JEE Main 2026 (Online) 4th April Evening Shift

The shortest distance between the lines
\(\vec{r}=\left(\frac{1}{3} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\frac{8}{3} \hat{\mathrm{k}}\right)+\lambda(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})\)
and $\vec{r}=\left(-\frac{2}{3...

If $\left(2 \alpha+1, \alpha^2-3 \alpha, \frac{\alpha-1}{2}\right)$ is the image of $(\alpha, 2 \alpha, 1)$ in the line $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}$, then the possible value(s) of $\alpha$ is (are)

The area of the region bounded by the curves $x+3 y^2=0$ and $x+4 y^2=1$ is equal to :

In the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x>0$, if the term independent of $x$ is (221)k, then k is equal to:

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

The integral $\int\limits_0^1 \cot ^{-1}\left(1+x+x^2\right) d x$ is equal to :

If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation
$x^3+a x^2+b x+c=0, a, b, \in \mathbb{R}$, then $\int_{-1}^1\left(x^3+a x^2+b x+c\right) d x$ is equal to:

Let $y=y(x)$ be the solution of the differential equation :
\(\frac{d y}{d x}+\left(\frac{6 x^2+\left(3 x^2+2 x^3+4\right) e^{-2 x}}{\left(x^3+2\right)\left(2+e^{-2 x}\right)}\right) y=2+e^{-2 x}\)
$x \in(-1,2)$, satisfying $y(0)=\frac{3}...

Let $f$ be a twice differentiable function such that
\(f(x)=\int_0^x \tan (t-x) d t-\int_0^x f(t) \tan t d t, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
Then $f^{\prime \prime}\left(\frac{\pi}{6}\right)+12 f^{\prime}\left(-\frac{\p...

Let $\mathrm{P}(3 \cos \alpha, 2 \sin \alpha), \alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1, \mathrm{Q}$ be a point on the circle $x^2+y^2-14 x-14 y+82=0$ and R be a point on the line $x+y=5$ such that the centro...

For the function $f:[1, \infty) \rightarrow[1, \infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements:
(I) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x)\right\}$ contains exactly two elements, and
(II) The set $\mathrm...

Let for some $\alpha \in \mathbb{R}, f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2 y^2+y+\alpha x y$ for all $x, y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum\limits_{n=1}^5(\alpha+f(n...

Let $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\frac{8}{3}$. If the line $x=\alpha$ intersects the hyperbola H at the points A and...

Let $f(x)=\left\{\begin{array}{cc}e^{x-1} & , x<0 \\ x^2-5 x+6 & , x \geq 0\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, th...

Let $\alpha=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty$ and
$\beta=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots \infty$. Then the value of
$(0.2)^{\log _{\sqrt{5}}(\alpha)}+(0.04)^{\log _5(\beta)}$ is equal to :

Let $A=\left[\begin{array}{ccc}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{array}\right]$ and $\operatorname{det}(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2...

If the system of equations :
$$ \begin{aligned} & x+y+z=5 \\ & x+2 y+3 z=9 \\ & x+3 y+\lambda z=\mu \end{aligned} $$
has infinitely many solutions, then the value of $\lambda+\mu$ is :

Let $\mathrm{A}, \mathrm{B}$ and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16 x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle A B C$ be a co...

Let
$\mathrm{A}=\{(a, b, c): a, b, c$ are non-negative integers and $a+b+2 c=22\}$.
Then $n(\mathrm{~A})$ is equal to :

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\frac{a}{b}$, where $a, b \in$ N and $\operatorname{gcd}(a, b)=1$, then $a+b$ is equal to $\_\_\_\...

If the quadratic equation $(\lambda+2) x^2-3 \lambda x+4 \lambda=0, \lambda \neq-2$, has two positive roots, then the number of possible integral values of $\lambda$ is :

For 10 observations $x_1, x_2, \ldots, x_{10}$, if $\sum\limits_{i=1}^{10}\left(x_i+2\right)^2=180$ and $\sum\limits_{i=1}^{10}\left(x_i-1\right)^2=90$, then their standard deviation is :

Let $\mathrm{A}, \mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha, \alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $A B$ meets the angle bisector of the given half-lines at the poi...

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u} \times \hat{v}|=\frac{\sqrt{3}}{2}$. If $\overrightarrow{\mathrm{A}}=\lambda \hat{u}+\hat{v}+(\hat{u} \times \hat{v})$, then $\lambda$ is equal to: