JEE Main 2026 Apr 5 Shift 2 — Mathematics

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of the equation $x^2 - 4x + q = 0$; $p, q \in \mathbf{Z}$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then $|p + q|$ equals :

Let $z_1, z_2 \in \mathbf{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to :

If $f : \mathbf{N} \to \mathbf{Z}$ is defined by

$$f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}, \; k \in \mathbf{N},$$

and $\displaystyle\sum_{n=1}^{k} f(n) = 98$, then $k$ is equal to :

Let $M$ be a $3 \times 3$ matrix such that

$$M\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \; M\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \text{ and } M\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}. \text{ If } M\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 7 \\ 11 \end{bmatrix},$$

then $x + y + z$ equals :

If the sum of the first 10 terms of the series $\dfrac{1}{1 + 1^4 \times 4} + \dfrac{2}{1 + 2^4 \times 4} + \dfrac{3}{1 + 3^4 \times 4} + \dfrac{4}{1 + 4^4 \times 4} + \ldots$ is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to :

Let $A_1, A_2, A_3, \ldots, A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to :

The coefficient of $x^2$ in the expansion of $\left(2x^2 + \dfrac{1}{x}\right)^{10}$, $x \neq 0$, is :

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is :

A variable $X$ takes values $0, 0, 2, 6, 12, 20, \ldots, n(n-1)$ with frequencies $^{n}C_0, {}^{n}C_1, {}^{n}C_2, {}^{n}C_3, {}^{n}C_4, {}^{n}C_5, \ldots, {}^{n}C_n$, respectively. If the mean of this data is 60, then its median is :

Let the point $P$ be the vertex of the parabola $y = x^2 - 6x + 12$. If a line passing through the point $P$ intersects the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ at the points $R$ and $S$, then the maximum value of $(PR + PS)^2$ is :

Let the directrix of the parabola $P : y^2 = 8x$, cut $x$-axis at the point $A$. Let $B(\alpha, \beta)$, $\alpha > 1$, be a point on $P$ such that the slope of $AB$ is $3/5$. If $BC$ is a focal chord of $P$, then six times the area of $\Delta ABC$ is :

Let the eccentricity $e$ of a hyperbola satisfy the equation $6e^2 - 11e + 3 = 0$. If the foci of the hyperbola are $(3, 5)$ and $(3, -4)$, then the length of its latus rectum is :

Let a triangle $PQR$ be such that $P$ and $Q$ lie on the line $\dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2}$ and are at a distance of 6 units from $R\,(1, 2, 3)$. If $(\alpha, \beta, \gamma)$ is the centroid of $\Delta PQR$, then $\alpha + \beta + \gamma$ is equal to :

If the distance of the point $(a, 2, 5)$ from the image of the point $(1, 2, 7)$ in the line $\dfrac{x}{1} = \dfrac{y-1}{1} = \dfrac{z-2}{2}$ is 4, then the sum of all possible values of $a$ is equal to :

Let $O$ be the origin, $\overrightarrow{OP} = \vec{a}$ and $\overrightarrow{OQ} = \vec{b}$. If $R$ is the point on $\overrightarrow{OP}$ such that $\overrightarrow{OP} = 5\,\overrightarrow{OR}$, and $M$ is the point such that $\overrightarrow{OQ} = 5\,\overrightarrow{RM}$, then $\overrightarrow{PM}$ is equal to :

Let $f(x) = \displaystyle\lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x) = \sin x$, $x \in \mathbf{R}$ is :

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\displaystyle\int_{\log_e(\alpha - 1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}$ is :

Let $f : [1, \infty) \to \mathbf{R}$ be a differentiable function defined as $f(x) = \displaystyle\int_1^x f(t)\,dt + (1 - x)(\log_e x - 1) + e$. Then the value of $f(f(1))$ is :

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbf{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements, in the relation $R = \left\{\left((a_1, b_1), (a_2, b_2)\right) \in ((A \times B) \times (A \times B)) : a_1 + b_2 \text{ divides } a_2 + b_1\right\}$ is __________.

From the point $(-1, -1)$, two rays are sent making angles of $45°$ with the line $x + y = 0$. These rays get reflected from the mirror $x + 2y = 1$. If the equations of the reflected rays are $ax + by = 9$ and $cx + dy = 7$, $a, b, c, d \in \mathbf{Z}$, then the value of $ad + bc$ is __________.

If $S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\dfrac{5\theta}{2} = \cos 7\theta \cos\dfrac{7\theta}{2}\right\}$, then $n(S)$ is equal to __________.

Let $f : \mathbf{R} \to \mathbf{R}$ be a function such that $f(x) + 3f\left(\dfrac{\pi}{2} - x\right) = \sin x$, $x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3$, $\beta > 0$, is $\alpha^2$, then $30\beta^3$ is equal to __________.

Let $y = y(x)$ be the solution of the differential equation

$$(\tan x)^{1/2}\,dy = \left(\sec^3 x - (\tan x)^{3/2}\,y\right)dx, \; 0 < x < \frac{\pi}{2}, \; y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}.$$

If $y\left(\dfrac{\pi}{3}\right) = \dfrac{4}{5}\,\alpha$, then $\alpha^4$ equals __________.