JEE Main 2026 (Online) 5th April Evening Shift

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

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

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)+3 f\left(\frac{\pi}{2}-x\right)=\sin x, x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g...

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

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

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

Let $\left(2^{1-\mathrm{a}}+2^{1+\mathrm{a}}\right), f(\mathrm{a}),\left(3^{\mathrm{a}}+3^{-\mathrm{a}}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(\mathrm{a})$. Then the value of the integral $\int\limits_{\log _e(\alpha-1)}...

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1 / 2} \mathrm{~d} y=\left(\sec ^3 x-(\tan x)^{3 / 2} y\right) \mathrm{d} x, 0 < x <\frac{\pi}{2}, y\left(\frac{\pi}{4}\right)=\frac{6 \sqrt{2}}{5}$. If $y\left(\frac{\pi}...

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

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

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

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

If $f: \mathbf{N} \rightarrow \mathbf{Z}$ is defined by
$$ f(n)=\left|\begin{array}{ccc} n & -1 & -5 \\ -2 n^2 & 3(2 k+1) & 2 k+1 \\ -3 n^3 & 3 k(2 k+1) & 3 k(k+2)+1 \end{array}\right|, k \in N, $$
and $\sum\limits_{n=1}^k f(n)=98$, then $k...

Let M be a $3 \times 3$ matrix such that $\mathrm{M}\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right), \mathrm{M}\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l...

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

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 :

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 t...

Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3, \ldots \ldots . ., \mathrm{A}_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $\mathrm{A}_{25}, \mathrm{~A}_{28}, \mathrm{~A}_{31}$ and $\mathrm{A}_{36}$ is equa...

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

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

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

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 :

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$ and $c x+d y=7, a,...

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

Let $O$ be the origin, $\overrightarrow{O P}=\vec{a}$ and $\overrightarrow{O Q}=\vec{b}$. If $R$ is the point on $\overrightarrow{O P}$ such that $\overrightarrow{O P}=5 \overrightarrow{O R}$, and $M$ is the point such that $\overrightarrow...