$\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

1

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text{th}}$ roll than…

Topic: JEE Main 2024 (9 April Evening Shift)

2

If $\log_{e} y = 3\sin^{-1}x$, then $,(1-x^{2})y''-xy',$ at $x=\dfrac{1}{2}$ is equal to:

Topic: JEE Main 2024 (9 April Evening Shift)

3

Let $\alpha,\beta;\ \alpha>\beta,$ be the roots of the equation $x^{2}-\sqrt{2},x-\sqrt{3}=0$. Let $P_{n}=\alpha^{n}-\b…

Topic: JEE Main 2024 (9 April Evening Shift)

4

The integral $\displaystyle \int_{1/4}^{3/4} \cos\left( 2\cot^{-1}\sqrt{\frac{1-x}{1+x}} \right),dx$ is equal to:

Topic: JEE Main 2024 (9 April Evening Shift)

5

Two vertices of a triangle $ABC$ are $A(3,-1)$ and $B(-2,3)$, and its orthocentre is $P(1,1)$. If the coordinates of $C…

Topic: JEE Main 2024 (9 April Evening Shift)

6

Let $\displaystyle \int_{0}^{x}\sqrt{1-\big(y'(t)\big)^{2}},dt=\int_{0}^{x}y(t),dt,\ 0\le x\le 3,\ y\ge0,\ y(0)=0$. The…

Topic: JEE Main 2024 (9 April Evening Shift)

7

Let the range of the function $f(x)=\dfrac{1}{2+\sin3x+\cos3x},\ x\in\mathbb{R}$ be $[a,b]$. If $\alpha$ and $\beta$ ar…

Topic: JEE Main 2024 (9 April Evening Shift)

8

Let $z$ be a complex number such that the real part of $\displaystyle \frac{z-2i}{z+2i}$ is zero. Then, the maximum val…

Topic: JEE Main 2024 (9 April Evening Shift)

9

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $\displaystyle E:\ \frac{(x-1)^{2}}{100}+\frac{(y…

Topic: JEE Main 2024 (9 April Evening Shift)

1

The value of the integral $\displaystyle \int_{-1}^{2} \log_e \big(x + \sqrt{x^2 + 1}\big),dx$ is

Topic: JEE Main 2024 (9 April Evening Shift)

2

Between the following two statements: Statement I: Let $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = 2\hat{…

Topic: JEE Main 2024 (9 April Evening Shift)

3

If the variance of the frequency distributionis $160$, then the value of $c\in\mathbb{N}$ is:

Topic: JEE Main 2024 (9 April Evening Shift)

4

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^…

Topic: JEE Main 2024 (9 April Evening Shift)

5

$\displaystyle \lim_{x\to 0}\ \frac{e^{-(1+2x)^{\tfrac{1}{2x}}}}{x}$ is equal to:

Topic: JEE Main 2024 (9 April Evening Shift)

6

Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\dfrac{11}{3…

Topic: JEE Main 2024 (9 April Evening Shift)

7

The area (in square units) of the region enclosed by the ellipse $x^{2}+3y^{2}=18$ in the first quadrant below the line…

Topic: JEE Main 2024 (9 April Evening Shift)

8

Let $a,ar,ar^{2},\ldots$ be an infinite G.P. If $\displaystyle \sum_{n=0}^{\infty} a r^{n}=57$ and $\displaystyle \sum_…

Topic: JEE Main 2024 (9 April Evening Shift)

9

The sum of the coefficients of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of $\big(x^{2/3}+\tfrac{1}{2}x^{-2/5}…

Topic: JEE Main 2024 (9 April Evening Shift)

10

Let $\vec a=2\hat i+\alpha\hat j+\hat k,\ \vec b=-\hat i+\hat k,\ \vec c=\beta\hat j-\hat k$, where $\alpha,\beta$ are …

Topic: JEE Main 2024 (9 April Evening Shift)