21.

If a, b and c are three non-coplanar vectors and p, q and r are vectors defined by $\mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}, \mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]} \mathrm{and} \mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, then the value of (a + b) . p + (b + c) . q + (c + a) . r is equal to

• 0

• 1

• 2

• 3

If (a $\times$ b)² + (a . b)² = 144 and $\left|\mathrm{a}\right|$ = 4, $\left|\mathrm{b}\right|$ is equal to

• 16

• 8

• 3

• 12

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{{\sin }^3\left(\mathrm{x}\right)}{{\sin }^3\left(\mathrm{x}\right) + {\cos }^3\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{6}$

If i + j - k and 2i - 3j + k are adjacent sides of a parallelogram, then the lengths of its diagonals are

• $\sqrt{3}, \sqrt{14}$

• $\sqrt{13}, \sqrt{14}$

• $\sqrt{21}, \sqrt{3}$

• $\sqrt{21}, \sqrt{13}$

If the volume of the parallelepiped formed by three non-coplanar vectors a, b and c is 4 cu units, then [a x b b x c c x a] is equal to

• 64

• 16

• 4

• 8

If [x] is the greatest integer function not greater than x, then  ${\int }_0^{11}\left[\mathrm{x}\right]\mathrm{dx}$ is equal to

• 45

• 66

• 35

• 55

If $\mathrm{n} \in \mathrm{N}$ and I_n = $\int {\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}\mathrm{dx}$, then I_n + nI_{n - 1} is equal to

• $\frac{{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n} + 1}}{\mathrm{n} +\hspace{0.17em}1}$

• $\mathrm{x}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}} +\hspace{0.17em}\mathrm{C}$

• ${\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n} - 1}$

• $\frac{{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}}{\mathrm{n}}$

Solution of e^dy/dx = x when x = 1 and y = 0 is

• y = x(log(x - 1) + 4

• y = x(log(x) - 1) + 3

• y = x(log(x) + 1) + 1

• y = x(log(x) - 1) + 1