Question

To promote the making of toilets for women, as organisation tried to generate awareness through (i) house calls (ii) letters and (iii) announcements. The cost for each mode per attempt is given below : (i) ₹ 50 (ii) ₹ 20 (iii) ₹ 40 The number of attempts made in three villages X, Y and Z and given below :

Solve the following differential equation : $x{\frac{d y}{d x}}\ +y-x+x y\cot x=0,x\neq0$

Find : $\large\textstyle\int\!{\large\frac{x^{2}+x+1}{(x^{2}+1)\left(x+2\right)}}\,d x.$

Let f : N → R be a function defined as f(x) = 4x² + 12x + 15. Then show that f : N → S, where S is range of f, is invertible. Also find the inverse of f.

Using properties of determinants, prove that $\begin{vmatrix}1 & 1 & 1+3x \\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}$ = $9\bigl(3x y z+x y+y z+z x\bigr)$

Solve the differential equation $(\tan^{-1}x-y)d x=(1+x^{2})\,d y$

Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.