Engineering-Mathematics
Question 1 |
Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?
A | AA′ = 1 |
B | A = A-1 |
C | AB = BA |
D | (AB)' = BA |
Question 2 |
Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).
A | yn+1 = yn + hf(xn, yn) |
B | yn+1 = yn + hf(xn+1, yn+1) |
C | yn+1 = yn-1 + 2hf(xn, yn) |
D | yn+1 = (1 + h) f(xn+1, yn+1) |
With initial value y(x0) = y0. Here the function f and the initial data x0 and y0 are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y0, y1, y2, ....... such that yn approximates y(x0 + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
yn+1 = yn + hf(x n+1, yn+1)
Question 3 |
Let A and B be any two arbitrary events, then, which one of the following is true?
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) |
(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 4 |
The number of distinct simple graphs with upto three nodes is
A | 15 |
B | 10 |
C | 7 |
D | 9 |
Question 5 |
The tank of matrix is:
A | 0 |
B | 1 |
C | 2 |
D | 3 |
Question 6 |
Some group (G,o) is known to be abelian. Then, which one of the following is true for G?
A | g = g-1 for every g ∈ G |
B | g = g2 for every g ∈ G |
C | (goh)2 = g2oh2 for every g,h ∈ G |
D | G is of finite order |
For Abelian group, commutative also holds
i.e., (aob) = (boa)
Consider option (C):
(goh)2 = (goh)o(gog)
= (hog)o(goh)
= (ho(gog)oh)
= ((hog2)oh)
= (g2oh)oh
= g2o(hoh)
= g2oh2 [True]
Question 7 |
In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, non-zero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)th element of the lower triangular matrix in this new representation is:
A | i + j |
B | i + j - 1 |
C | j + i(i-1)/2 |
D | i + j(j-1)/2 |
If we assume array index starting from 1 then, ith row contains i number of non-zero elements. Before ith row there are (i-1) rows, (1 to i-1) and in total these rows has 1+2+3......+(i-1) = i(i-1)/2 elements.
Now at ith row, the jth element will be at j position.
So the index of (i, j)th element of lower triangular matrix in this new representation is
j = i(i-1)/2
Question 8 |
The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is
A | n |
B | n2 |
C | n(n-1)/2 |
D | n(n+1)/2 |
n = 1
(n-1) = 2
(n-2) = 3
So, Total = n(n+1)/2
Question 9 |
On the set N of non-negative integers, the binary operation __________ is associative and non-commutative.
A | fog |
(fog)(x) = f(g(x))
It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.
Note that if fog exists then gof might not even exists.
Question 10 |
Amongst the properties {reflexivity, symmetry, anti-symmetry, transitivity} the relation R = {(x,y) ∈ N2 | x ≠ y } satisfies __________.
A | symmetry |
It is symmetric as if xRy then yRx.
It is not antisymmetric as xRy and yRx are possible and we can have x≠y.
It is not transitive as if xRy and yRz then xRz need not be true. This is violated when x=x.
So, symmetry is the answer.
Question 11 |
The number of subsets {1, 2, ... n} with odd cardinality is __________.
A | 2n-1 |
And so, no. of subsets with odd cardinality is half of total no. of subsets = 2n /n = 2n-1
Question 12 |
The number of edges in a regular graph of degree d and n vertices is _________.
A | d*n/2 |
d * n = 2 * |E|
∴ |E| = d*n/2
Question 13 |
The probability of an event B is P1. The probability that events A and B occur together is P2 while the probability that A and occur together is P3. The probability of the event A in terms of P1, P2 and P3 is __________.
A | P2 + P3 |
P3 = P(A) - P2
P(A) = P2 + P3
Question 14 |
Let A, B and C be independent events which occur with probabilities 0.8, 0.5 and 0.3 respectively. The probability of occurrence of at least one of the event is __________
A | 0.93 |
Since all the events are independent, so we can write
P(A∪B∪C) = P(A) + P(B) + P(C) - P(A)P(B) - P(B)P(C) - P(A)P(C) + P(A)P(B) P(C)
= 0.8 + 0.5 + 0.3 - 0.4 - 0.5 - 0.24 + 0.12
= 0.93
Question 15 |
The Hasse diagrams of all the lattices with up to four elements are __________ (write all the relevant Hasse diagrams).
A |
We can't draw lattice with 1 element.
For 2 element:
For 3 element:
For 4 element:
Question 16 |
Match the following items
A | (i) - (b), (ii) - (c), (iii) - (d), (iv) - (a) |
Question 17 |
Find the inverse of the matrix
A | |
B | |
C | |
D |
-λ3 + 2λ2 - 2 = 0
Using Cayley-Hamiltonian theorem
-A3 + 2A2 - 2I = 0
So, A-1 = 1/2 (2A - A2)
Solving we get,
Question 18 |
Let p and q be propositions. Using only the truth table decide whether p ⇔ q does not imply p → q is true or false.
A | True |
B | False |
So, "imply" is False making "does not imply" True.
Question 19 |
(a) Let * be a Boolean operation defined as
If C = A * B then evaluate and fill in the blanks:
(i) A * A = _______
(ii) C * A = _______
(b) Solve the following boolean equations for the values of A, B and C:
A | Theory Explanation. |
Question 20 |
A 3-ary tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a 3-ary tree with n interval nodes is 2(n-1)+3.
A | Theory Explanation. |
Question 21 |
Every element a of some ring (R,+,0) satisfies the equation aoa = a.
Decide whether or not the ring is commutative.
A | Theory Explanation. |
Question 22 |
The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 7 is:
A | 16/25 |
B | (9/10)3 |
C | 27/75 |
D | 18/25 |
Question 23 |
Let R be a symmetric and transitive relation on a set A. Then
A | R is reflexive and hence an equivalence relation |
B | R is reflexive and hence a partial order
|
C | R is reflexive and hence not an equivalence relation |
D | None of the above |
i) Symmetric
ii) Reflexive
iii) Transitive
If a relation is said to be symmetric and transitive then we can't say the relation is reflexive and equivalence.
Question 24 |
The number of elements in he power set P (S) of the set S = {(φ), 1, (2, 3)} is:
A | 2 |
B | 4 |
C | 8 |
D | None of the above |
P(S) = {φ, {{φ}}, {1}, {{2, 3}}, {{φ}, 1}, {1, {2, 3}}, {{φ}, 1, {2, 3}}}
In P(S) it contains 8 elements.
Question 25 |
In the interval [0, π] the equation x = cos x has
A | No solution |
B | Exactly one solution |
C | Exactly two solutions |
D | An infinite number of solutions |
x & cos(x) are intersecting at only one point.
Question 26 |
If at every point of a certain curve, the slope of the tangent equals −2x/y the curve is
A | a straight line |
B | a parabola |
C | a circle |
D | an ellipse |
Question 27 |
The value of k for which 4x2 - 8xy + ky2 = 0 does not represent a pair of straight lines (both passing through the origin) is:
A | 0 |
B | 2 |
C | 9 |
D | 3 |
Question 28 |
The rank of the following (n + 1)×(n+1) matrix, where a is a real number is
A | 1 |
B | 2 |
C | n |
D | Depends on the value of a |
Question 29 |
The minimum number of edges in a connected cyclic graph on n vertices is:
A | n - 1 |
B | n |
C | n + 1 |
D | None of the above |
In cyclic graph:
No. of edges = No. of vertices
⇒ n = n
Question 30 |
If the cube roots of unity are 1, ω and ω2, then the roots of the following equation are (x - 1)3 + 8 = 0
A | -1, 1 + 2ω, 1 + 2ω2 |
B | 1, 1 - 2ω, 1 - 2ω2 |
C | -1, 1 - 2ω, 1 - 2ω2 |
D | -1, 1 + 2ω, -1 + 2ω2 |
Question 31 |
A language with string manipulation facilities uses the following operations
head(s): first character of a string tail(s): all but the first character of a string concat(s1,s2):s1 s2 for the string acbc what will be the output of concat(head(s), head(tail(tail(s))))
A | ac |
B | bc |
C | ab |
D | cc |
concat (a, head (tail (cbc)))
concat (a, head (bc))
concat (a, b)
ab
Question 32 |
A unit vector perpendicular to both the vectors a = 2i - 2j + k and b = 1 + j - 2k is:
A | 1/√3 (1+j+k) |
B | 1/3 (1+j-k) |
C | 1/3 (1-j-k) |
D | 1/√3 (1+j-k) |
E | None of the above. |
Question 33 |
A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is:
A | 2/3 |
B | 4/5 |
C | 1/2 |
D | 2/1 |
Probability of first ball black and second one white is,
Question 34 |
The iteration formula to find the square root of a positive real number b using the Newton Raphson method is
A | |
B | |
C | |
D | None of the above |
Question 35 |
Let A be the set of all non-singular matrices over real number and let* be the matrix multiplication operation. Then
A | A is closed under* but < A, *> is not a semigroup |
B | |
C | |
D |
Question 36 |
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
A | C1ex + C2e2x |
B | C1e-x + C2e3x |
C | C1e-x + C2e-2x |
D | C1e-2x + C22-x |
Question 37 |
If the proposition ¬p ⇒ ν is true, then the truth value of the proposition ¬p ∨ (p ⇒ q), where ¬ is negation, ‘∨’ is inclusive or and ⇒ is implication, is
A | true |
B | multiple valued |
C | false |
D | cannot be determined |
So, either p or q must be True.
Now,
¬p ∨ (p → q)
= ¬p ∨ (¬p ∨ q)
= ¬p ∨ q
Since nothing c an be said about the truth values of p, it implies that ¬p ∨ q can also be True or False. Hence, the value cannot be determined.
Question 38 |
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion
A | Theory Explanation. |
Question 39 |
Obtain the principal (canonical) conjunctive normal form of the propositional formula
(p ∧ q) V (¬q ∧ r)
Where ‘∧’ is logical and, ‘v’ is inclusive or and ¬ is negation.
A | Theory Explanation. |
Question 40 |
Let G1 and G2 be subgroups of a group G.
(a) Show that G1 ∩ G2 is also a group of G.
(b) Is G1 ∪ G2 always a subgroup of G?
A | Theory Explanation. |
Question 41 |
Prove using mathematical induction for n≥5, 2n > n2
A | Theory Explanation. |
Question 42 |
Prove that in finite graph, the number of vertices of odd degree is always even.
A | Theory Explanation. |
Question 43 |
(a) Find the minimum value of 3 - 4x + 2x2.
(b) Determine the number of positive integers (≤ 720) which are not divisible by
any of numbers 2, 3, and 5.
A | Theory Explanation. |
Question 44 |
Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.
A | 7 |
If ‘H” is a subgroup of finite group (G,*) then O(H) is the divisor of O(G).
Given that the order of group is 35. Its divisors are 1,5,7,35.
It is asked that the size of largest possible subgroup other than G itself will be 7.
Question 45 |
Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.
A | 0.125 |
The number of reflexive relations is 2^(n^2-n).
The total number of relations on a set with n elements is 2^ (n^2).
The probability of choosing the reflexive relation out of set of relations is
= 2^(n^2-n) /2^ (n^2)
= 2^( n^2-n- n^2)
= 2^(-n)
Given n=3, the probability will be 2-n = ⅛ = 0.125
Question 46 |
Consider the functions
- I. e-x
II. x2-sin x
III. √(x3+1)
Which of the above functions is/are increasing everywhere in [0,1]?
A | II and III only |
B | III only |
C | II only |
D | I and III only |
I. e-x
II. f'(x) = -e-x
f'(x)<0 on the interval [0,1] so this is not an increasing function.
II. x2-sinx
f'(x) = 2x - cosx
at x=0, f'(0) = 2(0) - 1 = -1 < 0
f(x) = x2 - sinx is decreasing over some interval, increasing over some interval as cosx is periodic.
As the question is asked about increasing everywhere II is false.
III. √(x3+1) = (x3+1)1/2
f'(x) = 1/2(3x2/√(x3+1))>0
f(x) is increasing over [0,1].
Question 47 |
For n>2, let a ∈ {0,1}n be a non-zero vector. Suppose that x is chosen uniformly at random from {0,1}n. Then, the probability that is an odd number is _____.
A | 0.5 |
‘x’ is a vector chosen randomly from {0,1}n
‘a’ can have 2(n-1) possibilities, x can have 2n possibilities.
∑aixi have (2n-1)(2n) possibilities, which is an even number of outcomes.
The probability of https://solutionsadda.in/wp-content/uploads/2020/02/41.jpg is odd is ½.
For example:
Take n=3
a = {001, 010, 100, 011, 101, 111}
x = {000, 001, 010, 011, 100, 101, 110, 111}
Computed as [001]×[000] = 0+0+0 = 0 Output = even
[001]×[001] = 0+0+1 = 0 Output = odd
Similarly, there could be 28 even, 28 odd outputs for the a(size=7), x(size=8) of total 56 outputs.
Question 48 |
Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colours required to edge-colour G is _____.
A | 7 |
The vertex in the set of size 3 has 4 edges connected to 4 vertices on other set. So, edge color of G is max(3,4) i.e. 4.
When a vertex is added to the graph with 7 vertices ( K3x4 has 7 vertices), there would be 7 edges associated to that new vertex. As per the edge coloring “no two adjacent edges have same color).
As the new vertex with 7 edges need to be colored with 7 colors, the edge color of graph G is 7.
Question 49 |
Which one of the following predicate formulae is NOT logically valid?
Note that W is a predicate formula without any free occurrence of x.
A | ∃x(p(x) → W) ≡ ∀x p(x) → W |
B | ∀x(p(x) → W) ≡ ∀x p(x) → W
|
C | ∃x(p(x) ∧ W) ≡ ∃x p(x) ∧ W |
D | ∀x(p(x) ∨ W) ≡ ∀x p(x) ∨ W |
~p→q ≡ ~p∨q
Demorgan laws:
~(∀x(a(x)) ≡ ∃x~a(x)
~(∃x(a(x)) ≡ ∀x~a(x)
(A) ∃x(p(x)→w) ≡ ∀x p(x)→w
LHS: ∃x(p(x)→w) ≡ ∃x(~p(x)∨w)
≡ ∃x(~p(x))∨w
Demorgan’s law:
~(∀x(a(x)) = ∃x ~ a(x)
≡ ~(∀x P(x)) ∨ w
≡ (∀x) P(x) → w ≡ RHS
It’s valid.
(B) ∀x(P(x) → w) ≡ ∀x(~P(x) ∨ w)
≡ ∀x(~P(x)) ∨ w
≡ ~(∃x P(x)) ∨ w
≡ ∃x P(x) → w
This is not equal to RHS.
(C) ∃x(P(x) ∧ w) ≡ ∃x P(x) ∧ w
‘w’ is not a term which contains x.
So the quantifier does not have any impact on ‘w’.
Thus it can be written as
∃x(P(x)) ∧ w) ≡ ∃x P(x) ∧ w
(D) ∀(x)(P(x) ∨ w) ≡ ∀x P(x) ∨ w
‘w’ is not a term which contains ‘x’.
So the quantifier does not have an impact on ‘w’.
Thus ∀(x)(P(x) ∨ w) ≡ ∀x P(x) ∨ w
Question 50 |
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is _______.
A | 12 |
― ― ― ― ―
Given: L I L A C
The derangement formula ⎣n!/e⎦ cannot be directly performed as there are repeated characters.
Let’s proceed in regular manner:
The L, L can be placed in other ‘3’ places as
(1) Can be arranged such that A, I, C be placed in three positions excluding ‘C’ being placed at its own position, which we get only 2×2×1 = 4 ways.
Similarly (2) can be filled as A, I, C being placed such that 4th position is not filled by A, so we have 2×2×1 = 4 ways. Similarly with (3).
Totally, we get 4+4+4 = 12 ways.