JEE MAIN 2022 Previous Year Questions (PYQs) – Page 7 of 31
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Let $x * y = {x^2} + {y^3}$ and $(x * 1) * 1 = x * (1 * 1)$.
Then a value of $2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)$ is :
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Solution
Given \( x*y=x^2+y^3 \)
\(\therefore (x*1)*1=x*(1*1)\)
\((x^2+1)*1=x*(2)\)
\((x^2+1)^2+1=x^2+8\)
\(x^4+2x^2+1+1=x^2+8\)
\(x^4+x^2-6=0\)
\((x^2+3)(x^2-2)=0\)
\(x^2=2\)
Now,
\(2\sin^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)\)
\(=2\sin^{-1}\left(\frac{4+2-2}{4+2+2}\right)\)
\(=2\sin^{-1}\left(\frac{1}{2}\right)\)
\(=2\cdot\frac{\pi}{6}\)
\(=\frac{\pi}{3}\)
\(\therefore (x*1)*1=x*(1*1)\)
\((x^2+1)*1=x*(2)\)
\((x^2+1)^2+1=x^2+8\)
\(x^4+2x^2+1+1=x^2+8\)
\(x^4+x^2-6=0\)
\((x^2+3)(x^2-2)=0\)
\(x^2=2\)
Now,
\(2\sin^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)\)
\(=2\sin^{-1}\left(\frac{4+2-2}{4+2+2}\right)\)
\(=2\sin^{-1}\left(\frac{1}{2}\right)\)
\(=2\cdot\frac{\pi}{6}\)
\(=\frac{\pi}{3}\)
Let a vector $\vec{a}$ has magnitude $9$. Let a vector $\vec{b}$ be such that for every $(x,y)\in\mathbb{R}\times\mathbb{R}-{(0,0)}$, the vector $(x\vec{a}+y\vec{b})$ is perpendicular to the vector $(6y\vec{a}-18x\vec{b})$. Then the value of $|\vec{a}\times\vec{b}|$ is equal to:
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Solution
The sum of all the real roots of the equation $({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$ is
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Solution
For $t\in(0,2\pi)$, if $\triangle ABC$ is an equilateral triangle with vertices $A(\sin t,-\cos t)$, $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\tfrac{1}{3}\right)$, then $(a^{2}-b^{2})$ is equal to:
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Solution
Let the system of linear equations
$x + y + \alpha z = 2$,
$3x + y + z = 4$,
$x + 2z = 1$
have a unique solution $(x^*, y^*, z^*)$. If $(\alpha, x^*)$, $(y^*, \alpha)$ and $(x^*, -y^*)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is ?
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Solution
For $\alpha \in \mathbb{N}$, consider a relation $R$ on $\mathbb{N}$ given by
$R={(x,y):3x+\alpha y \text{ is a multiple of } 7}$.
The relation $R$ is an equivalence relation if and only if:
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Solution
Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is
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Solution
Out of $60%$ female and $40%$ male candidates appearing in an exam, $60%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability that the chosen candidate is a female, is:
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Solution
Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.$$
where [t] denotes greatest integer $$\le$$ t. If m is the number of points where $$f$$ is not continuous and n is the number of points where $$f$$ is not differentiable, then the ordered pair (m, n) is :
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Solution
If $y=y(x),\; x\in(0,\pi/2)$ be the solution curve of the differential equation
$$(\sin^{2}2x)\dfrac{dy}{dx}+(8\sin^{2}2x+2\sin 4x)y=2e^{-4x}(2\sin 2x+\cos 2x),$$
with $y(\pi/4)=e^{-\pi}$, then $y(\pi/6)$ is equal to :
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Solution
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