JEE Main 2026 (Online) 5th April Morning Shift

The square of the distance of the point $\mathrm{P}(5,6,7)$ from the line $\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4}$ is equal to:

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0$ and $\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})$ from the origin is :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f^{\prime}(0)=3$. Then the minimum value of the function $g(x)=3+e^x f(x...

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 27,1 \leq y \leq x^2\right\}$ is equal to :

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3}-x^4\right)^n, x \neq 0$, is zero, then the value of $n$ is $\_\_\_\_$ .

Let P be a moving point on the circle $x^2+y^2-6 x-8 y+21=0$. Then, the maximum distance of P from the vertex of the parabola $x^2+6 x+y+13=0$ is equal to:

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2+a x+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3 i \sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

The value of the integral $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{4-\operatorname{cosec}^2 x}{\cos ^4 x}\right) d x$ is :

The value of the integral $\int\limits_0^{\infty} \frac{\log _e(x)}{x^2+4} d x$ is:

Let $y=y(x)$ be the solution of the differential equation $x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}$ and let $\alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)$. T...

Let a focus of the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $\mathrm{S}(4,0)$ and its eccentricity be $\frac{4}{5}$. If the point $\mathrm{P}(3, \alpha)$ lies on E and O is the origin, then the area of $\triangle \mathrm{P...

Let $\mathrm{A}=\{1,2,3,4,5,6\}$. The number of one-one functions $f: \mathrm{A} \rightarrow \mathrm{A}$ such that $f(1) \geq 3, f(3) \leq 4$ and $f(2)+f(3)=5$, is $\_\_\_\_$ .

If $\frac{\pi}{4}+\sum\limits_{p=1}^{11} \tan ^{-1}\left(\frac{2^{p-1}}{1+2^{2 p-1}}\right)=\alpha$, then $\tan \alpha$ is equal to $\_\_\_\_$ .

The product of all possible values of $\alpha$, for which
$\lim \limits_{x \rightarrow 0}\left(\frac{1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x)}{\sin ^2((\alpha+1) x)}\right)=2$, is :

Consider the system of linear equations in $x, y, z$ :
$$ \begin{aligned} & x+2 y+t z=0 \\ & 6 x+y+5 t z=0 \\ & 3 x+t^2 y+f(t) z=0 \end{aligned} $$
where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system ha...

Let A be a $3 \times 3$ matrix such that
$$ \mathrm{A}^{\mathrm{T}}\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]=\left[\begin{array}{l} 5 \\ 2 \\ 2 \end{array}\right], \mathrm{A}^{\mathrm{T}}\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{a...

Two players A and B play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player A wins the series is $\_\_\_\_$ .

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

Let the sum of the first $n$ terms of an A.P. be $3 n^2+5 n$. Then the sum of squares of the first 10 terms of the A.P. is:

$\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:

The mean deviation about the mean for the data
$$ \begin{array}{|c|c|c|c|c|c|c|} \hline x_i & 5 & 7 & 9 & 10 & 12 & 15 \\ \hline f_i & 8 & 6 & 2 & 2 & 2 & 6 \\ \hline \end{array} $$
\(\text { is equal to: }\)

In an equilateral triangle $P Q R$, let the vertex $P$ be at $(3,5)$ and the side $Q R$ be along the line $x+y=4$. If the orthocentre of the triangle PQR is $(\alpha, \beta)$, then $9(\alpha+\beta)$ is equal to:

The sum of all the integral values of $p$ such that the equation $3 \sin ^2 x+12 \cos x-3=p, x \in \mathbb{R}$, has at least one solution, is:

Let $\tan A, \tan B$, where $A, B \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2-2 x-5=0$. Then $20 \sin ^2\left(\frac{A+B}{2}\right)$ is equal to:

Let $\vec{a}=\sqrt{7} \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+2 \hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}+\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{r} \cdot \vec{a}=0$, then $|3 \vec{r}|^2$ is equ...