The volume of a sphere is increasing at the rate of 8 cm³/sec. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

Show that the function f given by f(x) = tan⁻¹ (sin x+ cos x) is decreasing for all $x\in\left({\frac{\pi}{4}},{\frac{\pi}{2}}\right).$

Show that the function f(x) = x³ – 3x² + 6x – 100 is increasing on R.

If x changes from 4 to 4·01, then find the approximate change in log x.

Find the general solution of the differential equation ${\frac{d y}{d x}}+2y=e^{3x}$

Find the general solution of the differential equation ${\frac{d y}{d x}}+{\frac{2}{x}}y=x$

Find the integrating factor of the differential equation ${\frac{d y}{d x}}+y\ =\ {\frac{1+y}{x}}$

If A = $\left(\begin{array}{l l }3 & 1 \\ -1 & 2\end{array}\right)$ and I = $\left(\begin{array}{l l }1 & 0 \\ 0 & 1\end{array}\right)$ find k so that A² = 5A + kI.

Find : $\textstyle\int$$\frac{1}{\sqrt{x^{2}-4\,x}}\,d x$

Find : $\textstyle\int$$\frac{d x}{\sqrt{3-2\,x-x^{2}}}$

Find : $\textstyle\int$$\frac{x}{\sqrt{32-x^{2}}}\,d x$

if $|\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}|$ =60, $|\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}|$=40 and $\stackrel{\rightarrow}{a}\$=22,then find $\stackrel{\rightarrow}{b}\$.

The position vectors of points A, B and C are $λ\stackrel{\wedge}{i}+3\stackrel{\wedge}{j}$,$12\stackrel{\wedge}{i}+μ\stackrel{\wedge}{j}$ and $11\stackrel{\wedge}{i}-3\stackrel{\wedge}{j}$ respectively. If C divides the line segment joining A and B in the ratio 3 : 1, find the values of λ and μ.

Find : $\textstyle\int$${\sqrt{x^{2}-2x}}\ d x$

Find : $\textstyle\int e^{x}\,{\frac{\sqrt{1+\sin2x}}{1+\cos2\,x}}\,d x$.