Question

Consider the following statements:

1. The equation 2 sin² θ - cos θ + 4 = 0 is possible for all θ

2. tan θ + cot θ cannot be less than 2, where 0 < θ < \(\frac{\pi}{{2}}\)

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 2 : 2 only

Calculations :

Statement 1

2 sin2 θ - cos θ + 4 = 0

⇒ 2 (1 - cos² θ) - cos θ + 4 = 0 

⇒ 2 - 2 cos² θ - cos θ + 4 = 0 

⇒ 2 cos² θ + cos θ - 6 = 0 

⇒ 2 cos² θ + 4 cos θ - 3 cos θ - 6 = 0 

⇒ 2 cos θ (cos θ + 2) - 3 (cos θ + 2) = 0 

⇒ (2 cos θ - 3)(cos θ + 2) = 0 

⇒ cos θ = 3/2 or -2  

This statement is incorrect as cos θ lies between -1 and 1 

Now statement 2

tan θ + cot θ is more than or equal to 2 

(tan θ + cot θ)/2  ≥ √(tan θ .cot θ)       (Arithmetic mean ≥ Geometric mean)

(tan θ + cot θ)/2 ≥ 1

Tan θ + cot θ ≥ 2  

We can conclude that only statement 2 is correct. 

∴ Option 2 is the correct choice.