Question

If A = $\left(\begin{array}{l l l }1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{array}\right)$, find $\left(A^{\prime}\right)^{-1}.$

Using properties of determinants, prove that : $\begin{vmatrix}b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y\end{vmatrix}$ = 2 $\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}$ .

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 ?

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

A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ? Do you think that statement of B is always true ?

Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?

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