JEE MAIN 2022 Previous Year Questions (PYQs) – Page 11 of 31

$ \text{Let } y=y(x) \text{ be the solution curve of the differential equation } \dfrac{dy}{dx}+\dfrac{1}{x^{2}-1},y=\left(\dfrac{x-1}{x+1}\right)^{1/2},; x>1,\ \text{passing through the point } \left(2,\sqrt{\tfrac{1}{3}}\right). \text{ Then } \sqrt{7},y(8) \text{ is equal to:} $

Let E₁ and E₂ be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$, $P({E_2}|{E_1}) = {3 \over 4}$ and $P({E_1} \cap {E_2}) = {1 \over 8}$. Then :

Let the hyperbola $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ pass through the point $(2\sqrt{2}, -2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?

Let $A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$. If M and N are two matrices given by $M = \sum\limits_{k = 1}^{10} {{A^{2k}}} $ and $N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} $ then MN² is :

$ \text{Let } S \text{ be the set of all } a \in \mathbb{R} \text{ for which the angle between the vectors } \vec{u}=a(\log_{e} b),\hat{i}-6\hat{j}+3\hat{k} \text{ and } \vec{v}=(\log_{e} b),\hat{i}+2\hat{j}+2a(\log_{e} b),\hat{k},\ (b>1), \text{ is acute. Then } S \text{ is equal to:} $

Let $g:(0,\infty ) \to R$ be a differentiable function such that $\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c} $, for all x > 0, where c is an arbitrary constant. Then :

$ \text{Let A and B be two events such that } P(B|A)=\frac{2}{5}, P(A|B)=\frac{1}{7},; \text{and } P(A\cap B)=\frac{1}{9}. $ Consider:(S1) $P(A' \cup B)=\frac{5}{6}$ (S2) $P(A' \cap B')=\frac{1}{18}$

Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$. Then, for which of the following range of $\alpha$, the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?

Let $R$ be a relation from the set ${1,2,3,\dots,60}$ to itself such that R={(a,b):b=pq,    where p,q≥3 are prime numbers}.R = \{(a,b) : b = pq, \;\; \text{where $p,q \geq 3$ are prime numbers} \}.R={(a,b):b=pq,where p,q≥3 are prime numbers}. Then, the number of elements in $R$ is :

Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} > 0$, $i = 1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\overrightarrow a $ on the vector $3\widehat i + 4\widehat j$ be 7. Let $\overrightarrow b $ be a vector obtained by rotating $\overrightarrow a $ with 90$^\circ$. If $\overrightarrow a $, $\overrightarrow b $ and x-axis are coplanar, then projection of a vector $\overrightarrow b $ on $3\widehat i + 4\widehat j$ is equal to: