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 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 ___________.