JEE MAIN 2019 Previous Year Questions (PYQs) – Page 5 of 34
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A circle touching the $x$-axis at $(3,0)$ and making an intercept of length $8$ on the $y$-axis passes through the point:
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Solution
Let $\mathbb{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:\mathbb{N}\to\mathbb{N}$ such that
$$
f(n)=
\begin{cases}
\dfrac{n+1}{2}, & \text{if $n$ is odd},\\[4pt]
\dfrac{n}{2}, & \text{if $n$ is even},
\end{cases}
\qquad
g(n)=n-(-1)^n.
$$
Then $f\circ g$ is –
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Solution
The outcome of each of 30 items was observed; 10 items gave an outcome $\dfrac{1}{2}-d$ each, 10 items gave outcome $\dfrac{1}{2}$ each and the remaining 10 items gave outcome $\dfrac{1}{2}+d$ each. If the variance of this outcome data is $\dfrac{4}{3}$ then $|d|$ equals :
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Solution
If $x\log_e(\log_e x)-x^2+y^2=4\ (y>0)$, then $\left.\dfrac{dy}{dx}\right|_{x=e}$ is equal to :
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Solution
The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is :
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Solution
Let $A=\begin{pmatrix}
0 & 2q & r\\
p & q & -r\\
p & -q & r
\end{pmatrix}$. If $AA^{T}=I_{3}$, then $|p|$ is :
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Solution
Let $[x]$ denote the greatest integer less than or equal to $x$. Then
$\displaystyle \lim_{x\to 0}\frac{\tan(\pi\sin^{2}x)+\left(|x|-\sin(x[x])\right)^{2}}{x^{2}}$ :
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Solution
If $y(x)$ is the solution of the differential equation
$\dfrac{dy}{dx}+\left(\dfrac{2x+1}{x}\right)y=e^{-2x},\ x>0,$ where $y(1)=\dfrac{1}{2}e^{-2}$, then
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Solution
A square is inscribed in the circle $x^{2}+y^{2}-6x+8y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :
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Solution
If one real root of the quadratic equation $81x^{2}+kx+256=0$ is cube of the other root, then a value of $k$ is :
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Solution
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