Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat i+2\hat j+\hat k$ and $2\hat i+4\hat j+4\hat k$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is:

1

If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\b…

Topic: JEE Main 2024 (29 January Morning Shift)

2

Let $R$ be a relation on $\mathbb{Z}\times\mathbb{Z}$ defined by $(a,b)R(c,d)$ iff $ad-bc$ is divisible by $5$. Then $R…

Topic: JEE Main 2024 (29 January Morning Shift)

3

Let $A=\begin{bmatrix} 1&0&0\\ 0&\alpha&\beta\\ 0&\beta&\alpha \end{bmatrix}$ and $\;|2A|^{3}=2^{21}$ where $\alpha,\b…

Topic: JEE Main 2024 (29 January Morning Shift)

4

In an A.P., the sixth term $a_6=2$. If the product $a_1a_4a_5$ is the greatest, then the common difference of the A.P. …

Topic: JEE Main 2024 (29 January Morning Shift)

5

A fair die is thrown until $2$ appears. Then the probability that $2$ appears in an even number of throws is:

Topic: JEE Main 2024 (29 January Morning Shift)

6

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P., then the common…

Topic: JEE Main 2024 (29 January Morning Shift)

7

If $\alpha,\;-\dfrac{\pi}{2}

Topic: JEE Main 2024 (29 January Morning Shift)

8

Let $\left(5,\dfrac{9}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\!\left(\df…

Topic: JEE Main 2024 (29 January Morning Shift)

9

Let $\vec a,\vec b,\vec c$ be three non-zero vectors such that $\vec b$ and $\vec c$ are non-collinear. If $\vec a+5\ve…

Topic: JEE Main 2024 (29 January Morning Shift)

10

Suppose $f(x)=\dfrac{(2^{x}+2^{-x})\tan x\,\sqrt{\tan^{-1}(x^{2}-x+1)}}{(7x^{2}+3x+1)^{3}}$. Then the value of $f'(0)$ …

Topic: JEE Main 2024 (29 January Morning Shift)

1

In $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is…

Topic: JEE Main 2024 (29 January Morning Shift)

2

For $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\displaystyle\int \frac{\csc x+\sin x}{\csc x\sec x+\ta…

Topic: JEE Main 2024 (29 January Morning Shift)

3

Let $A$ be a square matrix such that $AA^{\mathrm T}=I$. Then $\dfrac12\,A\Big[(A+A^{\mathrm T})^{2}+(A-A^{\mathrm T})^…

Topic: JEE Main 2024 (29 January Morning Shift)

4

If $z=\dfrac{1}{2}-2i$ is such that $|z+1|=\alpha z+\beta(1+i)$, $i=\sqrt{-1}$ and $\alpha,\beta\in\mathbb{R}$, then $\…

Topic: JEE Main 2024 (29 January Morning Shift)

5

Consider the function $f:\left[\dfrac{1}{2},1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt{2}\,x^{3}-3\sqrt{2}\,x-1$. C…

Topic: JEE Main 2024 (29 January Morning Shift)

6

Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the midpoint of $PQ$. The distance of the centroid of $…

Topic: JEE Main 2024 (29 January Morning Shift)

7

$\displaystyle \lim_{x\to\frac{\pi}{2}} \left( \frac{1}{(x-\frac{\pi}{2})^{2}}\, \frac{\left(\frac{\pi}{3}\right)^{3}}{…

Topic: JEE Main 2024 (29 January Morning Shift)

8

A function $y=f(x)$ satisfies $f(x)\sin 2x+\sin x-(1+\cos^2x)\,f'(x)=0$ with condition $f(0)=0$. Then $f\!\left(\dfrac{…

Topic: JEE Main 2024 (29 January Morning Shift)

9

If the value of the integral $\displaystyle \int_{-\pi/2}^{\pi/2} \left( \dfrac{x^{2}\cos x}{1+x^{2}} +\dfrac{1+\sin^{2…

Topic: JEE Main 2024 (29 January Morning Shift)