First slide

Introduction to ITF

Question

If $x (3 - x) \geq 2$ then $\sin^{- 1} (x) + \sin^{- 1} (x^{2}) + ............ + \sin^{- 1} (x^{10}) =$

Moderate

A

$\frac{π}{2}$
B

$2 π$
C

$5 π$
D

$10 π$

Solution

$W e h a v e x (3 - x) \geq 0 \Rightarrow 3 x - x^{2} \geq 2 \Rightarrow x^{2} - 3 x + 2 \leq 0 \Rightarrow (x - 1) (x - 2) \leq 0 \Rightarrow 1 \leq x \leq 2 \Rightarrow x = 1 (∵ d o m a i n o f \sin^{- 1} x i s [- 1, 1])$
$∴ \sin^{- 1} (x) + \sin^{- 1} (x^{2}) + . . . . . . + s {in}^{- 1} (x^{10}) = \sin^{- 1} (1) + \sin^{- 1} (1) + . . . . . . + s {in}^{- 1} (1)$
$= \frac{π}{2} + \frac{π}{2} . . . . . . + \frac{π}{2} (10 t i m e s) = 5 π$

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