JEE PYQ: Trigonometric Ratios And Identities Question 1
Question 1 - 2021 (18 Mar Shift 2)
If $15 \sin^4 \alpha + 10 \cos^4 \alpha = 6$, for some $\alpha \in \mathbb{R}$, then the value of $27 \sec^6 \alpha + 8 \csc^6 \alpha$ is equal to:
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(1) 350
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(2) 500
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(3) 400
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(4) 250
Show Answer
Answer: (4) 250
Solution
$15 \sin^4 \alpha + 10 \cos^4 \alpha = 6(\sin^2 \alpha + \cos^2 \alpha)^2 \Rightarrow (3\sin^2\alpha - 2\cos^2\alpha)^2 = 0$. So $\tan^2\alpha = \frac{2}{3}$, $\cot^2\alpha = \frac{3}{2}$. Then $27(1+\frac{2}{3})^3 + 8(1+\frac{3}{2})^3 = 250$.
BETA
