Consider the following two statement. Statement p : The value of … - Vidyadip

MCQ

Consider the following two statement.

Statement $p$ : The value of $sin\,120^o$ can be divided by taking $\theta\, = 240^o$ in the equation $2\,\sin \frac{\theta }{2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $

Statement $q$ : The angles $A, B, C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation $\cos \left( {\frac{1}{2}\left( {A + C} \right)} \right) + \cos \left( {\frac{1}{2}\left( {B + D} \right)} \right) = 0$

Then the truth values of $p$ and $q$ are respectively.

✓
$F, T$
B
$T, T$
C
$F, F$
D
$T, F$

✓

Answer

Correct option: A.

$F, T$

a
For statement $p$: $\sin {120^ \circ } = \frac{{\sqrt 3 }}{2}$ $ \Rightarrow 2\sin {120^ \circ } = \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 (\frac{{A + C}}{2}) + \cos (\frac{{B + D}}{2}) = 0$

So statement $p$ is False and statement $q$ is True

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