JEE MAIN 2023 Previous Year Questions (PYQs) – Page 32 of 34

A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$. If $2 A_{1}+3 A_{2}$ is minimum then $\left(\pi l_{1}\right): l_{2}$ is equal to :

Let $\alpha\in(0,1)$ and $\beta=\log_{e}(1-\alpha)$. Let $P_{n}(x)=x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}+\cdots+\dfrac{x^{n}}{n},\ x\in(0,1)$. Then the integral $\displaystyle \int_{0}^{\alpha}\frac{t^{50}}{1-t}\,dt$ is equal to

Let $\vec a=2\hat{\imath}+\hat{\jmath}+\hat{k}$, and $\vec b,\vec c$ be two nonzero vectors such that $\left\lvert \vec a+\vec b+\vec c \right\rvert=\left\lvert \vec a+\vec b-\vec c \right\rvert$ and $\vec b\cdot\vec c=0$. Consider the statements: (A) $\left\lvert \vec a+\lambda\vec c \right\rvert \ge \lvert \vec a\rvert \text{ for all } \lambda\in\mathbb{R}$. (B) $\vec a$ and $\vec c$ are always parallel. Then,

both (A) and (B) are correct  

only (A) is correct  

 neither (A) nor (B) is correct

Let $A=\begin{pmatrix}1&0&0\\[2pt]0&4&-1\\[2pt]0&12&-3\end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:

For all $z\in\mathbb{C}$ on the curve $\mathcal{C}_1:\ |z|=4$, let the locus of the point $z+\dfrac{1}{z}$ be the curve $\mathcal{C}_2$. Then:

$ \text{The value of } \displaystyle \int_{\pi/3}^{\pi/2} \frac{2+3\sin x}{\sin x\,(1+\cos x)}\,dx \text{ is equal to:} $

Let $\vec a=\hat{\imath}+2\hat{\jmath}+3\hat{k}$, $\vec b=\hat{\imath}-\hat{\jmath}+2\hat{k}$ and $\vec c=5\hat{\imath}-3\hat{\jmath}+3\hat{k}$ be three vectors. If $\vec r$ is a vector such that $\vec r\times\vec b=\vec c\times\vec b$ and $\vec r\cdot\vec a=0$, then $25\lvert\vec r\rvert^{2}$ is equal to:

Let $f:\mathbb{R}-\{2,6\}\to\mathbb{R}$ be the real-valued function defined as $f(x)=\dfrac{x^{2}+2x+1}{x^{2}-8x+12}$. Then the range of $f$ is:

$ \text{The equation } e^{4x}+8e^{3x}+13e^{2x}-8e^{x}+1=0,\ x\in\mathbb{R}\ \text{ has:} $

Let $(a,b)\subset(0,2\pi)$ be the largest interval for which $\sin^{-1}(\sin\theta)-\cos^{-1}(\sin\theta)>0,\ \theta\in(0,2\pi)$, holds. If $\alpha x^{2}+\beta x+\sin^{-1}(x^{2}-6x+10)+\cos^{-1}(x^{2}-6x+10)=0$ and $\alpha-\beta=b-a$, then $\alpha$ is equal to: