GATE 2025 Engineering Maths Medium Questions

Consider the given function $f(x)$:

$f(x)=\{\begin{array}{ll}ax+b& \text{for }x<1\\ x^3+x^2+1& \text{for }x\ge 1\end{array}$

If the function is differentiable everywhere, the value of $b$ must be _______. (rounded off to one decimal place)

Let S be the set of all ternary strings defined over the alphabet $\{a,b,c\}$. Consider all strings in S that contain at least one occurrence of two consecutive symbols, that is, "aa", "bb" or "cc". The number of such strings of length $5$ that are possible is ______. (Answer in integer)

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)

If $A=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right]$, then which ONE of the following is $A^8$ ?

The value of $x$ such that $x>1$, satisfying the equation ${\int }_1^{x}t\ln tdt=\frac{1}{4}$ is

Let $L,M,$ and $N$ be non-singular matrices of order 3 satisfying the equations

$L^2=L^{-1},M=L^8$ and $N=L^2$.

Which ONE of the following is the value of the determinant of $(M-N)$?

Let $P(x)$ be an arbitrary predicate over the domain of natural numbers.

Which ONE of the following statements is TRUE?

Consider a system of linear equations $PX=Q$ where $P\in {\mathbb{R}}^{3\times 3}$ and $Q\in {\mathbb{R}}^{3\times 1}$. Suppose $P$ has an LU decomposition, $P=LU$, where

$L=\left[\begin{array}{ccc}1& 0& 0\\ l_{21}& 1& 0\\ l_{31}& l_{32}& 1\end{array}\right]$ and $U=\left[\begin{array}{ccc}{u}_{11}& u_{12}& u_{13}\\ 0& u_{22}& u_{23}\\ 0& 0& u_{33}\end{array}\right]$.

Which of the following statement(s) is/are TRUE?

A quadratic polynomial $(x-\alpha )(x-\beta )$ over complex numbers is said to be square invariant if $(x-\alpha )(x-\beta )=(x-{\alpha }^2)(x-{\beta }^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.

The probability that the roots of the chosen polynomial are equal is ________. (rounded off to one decimal place)

An application executes $6.4\times 10^8$ number of instructions in $6.3$ seconds. There are four types of instructions, the details of which are given in the table. The duration of a clock cycle in nanoseconds is ________. (rounded off to one decimal place)