| 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 \[\cos \left( \frac{1}{2}(A+C) \right)+\cos \left( \frac{1}{2}(B+D) \right)=0\]. Then the truth values of p and q are respectively. |
A) F, T
B) T, T
C) F, F
D) T, F
Correct Answer: A
Solution :
For statement p: \[\sin 120{}^\circ =\frac{\sqrt{3}}{2}\Rightarrow 2\sin 120{}^\circ =\sqrt{3}\] \[\sqrt{1+\sin 240{}^\circ }-\sqrt{1-\sin 240{}^\circ }=\sqrt{\frac{1-\sqrt{3}}{2}}-\sqrt{\frac{1+\sqrt{3}}{2}}\ne \sqrt{3}\] For statement q: \[\frac{A+C}{2}+\frac{B+D}{2}=\pi \Rightarrow \cos (\frac{A+C}{2})+\cos (\frac{B+D}{2})=0\] So statement p is False and statement q is True. So the correct answer is option A.
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