Question

Consider the following two statements:
Statement $p$:
The value of $\sin {120}^{\circ }$ can be divided by taking $\theta ={240}^{\circ }$ in the equation $2\sin \frac{\theta }{2}=\sqrt{1+\sin \theta }-\sqrt{1-2\theta }$.
Statement $q$:
The angles $A,B,C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation $\cos \left(\frac{1}{2}\right(A+C\left)\right)+\cos \left(\frac{1}{2}\right(B+D\left)\right)=0$
Then the truth values of $p$ and $q$ are respectively.

• A. F, T
• B. T, T
• C. F, F
• D. T, F

Answer 1

The correct option is A F, T

For statement p:

$\sin {120}^o=\frac{\sqrt{3}}{2}\Rightarrow 2\sin {120}^o=\sqrt{3}$

$\sqrt{1+\sin {240}^o}-\sqrt{1-\sin {240}^o}=\sqrt{\frac{1-\sqrt{3}}{2}}-\sqrt{\frac{1+\sqrt{3}}{2}}\ne \sqrt{3}$

For statement q:

$\frac{A+C}{2}+\frac{B+D}{2}=\pi \Rightarrow \cos \left(\frac{A+C}{2}\right)+\cos \left(\frac{B+D}{2}\right)=0$

So statement p is False and statement q is True. So the correct answer is option A.