If a = 2, b = 3 and c = 5in $∆$BC, then $\angle$C is to equal

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• None of these

If f : $\mathrm{R} \to \mathrm{R}$ be the signum function and g : $\mathrm{R} \to \mathrm{R}$ be the greatest integer function, then $\sin \left\{\mathrm{\pi }\left(\left(\mathrm{fog}\right)\left(\frac{1}{2}\right)\right)\right\}$ is equal to

• 1

• $\frac{\sqrt{3}}{2}$

• 0

• $\frac{1}{\sqrt{2}}$

AB is a vertical pole. The end A is on the ground level. C is the middle point of AB and P is a point on the ground level. The portion BC subtends an angle $\mathrm{\beta }$ at P. If AP = nAB, then $\tan \left(\mathrm{\beta }\right)$ is equal to

• $\frac{\mathrm{n}}{2{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{\mathrm{n} + 1}$

• None of these

From a point on the level ground, the angle of elevation of top of a pole is $30°$ on moving 20metres nearer,the angle of elevation is $45°$. Then,the height of the pole $\left(\mathrm{in} \mathrm{metres}\right)$, is

• $10\left(\sqrt{3} - 1\right)$

• $10\left(\sqrt{3} - 1\right)$

• 15

• 20

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

• $(\frac{3}{4}, 1]$

• $[\frac{3}{4}, 1)$

• $\left[\frac{3}{4}, 1\right]$

• $\left(\frac{3}{4}, 1\right)$

169.

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

• 0

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $\frac{3}{2}$

${\cos }^2\left(76°\right) + {\cos }^2\left(16°\right) - \cos \left(76°\right)\cos \left(16°\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• $\frac{1}{2}$

• $-\frac{1}{ 2}$

• $\frac{3}{4}$