Question

Find the equations of the tangent and normal to the curve ${\frac{x^{2}}{a^{2}}}-{\frac{y^{2}}{b^{2}}}=1$ at the point $({\sqrt{2}}a,b)$ .

Find the general solution of the differential equation: ${\frac{d x}{d y}}\;=\;{\frac{y\tan y-x\tan y-x y}{y\tan y}}$

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$

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 ?

Find : $\large\textstyle\int(3x+1)\large{\sqrt{4-3x-2x^{2}}}d x\,.$

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

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