The value of $a$ for which $a x^{2} + s i n^{- 1} (x^{2} - 2 x + 2) + c o s^{- 1} (x^{2} - 2 x + 2) = 0$ has a real solution is

\frac{π}{2}

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- \frac{π}{2}

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\frac{2}{π}

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- \frac{2}{π}

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Solution

The correct option is B $- \frac{π}{2}$
$a x^{2} + s i n^{- 1} ((x - 1)^{2} + 1) + c o s^{- 1} ((x - 1)^{2} + 1) = 0$
$a x^{2} + \frac{π}{2} = 0$
$x^{2} = \frac{- \frac{π}{2}}{a}$
$x^{2} = \frac{- π}{2 a}$ ...(i)
And
$- 1 \leq (x - 1)^{2} + 1 \leq 1$
$- 2 \leq (x - 1)^{2} \leq 0$
$(x - 1)^{2} \leq 0$
There is only one possibility
$(x - 1)^{2} = 0$
$x = 1$
Substituting in 1, we get
$a = - \frac{π}{2}$

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