Question

Consider the following statements:

1. sin θ = x + \(\frac{1}{x}\) is possible for some real value of x.
2. cos θ = x + \(\frac{1}{x}\) is possible for some real value of x.

Which of the above statements is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2 
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 4 : Neither 1 nor 2

Calculations :

Staement 1

sin θ = x + (1/x) 

⇒ sin θ = (x² + 1)/x

⇒ x.sin θ = x² + 1

⇒ x² - x.sin θ + 1 = 0 

For any quadratic equation(ax² + bx + c) real values exist when b² > 4ac

So, 

(- sin θ)² > 4 × 1 × 1 

⇒ sin² θ > 4 

⇒ sin θ > 2 

We know that -1 ≤ sin θ ≤ 1

So no value exist for some real value of x.

Statement 2 

cos θ = x + (1/x) 

⇒ cos θ = (x2 + 1)/x

⇒ x.cos θ = x2 + 1

⇒ x2 - x.cos θ + 1 = 0 

For any quadratic equation(ax2 + bx + c) real values exist when b2 > 4ac

So, 

(- cos θ)2 > 4 × 1 × 1 

⇒ cos2 θ > 4 

⇒ cos θ > 2 

We know that -1 ≤ cos θ ≤ 1

So no value exist for some real value of x. 

So no statement is following. 

∴ Option 4 is the correct choice.