JEE Main 2026 Apr 6 Shift 2 — Mathematics

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3}$. Then $f$ is :

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0,\ n \in \mathbb{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

Let $S = \left\{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0\right\}$. Then $\sum_{z \in S} z^8$ is equal to :

The sum of all possible values of $\theta \in [0, 2\pi]$, for which the system of equations : $$x\cos 3\theta - 8y - 12z = 0$$ $$x\cos 2\theta + 3y + 3z = 0$$ $$x + y + 3z = 0$$ has a non-trivial solution, is equal to :

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $B = [b_{ij}],\ 1 \le i, j \le 3$. If $B = A^{99} - I$, then the value of $\frac{b_{31} - b_{21}}{b_{32}}$ is :

The sum $1 + \frac{1}{2}\left(1^2 + 2^2\right) + \frac{1}{3}\left(1^2 + 2^2 + 3^2\right) + \ldots$ upto 10 terms is equal to :

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

Let the mean and the variance of seven observations $2, 4, \alpha, 8, \beta, 12, 14$, $\alpha < \beta$, be 8 and 16 respectively. Then the quadratic equation whose roots are $3\alpha + 2$ and $2\beta + 1$ is :

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

Let C be a circle having centre in the first quadrant and touching the $x$-axis at a distance of 3 units from the origin. If the circle C has an intercept of length $6\sqrt{3}$ on $y$-axis, then the length of the chord of the circle C on the line $x - y = 3$ is :

The eccentricity of an ellipse E with centre at the origin O is $\frac{\sqrt{3}}{2}$ and its directrices are $x = \pm \frac{4\sqrt{6}}{3}$. Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :

Let $x = 9$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $P(\alpha, 0)$, $\alpha > 0$, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

If $\sin\left(\tan^{-1}\left(x\sqrt{2}\right)\right) = \cot\left(\sin^{-1}\sqrt{1 - x^2}\right)$, $x \in (0, 1)$, then the value of $x$ is :

The shortest distance between the lines $\frac{x-4}{1} = \frac{y-3}{2} = \frac{z-2}{-3}$ and $\frac{x+2}{2} = \frac{y-6}{4} = \frac{z-5}{-5}$ is :

Let $\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}$ and $\vec{b} = 6\hat{i} + 3\hat{j} + 3\hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $\left(2\vec{a} + 3\vec{b}\right)$ and $\left(\vec{a} - \vec{b}\right)$ is :

Let $\lim_{x \to 2} \frac{(\tan(x - 2))\left(rx^2 + (p - 2)x - 2p\right)}{(x - 2)^2} = 5$ for some $r, p \in \mathbb{R}$. If the set of all possible values of $q$, such that the roots of the equation $rx^2 - px + q = 0$ lie in $(0, 2)$, be the interval $(\alpha, \beta]$, then $4(\alpha + \beta)$ equals :

Let $A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$ be a singular matrix. Let $f(x) = \int_0^x \left(t^2 + 2t + 3\right) dt$, $x \in [1, \alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$, then $3(M - m)$ is equal to :

Let $f: \mathbb{R} \to \mathbb{R}$ be such that $f(xy) = f(x)f(y)$, for all $x, y \in \mathbb{R}$ and $f(0) \neq 0$. Let $g : [1, \infty) \to \mathbb{R}$ be a differentiable function such that $$x^2 g(x) = \int_1^x \left(t^2 f(t) - t\,g(t)\right) dt.$$ Then $g(2)$ is equal to :

The area of the region $\left\{(x, y) : x^2 - 8x \le y \le -x\right\}$ is :

The value of the integral $\int_{-1}^{1} \left(\frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}\right) dx$ is equal to :

Let $R = \left\{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e (x + y) \le 2\right\}$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________.

If $(1 - x^3)^{10} = \sum_{r=0}^{10} a_r x^r (1 - x)^{30 - 2r}$, then $\frac{9a_9}{a_{10}}$ is equal to ________.

Let the line $x - y = 4$ intersect the circle $C : (x - 4)^2 + (y + 3)^2 = 9$ at the points Q and R. If $P(\alpha, \beta)$ is a point on C such that $PQ = PR$, then $(6\alpha + 8\beta)^2$ is equal to ________.

Let the image of the point $P(0, -5, 0)$ in the line $\frac{x - 1}{2} = \frac{y}{1} = \frac{z + 1}{-2}$ be the point R and the image of the point $Q\left(0, \frac{-1}{2}, 0\right)$ in the line $\frac{x - 1}{-1} = \frac{y + 9}{4} = \frac{z + 1}{1}$ be the point S. Then the square of the area of the parallelogram PQRS is ________.

Let $f(x) = \begin{cases} x^3 + 8\ ; & x < 0, \\ x^2 - 4\ ; & x \ge 0, \end{cases}$ and $g(x) = \begin{cases} (x - 8)^{1/3}\ ; & x < 0, \\ (x + 4)^{1/2}\ ; & x \ge 0. \end{cases}$ Then the number of points, where the function $g \circ f$ is discontinuous, is ________.