Question

Find the angle between the vectors ${\vec{a}}={\hat{i}}+{\hat{j}}-{\hat{k}}$ and ${\vec{b}}={\hat{i}}-{\hat{j}}+{\hat{k}}$

Find : $\textstyle\int$$\frac{d x}{5-8\,x-x^{2}}$

Find : $\textstyle\int$$\frac{x}{\sqrt{32-x^{2}}}\,d x$

One bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is red.

Prove that if E and F are independent events, then the events E and F' are also independent.

If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|.

Obtain the differential equation of the family of circles passing through the points (a, 0) and (– a, 0).