If the axes are rotated through an angle 45° in the positive direction without changing the origin,then the co-ordinates of the point ($\sqrt{2}$, 4) in the old system are

• $\left(1 - 2\sqrt{2}, 1 + 2\sqrt{2}\right)$

• $\left(1 + 2\sqrt{2}, 1 - 2\sqrt{2}\right)$

• $\left(2\sqrt{2}, \sqrt{2}\right)$

• $\left(\sqrt{2}, 2\right)$

• ${\left({\mathrm{x}}^2\mathrm{y}\right)}^{2/3} + ({\mathrm{xy}}^2{)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = 1$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right) = 4\mathrm{xy}$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right) = 12\mathrm{xy}$

• ${\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)}^2 = 16\mathrm{xy}$

$\mathrm{f}\left(\mathrm{x}\right) = \frac{{\cos }^2\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}{{\sin }^2\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}, \mathrm{for} \mathrm{x} \in \mathrm{R}$, then f(2002) is equal to

• 1

• 2

• 3

• 4

The function f : R $\to$ R is defined by f(x) = ${\cos }^2\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right) \mathrm{for} \mathrm{x} \in \mathrm{R}$, then f(R) is equal to

• $(\frac{3}{4}, 1]$

• $[\frac{3}{4}, 1)$

• $\left[\frac{3}{4}, 1\right]$

• $\left(\frac{3}{4}, 1\right)$

$\mathrm{If} {\mathrm{x}}_{\mathrm{n}} = \cos \left(\frac{\mathrm{\pi }}{4^{\mathrm{n}}}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{4^{\mathrm{n}}}\right), \mathrm{then} {\mathrm{x}}_1, {\mathrm{x}}_2, {\mathrm{x}}_3...\infty \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1 + \mathrm{i}\sqrt{3}}{2}$

• $\frac{- 1 + \mathrm{i}\sqrt{3}}{2}$

• $\frac{1 - \mathrm{i}\sqrt{3}}{2}$

• $\frac{- 1 - \mathrm{i}\sqrt{3}}{2}$

$\mathrm{If} \mathrm{z} = 3 + 5\mathrm{i}, \mathrm{then} {\mathrm{z}}^3 + \overline{\mathrm{z}} + 198 \mathrm{is} \mathrm{equal} \mathrm{to}$

• - 3 - 5i

• - 3 + 5i

• 3 - 5i

• 3 + 5i

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\sin }^2\left(\frac{\mathrm{\pi }}{8} + \frac{\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{8} - \frac{\mathrm{x}}{2}\right), \mathrm{then} \mathrm{the} \mathrm{period} \mathrm{of} \mathrm{f} \mathrm{is}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $\frac{56}{33}$

• $\frac{33}{56}$

• $\frac{16}{65}$

• $\frac{60}{61}$

189.

$\sum _{\mathrm{k} = 1}^3 {\cos }^2\left(\left(2\mathrm{k} - 1\right)\frac{\mathrm{\pi }}{12}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $\frac{3}{2}$

$\mathrm{If} \frac{3 + 2\mathrm{isin}\left(\mathrm{\theta }\right)}{1 - 2\mathrm{isin}\left(\mathrm{\theta }\right)} \mathrm{is} \mathrm{a} \mathrm{real} \mathrm{number} \mathrm{and} 0 < \mathrm{\theta } < 2\mathrm{\pi }, \mathrm{then} \mathrm{\theta } \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$