Mathematics Gate Previous Year Questions

A function y(x) is defined in the interval [0, 1] on the 𝑥-axis as

a) 5/6
b) 6/5
c) 13/6
d) 6/13

The value of the following limit is ____________

The number of arrangements of six identical balls in three identical bins is______.

a) 36
b) 21
c) 12
d) 7

Consider the following two statements with respect to the matrices A_m×n , B_n×m , C_n×n and D_n×n

Statement 1: tr(AB) = tr(BA)
Statement 2: tr(CD) = tr(DC)
where tr() represents the trace of a matrix. Which one of the following holds?

a) Statement 1 is correct and Statement 2 is wrong.
b) Statement 1 is wrong and Statement 2 is correct.
c) Both Statement 1 and Statement 2 are correct.
d) Both Statement 1 and Statement 2 are wrong.

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is __________ .

a) 0.3024
b) 0.4235
c) 0.6976
d) 0.8125

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A)*rank (B)
II. det(AB) = det(A)*det(B)
III. rank(A+B) ≤ rank(A) + rank(B)
IV. det(A+B) ≤ det(A) + det(B)

Which of the above statements are TRUE ?

a) I and II only
b) I and IV only
c) II and III only
d) III and IV only

Consider the functions

I. e^-x
II. x² - sin x
III. √x^3+1

Which of the above functions is/are increasing everywhere in [0, 1] ?

a) Ⅲ only
b) Ⅱ only
c) Ⅱ and Ⅲ only
d) Ⅰ and Ⅲ only

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 Σⁱ⁼¹_i=n a_ix_i is an odd number is _________

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

a) I implies II; II does not imply I
b) II implies I; I does not imply II
c) I does not imply II; II does not imply I
d) I and II are equivalent statements

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 a quadratic equation x² - 13x + 36 = 0 with coefficients in a base b. The solutions of this equation in the same base b are x = 5 and x = 6. Then b = ______.

The representation of the value of a 16-bit unsigned integer X in a hexadecimal number system is BCA9. The representation of the value of X in octal number system is:

a) 571244
b) 736251
c) 571247
d) 136251

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

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

a) 2
b) 1
c) 1/2
d) -1/2

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 _____.

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

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

a) 4
b) 13
c) 8
d) 16

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?

a) I, II and III are true
b) Only II and III are true
c) Only III is true
d) None of them is true

Consider the following experiment.

Step 1. Flip a fair coin twice.
Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step 3. If the outcomes are either (HEADS, HEAD) or (HEADS, TAILS), then output N and stop.
Step 4. 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).

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

properties:
f(n) = f(n/2) if nis even
f(n) = f(n+5) if nis odd

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

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

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

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 :

If f(x) = 2x⁷ + 3x - 5 Which of the following is a factor of f(x)?

a) (x³ + 8)
b) (x - 1)
c) (2x - 5)
d) (x + 1)

A function f(x) is linear and has a value of 29 at x = –2 and 39 at x = 3. Find its value at x = 5.

a) 59
b) 45
c) 43
d) 35

The number of divisors of 2100 is _________.

Consider a function f(x) = 1 – |x| on –1 ≤ x ≤ 1. The value of x at which the function attains a maximum and the maximum value of the function are:

a) 0, –1
b) –1, 0
c) 0, 1
d) –1, 2

The base (or radix) of the number system such that the following equation holds is____________.

³¹²/₂₀= 13.1

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 ____________

Consider the following system of equations:

3x + 2y = 1
4x + 7z = 1
x + y + z = 3
x – 2y + 7z = 0

The number of solutions for this system is __________

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________