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Number of solutions of tan^-1(root a) + sin^-1(root(a+1)) = a given angle

This type of question, asked 2 times: 2021 Paper 11, 31 Jan 2023. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 11

    55
    The number of real roots of the equation tan⁡−1x(x+1)+sin⁡−1x2+x+1=π4\tan^{- 1}\sqrt{x(x + 1)}+\sin^{- 1}\sqrt{x^{2}+ x + 1}=\frac{\pi}{4} is :

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  2. Question 2

    31 Jan 2023

    68
    Let y=f(x)y = f(x) represent a parabola with focus (−12,0)\left( -\frac{1}{2},0 \right) and directrix y=−12y = -\frac{1}{2}. Then S={x∈R:tan⁡−1(f(x))+sin⁡−1(f(x)+1)=π2}S =\left\{ x\in R:\tan^{- 1}(\sqrt{f(x)}) +\sin^{- 1}(\sqrt{f(x) + 1}) =\frac{\pi}{2} \right\} :

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