Q64 / 160 · AP EAMCET 2024 (20 May Shift 2) · Mathematics

If $\cos ^{-1} 2 x+\cos ^{-1} 3 x=\frac{\pi}{3}$ and $4 x^2=\frac{a}{b}$ then $a+b=$

  • A
    12
  • B
    11
  • C
    31
  • Correct answer
    D
    10

Solution

$\cos ^{-1} 3 x=\frac{\pi}{3}-\cos ^{-1} 2 x$ $\begin{aligned} & 3 x=\cos \frac{\pi}{3} \times 2 x+\sin \frac{\pi}{3} \sqrt{1-4 x^2} \\ & 2 x=\frac{\sqrt{3}}{2} \sqrt{1-4 x^2} \\ & 16 x^2=3\left(1-4 x^2\right) \\ & 28 x^2=3 \Rightarrow 4 x^2=\frac{3}{7}=\frac{a}{b} \\ & a=3, b=7 \\ & a+b=10\end{aligned}$

Hence, the correct answer is option: 'D'.

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