JEE MAIN 2023 Previous Year Questions (PYQs) – Page 33 of 34

Let $\alpha>0$. If $\displaystyle \int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}\,dx=\dfrac{16+20\sqrt{2}}{15}$, then $\alpha$ is equal to:

Let the mean and standard deviation of marks of class $A$ of $100$ students be respectively $40$ and $\alpha\ (>\,0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+n$ students are respectively $50$ and $350$, then the sum of variances of classes $A$ and $B$ is:

$ \displaystyle \lim_{x\to\infty} \left\{ \frac{\big(\sqrt{3x+1}+\sqrt{3x-1}\big)^{6}+\big(\sqrt{3x+1}-\sqrt{3x-1}\big)^{6}}{\big(x+\sqrt{x^{2}-1}\big)^{6}+\big(x-\sqrt{x^{2}-1}\big)^{6}} - x^{3} \right\} $ is:

Let $y=y(x)$ be the solution of the differential equation $(3y^{2}-5x^{2})\,y\,dx+2x\,(x^{2}-y^{2})\,dy=0$ such that $y(1)=1$. Then $\left|(y(2))^{3}-12y(2)\right|$ is equal to:

The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P\!\left(\dfrac{1+a}{2},\,\dfrac{1-a}{2}\right)$ on the circle $2x^{2}+2y^{2}-(1+a)x-(1-a)y=0$, is equal to:

The value of $\displaystyle \frac{1}{1\cdot 50!}+\frac{1}{3\cdot 48!}+\frac{1}{5\cdot 46!}+\cdots+\frac{1}{49\cdot 2!}+\frac{1}{51\cdot 1!}$ is:

Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x)-2\cos^{-1}\!\big(\sqrt{1-x^{2}}\big)=\pi,\ x\in\left[-\dfrac{1}{2},\,\dfrac{1}{2}\right]$. Then $\displaystyle \sum_{x\in S} 2\sin^{-1}(x^{2}-1)$ is equal to:

Let $R$ be a relation on $\mathbb{R}$, given by $R=\{(a,b):\,3a-3b+\sqrt{7}\text{ is an irrational number}\}$. Then $R$ is:

The shortest distance between the lines $\dfrac{x-5}{1}=\dfrac{y-2}{2}=\dfrac{z-4}{-3}$ and $\dfrac{x+3}{1}=\dfrac{y+5}{4}=\dfrac{z-1}{-5}$ is:

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z = 1$ $x + \lambda y + z = 1$ $x + y + \lambda z = 1$ is inconsistent, then $\displaystyle \sum_{\lambda \in S}\big(|\lambda|^{2}+|\lambda|\big)$ is equal to: