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.

Solve the differential equation $x\log x\ {\frac{d y}{d x}}+y\ =$${\frac{2}{x}}\mathrm{log}\,x.$

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Find the equations of the tangent and normal to the curve x = a sin³θ, y = b cos³θ at $\mathbf{\theta}={\frac{\pi}{4}}\,.$

Find the particular solution of the following differential equation : $x\left(x^{2}-1\right){\frac{d y}{d x}}\ =1;$ y = 0, when x = 2.

Evaluate : $\large\textstyle\int\!{\frac{x+2}{\sqrt{x^{2}+5x+6}}}d x~.$

Find : $\textstyle\int$${\frac{x^{2}}{x^{4}+x^{2}-2}}d x$