JEE MAIN Application Of Derivatives Previous Year Questions (PYQs) – Page 1 of 3

If $m$ is the minimum value of $k$ for which the function $f(x)=x\sqrt{kx-x^{2}}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m$, then the ordered pair $(m,M)$ is equal to:

A helicopter is flying along the curve given by $y - x^{3/2} = 7,\ (x \ge 0)$. A soldier positioned at the point $\left(\dfrac{1}{2},\,7\right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is –

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}$. The surface area (in $\mathrm{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is :

The remainder when $\big((64)^{(64)}\big)^{(64)}$ is divided by $7$ is:

The real number $k$ for which the equation $2x^{3}+3x+k=0$ has two distinct real roots in $[0,1]$

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_{1}$ be twice the area $A_{2}$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point:

For the function $f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$, which one of the following is NOT correct?

Consider the region $R = {(x, y) : x \le y \le 9 - \dfrac{11}{3}x^2, , x \ge 0}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$ is

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola $y=12-x^{2}$, such that the rectangle lies inside the parabola, is: