Q42 / 125 · BITSAT 2010 · Mathematics
If \(12 \cot ^2 \theta-31 \operatorname{coses} \theta+32=0\), then the value of \(\sin \theta\) is
- Correct answerA\(\frac{3}{5}\) or 1
- B\(\frac{2}{3}\) or \(\frac{-2}{3}\)
- C\(\frac{4}{5}\) or \(\frac{3}{4}\)
- D\(\pm \frac{1}{2}\)
Solution
\(\begin{aligned}
& 12 \cot ^2 \theta-31 \operatorname{cosec} \theta+32=0 \\
& \Rightarrow 12\left(\operatorname{cosec}^2 \theta-1\right)-31 \operatorname{cosec} \theta+32=0 \\
& \Rightarrow \quad 12 \operatorname{cosec}^2 \theta-31 \operatorname{cosec} \theta+20=0 \\
& \Rightarrow \quad 12 \operatorname{cosec}^2 \theta-16 \operatorname{cosec} \theta-15 \operatorname{cosec} \theta+20=0 \\
& \Rightarrow \quad(4 \operatorname{cosec} \theta-5)(3 \operatorname{cosec} \theta-4)=0 \\
& \Rightarrow \quad \operatorname{cosec} \theta=\frac{5}{4}, \frac{4}{3} ; \therefore \sin \theta=\frac{4}{5}, \frac{3}{4}
\end{aligned}\)
Hence, the correct answer is option: 'A'.