Let a differentiable function $f$ satisfy the equation $\int_0^{36} f\left(\frac{tx}{36}\right)dt=4\alpha f(x)$ If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4,\beta)$, then $\beta^\alpha$ is equal to ______.

1

The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is:

Topic: JEE Main 2026 (24 January Morning Shift)

2

Let $A_1$ be the bounded area enclosed by the curves $y=x^2+2,; x+y=8$ and y-axis lies in the first quadrant. Let $A_2$…

Topic: JEE Main 2026 (24 January Morning Shift)

3

The mean and variance of a data of $10$ observations are $10$ and $2$, respectively. If an observation $\alpha$ in the…

Topic: JEE Main 2026 (24 January Morning Shift)

4

From a lot containing $10$ defective and $90$ non-defective bulbs, $8$ bulbs are selected one by one with replacement. …

Topic: JEE Main 2026 (24 January Morning Shift)

5

Let \( \alpha, \beta \in \mathbb{R} \) be such that the function \( f(x)= \begin{cases} 2\alpha(x^2-2)+2\beta x, & x

Topic: JEE Main 2026 (24 January Morning Shift)

6

Consider an A.P.: $a,a_2,\ldots,a_n$ If $a_2-a_1=-\frac{3}{4},; a_n-a_{n-1}=\frac{1}{4}$ and $\sum a_i=\frac{525}{2}$…

Topic: JEE Main 2026 (24 January Morning Shift)

7

Let $S=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\cdots$ (13 terms) If $13S=\frac{2^k}{n!}$, then $n+k$ is equal …

Topic: JEE Main 2026 (24 January Morning Shift)

8

Let $A(1,0),; B(2,-1)$ and $C\left(\frac{7}{3},\frac{4}{3}\right)$ be three points. If equation of bisector of angle $…

Topic: JEE Main 2026 (24 January Morning Shift)

9

Let $R$ be a relation defined on ${1,2,3,4}\times{1,2,3,4}$ by $R={((a,b),(c,d)):2a+3b=3c+4d}$ Then the number of ele…

Topic: JEE Main 2026 (24 January Morning Shift)

10

Let $f(t)=\int \frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}dt,; t>1$ If $f(e^{\pi/2})=-e^{-\pi/2}$ and $f(e^{\pi/4})=\al…

Topic: JEE Main 2026 (24 January Morning Shift)

1

Let a line $L$ passing through the point $P(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z…

Topic: JEE Main 2026 (24 January Morning Shift)

2

The number of numbers greater than $5000$, less than $9000$ and divisible by $3$, that can be formed using digits $0,1,…

Topic: JEE Main 2026 (24 January Morning Shift)

3

Let $(2\alpha,\alpha)$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^2},; t2$ is ______.

Topic: JEE Main 2026 (24 January Morning Shift)

4

The number of $3\times2$ matrices $A$, which can be formed using the elements of the set ${-2,-1,0,1,2}$ such that the …

Topic: JEE Main 2026 (24 January Morning Shift)