If ${\left(\frac{1 + \sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)}{ 1 + \sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)}\right)}^2 = \mathrm{\lambda }\left(\frac{1 - \cos \left(\mathrm{\theta }\right)}{1 +\hspace{0.17em}\cos \left(\mathrm{\theta }\right)}\right)$, then $\mathrm{\lambda }$ equals

• - 1

• 1

• 2

• - 2

If a line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ with x-axis, y-axis and z-axis respectively, then ${\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ equals

• 1

• 2

• 3

• - 1

A man 2 m tall walks at a uniform speed of 5 km/h away from a lamp post 6 m high. The rate at which the length of his shadow increases, is

• 2.5km/h

• 5km/h

• 15km/h

• $\frac{5}{3} \mathrm{km}/\mathrm{h}$

If b + c = 3a, then the value of $\cot \left(\frac{\mathrm{B}}{2}\right)\cot \left(\frac{\mathrm{C}}{2}\right)$ is

• $\sqrt{2}$

• $\sqrt{3}$

• 2

• 1

The value of $\tan \left(57°\right) - \tan \left(12°\right) - \tan \left(57°\right)\tan \left(12°\right)$ is

• $\tan \left(69°\right)$

• $\tan \left(45°\right)$

• 0

• None of these

If a and b are two vectors of magnitude 2 inclined at an angle 60°, then the angle between a and a + b is

• 30°

• 60°

• 45°

• None of these

The angle of elevation of the Sun, when the shadow of the pole is $\sqrt{3}$ times the height of the pole, is

• 60°

• 45°

• 15°

• 30°

148.

The curves x = y² and xy = k cut at right angles, if k² is equal to

• 1

• 0

• $\frac{1}{8}$

• 8

If $\sin \left(\mathrm{\theta }\right) \frac{\sqrt{3}}{2}$, then the general value of $\mathrm{\theta }$ is

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{3}$

• $2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{6}$

• $2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{3}$

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{6}$

The value of $\tan \left(9°\right) - \tan \left(27°\right) - \tan \left(63°°\right) + \tan \left(31°\right)$ is

• 3

• 2

• 8

• 4