Question

Find : $\textstyle\int$$\textstyle\left\{{\frac{\left(x+\sin^{2}x\right)\sec^{2}x}{1+x^{2}}}\right\}d x~.$

The radius r of a right circular cylinder is decreasing at the rate of 3 cm/min. and its height h is increasing at the rate of 2 cm/min. When r =7 cm and h = 2 cm, find the rate of change of the volume of cylinder. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

Find the area of the parallelogram whose diagonals are represented by the vectors $\stackrel{\rightarrow}{a}\$ = $2{\hat{i}}-3{\hat{j}}+4{\hat{k}}$ and ${\vec{b}}=2{\hat{i}}-{\hat{j}}+2{\hat{k}}$

If E and F be two events such that P(E) = 1/3, P(F) = 1/4, find P(E U F) if E and F are independent events.

For the curve y = 5x – 2x³, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3.

The radius r of a right circular cone is decreasing at the rate of 3 cm/minute and the height h is increasing at the rate of 2 cm/minute. When r = 9 cm and h = 6 cm, find the rate of change of its volume.

Find the sum of the order and the degree of the following differential equations : ${\frac{d^{2}y}{d x^{2}}}+{3{\sqrt\frac{{d y}}{d x}}}+(1+x)=0$