First slide

Introduction to ITF

Question

If $\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}$ and f(1) = 1, f(p + q) = f(p). f(q) $\forall$ p, q ∈ R then, $x^{f (1)} + y^{(2)} + z^{f (3)} - \frac{x + y + z}{x^{f (1)} + y^{f (2)} + z^{f (3)}} =$

Moderate

A

0
B

1
C

2
D

3

Solution

Since, $- \frac{π}{2} \leq \sin^{- 1} x \leq \frac{π}{2}$
$\begin{array}{l} ∴ \sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2} \\ \Rightarrow \sin^{- 1} x = \sin^{- 1} y = \sin^{- 1} z = \frac{π}{2} \\ \Rightarrow x = y = z = 1 \end{array}$
Also, f(p + q) = f(p). f(q) ∀ p, q ∈ R …. (1)
Given, f(1) = 1
From (1),
f(1 + 1) = f(1). f(1)
⇒ f(2) =1² = 1 ………. (2)
From (2), f(2 + 1) = f(2) . f(1)
⇒ f(3) = 1² .1 = 1³ = 1
Now, given expression $= 3 - \frac{3}{3} = 2$

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