JEE MAIN 2022 Previous Year Questions (PYQs) – Page 13 of 31

Let the solution curve $y = y(x)$ of the differential equation $\left(1 + e^{2x}\right)\left(\dfrac{dy}{dx} + y\right) = 1$ pass through the point $\left(0, \dfrac{\pi}{2}\right)$. Then, $\lim_{x \to \infty} e^x y(x)$ is equal to :

Let a, b $\in$ R be such that the equation $a{x^2} - 2bx + 5 = 0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - 2bx + 21 = 0$, then ${\alpha ^2} + {\beta ^2}$ is equal to :

$ \text{Let the solution curve } y=y(x) \text{ of the differential equation } (1+e^{2x})!\left(\dfrac{dy}{dx}+y\right)=1 \text{ pass through the point } \left(0,\dfrac{\pi}{2}\right). $ $ \text{Then } \lim_{x\to\infty} e^{x}y(x) \text{ is equal to:} $

Let z₁ and z₂ be two complex numbers such that ${\overline z _1} = i{\overline z _2}$ and $\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi $. Then :

Let a line L pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$. If the line $\mathrm{L}$ also passes through the point $(1,1)$ and touches the circle $17\left(x^{2}+y^{2}\right)=16$, then the eccentricity of the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$ is :

The system of equations

$ - kx + 3y - 14z = 25$

$ - 15x + 4y - kz = 3$

$ - 4x + y + 3z = 4$

is consistent for all k in the set

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$\left(k_{1}, k_{2}\right)$, then $k_{1}+k_{2}$ is equal to :

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)$ is equal to

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))$, then the value of $164 \,\cos ^{2} \theta$ is equal to :

The area of the region enclosed between the parabolas y² = 2x $-$ 1 and y² = 4x $-$ 3 is