JEE Main 2026 Apr 4 Shift 2 — Mathematics

For the function $f:[1,\infty) \to [1,\infty)$ defined by $f(x) = (x-1)^4 + 1$, among the two statements:

(I) The set $S = \{x \in [1,\infty) : f(x) = f^{-1}(x)\}$ contains exactly two elements, and

(II) The set $S = \{x \in [1,\infty) : f(x) = f^{-1}(x+1)\}$ is an empty set,

Let $S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$. Then $\displaystyle\sum_{z \in S} |z + \sqrt{3}\,i|^2$ is equal to:

If the system of equations:

$$x + y + z = 5$$

$$x + 2y + 3z = 9$$

$$x + 3y + \lambda z = \mu$$

has infinitely many solutions, then the value of $\lambda + \mu$ is:

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

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $\det(A - \alpha I) = 0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x - p)^2 + (y - 2p)^2 = 320$, intersects the co-ordinate axes at

Let $\alpha = \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \ldots\infty$ and $\beta = \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \ldots\infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_5(\beta)}$ is equal to:

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

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

Let P $(3\cos\alpha, 2\sin\alpha)$, $\alpha \neq 0$, be a point on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$, Q be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$ and R be a point on the line $x + y = 5$ such that the centroid of the triangle PQR is $\left(2 + \cos\alpha,\ 3 + \dfrac{2}{3}\sin\alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\dfrac{8}{3}$. If the line $x = \alpha$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $4\sqrt{15}$, where O is the origin, then $\alpha^2$ equals

$\displaystyle\max_{0 \le x \le \pi}\left(16\sin\left(\frac{x}{2}\right)\cos^3\left(\frac{x}{2}\right)\right)$ is equal to:

The shortest distance between the lines

$$\vec{r} = \left(\frac{1}{3}\hat{i} + 2\hat{j} + \frac{8}{3}\hat{k}\right) + \lambda\left(2\hat{i} - 5\hat{j} + 6\hat{k}\right)$$

and $\vec{r} = \left(-\dfrac{2}{3}\hat{i} - \dfrac{1}{3}\hat{k}\right) + \mu\left(\hat{j} - \hat{k}\right)$, $\lambda, \mu \in \mathbb{R}$, is:

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

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

Let for some $\alpha \in \mathbb{R}$, $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+y) = f(x) + 2y^2 + y + \alpha xy$ for all $x, y \in \mathbb{R}$. If $f(0) = -1$ and $f(1) = 2$, then the value of $\displaystyle\sum_{n=1}^{5}\left(\alpha + f(n)\right)$ is:

Let

$$A = \{(a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22\}.$$

Then $n(A)$ is equal to:

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

Let $y = y(x)$ be the solution of the differential equation:

$$\frac{dy}{dx} + \left(\frac{6x^2 + \left(3x^2 + 2x^3 + 4\right)e^{-2x}}{\left(x^3 + 2\right)\left(2 + e^{-2x}\right)}\right)y = 2 + e^{-2x},$$

$x \in (-1, 2)$, satisfying $y(0) = \dfrac{3}{2}$. If $y(1) = \alpha(2 + e^{-2})$, $\alpha$ is equal to:

The integral $\displaystyle\int_{0}^{1} \cot^{-1}\left(1 + x + x^2\right)dx$ 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 $\dfrac{a}{b}$, where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$, then $a + b$ is equal to __________

Let $f(x) = \begin{cases} e^{x-1} & , \ x < 0 \\ x^2 - 5x + 6 & , \ x \ge 0 \end{cases}$ 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, then $\alpha + \beta$ is equal to __________

Let A, 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 AB meets the angle bisector of the given half-lines at the point Q. If $PQ = \dfrac{9}{2}$ and R is the radius of the circumcircle of $\triangle PAB$, then $\dfrac{\alpha^2}{R}$ is equal to __________

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex B containing the right angle be $(4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of latus rectum of $C_o$ is __________

Let $f$ be a twice differentiable function such that

$$f(x) = \int_{0}^{x} \tan(t - x)\,dt - \int_{0}^{x} f(t)\tan t\,dt, \quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right).$$

Then $f''\left(\dfrac{\pi}{6}\right) + 12\,f'\left(-\dfrac{\pi}{6}\right) + f\left(\dfrac{\pi}{6}\right)$ is equal to __________