JEE MATH APEX

IAT 2026

Mathematics Questions

JEE MATH APEX
Let p(x) be a quadratic polynomial such that p(1) = p(-1) = 0. What is the coefficient of x in p(x)?
(A) -1
(B) 0
(C) 1
(D) 2
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JEE MATH APEX
Consider the following sets of points in the complex plane
A = \left\{\cos\left(\frac{2n\pi}{5}\right) + i \sin\left(\frac{2n\pi}{5}\right) : n \in \mathbb{Z}\right\} and
B = \left\{\cos\left(\frac{2n}{5}\right) + i \sin\left(\frac{2n}{5}\right) : n \in \mathbb{Z}\right\}.
Which of the following statements is TRUE?
(A) A is infinite but B is finite.
(B) A is infinite and B is also infinite.
(C) A is finite and B is also finite.
(D) A is finite but B is infinite.
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JEE MATH APEX
Let r, l be two integers such that r \geq l \geq 3. What is the total number of functions
f : \{1, 2, \dots, r\} \to \{1, 2, \dots, r\}
such that f(1), f(2), \dots, f(l) are all distinct?
(A) r(r - 1)(r - 2) \cdots (r - l + 1)
(B) r^r
(C) r^{r-l}(r - 1)(r - 2) \cdots (r - l + 1)
(D) r^{r-l+1}(r - 1)(r - 2) \cdots (r - l + 1)
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JEE MATH APEX
Let a_1, a_2, a_3, \dots be a geometric progression of positive integers such that a_1 = 3 and a_{n+2} - 2a_n = a_{n+1} for all positive integers n. What is the value of a_1 + a_2 + a_3 + a_4 + a_5?
(A) 255
(B) 99
(C) 93
(D) 120
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JEE MATH APEX
Let n = 20^{26}. What is the remainder when 49^n + 41^n + 10n is divided by 100?
(A) 1
(B) 49
(C) 90
(D) 2
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JEE MATH APEX
Consider the data of scores obtained by students in an examination. If the score of every student is increased by 2 marks, then which of the following statements is TRUE?
(A) The mean deviation about the median is increased by 2.
(B) The variance is increased by 2.
(C) The mean deviation about the mean is increased by 2.
(D) The mean deviation about the mean does not change.
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JEE MATH APEX
For a 2 \times 2 matrix A, whose elements are real numbers, denote by A^m the product \underbrace{AA \cdots A}_{m \text{ times}} where m is a positive integer.
Define x_0 = 0, x_1 = 1, x_n = x_{n-1} + x_{n-2}, for all n \geq 2 and
A_n = \begin{bmatrix} x_{n+1} & x_n \\ x_n & x_{n-1} \end{bmatrix}, for all n \geq 1.
Which of the following statements is TRUE for all m \geq 3?
(A) A_m - A_{m-1} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
(B) A_1^m = A_1^{m-1} + A_1^{m-2}
(C) \det(A_m) = -1
(D) A_1^m - A_1^{m-1} + \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
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JEE MATH APEX
Let f : \mathbb{R} \to \mathbb{R} be the function given by
f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1|.
What is the number of points where f is NOT differentiable?
(A) 2
(B) 0
(C) 1
(D) 3
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JEE MATH APEX
For real numbers a and b, consider the function f : \mathbb{R} \to \mathbb{R} given by
f(x) = \begin{cases} -ax - b & \text{if } x 1. \end{cases}
How many pairs (a, b) are there for which f is continuous at every point of \mathbb{R}?
(A) 2
(B) 1
(C) infinitely many
(D) 0
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JEE MATH APEX
Let \mathcal{C} be the set of all the circles in a plane. If
\mathcal{R} = \{(C_1, C_2) \in \mathcal{C} \times \mathcal{C} \mid C_1 \text{ and } C_2 \text{ intersect}\},
then which of the following statements is TRUE?
(A) \mathcal{R} is reflexive and transitive but not symmetric.
(B) \mathcal{R} is not a relation.
(C) \mathcal{R} is symmetric and transitive but not reflexive.
(D) \mathcal{R} is reflexive and symmetric but not transitive.
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JEE MATH APEX
Consider the function f : \mathbb{R} \to \mathbb{R} defined by f(x) = \sin^2(7x) - \sin^2(5x). Which of the following statements is NOT TRUE?
(A) f is increasing on \left(\frac{3\pi}{2}, 2\pi\right).
(B) f(x) > 0, for all x \in \left(0, \frac{\pi}{48}\right).
(C) f\left(x + \frac{\pi}{2}\right) + f(x) = 0, for all x \in \mathbb{R}.
(D) f\left(\frac{\pi}{12}\right) = 0
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JEE MATH APEX
What is the value of \int_{-1}^{2} \min\{1 - x, 1 - x^3\} dx?
(A) -1
(B) 0
(C) 1
(D) 2
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JEE MATH APEX
Consider the points A(4\hat{i} + \hat{j} + 3\hat{k}), B(2\hat{j}) and C(-4\hat{i} + 3\hat{j} - 3\hat{k}). Which of the following statements is TRUE?
(A) \overrightarrow{AB} \times \overrightarrow{BC} = \hat{i} + \hat{j} + \hat{k}.
(B) \overrightarrow{AB} + 3\overrightarrow{BC} is perpendicular to \overrightarrow{AC}.
(C) \overrightarrow{AB}, \overrightarrow{BC} and \overrightarrow{CA} are mutually perpendicular.
(D) A, B and C are collinear.
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JEE MATH APEX
Let l_1 be the line joining (1, 1, 1) and (3, 1, 3) and let l_2 be the line joining (0, 2, -1) and (2, 0, 3). What is the angle between l_1 and l_2?
(A) 60^\circ
(B) 45^\circ
(C) 30^\circ
(D) 90^\circ
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JEE MATH APEX
Suppose there are two boxes B_1 and B_2, each having $3$ red and $4$ black balls. One ball is drawn at random from B_1. If it is red, $4$ red balls are put into B_2, otherwise $3$ black balls are put into B_2. Then one ball is randomly drawn from B_2. If this ball is red, what is the conditional probability that the ball drawn from B_1 was also red?
(A) \frac{35}{57}
(B) \frac{3}{7}
(C) \frac{33}{53}
(D) \frac{99}{257}
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