Question

Find the equation of the normal at the point (am², am³) for the curve ay² = x³.

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

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.

Can $y=a x+{\frac{b}{a}}$ be a solution of the following differential equation ? $y=x{\frac{d y}{d x}}+{\frac{b}{\frac{d y}{d x}}}$ If no, find the solution of the D.E.

Solve the differential equation: $x\,{\frac{d y}{d x}}\ +y=x\cos x+\sin x,$ given $y\left({\frac{\pi}{2}}\right)=1.$

Using properties of determinants, prove that $\begin{vmatrix}x+λ & 2x & 2x \\ 2x & x+λ & 2x \\ 2x & 2x & x+λ\end{vmatrix}$ = $(5x+{\lambda})({\lambda}-x)^{2}.$

Evaluate : $\large\textstyle\int(x-3){\sqrt{x^{2}+3x-18}}\,d x\,.$