Question

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.

If A = $\left(\begin{array}{l l l }2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right)$ , then find value of A² - 3A + 2I.

Find the equation of tangent to the curve x² + 3y = 3, which is parallel to line y – 4x + 5 = 0.

Using properties of determinants, prove that : $\begin{vmatrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{vmatrix}$ = -2

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.

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

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