Let G be an arbitrary group. Consider the following relations on G:

      R1: ∀a,b ∈ G, aR1b if and only if ∃g ∈ G such that a = g-1bg
      R2: ∀a,b ∈ G, aR2b if and only if a = b-1

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.
II. Determinant of X is non-zero.

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

Let U = {1,2,...,n}. Let A = {(x,X)|x ∈ X, X ⊆ U}. Consider the following two statements on |A|.

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) _____.

∀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

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 __________.

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 N be the set of natural numbers. Consider the following sets,

P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0, 1} to N
R: Set of functions from N to {0, 1}
S: Set of finite subsets of N

Which of the above sets are countable?

Consider the following statements.

(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized

Which one of the following options is correct?

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

I. p ⇒ q
II. q ⇒ p
III. (¬q) ∨ p
IV. (¬p) ∨ q

Consider the first-order logic sentence F: ∀x(∃yR(x,y)). Assuming non-empty logical domains, which of the sentences below are implied by F?

I. ∃y(∃xR(x,y))
II. ∃y(∀xR(x,y))
III. ∀y(∃xR(x,y))
IV. ¬∃x(∀y¬R(x,y))

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?

(i)  One eigenvalue must be in [-5, 5].
(ii) The eigenvalue with the largest magnitude 
     must be strictly greater than 5.

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

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 _________.

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 ___________.

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

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 p,q,r,s represent the following propositions.

p: x ∈ {8,9,10,11,12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number

The integer x≥2 which satisfies ¬((p ⇒ q) ∧ (¬r ∨ ¬s))  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 ___________.

A function f:N⁺ → N⁺, defined on the set of positive integers N⁺, satisfies the following properties:

f(n) = f(n/2)           if n is even
f(n) = f(n+5)           if n is odd

Let R = {i|∃j: f(j)=i} be the set of distinct values that f takes. The maximum possible size of R is __________.

Consider the following experiment.

Step1. Flip a fair coin twice.
Step2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.
Step4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) ________.

Consider the following expressions:

(i) false
(ii) Q
(iii) true
(iv) P ∨ Q
(v) ¬Q ∨ P

The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) 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 ________.

Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

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 _________.

A binary relation R on ℕ × ℕ is defined as follows: (a,b)R(c,d) if a≤c or b≤d. Consider the following propositions:

P: R is reflexive
Q: Ris transitive

Which one of the following statements is TRUE?

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

Consider a set U of 23 different compounds in a Chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U. Consider the following statements:

I. Each compound in U\S reacts with an odd number of compounds.
II. At least one compound in U\S reacts with an odd number of compounds.
III. Each compound in U\S reacts with an even number of compounds.

Which one of the above statements is ALWAYS TRUE?

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

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