Question

Solve the following differential equation : $\mathrm{cosec}\,x\,\mathrm{log}\,y\,\frac{d y}{d x}+x^{2}y^{2}=0\,.$

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}-y=\cos x\ {\mathrm{~for~}}x=0,y=1.$

Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem, independently, then find the probability that (i) the problem is solved (ii) exactly one of them solves the problem

Solve the following differential equation. (1 + y²) (1 + log |x|) dx + x dy = 0

Using properties of determinants, prove that : $\begin{vmatrix}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}$ = $a b c+b c+c a+a b.$

Sand is pouring from the pipe at the rate of 12 cm³/s. The falling sand forms a cone on a ground in such a way that the height of cone is always one-sixth of radius of the base. How fast is the height of sand cone increasing when the height is 4 cm ?

Let f : W → W be defined as $f(n)={\binom{n+1,{\mathrm{if~}}~n{\mathrm{~is~even}}}{n-1,{\mathrm{~if~}}~n{\mathrm{~is~odd}}}}$ show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.