Question

If A = $\begin{vmatrix}2 & 3 & -1 \\ 4 & 1 & 0 \\ 3 & 3 & 2\end{vmatrix}$ , find $M_{12}\times M_{21}+C_{21}\times C_{12}$ when $M_{i j}$ is called minor and $C_{i j}$ is called co-factors of A.

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

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

The radius r of a right circular cylinder is increasing at the rate of 5 cm/min and its height h, is decreasing at the rate of 4 cm/min. When r= 8 cm and h = 6 cm, find the rate of change of the volume of cylinder.

A box contains 50 bolts and 50 nuts. Half of the bolts and nuts are rusted. If two items are drawn with replacement, what is the probability that either both are rusted or both are bolts.

If P(E) = 1/2, and P(F) = 1/5 find P(E U F)' if E and F are independent events.

A balloon, which always remains spherical, has a variable diameter ${\frac{2}{3}}(3x+1)$ . Find the rate of change of its volume with respect to x.