First slide

Introduction to ITF

Question

The number of real solutions of the equations $\tan^{- 1} \sqrt{x^{2} - 3 x + 2} + \cos^{- 1} \sqrt{4 x - x^{2} - 3} = π$ is

Moderate

A

one
B

two
C

zero
D

infinite

Solution

Since $\sqrt{x^{2} - 3 x + 2} \geq 0 \Rightarrow \tan^{- 1} \sqrt{x^{2} - 3 x + 2} < \frac{π}{2}$
Since $\sqrt{4 x - x^{2} - 3} \geq 0 \Rightarrow 0 < \cos^{- 1} \sqrt{4 x - x^{2} - 3} \leq \frac{π}{2}$
⇒ 0 < L.H.S. < π
⇒ The given equation has no solution

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