JEE MAIN 2023 Previous Year Questions (PYQs) – Page 28 of 34

The set of all values of \(\lambda\) for which the equation \[ \cos^{2}(2x)-2\sin^{4}x-2\cos^{2}x=\lambda \] has a real solution \(x\), is:

Let $S$ be the set of all values of $\lambda$ for which the shortest distance between the lines $\dfrac{x-\lambda}{0}=\dfrac{y-3}{4}=\dfrac{z+6}{1}$ and $\dfrac{x+\lambda}{3}=\dfrac{y}{-4}=\dfrac{z-6}{0}$ is $13$. Then $8\Big|\displaystyle\sum_{\lambda\in S}\lambda\Big|$ is equal to:

The area of the region \[ A=\{(x,y): |\,\cos x - \sin x\,| \le y \le \sin x,\; 0 \le x \le \tfrac{\pi}{2}\} \] is:

If the set $\left\{\operatorname{Re}\!\left(\dfrac{z-\overline{z}+z\overline{z}}{\,2-3z+5\overline{z}\,}\right): z\in\mathbb{C},\ \operatorname{Re}(z)=3\right\}$ is equal to the interval $(\alpha,\beta]$, then $24(\beta-\alpha)$ is equal to:

Let \(\vec{a} = 4\hat{i} + 3\hat{j}\) and \(\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}\). If \(\vec{c}\) is a vector such that \[ \vec{c}\cdot(\vec{a}\times\vec{b}) + 25 = 0,\qquad \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k}) = 4, \] and the projection of \(\vec{c}\) on \(\vec{a}\) is \(1\), then the projection of \(\vec{c}\) on \(\vec{b}\) equals:

The value of the integral \(\displaystyle \int_{1}^{2} \left(\frac{t^{4}+1}{t^{6}+1}\right) dt\) is:

Let \(K\) be the sum of the coefficients of the odd powers of \(x\) in the expansion of \((1+x)^{99}\). Let \(a\) be the middle term in the expansion of \(\left(2+\frac{1}{\sqrt{2}}\right)^{200}\). If \(\displaystyle \frac{\binom{200}{99} \, K}{a} = \frac{2^{\,\ell} \, m}{n}\), where \(m\) and \(n\) are odd numbers, then the ordered pair \((\ell,n)\) is:

The value of the integral \(\displaystyle \int_{1/2}^{2} \frac{\tan^{-1}x}{x}\,dx\) is equal to:

The shortest distance between the lines \[ \frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5} \quad\text{and}\quad \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3} \] is:

The functions $f$ and $g$ are twice differentiable on $\mathbb{R}$ such that $f''(x) = g''(x) + 6x$ $f'(1) = 4g'(1) - 3 = 9$ $f(2) = 3g(2) = 12$ Then which of the following is NOT true?