Question

The equation $2^{\frac{2\pi}{\cos^{- 1}x}} - \left( a + \frac{1}{2} \right)2^{\frac{\pi}{\cos^{- 1}x}}$– a² = 0 has only one real root, then-

• A. 1 ≤ a ≤ 3
• B. a ≤ –3 or a ≥ 1
• C. 1 < a < 3
• D. None

Answer 1

Answer: (B)

a ≤ –3 or a ≥ 1

Answer 2

1 ≤ $\frac{\pi}{\cos^{- 1}x}$ < ∞ ⇒ 2 ≤ $2^{\frac{\pi}{\cos^{- 1}x}}$ < ∞

Let t = $2^{\frac{\pi}{\cos^{- 1}x}}$

So t² – $\left( a + \frac{1}{2} \right)$t – a² = 0

∴ f (2) < 0

4 – (2a + 1) – a² ≤ 0

a² + 2a – 3 ≥ 0

a ≤ –3, a ≥ 1