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Q.

The number of solutions of the equation $\sin^{- 1} [x^{2} + \frac{1}{3}] + \cos^{- 1} [x^{2} - \frac{2}{3}] = x^{2},$ for
$x \in [- 1, 1]$ and [x] denotes the greatest integer less than or equal to x, is:

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a

4

b

2

c

Infinite

d

0

answer is D.

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Detailed Solution

$[x^{2} - \frac{2}{3}] = [x^{2} + \frac{1}{3} - 1] = [x^{2} + \frac{1}{3}] - 1 - 1 \leq [x^{2} + \frac{1}{3}] \leq 1 \Rightarrow [x^{2} + \frac{1}{3}] = - 1, 0, 1 [x^{2} + \frac{1}{3}] = - 1 \Rightarrow not possible [x^{2} + \frac{1}{3}] = 0 \Rightarrow 0 + π = x^{2} \Rightarrow no solution [x^{2} + \frac{1}{3}] = 0 \Rightarrow \frac{π}{2} + \frac{π}{2} = x^{2} \Rightarrow no solution$

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The number of solutions of the equation sin-1x2+13+cos-1x2-23=x2, forx∈-1,1 and [x] denotes the greatest integer less than or equal to x, is: