Let $(\alpha,\beta)$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of the equation:

1

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2}\,x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to:

Topic: JEE Main 2023 (13 April Evening Shift)

2

The coefficient of $x^5$ in the expansion of $\left(2x^3-\dfrac{1}{3x^2}\right)^5$ is:

Topic: JEE Main 2023 (13 April Evening Shift)

3

The line that is coplanar to the line $\dfrac{x+3}{-3}=\dfrac{y-1}{1}=\dfrac{z-5}{5}$ is:

Topic: JEE Main 2023 (13 April Evening Shift)

4

If $\displaystyle \lim_{x\to0}\frac{e^{ax}-\cos(bx)-\dfrac{e^{x}-e^{-x}}{2}}{1-\cos(2x)}=17$, then $5a^2+b^2$ is equal …

Topic: JEE Main 2023 (13 April Evening Shift)

5

Let $S=\{\,z\in\mathbb{C}:\ \overline{z}=i\big(z^2+\operatorname{Re}(\overline{z})\big)\,\}$. Then $\displaystyle \sum_…

Topic: JEE Main 2023 (13 April Evening Shift)

6

Let $|\vec a|=2$, $|\vec b|=3$ and the angle between the vectors $\vec a$ and $\vec b$ be $\dfrac{\pi}{4}$. Then $|(\ve…

Topic: JEE Main 2023 (13 April Evening Shift)

7

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a …

Topic: JEE Main 2023 (13 April Evening Shift)

8

Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be …

Topic: JEE Main 2023 (13 April Evening Shift)

1

If the system of equations $2x+y-z=5$ $2x-5y+\lambda z=\mu$ $x+2y-5z=7$ has infinitely many solutions, then $(\lambda+\…

Topic: JEE Main 2023 (13 April Evening Shift)

2

The area of the region $\{(x,y): x^2 \le y \le |x^2-4|,\ y \ge 1\}$ is:

Topic: JEE Main 2023 (13 April Evening Shift)

3

The value of $\dfrac{e^{-\pi/4}+\displaystyle\int_{0}^{\pi/4} e^{-x}\tan^{50}x\,dx}{\displaystyle\int_{0}^{\pi/4} e^{-x…

Topic: JEE Main 2023 (13 April Evening Shift)

4

The range of $f(x)=4\sin^{-1}\!\left(\dfrac{x^2}{x^2+1}\right)$ is:

Topic: JEE Main 2023 (13 April Evening Shift)

5

For a triangle $ABC$, $\overrightarrow{AB}=-2\hat i+\hat j+3\hat k$ $\overrightarrow{CB}=\alpha\hat i+\beta\hat j+\g…

Topic: JEE Main 2023 (13 April Evening Shift)