JEE MAIN 2022 Previous Year Questions (PYQs) – Page 3 of 31

Let $A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^2 + \beta A = 2I$. Then $\alpha + \beta$ is equal to :

$ \text{Suppose } a_1, a_2, \ldots, a_n, \ldots \text{ be an arithmetic progression of natural numbers. If } \dfrac{S_5}{S_9} = \dfrac{5}{17} \text{ and } 110 < a_{15} < 120, \text{ then the sum of the first ten terms of the progression is equal to:} $

$ \text{Let } f:\mathbb{R}\to\mathbb{R} \text{ be a function defined as } f(x)=a\sin\!\left(\frac{\pi\lfloor x\rfloor}{2}\right)+\lfloor 2-x\rfloor,\ a\in\mathbb{R}, \text{ where } \lfloor t\rfloor \text{ is the greatest integer } \le t. \text{ If } \lim_{x\to -1} f(x) \text{ exists, then the value of } \int_{0}^{4} f(x)\,dx \text{ is equal to:}$

$ I = \int_{\pi/4}^{\pi/3} \left( \frac{8 \sin x - \sin 2x}{x} \right) dx. \ \text{ Then} $

The area of the smaller region enclosed by the curves $y^2 = 8x + 4$ and $x^2 + y^2 + 4\sqrt{3}x - 4 = 0$ is equal to

Let $y = y_{1}(x)$ and $y = y_{2}(x)$ be two distinct solutions of the differential equation $\dfrac{dy}{dx} = x + y$, with $y_{1}(0) = 0$ and $y_{2}(0) = 1$ respectively. Then, the number of points of intersection of $y = y_{1}(x)$ and $y = y_{2}(x)$ is

Let $\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}$ be two vectors, such that $\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12\hat{k}$. Then the projection of $\vec{b} - 2\vec{a}$ on $\vec{b} + \vec{a}$ is equal to:

Let $S$ be the sample space of all five digit numbers. It $p$ is the probability that a randomly selected number from $S$, is a multiple of $7$ but not divisible by $5$, then $9p$ is equal to :

If the circle $x^{2} + y^{2} - 2gx + 6y - 19c = 0,; g,c \in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x - 2cy = 8$, then the length of intercept made by the circle on $x$-axis is :

$2\sqrt{23}$

Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, $P$ be a point on side $BC$, and $\Delta_1$ and $\Delta_2$ be the areas of triangles $APB$ and $ABC$, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines $AP$, $AC$ and the $x$-axis is: