Let $[ \cdot ]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left[\frac{k^2}{3^x}\right]$. Then $12\sum_{j=1}^{\infty} f(j)$ is equal to:

1

The largest $n \in \mathbb{N}$, for which $7^n$ divides $101!$, is:

Topic: JEE Main 2026 (21 January Evening Shift)

2

Let $z$ be the complex number satisfying $|z - 5| \le 3$ and having maximum positive principal argument. Then $34\left|…

Topic: JEE Main 2026 (21 January Evening Shift)

3

The system of equations$3x + y + 4z = 3$$2x + ay - z = -3$$x + 2y + z = 4$has no solution, then the value of $\alpha$ i…

Topic: JEE Main 2026 (21 January Evening Shift)

4

Let $A = {2,3,5,7,9}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $\ell$ be the nu…

Topic: JEE Main 2026 (21 January Evening Shift)

5

Let $y = y(x)$ be the solution of differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$, $x \in \left(-\frac{…

Topic: JEE Main 2026 (21 January Evening Shift)

6

For a triangle $ABC$, let $\vec{p} = \overrightarrow{BC},; \vec{q} = \overrightarrow{CA}$ and $\vec{r} = \overrightarro…

Topic: JEE Main 2026 (21 January Evening Shift)

7

For the matrices $A = \begin{bmatrix} 3 & -4 \ 1 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} -29 & 49 \ -13 & 18 \en…

Topic: JEE Main 2026 (21 January Evening Shift)

8

Let $A = {x : |x^2 - 10| \le 6}$ and $B = {x : |x - 2| > 1}$. Then

Topic: JEE Main 2026 (21 January Evening Shift)

9

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2},; \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. …

Topic: JEE Main 2026 (21 January Evening Shift)

10

If the area of the region ${(x,y) : 1 - 2x \le y \le 4 - x^2,; x \ge 0,; y \ge 0}$ is $\frac{\alpha}{\beta}$, $\alpha, …

Topic: JEE Main 2026 (21 January Evening Shift)

1

If $\displaystyle \int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^2),dx = a\tan^{-1}(2) - b\log_e(5)$, where $a,b \in \mathbb{N}$, …

Topic: JEE Main 2026 (21 January Evening Shift)

2

Let the maximum value of $(\sin^{-1}x)^2 + (\cos^{-1}x)^2$ for $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\rig…

Topic: JEE Main 2026 (21 January Evening Shift)

3

If $\displaystyle \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\ri…

Topic: JEE Main 2026 (21 January Evening Shift)

4

If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = …

Topic: JEE Main 2026 (21 January Evening Shift)