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

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

For the following matrices A and B, verify that [AB]’ = B’A’ where, A = $\left(\begin{array}{l }1 \\ -4 \\ 3\end{array}\right)$ , B = $\left(\begin{array}{l l l }-1 & 2 & 1\end{array}\right)$

If A = $\left(\begin{array}{l l l }2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right)$ , find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = 0.

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) =${\frac{x-2}{x-3}},~~\forall~x\in~A.$ Show that f is bijective. Also, find (i) x, if $f^{-1}(x)$ = 4 (ii) $f^{-1}(7)$

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.

Solve the differential equation : $3e^{x}\tan y\,d x\ +(2-e^{x})\,\mathrm{sec}^{2}\,y\,d y=0,$ given that when x = 0, y = ${\frac{\pi}{4}}.$ 

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

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

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

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$

Evaluate : $\large\textstyle\int\!{\frac{\ d x}{x(x^{5}+3)}}\,.$

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+1)}}\,.$

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+8)}}\,.$

Evaluate : ($\large\textstyle\int(3x-2){\sqrt{x^{2}+x+1}}\,d x~.$