List of top Mathematics Questions asked in GATE Computer Science and Information Technology

Consider a probability distribution given by the density function \( P(x) \).

\[ P(x) = \begin{cases} Cx^2, & \text{for } 1 \leq x \leq 4, \\ 0, & \text{for } x<1 \text{ or } x>4. \end{cases} \]
The probability that \( x \) lies between 2 and 3, i.e., \( P(2 \leq x \leq 3) \), is ______. (rounded off to three decimal places)

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ___________. (rounded off to three decimal places)

Let \( A = \{0, 1, 2, 3, \dots\} \) be the set of non-negative integers. Let \( F \) be the set of functions from \( A \) to itself. For any two functions, \( f_1, f_2 \in F \), we define \[ (f_1 \circ f_2)(n) = f_1(n) + f_2(n) \] for every number \( n \in A \). Which of the following is/are CORRECT about the mathematical structure \( (F, \circ) \)?
Consider the following four variable Boolean function in sum-of-product form \[ F(b_3, b_2, b_1, b_0) = \sum(0, 2, 4, 8, 10, 11, 12), \] where the value of the function is computed by considering \( b_3 b_2 b_1 b_0 \) as a 4-bit binary number, where \( b_3 \) denotes the most significant bit and \( b_0 \) denotes the least significant bit. Note that there are no don’t care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for \( F \)?
Let \( A \) be a \( 2 \times 2 \) matrix as given:

\[ A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \]
What are the eigenvalues of the matrix \( A^{13} \)?
A box contains 5 coins: 4 regular coins and 1 fake coin.

- When a regular coin is tossed, the probability \( P(\text{head}) = 0.5 \).
- For a fake coin, \( P(\text{head}) = 1 \).

You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is ______.
Let \(X\) be a 3-variable Boolean function that produces output as ‘1’ when at least two of the input variables are ‘1’. Which of the following statement(s) is/are CORRECT, where \(a, b, c, d, e\) are Boolean variables?
g(periodcentered) is a function from A to B, f(periodcentered) is a function from B to C, and their composition defined as f(g(periodcentered)) is a mapping from A to C. If f(periodcentered) and f(g(periodcentered)) are onto (surjective) functions, which ONE of the following is TRUE about the function g(periodcentered)?
A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?