JEE Main 2026 (Online) 22nd January Evening Shift

Let L be the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$ and let S be the set of all points $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ on L , whose distance from the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$ along the line $L$ is 7 ...

The area of the region $\mathrm{A}=\left\{(x, y): 4 x^2+y^2 \leqslant 8\right.$ and $\left.y^2 \leqslant 4 x\right\}$ is:

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $(1+x)^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}, 0 \leq \mathrm{r} \leq \mathrm{n}$. If
$P_n=C_0-C_1+\frac{2^2}{3} C_2-\frac{2^3}{4} C_3+\...

Let $\mathrm{S}=\left\{z \in \mathbb{C}: 4 z^2+\bar{z}=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z|^2$ is equal to:

Let [.] be the greatest integer function. If $\alpha=\int\limits_0^{64}\left(x^{1 / 3}-\left[x^{1 / 3}\right]\right) \mathrm{d} x$, then $\frac{1}{\pi} \int\limits_0^{\alpha \pi}\left(\frac{\sin ^2 \theta}{\sin ^6 \theta+\cos ^6 \theta}\rig...

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$, then $2 \sin \alpha$ is equal to :

Let S and $\mathrm{S}^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $\mathrm{P}(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(\mathrm{SP})^2+\left(\mathrm{S}^{\prime} \mathrm{P}\right)^2-\...

Let $f(x)=[x]^2-[x+3]-3, x \in \mathbf{R}$, where [.] is the greatest integer funtion. Then

Let the domain of the function $f(x)=\log _3 \log _5\left(7-\log _2\left(x^2-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$. If $\sum\limits_{x=1}^{\mathrm{n}}\left(\frac{f(x)}{\mathrm{g}(x)}\right)=19607$, then n is equal to :

Let $\mathrm{P}(10,2 \sqrt{15})$ be a point on the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$, whose foci are S and $\mathrm{S}^{\prime}$. If the length of its latus rectum is 8 , then the square of the area of $\Delta...

If $\lim\limits_{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to :

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^2\right\}$.
Let $\mathrm{S}=\left\{x \in(-2,2)\right.$ : the function $\mathrm{g}(x)=|x|\left[x^2\right]$ is discontinuous at $\left.x\right\}$.
The...

Let n be the number obtained on rolling a fair die. If the probability that the system
$$ \begin{aligned} & x-\mathrm{n} y+z=6 \\ & x+(\mathrm{n}-2) y+(\mathrm{n}+1) z=8 \\ & \quad(\mathrm{n}-1) y+z=1 \end{aligned} $$
has a unique solution ...

If $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ is a solution of the system of equations $A X=B$, where $\operatorname{adj} A=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3\end{array}\right]$ and $\mathrm{B}=\left[...

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4 x$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

Let $\alpha, \beta$ be the roots of the quadratic equation $12 x^2-20 x+3 \lambda=0, \lambda \in \mathbf{Z}$. If $\frac{1}{2} \leqslant|\beta-\alpha| \leqslant \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $\mathrm{a}^2, 2 \mathrm{~b}^2, \mathrm{c}^2$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left(\mathrm{a}^2+\mathrm{b}^2+\m...

The number of elements in the relation $\mathrm{R}=\left\{(x, y): 4 x^2+y^2<52, x, y \in \mathbf{Z}\right\}$ is

Let S be the set of the first 11 natural numbers. Then the number of elements in $A=\{B \subseteq S: n(B) \geqslant 2$ and the product of all elements of $B$ is even $\}$ is $\_\_\_\_$ .

If the mean deviation about the median of the numbers $\mathrm{k}, 2 \mathrm{k}, 3 \mathrm{k}, \ldots ., 1000 \mathrm{k}$ is 500 , then $\mathrm{k}^2$ is equal to :

Among the statements
$(S 1)$ : If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$
and
(S2) : If positive numbers $2 a, b, c$ are three consecutive terms of an A.P., th...

Let $\cos (\alpha+\beta)=-\frac{1}{10}$ and $\sin (\alpha-\beta)=\frac{3}{8}$, where $0<\alpha<\frac{\pi}{3}$ and $0<\beta<\frac{\pi}{4}$. If $\tan 2 \alpha=\frac{3(1-r \sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r, s \in N$, then $r+s$ is equal to ...

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2 \hat{k}, \lambda \in \boldsymbol{Z}$ be two vectors. Let $\vec{c}=\vec{a} \times \vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in $y z$-plane. If $|\vec{c}|=\sqrt{53...

Let a vector $\overrightarrow{\mathrm{a}}=\sqrt{2} \hat{i}-\hat{j}+\lambda \hat{k}, \lambda>0$, make an obtuse angle with the vector $\overrightarrow{\mathrm{b}}=-\lambda^2 \hat{i}+4 \sqrt{2} \hat{j}+4 \sqrt{2} \hat{k}$ and an angle $\theta...