Question

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

Let f : N → R be a function defined as f(x) = 4x² + 12x + 15. Then show that f : N → S, where S is range of f, is invertible. Also find the inverse of f.

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

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

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

If A = $\left(\begin{array}{l l l }1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{array}\right)$, find $\left(A^{\prime}\right)^{-1}.$

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