IAT 2025

Mathematics Questions

JEE MATH APEX

How many three digit numbers divisible by 5 are there in which no digits are repeated?

(A) 136

(B) 128

(D) 162

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JEE MATH APEX

Let $A$ be a $3 \times 3$ matrix with real entries such that
$A = \begin{bmatrix} 4 & -1 & \cos x \\ -1 & 5x & 25 \\ x^2 + 1 & 25 & 7 \end{bmatrix}$
For how many values of $x$ , the matrix $A$ is symmetric?

(A) 1

(B) 2

(D) infinitely many

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JEE MATH APEX

Let
$n = \sum_{r=0}^{10} (-1)^r \binom{10}{r} \left(\frac{3}{2}\right)^3 42^r 3^{20}$
Which one of the following statements is TRUE?

(A) $n$ is divisible by 5

(B) $n$ is divisible by 6

(D) $n$ is divisible by 9

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JEE MATH APEX

Let $f : \mathbb{R} \to \mathbb{R}$ be the function given by $f(x) = \cos (\tan^{-1} x)$ . Which one of the following statements is TRUE?

(A) $f$ is decreasing for $x > 0$

(B) $f$ is decreasing for $x < 0$

(D) $f$ is decreasing on the interval $(-1, 1)$

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JEE MATH APEX

Let
$A = \left\{ x \in \mathbb{R} \, \middle| \, -31 < \det\begin{bmatrix} 3x - 1 & 2 \\ -2 & 5 \end{bmatrix} \leq 29 \right\}$
Which one of the following statements is TRUE?

(A) $A = (-2, 2]$

(B) $A = (-2, 2)$

(D) $A = [-2, 2]$

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JEE MATH APEX

Let $z_1$ , $z_2$ , and $z_3$ be complex numbers satisfying:
$2 = |2z_1| = |z_2 - 1| = |z_3 + 1| = \left| \frac{1}{z_1} + \frac{1}{z_2 - 1} + \frac{1}{z_3 + 1} \right|$
What is the value of $|4z_1 + z_2 + z_3|$ ?

(A) 8

(B) 4

(D) $\frac{1}{8}$

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JEE MATH APEX

Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = |x^3 - 3x|\lfloor x \rfloor$ , where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ . Which one of the following statements is TRUE?

(A) Every non-zero integer is a point of discontinuity of $f$

(B) $f$ is continuous at every real number

(D) $f$ is continuous at every real number except for $0, \pm\sqrt{3}$

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JEE MATH APEX

Let $\ell$ be the tangent line to the ellipse $\frac{x^2}{4} + \frac{y^2}{0.25} = 1$ at $\left(\frac{3}{4}, \frac{\sqrt{3}}{4}\right)$ . What is the equation of the line perpendicular to $\ell$ passing through $(2, 0)$ ?

(A) $y = 4\sqrt{3}(x - 2)$

(B) $y = 2\sqrt{3}(x - 2)$

(D) $4\sqrt{3}y = (x - 2)$

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JEE MATH APEX

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a} + \vec{b}| = 15$ and
$\vec{a} \times (3\hat{i} - 4\hat{j} + 5\hat{k}) = (3\hat{i} - 4\hat{j} + 5\hat{k}) \times \vec{b}$ .
What is the value of $|(\vec{a} + \vec{b}) \cdot (2\hat{i} + 3\hat{j} + \hat{k})|$ ?

(A) $3\sqrt{2}$

(B) $0$

(D) $3$

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JEE MATH APEX

What is the derivative of $\log(\sin^2 x)$ with respect to $\sin x$ ?

(A) $2 \csc x$

(B) $\sin 2x$

(D) $\cot x \csc 2x$

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JEE MATH APEX

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \ldots$ . If $S_{n+3} - S_n = 13n + 7$ for all $n$ , what is the value of $a_{13} - a_{10}$ ?

(A) 13

(B) 137

(D) 12

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JEE MATH APEX

Five fair coins are tossed independently. What is the probability that at least two heads appear?

(A) $\frac{13}{16}$

(B) $\frac{7}{16}$

(D) $\frac{11}{16}$

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JEE MATH APEX

Let $f : \mathbb{R} \to \mathbb{R}$ be the function defined by
$f(x) = \begin{cases} x^2 - 4x - 5 & \text{if } x \geq 1 \\ 2x & \text{if } x < 1 \end{cases}$
Which one of the following statements is TRUE?

(A) $f$ is onto but not one-one

(B) $f$ is one-one but not onto

(D) $f$ is one-one and onto

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JEE MATH APEX

Which one of the following is the solution of the differential equation
$x^2 \frac{dy}{dx} + 9xy = x^4$ for $x > 0$ , given that $y = 0$ when $x = 1$ ?