Question

Using properties of determinants, prove that : $\begin{vmatrix}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}$ = $a b c+b c+c a+a b.$

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

Find the equation of tangent to curves x = sin 3t, y = cos 2t at $:t={\frac{\pi}{4}}\,.$

Consider f :$R_{+}\to[4,\infty)$ given by f(x) = x² + 4 Show that f is invertible with the inverse $f^{-1}$ of f given by $f^{-1}(y)$ = $\sqrt{y-4}$ , where $\textstyle R_{+}$ is the set of all nonnegative real numbers.

Solve the following differential equation : $(x^{2}-1){\frac{d y}{d x}}~+2x y={\frac{2}{x^{2}-1}},~~x\neq{\bf1}$

Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..

Using properties of determinants, prove that : $\begin{vmatrix}1 & 1+p & 1+p+q \\ 3 & 4+3p & 2+4p+3p \\ 4 & 7+4p & 2+7p+4q\end{vmatrix}$ = 1