EngineeringMathematics
Question 1 
The value of the above expression (rounded to 2 decimal places) is _______
A  0.25 
Question 2 
Which one of the following choices is correct?
A  Both S_{1}and _{S2} are tautologies. 
B  Neither _{S1}and _{S2} are tautology. 
C  _{S1}is not a tautology but _{S2}is a tautology. 
D  _{S1}is a tautology but _{S2}is not a tautology. 
A tautology is a formula which is "always true" . That is, it is true for every assignment of truth values to its simple components.
Method 1:
S1: (~p ^ (p Vq)) →q
The implication is false only for T>F condition.
Let's consider q as false, then
(~p ^ (p Vq)) will be (~p ^ (p VF)) = (~p ^ (p)) =F.
It is always F>F which is true for implication. So there are no cases that return false, thus its always True i.e. its Tautology.
S2:
q>(~p (p Vq))
The false case for implication occurs at T>F case.
Let q=T then (~p (p Vq)) = (~p (p VT))= ~p. (It can be false for p=T).
So there is a case which yields T>F = F. Thus its not Valid or Not a Tautology.
Method 2:
Question 3 
If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is _______
A  0.04 
Bayes theorem:
Probability of event A happening given that event B has already happened is
P(A/B) = P(B/A)*P(A) / P(B)
Here, it is asked that P( H transmitted / H received).
S can send signal to H with 0.1 probability, S can send signal to L with 0.9 probability.
The complete diagram can be
Probability that H Transmitted (H_t) given that H received (H_r)is
P( H_t / H_r) = P( H_r/ H_t) * P(H_t) / P(H_r)
P(Hr) = probability that H received = P( H received from H)+ P(H received from L)
It can be observed from the graph that H can receive in two ways (S to H to H) and (S to L to H)
The P(H_r) = 0.1*0.3 + 0.9*0.8= 0.03+0.72 = 0.75
P(H_received given that H_transmitted) =0.3
P(H transmitted ) = 0.1 i.e.
P( H_t / H_r) = P( H_r/ H_t) * P(H_t) / P(H_r)
= 0.3*0.1 / 0.75 = 0.04
Question 4 
A  11 
v  e + f = 2
v is number of vertices
e is number of edges
f is number of faces including bounded and unbounded
8e+5=2
=> 132 =e
The number of edges are =11
Question 5 
S1: There exist random variables X and Y such that
EXE(X)YE(Y)2>Var[X] Var[Y]
S2: For all random variables X and Y,
CovX,Y=EXE[X] YE[Y]
Which one of the following choices is correct?
A  S_{1}is false, but S_{2}is true. 
B  Both S_{1}and S_{2}are true. 
C  S_{1}is true, but S_{2}is false. 
D  Both S_{1}and S_{2} are false. 
For a dataset with single values, we have variance 0. EXE(X)YE(Y)2>Var[X] Var[Y]
This leads to inequance of 0>0 which is incorrect.
Its not xE(x). Thus S2 is also incorrect.
Question 6 
Which one of the following options is correct?
A  G is always cyclic, but H may not be cyclic. 
B  Both G and H are always cyclic. 
C  G may not be cyclic, but H is always cyclic. 
D  Both G and H may not be cyclic. 
If ‘G’ is a group with sides 6, its subgroups can have orders 1, 2, 3, 6.
(The subgroup order must divide the order of the group)
Given ‘H’ can be 1 to 6, but 4, 5 cannot divide ‘6’.
Then ‘H’ is not a subgroup.
G can be cyclic only if it is abelian. Thus G may or may not be cyclic.
The H can be cyclic only for the divisors of 6 and H cannot be cyclic for any non divisors of 6.
Question 7 
A  If a relation S is reflexive and circular, then S is an equivalence relation. 
B  If a relation S is transitive and circular, then S is an equivalence relation. 
C  If a relation S is circular and symmetric, then S is an equivalence relation. 
D  If a relation S is reflexive and symmetric, then S is an equivalence relation. 
Theorem: A relation R on a set A is an equivalence relation if and only if it is reflexive and circular.
For symmetry, assume that x, y ∈ A so that xRy, lets check for yRx.
Since R is reflexive and y ∈ A, we know that yRy. Since R is circular and xRy and yRy, we know that yRx. Thus R is symmetric.
For transitivity, assume that x, y, z ∈ A so that xRy and yRz. Check for xRz. Since R is circular and xRy and yRz, we know that zRx. Since we already proved that R is symmetric, zRx implies that xRz. Thus R is transitive.
Question 8 
A  3 
Question 9 
A  0.37 
Question 10 
Let M be the adjacency matrix of G.
Define graph G2on the same set of vertices with adjacency matrix N, where
Which one of the following statements is true?
A  
B  
C  
D 
Question 11 
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 12 
Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).
A  y_{n+1} = y_{n} + hf(x_{n}, y_{n}) 
B  y_{n+1} = y_{n} + hf(x_{n+1}, y_{n+1}) 
C  y_{n+1} = y_{n1} + 2hf(x_{n}, y_{n}) 
D  y_{n+1} = (1 + h) f(x_{n+1}, y_{n+1}) 
With initial value y(x_{0}) = y_{0}. Here the function f and the initial data x_{0} and y_{0} are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y_{0}, y_{1}, y_{2}, ....... such that y_{n} approximates y(x_{0} + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
y_{n+1} = y_{n} + hf(x _{n+1}, y_{n+1})
Question 13 
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(AB) = 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 14 
The number of distinct simple graphs with upto three nodes is
A  15 
B  10 
C  7 
D  9 
Question 15 
The tank of matrix is:
A  0 
B  1 
C  2 
D  3 
Question 16 
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 = g^{2} for every g ∈ G 
C  (goh)^{2} = g^{2}oh^{2} 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)
= ((hog^{2})oh)
= (g^{2}oh)oh
= g^{2}o(hoh)
= g^{2}oh^{2} [True]
Question 17 
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, nonzero 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(i1)/2 
D  i + j(j1)/2 
If we assume array index starting from 1 then, i^{th} row contains i number of nonzero elements. Before i^{th} row there are (i1) rows, (1 to i1) and in total these rows has 1+2+3......+(i1) = i(i1)/2 elements.
Now at i^{th} row, the j^{th} 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(i1)/2
Question 18 
The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is
A  n 
B  n^{2} 
C  n(n1)/2 
D  n(n+1)/2 
n = 1
(n1) = 2
(n2) = 3
So, Total = n(n+1)/2
Question 19 
On the set N of nonnegative integers, the binary operation __________ is associative and noncommutative.
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 20 
Amongst the properties {reflexivity, symmetry, antisymmetry, transitivity} the relation R = {(x,y) ∈ N^{2}  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 21 
The number of subsets {1, 2, ... n} with odd cardinality is __________.
A  2^{n1} 
And so, no. of subsets with odd cardinality is half of total no. of subsets = 2^{n} /n = 2^{n1}
Question 22 
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 23 
The probability of an event B is P_{1}. The probability that events A and B occur together is P_{2} while the probability that A and occur together is P_{3}. The probability of the event A in terms of P_{1}, P_{2} and P_{3} is __________.
A  P_{2} + P_{3} 
P_{3} = P(A)  P_{2}
P(A) = P_{2} + P_{3}
Question 24 
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 25 
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 26 
Match the following items
A  (i)  (b), (ii)  (c), (iii)  (d), (iv)  (a) 
Question 27 
Find the inverse of the matrix
A  
B  
C  
D 
λ^{3} + 2λ^{2}  2 = 0
Using CayleyHamiltonian theorem
A^{3} + 2A^{2}  2I = 0
So, A^{1} = 1/2 (2A  A^{2})
Solving we get,
Question 28 
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 29 
(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 30 
A 3ary tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a 3ary tree with n interval nodes is 2(n1)+3.
A  Theory Explanation. 
Question 31 
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 32 
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 33 
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 34 
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 35 
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 36 
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 37 
The value of k for which 4x^{2}  8xy + ky^{2} = 0 does not represent a pair of straight lines (both passing through the origin) is:
A  0 
B  2 
C  9 
D  3 
Question 38 
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 39 
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 40 
If the cube roots of unity are 1, ω and ω^{2}, then the roots of the following equation are (x  1)^{3} + 8 = 0
A  1, 1 + 2ω, 1 + 2ω^{2} 
B  1, 1  2ω, 1  2ω^{2} 
C  1, 1  2ω, 1  2ω^{2} 
D  1, 1 + 2ω, 1 + 2ω^{2} 
Question 41 
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 42 
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+jk) 
C  1/3 (1jk) 
D  1/√3 (1+jk) 
E  None of the above. 
Question 43 
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 44 
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 45 
Let A be the set of all nonsingular 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 46 
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
A  C_{1}e^{x} + C_{2}e^{2x} 
B  C_{1}e^{x} + C_{2}e^{3x} 
C  C_{1}e^{x} + C_{2}e^{2x} 
D  C_{1}e^{2x} + C_{2}2^{x} 
Question 47 
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 48 
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion
A  Theory Explanation. 
Question 49 
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 50 
Let G_{1} and G_{2} be subgroups of a group G.
(a) Show that G_{1} ∩ G_{2} is also a group of G.
(b) Is G_{1} ∪ G_{2} always a subgroup of G?
A  Theory Explanation. 