For any vector x, where i, j, k have their usual meanings the value of ${\left(\mathrm{x} \times \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{k}}\right)}^2 \mathrm{where} \hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ have their usual meanings, is equal to

• ${\left|\mathrm{x}\right|}^2$

• 2${\left|\mathrm{x}\right|}^2$

• 3${\left|\mathrm{x}\right|}^2$

• 4${\left|\mathrm{x}\right|}^2$

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

• $\sqrt{2}$ units

• 2 units

• $\sqrt{3}$ units

• $\sqrt{5}$ units

73.

Let a and B be the roots of x² + x + 1 = 0. If n be a positive integer, then ${\mathrm{\alpha }}^{\mathrm{n}} + {\mathrm{\beta }}^{\mathrm{n}}$ is

• $2\cos \left(\frac{2\mathrm{n\pi }}{3}\right)$

• $2\sin \left(\frac{2\mathrm{n\pi }}{3}\right)$

• $2\cos \left(\frac{\mathrm{n\pi }}{3}\right)$

• $2\sin \left(\frac{\mathrm{n\pi }}{3}\right)$

For real x, the greatest value of $\frac{{\mathrm{x}}^2 + 2\mathrm{x} + 4}{2{\mathrm{x}}^2 + 4\mathrm{x} + 9}$ is

• 1

• - 1

• $\frac{1}{2}$

• $\frac{1}{4}$

Let f be a non-constant continuous function for all x > 0. Let f satisfy the relation f(ax) f(a-x)=1 for some a $\in$ R*. Then, I = ${\int }_0^{\mathrm{a}}\frac{\mathrm{dx}}{1 + \mathrm{f}\left(\mathrm{x}\right)}$ is equal to

• a

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{a}}{2}$

• f(a)