JEE Main 2026 (Online) 4th April Morning Shift

A line with direction ratios $1,-1,2$ intersects the lines $\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3}$ and $\frac{x+1}{-1}=\frac{y-2}{1}=\frac{z}{4}$ at the points P and Q , respectively. If the length of the line segment PQ is $\alpha$, then $...

The square of the distance of the point $(-2,-8,6)$ from the line $\frac{x-1}{1}=\frac{y-1}{2}=\frac{z}{-1}$ along the line $\frac{x+5}{1}=\frac{y+5}{-1}=\frac{z}{2}$ is equal to:

The area of the region $\{(x, y): y \leq \pi-|x|, y \leq|x \sin x|, y \geq 0\}$ is:

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3+(1+x)^4+(1+x)^5+\ldots+(1+x)^{99}+(1+k x)^{100}, x \neq 0$, is $\left(43 n+\frac{101}{4}\right)\left({ }^{100} \mathrm{C}_3\right)$ for some $n \in...

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3 y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S , and the mid point of the line segment RS is $(\alpha, \beta)$, then $6(\alph...

Let $z$ be a complex number such that $|z+2|=|z-2|$ and arg $\left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^2$ is equal to:

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$, for all $x \in \mathbb{R}$. If $f(0)=0$, then $36\left(f^{\prime}(2)+f^{\prime \prime}(2)+\int_0^2 f(x) d x\right)$ is equal to:

Let $\int\limits_{-2}^2(|\sin x|+[x \sin x]) d x=2(3-\cos 2)+\beta$, where [ ⋅ ] is the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals:

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\left(1+x+x^2\right)\left(1-y+y^2\right), y(0)=\frac{1}{2}$. Then $(2 y(1)-1)$ is equal to

If $y=\tan ^{-1}\left(\frac{3 \cos x-4 \sin x}{4 \cos x+3 \sin x}\right)+2 \tan ^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right)$, then $\frac{d y}{d x}$ at $x=\frac{\sqrt{3}}{2}$ is equal to :

Consider the parabola $\mathrm{P}: y^2=4 k x$ and the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Let the line segment joining the points of intersection of P and E , be their latus rectums. If the eccentricity of E is $e$, the...

Let [•] denote the greatest integer function. If the domain of the function
$f(x)=\cos ^{-1}\left(\frac{4 x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha+\beta)$ is equal to :

The number of functions $f:\{1,2,3,4\} \rightarrow\{a, b, c\}$, which are not onto, is :

The number of points, at which the function $f(x)=\max \left\{6 x, 2+3 x^2\right\}+|x-1| \cos \left|x^2-\frac{1}{4}\right|, x \in(-\pi, \pi)$, is not differentiable, is
$\_\_\_\_$ .

Let $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5\end{array}\right]$. Then the sum of all elements of the matrix $\operatorname{adj}\left(\operatorname{adj}\left(2(\operatorname{adj} \mathrm{~A})^{-1}\right)\right)$ is equ...

Let $\mathrm{S}=\left\{\mathrm{A}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]: a, b, c, d \in\{0,1,2,3,4\}\right.$ and $\left.\mathrm{A}^2-4 \mathrm{~A}+3 \mathrm{I}=0\right\}$ be a set of $2 \times 2$ matrices. Then the number ...

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96 p$ is equal to $\_\_\_\_$ .

If the set of all solutions of $\left|x^2+x-9\right|=|x|+\left|x^2-9\right|$ is $[\alpha, \beta] \cup[\gamma, \infty)$, then ( $\alpha^2+\beta^2+\gamma^2$ ) is equal to:

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

Suppose that the mean and median of the non-negative numbers $21,8,17, a, 51,103, b, 13,67,(a>b)$, are 40 and 21 , respectively. If the mean deviation about the median is 26 , then $2 a$ is equal to :

Let the line $\mathrm{L}_1: x+3=0$ intersect the lines $\mathrm{L}_2: x-y=0$ and $\mathrm{L}_3: 3 x+y=0$ at the points A and B , respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ a...

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha, \beta)$, then $21(\alpha+\beta)$ is equal to :

If $\mathrm{A}=\frac{\sin 3^{\circ}}{\cos 9^{\circ}}+\frac{\sin 9^{\circ}}{\cos 27^{\circ}}+\frac{\sin 27^{\circ}}{\cos 81^{\circ}}$ and $\mathrm{B}=\tan 81^{\circ}-\tan 3^{\circ}$, then $\frac{\mathrm{B}}{\mathrm{A}}$ is equal to
$\_\_\_\_...

Let $\overrightarrow{a_k}=\left(\tan \theta_k\right) \hat{i}+\hat{j}$ and $\overrightarrow{b_k}=\hat{i}-\left(\cot \theta_k\right) \hat{j}$, where $\theta_k=\frac{2^{k-1} \pi}{2^n+1}$, for some $n \in \mathbb{N}, n>5$. Then the value of $\f...