CAT Quantitative Aptitude Questions | Geometry Questions for CAT - Trigonometry

The question is from CAT Geometry - Trigonometry. This is about trigonometric functions. We need to find out the number of possible values x can take. Trigonometry is an important topic for CAT Preparation for the CAT Exam. Trigonometric ideas can be completely opposite, as in, some questions test lot of common sense with direction, heights and distances, while others could test you on identities. CAT exam could test one on both these fronts. Make sure you master both in CAT - Trigonometry.

Question 2: Sin²⁰¹⁴x + Cos²⁰¹⁴x = 1, x in the range of [-5π, 5π], how many values can x take?

1. 0
2. 10
3. 21
4. 11

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Geometry - Trigonometry : Number x lies between 0 and 1, higher powers of the number will be smaller than lower powers. Think about that. And then think about what does this have to do with Trigonometry.

We know that Sin²x + Cos²x = 1 for all values of x.
If Sin x or Cos x is equal to –1 or 1, then Sin²⁰¹⁴x + Cos²⁰¹⁴x will be equal to 1.

Sin x is equal to –1 or 1 when x = –4.5π or –3.5π or –2.5π or –1.5π or –0.5π or 0.5π or 1.5π or 2.5π or 3.5π or 4.5π.

Cosx is equal to –1 or 1 when x = –5π or –4π or –3π or –2π or –π or 0 or π or 2π or 3π or 4π or 5π.

For all other values of x, Sin²⁰¹⁴ x will be strictly lesser than Sin²x.
For all other values of x, Cos²⁰¹⁴ x will be strictly lesser than Cos²x.

We know that Sin²x + Cos²x is equal to 1. Hence, Sin²⁰¹⁴x + Cos²⁰¹⁴x will never be equal to 1 for all other values of x. Thus there are 21 values.

The question is "How many values the 'x' can take?"

Hence, the answer is 21.

Choice C is the correct answer.