints are x ≥ 1, y ≥ 2, z ≥ 3 and u ≥ 0
-- Possible solutions?
Step-1: Combine all constraints (1+2+3+0) =6
Step-2: Subtract all constraints values with total equation number 29-6 =23.
Possibility to get repetition of a numbers of x,y and z but no chance for ‘u’ because its value is 0.
Step-3: So, Subtract repeated values into total equation value
= 29-3
= 26
Step-4: Possible solutions= 26 C​ 23
= 2600
Question 257
How many strings of 5 digits have the property that the sum of their digits is 7?
A
66
B
330
C
495
D
99
       Engineering-Mathematics       Combinatorics       UGC NET CS 2015 Jun- paper-2
Question 257 Explanation: 
→ From the given question, there should be string with 5 digits and sum of that digits should be 7
→ There is no specification about positive,negative digits and also repetition of digits.
→ We are assuming the positive digits which is greater than or equal to 0.
→ The starting digit of the string should not be zero. We will also consider the repetition of digits.
→ Example of such strings are 70000,61000,60100,60010..,
→ The possible number of strings are C((n+r-1),(r-1))
→ From the given data n=7,r=5 then We have to fine C(11,4) which is equal to 330
Question 258
Consider an experiment of tossing two fair dice, one black and one red. What is the probability that the number on the black die divides the number on red die?
A
22 / 36
B
12 / 36
C
14 / 36
D
6 / 36
       Engineering-Mathematics       Probability       UGC NET CS 2015 Jun- paper-2
Question 258 Explanation: 
→ From the given data, there are two dice and each dice the possible ways are 6. So total possible ways are 6*6=36 ways.
→ The possible ways are 1,2,3,4,5,6
→ The possible ways of one number divides another number are (1,1) (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5) and (6,6).
→ The probability is the number of outcomes / total outcomes = 14/36
Question 259
In how many ways can 15 indistinguishable fish be placed into 5 different ponds, so that each pond contains at least one fish ?
A
1001
B
3876
C
775
D
200
       Engineering-Mathematics       Combinatorics       UGC NET CS 2015 Jun- paper-2
Question 259 Explanation: 
→ We know that if we have “n” identical items which will be distributed in “r” distinct groups where each must get at least one then the number of way is C(n−1,r−1)
→ From the given question, n=15, r=5 We need to calculate C(14,4) and it’s value is 1001.
Question 260
Consider a Hamiltonian Graph (G) with no loops and parallel edges. Which of the following is true with respect to this Graph (G) ?
(a) deg (v) ≥ n / 2 for each vertex of G
(b) |E(G)| ≥ 1 / 2 (n - 1) (n - 2) + 2 edges
(c) deg (v) + deg (w) ≥ n for every n and v not connected by an edge.
A
(a) and (b)
B
(b) and (c)
C
(a) and (c)
D
(a), (b) and (c)
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2015 Jun- paper-2
Question 260 Explanation: 
→ According to Dirac's theorem on Hamiltonian cycles, the statement that an n-vertex graph in which each vertex has degree at least n/2 must have a Hamiltonian cycle.
→ According to ore’s theorem,Let G be a (finite and simple) graph with n ≥ 3 vertices. We denote by deg v the degree of a vertex v in G, i.e. the number of incident edges in G to v. Then, Ore's theorem states that if deg v + deg w ≥ n for every pair of distinct non adjacent vertices v and w of G, then G is Hamiltonian.
→ A complete graph G of n vertices has n(n−1)/2 edges and a Hamiltonian cycle in G contains n edges. Therefore the number of edge-disjoint Hamiltonian cycles in G cannot exceed (n − 1)/2. When n is odd, we show there are (n − 1)/2 edge-disjoint Hamiltonian cycles.
So the statement(b) is false and statement a and c are true.
Question 261
“If my computations are correct and I pay the electric bill, then I will run out of money. If I don’t pay the electric bill, the power will be turned off. Therefore, if I don’t run out of money and the power is still on, then my computations are incorrect.”
Convert this argument into logical notations using the variables c, b, r, p for propositions of computations, electric bills, out of money and the power respectively. (Where ¬ means NOT)
A
if (c Λ b)→r and ¬b→p, then (¬r Λ p)→¬c
B
if (c ∨ b)→r and ¬b→¬p, then (r Λ p)→c
C
if (c Λ b)→r and ¬p→b, then (¬r ∨ p)→¬c
D
if (c ∨ b)→r and ¬b→¬p, then (¬r Λ p)→¬c
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2015 Jun- paper-2
Question 261 Explanation: 
We can represent ,
“c” for my computations are correct
“b” for I pay the electric bill.
“r” for I will run out of money
“p” for the power is on.
(c Λ b) means my computations are correct and I pay the electric bill.
(¬r Λ p) means I don’t run out of money and the power is still on.
According to the statement , the option -(A) is correct.
Question 262
Match the following:
A
(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
B
(a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)
C
(a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)
D
(a)-(iv), (b)-(ii), (c)-(iii), (d)-(i)
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2015 Jun- paper-2
Question 262 Explanation: 
→ In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of P → Q is thus as ~Q → ~P.
→ The equivalence relation translates verbally into "if and only if" and is symbolized by a double-lined, double arrow pointing to the left and right (↔). If A and B represent statements, then A ↔ B means "A if and only if B."
→ The exportation rule is a rule in logic which states that "if (P and Q), then R" is equivalent to "if P then (if Q then R)".
→ The exportation rule may be formally stated as: (P ∧ Q) → R is equivalent to P → (Q →R)
→ A mode of argumentation or a form of argument in which a proposition is disproven by following its implications logically to an absurd conclusion. Arguments that use universals such as, “always”, “never”, “everyone”, “nobody”, etc., are prone to being reduced to absurd conclusions. The fallacy is in the argument that could be reduced to absurdity -- so in essence, reductio ad absurdum is a technique to expose the fallacy.
Question 263
Consider a proposition given as :
“ x ≥ 6, if x​ 2​ ≥ 5 and its proof as:
If x ≥ 6, then x​ 2​ = x.x ≥ 6.6 = 36 ≥ 25
Which of the following is correct w.r.to the given proposition and its proof?
(a)The proof shows the converse of what is to be proved.
(b)The proof starts by assuming what is to be shown.
(c)The proof is correct and there is nothing wrong.
A
(a) only
B
(c) only
C
(a) and (b)
D
(b) only
       Engineering-Mathematics       Combinatorics       UGC NET CS 2015 Jun- paper-2
Question 263 Explanation: 
The proof which is described in the question is wrong because for a given “x” value , x​ 2​ >=5 but in the proof they mentioned 36>=25 which is wrong.
Question 264
Match the following:
A
(a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)
B
(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
C
(a)-(iv), (b)-(i), (c)-(iii), (d)-(ii)
D
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
       Engineering-Mathematics       Linear-Algebra       UGC NET CS 2015 Jun- paper-2
Question 264 Explanation: 
Forward Reference Table → Uses linked list data structure
Mnemonic Table → Contains machine OP code
Segment Register Table → Uses array data structure
EQU → Assembler directive.
The EQU directive gives a symbolic name to a numeric constant, a register-relative value or a PC-relative value.
Question 265
AVA=A is called :
A
Identity law
B
De Morgan’​ s law
C
Idempotent law
D
Complement law
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2004 Dec-Paper-2
Question 265 Explanation: 
→ ​ De Morgan’s Laws:
(i). (A ​ V ​ B)’ = A' ​ ∧ ​ B'
(ii). (A ​ ∧ ​ B)’ = A' ​ V ​ B'
→ ​ Identity Law​ :
(i). 1 AND A = A
(ii). 0 OR A = A
→ ​ Complement law:
(i). A AND A'=1
(ii). A OR A'=0
→ ​ Idempotent law:
The idempotence in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).
(i). A V A=A
(ii). A ∧ A=A
According to boolean algebra
Question 266
If f(x) = x+1 and g(x)=x+3 then f0 f0 f0 f is :
A
g
B
g+1
C
g​ 4
D
None of the above
       Engineering-Mathematics       Relations-and-Functions       UGC NET CS 2004 Dec-Paper-2
Question 266 Explanation: 
Given data,
f(x)=x+1
g(x)=x+3
Constraint is f0 f0 f0 f
Step-1: We can write into fo fo fo f is f(f(f(x+1)))
We can write into f(f(x+2)) and f(x+3).
Step-2: Above constraint is equal to "x+4" because f(x+3)+1
Step-3: We can also write into fog(x)=x+4 and gof(x)=x+4.
So, g+1 is appropriate answer.
Question 267
The following lists are the degrees of all the vertices of a graph :
(i) 1, 2, 3, 4, 5
(ii) 3, 4, 5, 6, 7
(iii) 1, 4, 5, 8, 6
(iv) 3, 4, 5, 6
then
A
(i) and (ii)
B
(iii) and (iv)
C
(iii) and (ii)
D
(ii) and (iv)
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2004 Dec-Paper-2
Question 267 Explanation: 
Every graph is following basic 2 properties:
1. Sum of degrees of the vertices of a graph should be even.
2. Sum of degrees of the vertices of a graph is equal to twice the number of edges.
Statement-(i) is violating property-1.
= 1+2+3+4+5
= 15 is odd number.
Statement-(ii) is violating property-1.
= 3+4+5+6+7
= 25 is odd number.
Statement-(iii) is violating property-1.
= 1+4+5+8+6
= 24 is even number
Statement-(iv) is violating property-1
= 3+4+5+6
= 18 is even number
Question 268
If I​ m denotes the set of integers modulo m, then the following are fields with respect to the operations of addition modulo m and multiplication modulo m :
(i) Z​ 23
(ii) Z​ 29
(iii) Z​ 31
(iv) Z​ 33
Then
A
(i) only
B
(i) and (ii) only
C
(i), (ii) and (iii) only
D
(i), (ii), (iii) and (iv)
       Engineering-Mathematics       Set-Theory       UGC NET CS 2004 Dec-Paper-2
Question 269
T is a graph with n vertices. T is connected and has exactly n-1 edges, then :
A
T is a tree
B
T contains no cycles
C
Every pairs of vertices in T is connected by exactly one path
D
All of these
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2005 Dec-Paper-2
Question 269 Explanation: 
This is little bit tricky question.
Step-1:
n= number of vertices
n-1 = number of edges
Example: n=5 vertices and n-1=4 edges

Step-2: The above graph T won’t have cycle then we are calling as tree. Here, every pairs of vertices in T is connected by exactly one path.
Note: The above properties is nothing but minimum spanning tree properties.
Question 270
If the proposition ​ ¬ ​ P ​ → ​ Q is true, then the truth value of the proportion ​ ¬ ​ PV (P ​ → ​ Q) is:
A
True
B
Multi - Valued
C
Flase
D
Can not determined
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2005 Dec-Paper-2
Question 270 Explanation: 
We can also write (¬p → q) into (p ∨ q)

Here, we can minimize the boolean form into
¬p ∨ (p → q)
= ¬p ∨ (¬p ∨ q)
= ¬p ∨ q
In question, they are not discussed about the truth values of p, it implies that ¬p ∨ q also be True or False. So, we cannot be determined.
Question 271
Let A and B be two arbitrary events, then :
A
P(A∩B) = P(A) P(B)
B
P(P∪B) = P(A) + P(B)
C
P(A∪B) ≤ P(A) + P(B)
D
P(A​ / ​ B) = P(A∩B) + P(B)
       Engineering-Mathematics       Probability       UGC NET CS 2005 Dec-Paper-2
Question 271 Explanation: 
Option-A is happens when A and B are independent.
Option-B is happens when A and B are mutually exclusive.
Option-C is not happens.
Option-D is P(A∪B) ≤ P(A) + P(B) is true because P(A∪B) = P(A) + P(B) - P(A∩B).
Question 272
The transitive closure of a relation R on set A whose relation matrix

is
A
B
C
D
       Engineering-Mathematics       Set-Theory       UGC NET CS 2005 june-paper-2
Question 272 Explanation: 
The transitive closure of R is obtained by repeatedly adding (a,c) to R for each (a,b) ∈ R and (b,c) ∈ R.
Question 273
Consider the relation on the set of non-negative integers defined by ​ x ​ ≡ ​ y if and only if :
A
x mod 3=3 mod y
B
3 mod x≡3 mod y
C
x mod 3=y mod 3
D
None of the above
       Engineering-Mathematics       Set-Theory       UGC NET CS 2005 june-paper-2
Question 273 Explanation: 
A relation R is an equivalence relation if and only if it is reflexive, symmetric, and transitive.
1. The relation is reflexive: x mod 3 = x mod 3
2. The relation is symmetric: if x mod 3 = y mod 3, then y mod 3 = x mod 3
3. The relation is transitive: if x mod 3 = y mod 3, and y mod 3 = z mod 3, then x mod 3 = z mod 3
Question 274
Minimum number of individual shoes to be picked up from a dark room ( containing 10 pair of shoes) if we have to get at least one proper pair :
A
2
B
20
C
11
D
None of these
       Engineering-Mathematics       Combinatorics       UGC NET CS 2005 june-paper-2
Question 274 Explanation: 
→ There are 10 pair of shoes available in the dark room which means 20 individual shoes are available in the room.
→ If you pick shoes from one to ten individual shoes, there may be chance of getting same individual shoes.
→ There is no guarantee that getting one proper pair from 10 individual shoes.
→ If You pick 11 shoes, then there may chance of 10 individual shoes of same type and one individual shoe of another type. So we will get at least one proper pair shoe from 11 individual shoes.
Question 275
The proposition ~q ∨ p is equivalent to :
A
Not given any option
B
Not given any option
C
Not given any option
D
Not given any option
E
~q ∨ p ≣ q → p
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2006 Dec-paper-2
Question 275 Explanation: 
Options are not given. Excluded for evaluation.
~q ∨ p ≣ q → p
Question 276
The number of edges in a complete graph with N vertices is equal to :
A
N (N−1)
B
2N−1
C
N−1
D
N(N−1)/2
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2006 Dec-paper-2
Question 276 Explanation: 
N =5
Question 277
Which of the following is not true ?
A
Not given any option
B
A – B = A ∩~B
C
Not given any option
D
Not given any option
       Engineering-Mathematics       Set-Theory       UGC NET CS 2006 Dec-paper-2
Question 277 Explanation: 
A−B= A−(A∩B)
= A-(A∩B)
= A∩(A∩B)’
= A∩(A’ ∩ B’)
= (A∩A’)U(A∩B’)
= U(A∩B’)
= ​ ∅ U (​ A ∩ B’)
= A ∩ B’
Question 278
If (a​ 2​ −b​ 2​ ) is a prime number where a and b ε N, then :
A
(a​ 2​ −b​ 2​ )​ =3
B
(a​ 2​ −b​ 2​ )​ =a−b
C
(a​ 2​ −b​ 2​ )​ =a+b
D
(a​ 2​ −b​ 2​ )​ =5
       Engineering-Mathematics       Set-Theory       UGC NET CS 2006 Dec-paper-2
Question 278 Explanation: 
→ For any given numbers a and b which belongs to natural numbers , the options (A) and (D) are false
→ The set of natural numbers, denoted N, can be defined in the following ways: N = {0, 1, 2, 3, ...}
→ A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
→ ‘a-b’ may be gives the non negative values which is not prime number but the a+b may give the prime value.
Question 279
For a complete graph with N vertices, the total number of spanning trees is given by :
A
2N-1
B
N​ N-1
C
N​ N-2
D
2N+1
       Engineering-Mathematics       Graph-Theory       UGC NET CS 2006 June-Paper-2
Question 279 Explanation: 
If a graph is complete, total number of spanning trees are N​ N-2
Example:

Formula to find total number of spanning trees are N​ N-2
=5​ 5-2
=5​ 3
=125
Question 280
The preposition( p→q) ∧ (~q ∨ p) is equivalent to :
A
q →p
B
p→ q
C
(q →p)∨(p→ q)
D
(p →q)∨(q→ p)
E
None of the above
       Engineering-Mathematics       Propositional-Logic       UGC NET CS 2006 June-Paper-2
Question 280 Explanation: 
Question 281
The logic of pumping lemma is a good example of :
A
pigeon hole principle
B
recursion
C
divide and conquer technique
D
iteration
       Engineering-Mathematics       Combinatorics       UGC NET CS 2006 June-Paper-2
Question 281 Explanation: 
→ A pumping lemma (or) pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.
→ The proofs of these lemmas typically require counting arguments such as the pigeonhole principle.
→ Hence, the logic of pumping lemma is a good example of the pigeonhole principle.
Question 282
Let A={ x | -1 < x < 1 }=B. The function f(x)=x/2 from A to B is :
A
injective