Question

If A = $\left(\begin{array}{l l }1 & -1 \\ 2 & -1\end{array}\right)$ , B = $\left(\begin{array}{l l }a & 1 \\ b & -1\end{array}\right)$ and (A + B)² = A²+ B², then find the values of a and b.

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 particular solution of the following differential equation : ${\frac{d y}{d x}}\;=1+x^{2}+y^{2}+x^{2}y^{2},$ given that y = 1, when x = 0.

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 $f^{-1}.$

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 : $\large\textstyle\int(x-3){\sqrt{x^{2}+3x-18}}\,d x\,.$

Solve the differential equation given as : $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{d x}{d y}\;=1,x\neq0$