The function $f(x)=\tan x-x$

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is 180 cm , then

The value of $C$ in $(0,2)$ satisfying the mean value theorem for the function $f(x)=x(x-1)^2, x \in[0,2]$ is equal to

For the function $f(x)=x^3-6 x^2+12 x-3$; $x=2$ is

The function $x^x ; x>0$ is strictly increasing at

The maximum volume of the right circular cone with slant height 6 units is

If $f(x)=x e^{x(1-x)}$, then $f(x)$ is

If $$u=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$$ and $$v=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)$$, then $$\frac{d u}{d v}$$ is

The distance '$$s$$' in meters travelled by a particle in '$$t$$' seconds is given by $$s=\frac{2 t^3}{3}-18 t+\frac{5}{3}$$. The acceleration when the particle comes to rest is :

A particle moves along the curve $$\frac{x^2}{16}+\frac{y^2}{4}=1$$. When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is

An enemy fighter jet is flying along the curve, given by $$y=x^2+2$$. A soldier is placed at $$(3,2)$$ wants to shoot down the jet when it is nearest to him. Then, the nearest distance is

A circular plate of radius $$5 \mathrm{~cm}$$ is heated. Due to expansion, its radius increase at the rate of $$0.05 \mathrm{~cm} / \mathrm{s}$$. The rate at which its area is increasing when the radius is $$5.2 \mathrm{~cm}$$ is

The function $$f(x)=\log (1+x)-\frac{2 x}{2+x}$$ is increasing on

The coordinates of the point on the $$\sqrt{x}+\sqrt{y}=6$$ at which the tangent is equally inclined to the axes is

The function $$f(x)=4 \sin ^3 x-6 \sin ^2 x +12 \sin x+100$$ is strictly

The cost and revenue functions of a product are given by $$c(x)=20 x+4000$$ and $$R(x)=60 x+2000$$ respectively, where $$\mathrm{x}$$ is the number of items produced and sold. The value of $$x$$ to earn profit is

A particle starts form rest and its angular displacement (in radians) is given by $$\theta=\frac{t^2}{20}+\frac{t}{5}$$. If the angular velocity at the end of $$t=4$$ is $$k$$, then the value of $$5 k$$ is

The function $$f(x)=x^2-2 x$$ is strictly decreasing in the interval

The maximum slope of the curve $$y=-x^3+3 x^2+2 x-27$$ is

If the curves $$2 x=y^2$$ and $$2 x y=K$$ intersect perpendicularly, then the value of $$K^2$$ is

If the side of a cube is increased by $$5 \%$$, then the surface area of a cube is increased by

The maximum value of $$\frac{\log _e x}{x}$$, if $$x>0$$ is

The interval in which the function $$f(x)=x^3-6 x^2+9 x+10$$ is increasing in

The sides of an equilateral triangle are increasing at the rate of $$4 \mathrm{~cm} / \mathrm{sec}$$. The rate at which its area is increasing, when the side is $$14 \mathrm{~cm}$$