Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x)-2\cos^{-1}\!\big(\sqrt{1-x^{2}}\big)=\pi,\ x\in\left[-\dfrac{1}{2},\,\dfrac{1}{2}\right]$. Then $\displaystyle \sum_{x\in S} 2\sin^{-1}(x^{2}-1)$ is equal to:

1

The value of $\displaystyle \frac{1}{1\cdot 50!}+\frac{1}{3\cdot 48!}+\frac{1}{5\cdot 46!}+\cdots+\frac{1}{49\cdot 2!}+…

Topic: JEE Main 2023 (1 February Morning Shift)

1

Let $R$ be a relation on $\mathbb{R}$, given by $R=\{(a,b):\,3a-3b+\sqrt{7}\text{ is an irrational number}\}$. Then $R$…

Topic: JEE Main 2023 (1 February Morning Shift)

2

The shortest distance between the lines $\dfrac{x-5}{1}=\dfrac{y-2}{2}=\dfrac{z-4}{-3}$ and $\dfrac{x+3}{1}=\dfrac{y+5}…

Topic: JEE Main 2023 (1 February Morning Shift)

3

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z = 1$ $x…

Topic: JEE Main 2023 (1 February Morning Shift)

4

Let $S=\left\{x:\ x\in\mathbb{R}\ \text{and}\ (\sqrt{3}+\sqrt{2})^{\,x^{2}-4}+(\sqrt{3}-\sqrt{2})^{\,x^{2}-4}=10\right\…

Topic: JEE Main 2023 (1 February Morning Shift)

5

If the center and radius of the circle $\left|\dfrac{z-2}{z-3}\right|=2$ are respectively $(\alpha,\beta)$ and $\gamma$…

Topic: JEE Main 2023 (1 February Morning Shift)

6

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes …

Topic: JEE Main 2023 (1 February Morning Shift)

7

Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_{e}\!\big(\sqrt{1+x^{2}}+x\big),\ x\in[0,3]$. Then

Topic: JEE Main 2023 (1 February Morning Shift)

8

Let $f(x)=\begin{vmatrix} 1+\sin^{2}x & \cos^{2}x & \sin 2x\\ \sin^{2}x & 1+\cos^{2}x & \sin 2x\\ \sin^{2}x & \cos^{2}x…

Topic: JEE Main 2023 (1 February Morning Shift)

9

The area enclosed by the closed curve $\mathcal{C}$ given by the differential equation $\dfrac{dy}{dx}+\dfrac{x+a}{\,y-…

Topic: JEE Main 2023 (1 February Morning Shift)

10

If the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha,\beta)$, then the quadra…

Topic: JEE Main 2023 (1 February Morning Shift)