List of top Mathematics Questions asked in JEE Advanced

Let \(S=\left\{a+b\sqrt2:a,b\in Z\right\},T_1=\left\{(-1+\sqrt2)^n:n\in \N\right\},\ \text{and}\ T_2=\left\{(1+\sqrt2)^n:n\in\N\right\}\). Then which of the following statements is (are) TRUE ?

Consider the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let S (p, q) be a point in the first quadrant such that \(\frac{p^2}{9}+\frac{q^2}{4}>1\). Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and Obe the center of the ellipse. If the area of the triangle △ORT is \(\frac{3}{2}\) , then which of the following options is correct ?

Let \(\frac{\pi}{2}<x<\pi\) be such that \(\cot x=\frac{-5}{\sqrt{11}}\). Then
\((\sin\frac{11x}{2})(\sin 6x-\cos6x)+(\cos\frac{11x}{2})(\sin 6x+\cos 6x)\)
is equal to

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\) . Also assume that the probability of the answer for a question being guessed, given that the student’s answer is correct, is \(\frac{1}{6}\) . Then the probability that the student knows the answer of a randomly chosen question is

Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and
\(\lim\limits_{t→x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1\)
for each x > 0. Then, for all x > 0, f(x) is equal to

Let \( S \) be the set of all \( (\alpha, \beta) \in \mathbb{R} \times \mathbb{R} \) such that \[ \lim_{x \to \infty} \frac{\sin(x^2)(\log_e x)^\alpha \sin\left(\frac{1}{x^2}\right)}{x^{\alpha\beta} (\log_e(1+x))^\beta} = 0. \] Then which of the following is (are) correct?

Let \( k \in \mathbb{R} \). If \[ \lim_{x \to 0^+} \frac{\left(\sin(\sin(kx)) + \cos x + x\right)^2}{x} = e^6, \] then the value of \( k \) is:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \[ f(x) = \begin{cases} x^2 \sin\left(\frac{\pi}{x^2}\right), & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases} \] Then which of the following statements is TRUE?

Let the function \( f : [1, \infty) \to \mathbb{R} \) be defined by:
\[ f(t) = \begin{cases} (-1)^{n+1}2, & \text{if } t = 2n-1, \, n \in \mathbb{N}, \\ \frac{(2n+1-t)}{2}f(2n-1) + \frac{(t-(2n-1))}{2}f(2n+1), & \text{if } 2n-1 < t < 2n+1, \, n \in \mathbb{N}. \end{cases} \]
Define \( g(x) = \int_1^x f(t) \, dt, \, x \in (1, \infty). \) Let \( \alpha \) denote the number of solutions of the equation \( g(x) = 0 \) in the interval \( (1, 8] \) and \( \beta = \lim_{x \to 1^+} \frac{g(x)}{x-1}. \) Then the value of \( \alpha + \beta \) is .........

If \( n(X) = \binom{m}{6} \), then the value of \( m \) is _____.

The value of \( 2 \int_0^{\pi/2} f(x)g(x) \, dx - \int_0^{\pi/2} g(x) \, dx \) is _____.

Let the function \( f : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \frac{\sin x}{e^{\pi x}} \cdot \frac{x^{2023} + 2024x + 2025}{(x^2 - x + 3)} + \frac{2}{e^{\pi x}} \cdot \frac{x^{2023} + 2024x + 2025}{(x^2 - x + 3)}. \] Then the number of solutions of \( f(x) = 0 \) in \( \mathbb{R} \) is \_\_\_\_\_.

Let \( \vec{p} = 2\hat{i} + \hat{j} + 3\hat{k} \) and \( \vec{q} = \hat{i} - \hat{j} + \hat{k} \). If for some real numbers \( \alpha, \beta, \gamma \), we have \[ 15\mathbf{i} + 10\hat{j} + 6\hat{k} = \alpha (2\vec{p} + \vec{q}) + \beta (\vec{p} - 2\vec{q}) + \gamma (\vec{p} \times \vec{q}), \] then the value of \( \gamma \) is \_\_\_\_.

A bag contains \( N \) balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For \( i = 1, 2, 3 \), let \( W_i \), \( G_i \), and \( B_i \) denote the events that the ball drawn in the \( i \)-th draw is a white ball, green ball, and blue ball, respectively. If the probability \( P(W_1 \cap G_2 \cap B_3) = \frac{2{5N} \) and the conditional probability \( P(B_3 \mid W_1 \cup G_2) = \frac{2}{9} \), then \( N \) equals \_\_\_\_\_.}

A straight line drawn from the point \( P(1, 3, 2) \), parallel to the line \[ \frac{x-2}{1} = \frac{y-4}{2} = \frac{z-6}{1}, \] intersects the plane \( L_1 : x - y + 3z = 6 \) at the point \( Q \). Another straight line passing through \( Q \) and perpendicular to the plane \( L_1 \) intersects the plane \( L_2 : 2x - y + z = -4 \) at the point \( R \). Then which of the following statements is (are) TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x + y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \) and \( g : R \to (0, \infty) \) be a function such that \( g(x + y) = g(x)g(y) \) for all \( x, y \in \mathbb{R} \). If \( f\left(-\frac{3}{5}\right) = 12 \) and \( g\left(-\frac{1}{3}\right) = 2 \), then the value of \[ f\left(\frac{1}{4}\right) + g(-2) - 8 \cdot g(0) \] is \_\_\_\_\_.

Let \[ S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x \geq 0, y \geq 0, y^2 \leq 4x, y^2 \leq 12 - 2x, \text{ and } 3y + \sqrt{8}x \leq 5\sqrt{8}\}. \] If the area of the region \( S \) is \( \alpha \sqrt{2} \), then \( \alpha \) is equal to:

Considering only the principal values of the inverse trigonometric functions, the value of \[ \tan\left( \sin^{-1}\left(\frac{3}{5}\right) - 2 \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \right) \] is:

Let S = {1, 2, 3, 4, 5, 6} and X be the set of all relations R from S to S that satisfy both the following properties :
i. R has exactly 6 elements.
ii. For each (a, b) ∈ R, we have |a - b| ≥ 2.
Let Y = {R ∈ X : The range of R has exactly one element} and
Z = {R ∈ X : R is a function from S to S}.
Let n(A) denote the number of elements in a set A.

Let \(f:[0,\frac{\pi}{2}]→[0,1]\) be the function defined by \(f(x)=\sin^2x\) and let \(g:[0,\frac{\pi}{2}]→[0,\infin)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}=x^2}\).