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Mathematics · 2023 · 2 marks
CBSE 2023 · Region 1 · Set 1 · Q23
If $\displaystyle \sin \theta+\cos \theta=\sqrt{3}$, then find the value of $\displaystyle \sin \theta \cdot \cos \theta$.If $\displaystyle \sin \alpha=\frac{1}{\sqrt{2}}$ and $\displaystyle \cot \beta=\sqrt{3}$, then find the value of $\displaystyle \operatorname{cosec} \alpha+\operatorname{cosec} \beta$.
If $\displaystyle \sin \theta+\cos \theta=\sqrt{3}$, then find the value of $\displaystyle \sin \theta \cdot \cos \theta$.
If $\displaystyle \sin \alpha=\frac{1}{\sqrt{2}}$ and $\displaystyle \cot \beta=\sqrt{3}$, then find the value of $\displaystyle \operatorname{cosec} \alpha+\operatorname{cosec} \beta$.
Marking-scheme solution
\(\displaystyle \sin \theta+\cos \theta=\sqrt{3}\) squaring both sides
\[\begin{aligned}
& \sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=3 \\
& \Rightarrow 1+2 \sin \theta \cos \theta=3 \\
& \Rightarrow \sin \theta \cos \theta=1
\end{aligned}
\]
\(\displaystyle \operatorname{cosec} \alpha=\frac{1}{\sin \alpha}=\sqrt{2}\)
\(\displaystyle \operatorname{cosec} \beta=\sqrt{1+\cot ^{2} \beta}=\sqrt{1+3}=2\)
\(\displaystyle \therefore \operatorname{cosec} \alpha+\operatorname{cosec} \beta=\sqrt{2}+2\) or \(\displaystyle \sqrt{2}(\sqrt{2}+1)\)
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CBSE Class 10 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.