Question

Evaluate : $\large\textstyle\int\!{\frac{\ d x}{x(x^{5}+3)}}\,.$

Using properties of determinants, prove that : $\begin{vmatrix}2y & y-z-x & 2y \\ 2z & 2z & z-x-y \\ x-y-z & 2x & 2x\end{vmatrix}$ = $(x+y+z)^{3}.$

P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact ?

Using properties of determinants, prove that : $\begin{vmatrix}1 & 1+p & 1+p+q \\ 3 & 4+3p & 2+4p+3p \\ 4 & 7+4p & 2+7p+4q\end{vmatrix}$ = 1

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?

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

Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.