Application of Derivatives

If the angle made by the tangent at the point $\left(x_0, y_0\right)$ on the curve $x=12(t+\sin t \cos t), y=12$ $(1+\sin t)^2, 0 \lt t \lt \frac{\pi}{2}$, with the positive x -axis is $\frac{\pi}{3}$, then $y_0$ is…

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{d p(t)}{d t}=0.5 \mathrm{p}(\mathrm{t})-450$. If $p(0)=850$, then the time at which the population becomes zero is :

The point of inflexion for the curve $y=(x-a)^n$, where n is odd integer and $n \geq 3$ is

If $f(x)=\cos x, g(x)=\cos 2 x, h(x)=\cos 3 x$ and $I(x)$ $=\tan x$, then which of the following option is correct?

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

The function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function in

The curve given by $x+y=e^{x y}$ has a tangent parallel to the Y-axis at the point

The interval in which the function $f(x)=$ $\frac{4 x^2+1}{x}$ is decreasing is :

A spherical balloon is filled with $4500 ~\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 ~\pi$ cubic meters per minute then the rate (in meters per minute) at which…

If the tangent at $P(1,1)$ on $y^{2}=x(2-x)^{2}$ meets the curve again at $Q$, then $Q$ is

$f(x)=\left(\frac{e^{2 x}-1}{e^{2 x}+1}\right)$ is

If the radius of a sphere is measured as $9 \mathrm{~cm}$ with an error of $0.03 \mathrm{~cm}$, then find the approximating error in calculating its volume.