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The following question consist of two statements, one labelled as the 'Assertion $(A)^{\prime}$ and the other as 'Reason $(R):$ You are to examine these two statements carefully and select the answer.
Let $X=\{\theta \in[0,2 \pi] \cdot \sin \theta=\cos \theta\}$
Assertion (A): The number of elements in $X$ is 2. Reason (R) : $\sin \theta$ and $\cos \theta$ are both negative both in second and fourth quadrants.
Options:
Let $X=\{\theta \in[0,2 \pi] \cdot \sin \theta=\cos \theta\}$
Assertion (A): The number of elements in $X$ is 2. Reason (R) : $\sin \theta$ and $\cos \theta$ are both negative both in second and fourth quadrants.
Solution:
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Verified Answer
The correct answer is:
$\mathbf{A}$ is true but $\mathbf{R}$ is false.
$\quad(\mathrm{A}): \mathrm{X}=\{\theta \in[0,2 \pi]: \sin \theta=\cos \theta\}$
Number of elements in $\mathrm{X}$ is 2 . Since, $\sin \theta=\cos \theta$ is possible at $\theta=45^{\circ}$ and $225^{\circ}$ Since, $\cos \theta$ is negative in IInd quadrant but $\sin \theta$ is positive, $R$ is false.
Number of elements in $\mathrm{X}$ is 2 . Since, $\sin \theta=\cos \theta$ is possible at $\theta=45^{\circ}$ and $225^{\circ}$ Since, $\cos \theta$ is negative in IInd quadrant but $\sin \theta$ is positive, $R$ is false.
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