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

$x^4dy + (4x^3y + 2\sin x)dx = 0,; x > 0,; y\left(\frac{\pi}{2}\right) = 0.$

Then $\pi^4 y\left(\frac{\pi}{3}\right)$ is equal to:

1

Let $A = {-2, -1, 0, 1, 2, 3, 4}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x + y \le 2$. Let $\e…

Topic: JEE Main 2026 (23 January Morning Shift)

2

Let the line $y - x = 1$ intersect the ellipse $\frac{x^2}{2} + \frac{y^2}{1} = 1$ at the points $A$ and $B$. Then the …

Topic: JEE Main 2026 (23 January Morning Shift)

3

A rectangle is formed by the lines $x = 0, y = 0, x = 3$ and $y = 4$. Let the line $L$ be perpendicular to $3x + y + 6 …

Topic: JEE Main 2026 (23 January Morning Shift)

4

Let $\vec{a} = -\hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = \hat{i} - \hat{j} - 3\hat{k}$, $\vec{c} = \vec{a} \times \vec…

Topic: JEE Main 2026 (23 January Morning Shift)

5

Let $f(x) = \displaystyle \int \frac{(2 - x^2)e^x}{\sqrt{1+x}(1-x)^{3/2}},dx$. If $f(0) = 0$, then $f\left(\frac{1}{2}\…

Topic: JEE Main 2026 (23 January Morning Shift)

6

Let the domain of the function $f(x) = \log_5\log_7(9x - x^2 - 13)$ be the interval $(m,n)$. Let the hyperbola $\dfrac{…

Topic: JEE Main 2026 (23 January Morning Shift)

1

If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation$(-2+\sqrt{3})\left(|\sqrt{x}-3|\right) + (x…

Topic: JEE Main 2026 (23 January Morning Shift)

2

The sum of all possible values of $n \in \mathbb{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion…

Topic: JEE Main 2026 (23 January Morning Shift)

3

The vertices $B$ and $C$ of a triangle $ABC$ lie on the line $\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$. The …

Topic: JEE Main 2026 (23 January Morning Shift)

4

Number of solutions of $\sqrt{3}\cos2\theta + 8\cos\theta + 3\sqrt{3} = 0,; \theta \in [-3\pi, 2\pi]$ is:

Topic: JEE Main 2026 (23 January Morning Shift)

5

Let $S = {z : 3 \le |2z - 3(1 + i)| \le 7}$ be a set of complex numbers. Then $\min_{z \in S} \left|z + \frac{1}{2}(5 +…

Topic: JEE Main 2026 (23 January Morning Shift)

6

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function$f(\theta) = 4\left(\sin^4\l…

Topic: JEE Main 2026 (23 January Morning Shift)

7

A building construction work can be completed by two masons $A$ and $B$ together in $22.5$ days. Mason $A$ alone can co…

Topic: JEE Main 2026 (23 January Morning Shift)

8

Let \( f(x) = \begin{cases} \dfrac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \ne -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac…

Topic: JEE Main 2026 (23 January Morning Shift)

9

The value of the integral $\displaystyle \int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt{3}\tan 2x}$ is:

Topic: JEE Main 2026 (23 January Morning Shift)

10

Among the statements: I : \( \begin{vmatrix} 1 & \cos\alpha & \cos\beta \\ \cos\alpha & 1 & \cos\gamma \\ \cos\beta & …

Topic: JEE Main 2026 (23 January Morning Shift)