JEE Main 2017 · 8 Apr · Shift 1 · Mathematics Q88

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MathematicsSingle correct+4 / -1Question 88 of 90

If $S=\left\{x\in[0,2\pi]:\begin{vmatrix}0&\cos x&-\sin x\\\sin x&0&\cos x\\\cos x&\sin x&0\end{vmatrix}=0\right\}$ , then $\displaystyle\sum_{x\in S}\tan\left(\frac\pi3+x\right)$ is equal to :

4+2\sqrt3

-2+\sqrt3

-2-\sqrt3

-4-2\sqrt3

Answer and worked solution

Official answer: D

The determinant is $\cos^3x-\sin^3x$ , which is zero when $\tan x=1$ : $x=\frac\pi4,\frac{5\pi}4$ . For both,
$\tan\left(\frac\pi3+\frac\pi4\right)=\frac{\sqrt3+1}{1-\sqrt3}=-(2+\sqrt3),$
and the sum is $-4-2\sqrt3$ .

Topics:Trigonometric equations Determinants and properties

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JEE Main 2017 · 8 Apr · Shift 1, Mathematics question 88