${\cos }^4\left(\frac{\mathrm{\pi }}{8}\right) + {\cos }^4\left(\frac{3\mathrm{\pi }}{8}\right) +\hspace{0.17em}{\cos }^4\left(\frac{5\mathrm{\pi }}{8}\right) + {\cos }^4\left(\frac{7\mathrm{\pi }}{8}\right)$ is equal to

• $\frac{3}{2}$

• $- \frac{2}{3}$

• - 1

• 1

Number of solutions of the equationan $\tan \left(\mathrm{x}\right) + \sec \left(\mathrm{x}\right) = 2\cos \left(\mathrm{x}\right)$ lying in the interval $\left[0, 2\mathrm{\pi }\right]$ is

• 0

• 1

• 2

• 3

133.

In a triangle ABC, if $\tan \left(\frac{\mathrm{A}}{2}\right) = \frac{5}{6} \mathrm{and} \tan \left(\frac{\mathrm{B}}{2}\right) = \frac{20}{37}$ , then a + c is equal to

• b

• 2b

• 3b

• 4b

The angle of elevation of a jet fighter from a point A on the ground is 60°. After a flight of 10s, the angle of elevation changes to 30°. If thejet is flying at a speed of 432 km/h. Find the constant height at which thejet is flying.

• $200\sqrt{3} \mathrm{m}$

• $400\sqrt{3} \mathrm{m}$

• $600\sqrt{3} \mathrm{m}$

• $800\sqrt{3} \mathrm{m}$

In a $∆\mathrm{ABC}, \mathrm{if} \tan \left(\frac{\mathrm{A}}{2}\right) = \frac{5}{6} \mathrm{and} \tan \left(\frac{\mathrm{C}}{2}\right) = \frac{2}{5}$, then the sides a, b, c are in

• AP

• GP

• HP

• None of these

The value of $\cos \left(\frac{\mathrm{\pi }}{5}\right)\cos \left(\frac{2\mathrm{\pi }}{5}\right)\cos \left(\frac{4\mathrm{\pi }}{5}\right)\cos \left(\frac{8\mathrm{\pi }}{5}\right)$ will be

• $\frac{1}{16}$

• $- \frac{1}{16}$

• 0

• $\frac{1}{2}$

The function f(x) = sin(x) + cos(x) will be

• an even function

• an odd function

• a constant function

• None of these

A balloon is coming down at the rate of4 m/min and its angle of elevation is 45° from a point on the ground which has been reduced to 30°, after 10 min. Balloonwill be on the ground at a distance ofhow many metres from the observer ?

• $20\sqrt{3} \mathrm{m}$

• $20\left(3 + \sqrt{3}\right) \mathrm{m}$

• $10\left(3 + \sqrt{3}\right) \mathrm{m}$

• None of these

If $\sin \left(\mathrm{\theta }\right) + \csc \left(\mathrm{\theta }\right) = 2, \mathrm{then} {\sin }^2\left(\mathrm{\theta }\right) + {\csc }^2\left(\mathrm{\theta }\right)$ is equal to

• 1

• 4

• 2

• None of these

The maximum value of $5\cos \left(\mathrm{\theta }\right) + 3\cos \left(\mathrm{\theta } + \frac{\mathrm{\pi }}{3}\right) + 3$ is

• 10

• - 10

• 14

• None of these