Let $y=y(x)$ be the solution curve of the differential equation $\displaystyle \frac{dy}{dx}=\frac{y}{x}\bigl(1+xy^{2}(1+\log_{e}x)\bigr),\ x>0,\ y(1)=3.$ Then $\displaystyle \frac{y^{2}(x)}{9}$ is equal to:

1

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in \mathbb{R}\setminus\{0\}$ for which the system of linear …

Topic: JEE Main 2023 (25 January Morning Shift)

2

Let $z_{1}=2+3i$ and $z_{2}=3+4i$. The set $S=\left\{\,z\in\mathbb{C}:\ |z-z_{1}|^{2}-|z-z_{2}|^{2}=|z_{1}-z_{2}|^{2}\,…

Topic: JEE Main 2023 (25 January Morning Shift)

3

The points of intersection of the line $ax+by=0,\ (a\ne b)$ and the circle $x^{2}+y^{2}-2x=0$ are $A(\alpha,0)$ and $B(…

Topic: JEE Main 2023 (25 January Morning Shift)

4

Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}$ $L_{2}:\ \d…

Topic: JEE Main 2023 (25 January Morning Shift)

5

Let $y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$. Then $y' - y''$ at $x=-1$ is equal to:

Topic: JEE Main 2023 (25 January Morning Shift)

6

The distance of the point $P(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with di…

Topic: JEE Main 2023 (25 January Morning Shift)

7

Let $f(x)=\displaystyle \int \frac{2x}{(x^{2}+1)(x^{2}+3)}\,dx$. If $f(3)=\dfrac{1}{2}(\log_{e}5-\log_{e}6)$, then $f…

Topic: JEE Main 2023 (25 January Morning Shift)

8

Let $M$ be the maximum value of the product of two positive integers when their sum is $66$. Let the sample space $S=…

Topic: JEE Main 2023 (25 January Morning Shift)

9

The value of $\displaystyle \lim_{n\to\infty} \frac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n} {\sqrt{2n^{4}+4n+3}-\sqrt{n^{…

Topic: JEE Main 2023 (25 January Morning Shift)

10

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of on…

Topic: JEE Main 2023 (25 January Morning Shift)

1

Let $f:(0,1)\to \mathbb{R}$ be a function defined by $f(x)=\dfrac{1}{1-e^{-x}}$, and $g(x)=\bigl(f(-x)-f(x)\bigr)$.&nbs…

Topic: JEE Main 2023 (25 January Morning Shift)