Test: Trigonometry- 2 - Railways MCQ

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. 
Answer choice (C)

Let the hexagon ABCDEF be of side ‘a’. Line AD = 2a. Let towers at B and D be B’B and D’D respectively. 
From the given data we know that ∠B´AB = 30° and ∠D´AB = 45°. Keep in mind that the Towers B’B and D´D are not in the same plane as the hexagon.

Given,
Cos A = 1- Cos²A
=) Cos A = Sin² A
=) Cos²A = Sin⁴A
=) 1 – Sin² A = Sin⁴ A
=) 1 = Sin⁴ A + Sin² AD
=) 1³ = (Sin⁴A + Sin²A)³
=) 1 = Sin¹² A + Sin⁶A + 3Sin⁸ A + 3Sin¹⁰ A
=) Sin¹² A + Sin⁶A + 3Sin⁸ A + 3Sin¹⁰ A – 1 = 0
On comparing, 
a = 1, b = 3 , c = 3 , d = 1

Option(B) is correct

If you are quick at observing , this question can be solved just by looking at the options as ,

We can draw the figure as :-


Answer choice (D)

Writing the equation as :-

(log sin 1° - log cos 89°) + (log sin 2° - log cos 88°) + (log sin 3° - log cos 87°)………… + log tan 1°. log tan 89° + log tan 2°. log tan 88° + …….

=) (log sin 1° - log sin 1°) +(log sin 2° - log sin 2°)+……..+ log tan 1°cot 1° + log tan 2°cot 2°

=) log 1 = 0
 

We know that,
h² = p² + b² Given, p and b are positive integer, so h2 will be sum of two perfect squares. We see
a) 72 + 52 = 74 
b) 62 + 42 = 52
c) 32 + 22 = 13
d) Can’t be expressed as a sum of two perfect squares
The question is "Which of the following value can h2 not take, given that p and b are positive integers? "
Hence, the answer is 23