The sum of the infinite series $\cot^{-1}\left(\dfrac{7}{4}\right)+\cot^{-1}\left(\dfrac{19}{4}\right)+\cot^{-1}\left(\dfrac{30}{4}\right)+\cot^{-1}\left(\dfrac{67}{4}\right)+\cdots$ is:

1

Let $A$ be the point of intersection of the lines $L_{1}:\ \dfrac{x-7}{1}=\dfrac{y-5}{0}=\dfrac{z-3}{-1}$ and $L_{2}:\ …

Topic: JEE Main 2025 (4 April Evening Shift)

2

If $1^{2}\cdot{^{15}C_{1}}+2^{2}\cdot{^{15}C_{2}}+3^{2}\cdot{^{15}C_{3}}+\cdots+15^{2}\cdot{^{15}C_{15}}=2^{m}\cdot3^{n…

Topic: JEE Main 2025 (4 April Evening Shift)

3

Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\al…

Topic: JEE Main 2025 (4 April Evening Shift)

4

A line passing through the point $A(-2,0)$ touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. Th…

Topic: JEE Main 2025 (4 April Evening Shift)

5

The centre of a circle $C$ is at the centre of the ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$. Let…

Topic: JEE Main 2025 (4 April Evening Shift)

6

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1…

Topic: JEE Main 2025 (4 April Evening Shift)

7

If the sum of the first $20$ terms of the series $\dfrac{4\cdot1}{4+3\cdot1^{2}+1^{4}}+\dfrac{4\cdot2}{4+3\cdot2^{2}+2^…

Topic: JEE Main 2025 (4 April Evening Shift)

8

If a curve $y=y(x)$ passes through the point $\left(1,\dfrac{\pi}{2}\right)$ and satisfies the differential equation $(…

Topic: JEE Main 2025 (4 April Evening Shift)

9

Let $a>0$. If the function $f(x)=6x^3-45ax^2+108a^2x+1$ attains its local maximum and minimum values at the points $x_1…

Topic: JEE Main 2025 (4 April Evening Shift)

10

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A…

Topic: JEE Main 2025 (4 April Evening Shift)

1

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and…

Topic: JEE Main 2025 (4 April Evening Shift)

2

Let $f(x)+2f!\left(\frac{1}{x}\right)=x^{2}+5$ and $2g(x)-3g!\left(\frac{1}{x}\right)=x$, $x>0$. If $\alpha=\displaysty…

Topic: JEE Main 2025 (4 April Evening Shift)

3

Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of $A$…

Topic: JEE Main 2025 (4 April Evening Shift)

4

Let the values of $p$, for which the shortest distance between the lines $\dfrac{x+1}{3}=\dfrac{y}{4}=\dfrac{z}{5}$ and…

Topic: JEE Main 2025 (4 April Evening Shift)

5

Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1,\ f'(2)=4$. Let $\displaystyle \lim_{x\to 0}\big…

Topic: JEE Main 2025 (4 April Evening Shift)