Statement A (Assertion) : A B C D is a rectangle such that C A B=… - Vidyadip

MCQ

Statement $A$ (Assertion) : $A B C D$ is a rectangle such that $\angle C A B=60^{\circ}$ and $A C=a$ units. The area of rectangle $A B C D$ is $\frac{\sqrt{3}}{2} a^2$ sq. units.
Statement $R$ (Reason) : The value of $\sin 60^{\circ}$ is $\frac{\sqrt{3}}{2}$ and $\cos 60^2$ is $\frac{1}{2}$.

A
Both assertion (A) and reason ( $R$ ) are true and reason $(R)$ is the correct explanation of assertion $(A)$.
B
Both assertion (A) and reason ( $R$ ) are true and reason $(R)$ is not the correct explanation of assertion (A).
C
Assertion $(A)$ is true but reason $(R)$ is false.
✓
Assertion (A) is false but reason $(R)$ is true.

✓

Answer

Correct option: D.

Assertion (A) is false but reason $(R)$ is true.

(d): Clearly, reason is true.
In $\triangle A B C, A C=a$ units, $\angle A=60^{\circ}$
$\therefore \quad \sin 60^{\circ}=\frac{B C}{A C}=\frac{B C}{a}$
$\Rightarrow \frac{\sqrt{3}}{2}=\frac{B C}{2} \Rightarrow B C=\frac{\sqrt{3} e}{2}$
Also, $\cos 60^{\circ}-\frac{A B}{A C} \Rightarrow \frac{1}{2}=\frac{A B}{a}$
$\Rightarrow A B=s / 2$
$\therefore$ Area of rectangle $A B C D=A B \times B C$
$
-\frac{\pi}{2} \times \frac{\sqrt{3} r }{2}-\frac{\sqrt{3} r^2}{4}
$
$\therefore$ Assertion is false.

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