Question

Find the particular solution of the differential equation $e^{x}{\sqrt{1-y^{2}}}d x+\left({\frac{y}{x}}\right)d x=0$ given that y = 1 when x = 0.

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.

Find the equation(s) of the tangent(s) to the curve y = (x³ – 1) (x – 2) points where the curve intersects the x-axis.

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

Without expanding the determinant at any stage; prove that : $\begin{vmatrix}0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0\end{vmatrix}$ = 0.

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

Find the equation of tangent to the curve $y={\frac{x-7}{x^{2}-5x+6}}$ at the point, where it cuts the x-axis.