Question

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Show that the relation R in the set N × N defined by (a, b) R (c, d) if a² + d² = b² + c² $\forall\;a,\;b,\;c,\;d\in N,$ is an equivalence relation.

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

Using properties of determinants, prove that $\begin{vmatrix}x+λ & 2x & 2x \\ 2x & x+λ & 2x \\ 2x & 2x & x+λ\end{vmatrix}$ = $(5x+{\lambda})({\lambda}-x)^{2}.$

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

Solve the following differential equation : $x{\frac{d y}{d x}}\ +y-x+x y\cot x=0,x\neq0$

If the function f : R → R is given by f(x) = $\frac{x+3}{3}$ and g : R → R is given by g(x) = 2x – 3, then find (i) fog (ii) gof Is f⁻¹ = g?