Let the lines \[ \ell_1:\ \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2} \quad\text{and}\quad \ell_2:\ 3x+2y+z-2=0\;=\;x-3y+2z-13 \] be coplanar. If the point $P(a,b,c)$ on $\ell_1$ is nearest to the point $Q(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to:

1

Let $y=y(x),\ y>0$, be a solution curve of the differential equation \[ (1+x^2)\,dy = y(x-y)\,dx. \] If $y(0)=1$ and $y…

Topic: JEE Main 2023 (12 April Morning Shift)

2

Let $\mathbf{A}=\begin{bmatrix} 1 & \tfrac{1}{51} \\[2pt] 0 & 1 \end{bmatrix}$. If $\mathbf{B}=\begin{bmatrix} 1 & 2 \…

Topic: JEE Main 2023 (12 April Morning Shift)

3

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the …

Topic: JEE Main 2023 (12 April Morning Shift)

1

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits 1, 3, 7 …

Topic: JEE Main 2023 (12 April Morning Shift)

2

Let $C$ be the circle in the complex plane with centre $z_0=\tfrac{1}{2}(1+3i)$ and radius $r=1$. Let $z_1=1+i$ and t…

Topic: JEE Main 2023 (12 April Morning Shift)

3

If the point $(\alpha, \dfrac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the li…

Topic: JEE Main 2023 (12 April Morning Shift)

4

Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\…

Topic: JEE Main 2023 (12 April Morning Shift)

5

Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\…

Topic: JEE Main 2023 (12 April Morning Shift)

6

Let $\mathrm{P}\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ …

Topic: JEE Main 2023 (12 April Morning Shift)

7

If the local maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\sin^{2}x},; x\in\left(0,\dfr…

Topic: JEE Main 2023 (12 April Morning Shift)

8

$ \text{The area of the region enclosed by the curve } y=x^{3} \text{ and its tangent at the point } (-1,-1) \text{ is:…

Topic: JEE Main 2023 (12 April Morning Shift)

9

Let $D$ be the domain of the function $f(x)=\sin^{-1}\!\left(\log_{3x}\!\left(\dfrac{6+2\log_{3}x}{-5x}\right)\right)$.…

Topic: JEE Main 2023 (12 April Morning Shift)