The position vectors of the vertices $A,B,C$ of a triangle are $2\hat i-3\hat j+3\hat k$, $2\hat i+2\hat j+3\hat k$ and $-\hat i+\hat j+3\hat k$ respectively. Let $l$ denote the length of the angle bisector $AD$ of $\angle BAC$ (where $D$ is on the line segment $BC$). Then $2l^{2}$ equals:

1

The values of $\alpha$ for which $\begin{vmatrix} 1 & \dfrac{3}{2} & \alpha+\dfrac{3}{2}\\[4pt] 1 & \dfrac{1}{3} & \alp…

Topic: JEE Main 2024 (27 January Evening Shift)

2

The integral $\displaystyle \int \frac{x^{5}-x^{2}}{(x^{2}+3x+1)\,\tan^{-1}\!\left(x^{3}+\dfrac{1}{x^{2}}\right)}\,dx$ …

Topic: JEE Main 2024 (27 January Evening Shift)

3

Let $\alpha = \dfrac{(4!)!}{(4!)^{4!}}$ and $\beta = \dfrac{(5!)!}{(5!)^{5!}}$. Then:

Topic: JEE Main 2024 (27 January Evening Shift)

4

Let $R$ be the interior region between the lines $3x - y + 1 = 0$ and $x + 2y - 5 = 0$ containing the origin. The set o…

Topic: JEE Main 2024 (27 January Evening Shift)

5

Let $g(x)=3f\!\left(\dfrac{x}{3}\right)+f(3-x)$ and $f''(x)>0$ for all $x\in(0,3)$. If $g$ is decreasing in $(0,\alpha)…

Topic: JEE Main 2024 (27 January Evening Shift)

6

If $2\tan^2\theta-5\sec\theta=1$ has exactly $7$ solutions in the interval $\left[0,\dfrac{n\pi}{2}\right]$, for the le…

Topic: JEE Main 2024 (27 January Evening Shift)

7

If $y=y(x)$ is the solution curve of the differential equation $(x^2-4)\,dy-(y^2-3y)\,dx=0,\ x>2,\ y(4)=\dfrac{3}{2}$ a…

Topic: JEE Main 2024 (27 January Evening Shift)

8

If $\displaystyle \lim_{x\to0}\frac{3+a\sin x+b\cos x+\log_e(1-x)}{3\tan^2 x}=\frac{1}{3}$, then $2a-b$ is equal to:

Topic: JEE Main 2024 (27 January Evening Shift)

9

Let $f:\mathbb{R}\setminus\{-\tfrac{1}{2}\}\to\mathbb{R}$ and $g:\mathbb{R}\setminus\{-\tfrac{5}{2}\}\to\mathbb{R}$ be …

Topic: JEE Main 2024 (27 January Evening Shift)

10

The $20^{\text{th}}$ term from the end of the progression $20,\ 19\dfrac{1}{4},\ 18\dfrac{1}{2},\ 17\dfrac{3}{4},\ldots…

Topic: JEE Main 2024 (27 January Evening Shift)

1

For $0 < \mathrm{a} < 1$, the value of the integral $\int_\limits0^\pi \frac{\mathrm{d} x}{1-2 \mathrm{a} \cos x+\mathr…

Topic: JEE Main 2024 (27 January Evening Shift)