Question

Find the particular solution of the differential equation ${\frac{d y}{d x}}\ =1+x+y+x y$ given that y = 0 when x = 1.

Let f, g : R → R be two functions defined as f (x) = |x| + x and g (x) = |x| – x for all x ∈ R.Then find fog and gof.

If f(x) =$\frac{4x+3}{6x-4}\,,\;\;x\;\;\neq\;\;\frac{2}{3}\,,$ then Show that fof(x) = x for all $x\neq{\frac{2}{3}}\,.$ What is the inverse of f?

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

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.

Using properties of determinants, prove that : $\begin{vmatrix}a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b\end{vmatrix}$ = $2(a+b+c)^{3}.$

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$