JEE Main 2026 (Online) 21st January Evening Shift

Let the line L pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(-2, r, 1)$ is $\sqrt{\frac{14}{3}}$, then the sum of all possible values of $r$ is :

Let the line $L_1$ be parallel to the vector $-3\hat{i} + 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point $(4, 3, 5)$. If t...

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$ and $f'(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^2 x - 2 \tan x + a),\ 0 < x < \frac{\pi}{2}$....

If the area of the region $\{(x, y) : 1-2x \leq y \leq 4-x^2,\; x \geq 0,\; y \geq 0 \}$ is $\dfrac{\alpha}{\beta}$, $\alpha, \beta \in \mathbb{N}, \gcd(\alpha,\beta)=1$, then the value of $(\alpha+\beta)$ is:

If $\left(\frac{1}{{ }^{15} \mathrm{C}_0}+\frac{1}{{ }^{15} \mathrm{C}_1}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_1}+\frac{1}{{ }^{15} \mathrm{C}_2}\right) \ldots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\r...

If $P$ is a point on the circle $x^2+y^2=4, Q$ is a point on the straight line $5 x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is $\_\_\_\_$ .

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument.Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to:

Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim\limits_{n \to \infty} \frac{1}{n^{3}} \sum\limits_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum\limits_{j=1}^{\infty} f(i)$ is equal to ________.

If $\int\limits_0^1 4 \cot ^{-1}\left(1-2 x+4 x^2\right) \mathrm{d} x=\mathrm{a\,tan}^{-1}(2)-\mathrm{b\,log}_{\mathrm{e}}(5)$, where $\mathrm{a}, \mathrm{b} \in \mathrm{N}$, then $(2 \mathrm{a}+\mathrm{b})$ is equal to $\_\_\_\_$ .

Let $y = y(x)$ be the solution of the differential equation $\sec x \dfrac{dy}{dx} - 2y = 2 + 3 \sin x$, $x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$,$y(0) = -\dfrac{7}{4}$. Then $y\left(\dfrac{\pi}{6}\right)$ is equal to :

Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$, $x \in \mathbb{R}$. Then the value of $f'(5)$ is :

If the line $\alpha x+4 y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3 x^2+4 y^2=1$ at the point P in the first quadrant, then one of the focal distances of $P$ is :

Let the maximum value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{\mathrm{m}}{\mathrm{n}} \pi^2$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1...

If the system of equations
$ 3x + y + 4z = 3 $
$ 2x + \alpha y - z = -3 $
$ x + 2y + z = 4 $
has no solution, then the value of $ \alpha $ is equal to:

For the matrices $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}$, if $(A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, then among th...

Let $y^2 = 12x$ be the parabola with its vertex at $O$. Let $P$ be a point on the parabola and $A$ be a point on the $x$-axis such that $\angle OPA = 90^\circ$. Then the locus of the centroid of such triangles $OPA$ is:

Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16,16)$. If $P(\alpha,\ \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to:

The largest $n \in \mathbb{N}$, for which $7^n$ divides $101!$, is :

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2 a x+(3 a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :

The positive integer n, for which the solutions of the equation $x(x+2) + (x+2)(x+4) + \cdots + (x+2n-2)(x+2n) = \frac{8n}{3}$ are two consecutive even integers, is :

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1 + a_2 + \ldots + a_{10} = 62$, then $a_1$ is equal to:

Let $A = \{x : |x^2 - 10| \leq 6\}$ and $B = \{x : |x - 2| > 1\}$. Then

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \leq 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric...

A random variable X takes values 0, 1, 2, 3 with probabilities $\frac{2a+1}{30}$, $\frac{8a-1}{30}$, $\frac{4a+1}{30}$, $b$ respectively, where $a, b \in \mathbb{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of...

For a triangle $A B C$, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and $\vec{r}=\overrightarrow{B A}$. If $|\vec{p}|=2 \sqrt{3},|\vec{q}|=2$ and $\cos \theta=\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\v...