If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)²] equals _________.

If the ordinary generating function of a sequence is , then a₃ - a₀ is equal to ________.

Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x² + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _____.

Consider the first order predicate formula φ:

∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))]

Here 'a|b' denotes that 'a divides b', where a and b are integers. Consider the following sets:

S1.  {1, 2, 3, ..., 100}
S2.  Set of all positive integers
S3. Set of all integers

Which of the above sets satisfy φ?

Consider the following matrix:

The absolute value of the product of Eigen values of R is ______.

The largest eigenvalue of A is ________

Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is __________.

The chromatic number of the following graph is _______.

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _________.

Which one of the following is a closed form expression for the generating function of the sequence {a_n}, where a_n = 2n+3 for all n = 0, 1, 2, …?

Assume that multiplying a matrix G₁ of dimension p×q with another matrix G₂ of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G₁G₂G₃…G_n can be done by parenthesizing in different ways. Define G_iG_i+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G₁G₂G₃G₄G₅G₆ using parenthesization(G₁(G₂G₃))(G₄(G₅G₆)), G₂G₃ and G₅G₆ are the only explicitly computed pairs.

Consider a matrix multiplication chain F₁F₂F₃F₄F₅, where matrices F₁, F₂, F₃, F₄ and F₅ are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F₁F₂F₃F₄F₅ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are

Consider the first-order logic sentence

where ψ(s,t,u,v,w,x,y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements.

Which one of the following statements is necessarily true?

Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(H_G) denote the probability that Guwahati has high temperature. Similarly, P(M_G) and P(L_G) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_D), P(M_D) and P(L_D) for Delhi.

The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (H_G) then the probability of Delhi also having a high temperature (H_D) is 0.40; i.e., P(H_D ∣ H_G) = 0.40. Similarly, the next two entries are P(M_D ∣ H_G) = 0.48 and P(L_D ∣ H_G) = 0.12. Similarly for the other rows.

If it is known that P(H_G) = 0.2, P(M_G) = 0.5, and P(L_G) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______ .

Let G be a graph with 100! vertices, with each vertex labeled by a distinct permutation of the numbers 1, 2, …, 100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G.

Then, y + 10z = ___________.

Let c¹, cⁿ be scalars not all zero. Such that the following expression holds:

where a_i is column vectors in Rⁿ. Consider the set of linear equations.

where A = [a₁.......a_n] and

Then, Set of equations has

Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of T are:

Let X be a Gaussian random variable with mean 0 and variance σ². Let Y = max(X, 0) where max(a,b) is the maximum of a and b. The median of Y is __________.

Let p, q and r be prepositions and the expression (p → q) → r be a contradiction. Then, the expression (r → p) → q is

Let u and v be two vectors in R² whose Euclidean norms satisfy ||u||=2||v||. What is the value of α such that w = u + αv bisects the angle between u and v?

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is _________.

If f(x) = Rsin(πx/2) + S, f'(1/2) = √2 and , then the constants R and S are, respectively.

Let p, q, r denote the statements “It is raining”, “It is cold”, and “It is pleasant”, respectively. Then the statement “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold” is represented by

Consider the set X = {a,b,c,d e} under the partial ordering

R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}.

The Hasse diagram of the partial order (X,R) is shown below.

The minimum number of ordered pairs that need to be added to R to make (X,R) a lattice is _________.

Then the rank of P+Q is _________.

G is an undirected graph with n vertices and 25 edges such that each vertex of G has degree at least 3. Then the maximum possible value of n is ___________.

P and Q are considering to apply for job. The probability that p applies for job is 1/4. The probability that P applies for job given that Q applies for the job 1/2 and The probability that Q applies for job given that P applies for the job 1/3.The probability that P does not apply for job given that Q does not apply for the job

For any discrete random variable X, with probability mass function P(X=j)=p_j, p_j≥0, j∈{0, ..., N} and , define the polynomial function . For a certain discrete random variable Y, there exists a scalar β∈[0,1] such that g_y(Z)=(1-β+βz)^N. The expectation of Y is

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ - 4λ² + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________.

Let a_n be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for a_n?

A probability density function on the interval [a,1] is given by 1/x² and outside this interval the value of the function is zero. The value of a is _________.

Two eigenvalues of a 3 × 3 real matrix P are (2 + √-1) and 3. The determinant of P is __________.

The coefficient of x¹² in (x³ + x⁴ + x⁵ + x⁶ + ...)³ is _________.

Consider the recurrence relation a₁ = 8, a_n = 6n² + 2n + a_n-1. Let a₉₉ = K × 10⁴. The value of K is ___________.

Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(-x)) is 10, then the degree of (g(x) - g(-x)) is __________.

The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is _________.

Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A^-1)^T is _________.

Which one of the following well-formed formulae in predicate calculus is NOT valid?

The value of the expression 13⁹⁹(mod 17), in the range 0 to 16, is ________.

In the LU decomposition of the matrix ,if the diagonal elements of U are both 1, then the lower diagonal entry l₂₂ of L is

If g(x) = 1 - x and h(x) , then is:

Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:

For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L³ = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let p_r be the probability that an element (x,y,z) ∈ L³ chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b?

Let G = (V, E) be a simple undirected graph, and s be a particular vertex in it called the source.  For x ∈ V, let d(x)denote the shortest distance in G from s to x. A breadth first search (BFS) is performed starting at s. Let T be the resultant BFS tree. If (u, v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u) - d(v) ?   

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________.

Consider the operations f(X, Y, Z) = X'YZ + XY' + Y'Z'  and  g(X′, Y, Z) = X′YZ + X′YZ′ + XY Which one of the following is correct?

Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true?

Consider the following statements:

S1: If a candidate is known to be corrupt, then he will not be elected.
S2: If a candidate is kind, he will be elected.

Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?

The larger of the two eigenvalues of the matrix is _________.

The cardinality of the power set of {0, 1, 2, … 10} is _________.

The number of divisors of 2100 is ______.

In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?

Consider six memory partitions of sizes 200 KB, 400 KB, 600 KB, 500 KB, 300 KB and 250 KB, where KB refers to kilobyte. These partitions need to be allotted to four processes of sizes 357 KB, 210KB, 468 KB and 491 KB in that order. If the best fit algorithm is used, which partitions are NOT allotted to any process?

The number of onto function (surjective function) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ______.

Perform the following operations on the matrix

(i) add the third row to the second row
(ii) Subtract the third column from the first column

The determinant of the resultant matrix is ________.

Which one of the following well formed formulae is a tautology?

A graph is self-complementary if it is isomorphic to its complement for all self-complementary graphs on n vertices, n is

The secant method is used to find the root of an equation f(x) = 0. It is started from two distinct estimates x_a and x_b for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(x_b) is very small and then x_b is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by? Which is the suitable expression that is to be put in place of? So that it follows all steps of the secant method?

Secant

Initialize: xa, xb, ε, N     // ε = convergence indicator
fb = f(xb) i = 0
while (i < N and |fb| > ε) do
   i = i + 1                 // update counter
   xt = ?                    // missing expression for
                             // intermediate value
   xa = xb                   // reset xa
   xb = xt                   // reset xb
   fb = f(xb)                // function value at new xb
end while
if |fb| > ε
  then // loop is terminated with i = N
  write “Non-convergence”
else
  write “return xb”
end if 

Let f(x) = x ^-(1/3) and A denote the area of the region bounded bu f(x) and the X-axis, when x varies from -1 to 1. Which of the following statements is/are TRUE?

I) f is continuous in [-1,1]
II) f is not bounded in [-1,1]
III) A is nonzero and finite

Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. Let f be randomly chosen from F. The probability of f being one-to-one is ______.

In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following:

“The result of the toss is head if and only if I am telling the truth.”

Which of the following options is correct?

Suppose U is the power set of the set S = {1, 2, 3, 4, 5, 6}. For any T ∈ U, let |T| denote the number of element in T and T' denote the complement of T. For any T, R ∈ U, let TR be the set of all elements in T which are not in R. Which one of the following is true?

The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is _____.

In the given matrix, one of the eigenvalues is 1. the eigenvectors corresponding to the eigenvalue 1 are

⎡ 1 -1  2 ⎤
⎢ 0  1  0 ⎥
⎣ 1  2  1 ⎦

Consider a machine with a byte addressable main memory of 2²⁰ bytes, block size of 16 bytes and a direct mapped cache having 2¹² cache lines. Let the addresses of two consecutive bytes in main memory be (E201F)₁₆ and (E2020)₁₆. What are the tag and cache line address (in hex) for main memory address (E201F)₁₆?

If for non-zero x, where a≠b then is

The velocity v (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time t(in minutes) as follows:

t   2  4  6  8  10  12  14  16 18 20
v  10  18 25 29 32  20  11  5  2  0

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is _________.

If the following system has non-trivial solution,

px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0

then which one of the following options is True?

Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) ∈ R if and only if p - s = q - r. Which one of the following is true about R?

Let G = (V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G?

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ________.

Let the function

where and f(θ) denote the derivative of f with respect to θ. Which of the following is/are TRUE?

(I) There exists such that such that f(θ)=0.
(II) There exists such that such that f(θ)≠0.

The function f(x) = x sinx satisfies the following equation: f''(x) + f(x) + tcosx = 0. The value of t is __________.

A function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is ___________.

A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1-pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10-pennants is ________.

Let S denote the set of all functions  f:{0,1}⁴ → {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of log₂log₂N is ______.

Consider an undirected graph where self-loops are not allowed. The vertex set of G is {(i,j): 1 ≤ i ≤ 12, 1 ≤ j ≤ 12}. There is an edge between (a,b) and (c,d) if |a - c| ≤ 1 and |b - d| ≤ 1. The number of edges in this graph is __________.

An ordered -tuple (d₁, d₂, ..., d_n) with d₁ ≥ d₂ ≥ ... d_n is called graphic if there exists a simple undirected graph with n vertices having degrees d₁, d₂, ..., d_n respectively. Which of the following 6-tuples is NOT graphic?

Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working.  Let the probability that the system is deemed functional be denoted by p. Then 100p =_____________.

Each of the nine words in the sentence "The quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.)

The maximum number of edges in a bipartite graph on 12 vertices is ______.

If the matrix A is such that

then the determinant of A is equal to

A non-zero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3. Which one of the following must be TRUE?

In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function

0.75x3 – 2x2 – 2x + 4 = 0 

Consider the statements

(I) x3 = 0.
(II) The method converges to a solution in a finite number of iterations.

Which of the following is TRUE?

The product of the non-zero eigenvalues of the matrix

1 0 0 0 1
0 1 1 1 0
0 1 1 1 0
0 1 1 1 0
1 0 0 0 1

is ______.

The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______.

The number of distinct positive integral factors of 2014 is _________.

Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U. Consider the following two statements:

S1: There is a subset of S that is larger than every other subset.
S2: There is a subset of S that is smaller than every other subset.

Which one of the following is CORRECT?

A cycle on n vertices is isomorphic to its complement. The value of n is _____.

Which one of the following Boolean expressions is NOT a tautology?

Consider the following statements:

P: Good mobile phones are not cheap
Q: Cheap mobile phones are not good
L: P implies Q
M: Q implies P
N: P is equivalent to Q

Which one of the following about L, M, and N is CORRECT?

Let X and Y be finite sets and f: X→Y be a function. Which one of the following statements is TRUE?

Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L ≠ G and that the size of L is at least 4. The size of L  is __________.

Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

If V₁ and V₂ are 4-dimensional subspace of a 6-dimensional vector space V, then the smallest possible dimension of V₁∩V₂   is ______.

If , then the value of k is equal to ________.

With respect to the numerical evaluation of the definite integral ,  where a  and b are given, which of the following statements is/are TRUE?

I) The value of obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II) The value of obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

The value of the integral given below is

Let S be a sample space and two mutually exclusive events A and B be such that A∪B = S. If P(∙) denotes the probability of the event, the maximum value of P(A)P(B) is __________.

Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:

P. For each such function it must be the case that for every i, f(i) = i.
Q. For each such function it must be the case that for some i, f(i) = i.
R. Each such function must be onto.

Which one of the following is CORRECT?

There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that

  x * x = y * y = x * y * x = y * x * y * x = e   

where e is the identity element. The maximum number of elements in such a group is ________.

If G is a forest with n vertices and k connected components, how many edges does G have?

Let δ denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with δ ≥ 3, which one of the following is TRUE?

The CORRECT formula for the sentence, “not all rainy days are cold” is

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x² + y². Which one of the following statements is TRUE about ⊕?

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Which one of the following does NOT equal to

Which one of the following functions is continuous at x = 3?

Function f is known at the following points:

The value of computed using the trapezoidal rule is

What is the logical translation of the following statement?

  "None of my friends are perfect."

The following code segment is executed on a processor which allows only register operands in its instructions. Each instruction can have atmost two source operands and one destination operand. Assume that all variables are dead after this code segment.

c = a + b;
   d = c * a;
   e = c + a;
   x = c * c;
   if (x > a) {
      y = a * a;
   }
   else {
     d = d * d;
     e = e * e;
  }

Suppose the instruction set architecture of the processor has only two registers. The only allowed compiler optimization is code motion, which moves statements from one place to another while preserving correctness. What is the minimum number of spills to memory in the compiled code?

Consider the following logical inferences.

I1: If it rains then the cricket match will not be played.
The cricket match was played.
Inference: There was no rain.
I2: If it rains then the cricket match will not be played.
It did not rain.
Inference: The cricket match was played.

Which of the following is TRUE?

Consider the function f(x) = sin(x) in the interval x ∈ [π/4, 7π/4]. The number and location(s) of the local minima of this function are

Let A be the 2×2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = -1. Then the eigenvalues of the matrix A¹⁹ are

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational”