JEE Main 2022 · 25 Jul · Shift 2 · Mathematics Q30

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MathematicsInteger+4 / -1Question 30 of 90

Let $x=\sin\left(2\tan^{-1}\alpha\right)$ and $y=\sin\left(\dfrac12\tan^{-1}\dfrac43\right)$ . If $S=\left\{\alpha\in\mathbb R:y^2=1-x\right\}$ , then $\displaystyle\sum_{\alpha\in S}16\alpha^3$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 130

y. $y^2=\sin^2\left(\frac12\tan^{-1}\frac43\right)=\frac{1-3/5}2=\frac15$ , so $x=\frac45$ .

α. $\dfrac{2\alpha}{1+\alpha^2}=\frac45$ : $2\alpha^2-5\alpha+2=0$ , $\alpha=2,\frac12$ .

Sum $16\left(8+\frac18\right)=130$ .

Answer: 130

Topics:Inverse trigonometric functions Roots and coefficients

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JEE Main 2022 · 25 Jul · Shift 2, Mathematics question 30