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JEE Mains | PYQ

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is:

Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$, $x + y = 5$ and $7x + 4y = 15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x = 0$, $y = 0$ and $x + y = 1$ is:

The value of $\int\limits_{-1}^{1} \frac{(1 + \sqrt{|x| - x})e^x + (\sqrt{|x| - x})e^{-x}}{e^x + e^{-x}} dx$ is equal to:

The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is:

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:

For an integer \( n \ge 2 \), if the arithmetic mean of all coefficients in the binomial expansion of \( (x + y)^{2n - 3} \) is \( 16 \), then the distance of point \( P(2n - 1, n^2 - 4n) \) from the line \( x + y = 8 \) is:

If \( \displaystyle \lim_{x \to 1^{+}} \frac{(x-1)\big(6+\lambda \cos(x-1)\big)+\mu \sin(1-x)}{(x-1)^3} = -1 \), where \( \lambda, \mu \in \mathbb{R} \), then \( \lambda + \mu \) is equal to:

Considering the principal values of the inverse trigonometric functions, \( \sin^{-1}\!\left(\dfrac{\sqrt{3}}{2}\,x + \dfrac{1}{2}\sqrt{1-x^2}\right),\; -\dfrac{1}{2} < x < \dfrac{1}{\sqrt{2}} \) is equal to:

Let \( A = \{1,6,11,16,\ldots\} \) and \( B = \{9,16,23,30,\ldots\} \) be the sets consisting of the first \( 2025 \) terms of two arithmetic progressions. Then \( n(A \cup B) \) is:

\( 1 + 3 + 5^2 + 7 + 9^2 + \ldots \) upto \( 40 \) terms is equal to:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(0) = 1 \) and \( f(2x) - f(x) = x \) for all \( x \in \mathbb{R} \). If \( \displaystyle \lim_{n \to \infty} \Big[ f(x) - f\!\left(\dfrac{x}{2^n}\right) \Big] = G(x) \), then \( \displaystyle \sum_{r=1}^{10} G(r^2) \) is equal to:

In the expansion of \( \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^{n} \), \( n \in \mathbb{N} \), if the ratio of \( 15^{\text{th}} \) term from the beginning to the \( 15^{\text{th}} \) term from the end is \( \frac{1}{6} \), then the value of \( {}^{n}C_{3} \) is:

Consider two vectors \( \vec{u} = 3\hat{i} - \hat{j} \) and \( \vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}, \ \lambda > 0 \). The angle between them is given by \( \cos^{-1}\!\left(\frac{\sqrt{5}}{2\sqrt{7}}\right) \). Let \( \vec{v} = \vec{v}_1 + \vec{v}_2 \), where \( \vec{v}_1 \) is parallel to \( \vec{u} \) and \( \vec{v}_2 \) is perpendicular to \( \vec{u} \). Then the value of \( \lvert \vec{v}_1 \rvert^2 + \lvert \vec{v}_2 \rvert^2 \) is equal to:

Consider the equation \( x^{2} + 4x - n = 0 \), where \( n \in [20,100] \) is a natural number. Then the number of all distinct values of \( n \), for which the given equation has integral roots, is equal to:

Let \( A \) and \( B \) be two distinct points on the line \( L : \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2} \). Both \( A \) and \( B \) are at a distance \( 2\sqrt{17} \) from the foot of perpendicular drawn from the point \( (1,2,3) \) on the line \( L \). If \( O \) is the origin, then \( \overrightarrow{OA}\cdot\overrightarrow{OB} \) is equal to:

Let \( f:[0,\infty) \to \mathbb{R} \) be a differentiable function such that \( f(x)=1-2x+\int_{0}^{x} e^{x-t}f(t)\,dt \) for all \( x \in [0,\infty) \). Then the area of the region bounded by \( y=f(x) \) and the coordinate axes is:

Let \( f,g:(1,\infty)\to \mathbb{R} \) be defined as \( f(x)=\frac{2x+3}{5x+2} \) and \( g(x)=\frac{2-3x}{1-x} \). If the range of the function \( f\circ g : [2,4]\to \mathbb{R} \) is \( [\alpha,\beta] \), then \( \frac{1}{\beta-\alpha} \) is equal to:

Let the shortest distance between the lines \( \dfrac{x-3}{3} = \dfrac{y-\alpha}{-1} = \dfrac{z-3}{1} \) and \( \dfrac{x+3}{-3} = \dfrac{y+7}{2} = \dfrac{z-\beta}{4} \) be \( 3\sqrt{30} \). Then the positive value of \( 5\alpha + \beta \) is:

Consider the sets \( A = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + y^{2} = 25\} \), \( B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + 9y^{2} = 144\} \), \( C = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : x^{2} + y^{2} \le 4\} \) and \( D = A \cap B \). The total number of one-one functions from the set \( D \) to the set \( C \) is:

Let \( A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \} \), \( B = \{ z \in \mathbb{C} : \operatorname{Re}(z - iz) = 2 \} \) and \( S = A \cap B \). Then \( \sum\limits_{z \in S} |z|^{2} \) is equal to ______.

If for some \( \theta \in (0,\pi) \), \( A^{2} = A^{T} \), then the sum of the diagonal elements of the matrix \( (A + I)^{3} + (A - I)^{3} - 6A \) is equal to ______.

Let \( C \) be the circle \( x^{2} + (y - 1)^{2} = 2 \), \( E_{1} \) and \( E_{2} \) be two ellipse whose centres lie at the origin and major axes lie on \( x \)-axis and \( y \)-axis respectively. Let the straight line \( x + y = 3 \) touch the curves \( C, E_{1} \) and \( E_{2} \) at \( P(x_{1},y_{1}), Q(x_{2},y_{2}) \) and \( R(x_{3},y_{3}) \) respectively. Given that \( P \) is the mid-point of the line segment \( QR \) and \( PQ = \dfrac{2\sqrt{2}}{3} \), the value of \( 9(x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}) \) is equal to ______.

If the area of the region \( \{(x,y) : |x - 5| \le y \le 4\sqrt{x} \} \) is \( A \), then \( 3A \) is equal to ______.

Let \( m \) and \( n \) be the number of points at which the function \( f(x) = \max \{ x, x^{3}, x^{5}, \ldots , x^{21} \} , \; x \in \mathbb{R}, \) is not differentiable and not continuous, respectively. Then \( m + n \) is equal to ________.