If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the solutions of $\mathrm{atan}\left(\mathrm{\theta }\right) + \mathrm{bsec}\left(\mathrm{\theta }\right) = \mathrm{c}$, then $\tan \left(\mathrm{\alpha } + \mathrm{\beta }\right)$ is equal

• $\frac{2\mathrm{ac}}{{\mathrm{a}}^2 - {\mathrm{c}}^2}$

• $\frac{2\mathrm{ac}}{{\mathrm{c}}^2 - {\mathrm{a}}^2}$

• $\frac{2\mathrm{ac}}{{\mathrm{a}}^2 + {\mathrm{c}}^2}$

• None of these

If ${\sin }^{-1}\left(\frac{\mathrm{x}}{5}\right) + {\csc }^{-1}\left(\frac{5}{4}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• 4

• 5

• 1

• 3

If $\frac{\sin \left(\mathrm{A}\right)}{\sin \left(\mathrm{C}\right)} = \frac{\sin \left(\mathrm{A} - \mathrm{B}\right)}{\sin \left(\mathrm{B} - \mathrm{C}\right)}$, then a², b², c² are in

• AP

• GP

• HP

• None of these

If A = $60°$, a = 5, b = 4$\sqrt{3}$ in $∆\mathrm{ABC}$, then B is equal to

• 30°

• 60°

• 90°

• None of these

If y = sin²(x)cos²(x), then y_n is equal to

• $\left(\frac{4^{\mathrm{n}}}{8}\right) . \cos \left(4\mathrm{x} + \frac{\mathrm{n\pi }}{2}\right)$

• $\left({\left(- 1\right)}^{\mathrm{n}} \frac{4^{\mathrm{n}}}{8}\right) . \cos \left(4\mathrm{x} + \frac{\mathrm{n\pi }}{2}\right)$

• $\left(- \frac{4^{\mathrm{n}}}{8}\right) . \cos \left(4\mathrm{x} + \frac{\mathrm{n\pi }}{2}\right)$

• None of the above

In a $∆$ABC, if cot(A) cot(B) cot(C) > 0, then the triangle is

• acute angled

• right angled

• obtuse angled

• does not exist

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

149.

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