Angle $\mathrm{\alpha }$ is divided into two parts A and B such that A - B = x and tan A : tan B = p : q. The value of sin(x) is equal to :

• $\frac{\left(p + q\right)\sin \left(\alpha \right)}{p - q}$

• $\frac{\mathrm{psin}\left(\mathrm{\alpha }\right)}{\mathrm{p} + \mathrm{q}}$

• $\frac{\mathrm{psin}\left(\mathrm{\alpha }\right)}{\mathrm{p} - \mathrm{q}}$

• $\frac{\left(\mathrm{p} - \mathrm{q}\right)\sin \left(\mathrm{\alpha }\right)}{\mathrm{p} + \mathrm{q}}$

In a triangle ABC, a - 2b + c = 0. The value of $\cot \left(\frac{\mathrm{A}}{2}\right)\cot \left(\frac{\mathrm{C}}{2}\right)$ is  :

• $\frac{9}{2}$

• 3

• $\frac{3}{2}$

• 1

$\sqrt{1 + \sin \left(\mathrm{A}\right)} = \sin \left(\frac{\mathrm{A}}{2}\right) + \cos \left(\frac{\mathrm{A}}{2}\right)$ is true if :

• $\frac{3\mathrm{\pi }}{2} < \mathrm{A} < \frac{5\mathrm{\pi }}{2} \mathrm{only}$

• $\frac{\mathrm{\pi }}{2} < \mathrm{A} < \frac{3\mathrm{\pi }}{2} \mathrm{only}$

• $\frac{3\mathrm{\pi }}{2} < \mathrm{A} < \frac{7\mathrm{\pi }}{2}$

• $0 < \mathrm{A} < \frac{3\mathrm{\pi }}{2}$

54.

In triangle ABC, if $\frac{{\sin }^2\left(\mathrm{A}\right) + {\sin }^2\left(\mathrm{B}\right) + {\sin }^2\left(\mathrm{C}\right)}{{\cos }^2\left(\mathrm{A}\right) +\mathbf{ }{\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right)}$ = 2 then the triangle is :

• right angled

• equilateral

• isosceles

• obtuse-angled

Given that $\tan \left(\mathrm{\alpha }\right) \mathrm{and} \tan \left(\mathrm{\beta }\right)$ are the roots of the equation. x² + bx + c = 0 with b $\ne$0. What is $\tan \left(\mathrm{\alpha } + \mathrm{\beta }\right) = ?$

• b(c - 1)

• c(b - 1)

• c(b - 1) ^{- 1}

• b(c - 1) ^{- 1}

Given that $\tan \left(\mathrm{\alpha }\right) \mathrm{and} \tan \left(\mathrm{\beta }\right)$ are the roots of the equation

x² + bx + c = 0 with b $\ne 0$

What is $\sin \left(\mathrm{\alpha } + \mathrm{\beta }\right)\sec \left(\mathrm{\alpha }\right)\sec \left(\mathrm{\beta }\right)$ = ?

• b

•  - b

• c

•  - c

Consider a triangle ABC in which

$\cos \left(\mathrm{A}\right) + \cos \left(\mathrm{B}\right) + \cos \left(\mathrm{C}\right) = \sqrt{3}\sin \left(\frac{\mathrm{\pi }}{3}\right)$

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \sin \left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{\mathrm{B}}{2}\right)\sin \left(\frac{\mathrm{C}}{2}\right) ?$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{16}$

Consider a triangle ABC in which

$\cos \left(\mathrm{A}\right) + \cos \left(\mathrm{B}\right) + \cos \left(\mathrm{C}\right) = \sqrt{3}\sin \left(\frac{\mathrm{\pi }}{3}\right)$

What is the value of $\cos \left(\frac{\mathrm{A} + \mathrm{B}}{2}\right)\cos \left(\frac{\mathrm{B} + \mathrm{C}}{2}\right)\cos \left(\frac{\mathrm{C} + \mathrm{A}}{2}\right) ?$

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{1}{16}$

• None of the above

Consider the equation $\mathrm{ksin}\left(\mathrm{x}\right) + \cos \left(2\mathrm{x}\right) = 2\mathrm{k} - 7$. If the equation possesses solution, then what is the maximum value of k ?

• 1

• 2

• 4

• 6

Consider the following statements :

1. If ABC is an equilateral triangle, then 3tan(A + B) tan (C) = 1
2. If ABC is a triangle in which A = 78°, A = 66°, then $\tan \left(\frac{\mathrm{A}}{2} + \mathrm{C}\right) < \tan \left(\mathrm{A}\right)$
3. ABC is any triangle, then 

$$\begin{gathered} \tan \left(\frac{\mathrm{A} +\hspace{0.17em}\mathrm{B}}{2}\right)\sin \left(\frac{\mathrm{C}}{2}\right) < \cos \left(\frac{\mathrm{C}}{2}\right) \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{statements} \mathrm{is}/\mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1 only

• 2 only

• 1 and 2

• 2 and 3