JEE MAIN 2022 Previous Year Questions (PYQs) – Page 31 of 31
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If the system of linear equations
$8x + y + 4z = -2$
$x + y + z = 0$
$\lambda x - 3y = \mu$
has infinitely many solutions, then the distance of the point $(\lambda, \mu, -\tfrac{1}{2})$ from the plane $8x + y + 4z + 2 = 0$ is :
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Solution
The odd natural number $a$, such that the area of the region bounded by $y=1$, $y=3$, $x=0$, $x=y^{a}$ is $\dfrac{364}{3}$, is equal to :
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Solution
Consider two G.P.s: $2, 2^{2}, 2^{3}, \ldots$ (of $60$ terms) and $4, 4^{2}, 4^{3}, \ldots$ (of $n$ terms).
If the geometric mean of all the $60+n$ terms is $(2)^{\tfrac{225}{8}}$, then $\displaystyle \sum_{k=1}^{n} k(n-k)$ is equal to:
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Solution
If the function $f(x) =
\begin{cases}
\dfrac{\log_e(1 - x + x^{2}) + \log_e(1 + x + x^{2})}{\sec x - \cos x}, & x \in \left( -\tfrac{\pi}{2}, \tfrac{\pi}{2} \right) \setminus \{0\} \\
k, & x = 0
\end{cases}$
is continuous at $x=0$, then $k$ is equal to:
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Solution
If $f(x)=
\begin{cases}
x+a, & x\le 0\\
|x-4|, & x>0
\end{cases}
\quad\text{and}\quad
g(x)=
\begin{cases}
x+1, & x<0\\
(x-4)^{2}+b, & x\ge 0
\end{cases}$
are continuous on $\mathbb{R}$, then $(g\circ f)(2)+(f\circ g)(-2)$ is equal to:
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Solution
Let $f(x)=
\begin{cases}
x^{3}-x^{2}+10x-7, & x\le 1,\\
-2x+\log_{2}(b^{2}-4), & x>1.
\end{cases}$
Then the set of all values of $b$ for which $f(x)$ has maximum value at $x=1$ is:
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Solution
A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is $14$.
Let $f(x,y)=0$ be the locus of $P$, which intersects the $x$-axis at the points $A,B$ and the $y$-axis at the points $C,D$.
Then the area of the quadrilateral $ACBD$ is equal to:
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Solution
The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line
$x+y-z=0 = x-2y+3z-5$ is :
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Solution
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