First slide

Introduction to ITF

Question

If $\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}$ , then $\frac{\sum_{k = 1}^{2} (x^{100 k} + y^{106 k})}{\sum x^{207} y^{207}}$ is

Moderate

A

$\frac{1}{3}$
B

$\frac{4}{3}$
C

$\frac{2}{3}$
D

none of these

Solution

We have, $\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}$ .
It is possible only when
$\begin{array}{l} \sin^{- 1} x = \frac{π}{2}, \sin^{- 1} y = \frac{π}{2} and \sin^{- 1} z = \frac{π}{2} \\ \Rightarrow x = 1, y = 1 and z = 1 \\ ∴ \frac{\sum_{k = 1}^{2} (x^{100 k} + y^{106 k})}{\sum x^{207} y^{207}} = \frac{(x^{100} + y^{106}) + (x^{200} + y^{212})}{x^{207} y^{207} + y^{207} z^{207} + z^{207} x^{207}} \\ = \frac{1 + 1 + 1 + 1}{1 + 1 + 1} = \frac{4}{3} \end{array}$

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