Integer Answer Type Questions: Trigonometric Functions & Equations - JEE Advanced

Q. 1. The number of all possible values of q where 0 < θ < π, for which the system of equations (y + z) cos 3θ = (xyz) sin 3θ

(xyz) sin 3θ = (y + 2z) cos 3θ  + y sin 3θ

have a solution (x₀, y₀, z₀) with y₀ z₀ ≠ 0, is (2010)

Ans. Sol.  (3) The given equations are
xyz sin 3θ = ( y+ z ) cos3θ - (1)
xyz sin 3θ = 2 z cos3θ+ 2 y sin3θ - (2)
xyz sin3θ = y + 2z cos3θ+ y sin3θ - (3)

Operating (1) - (2) and (3) - (1), we get
(cos 3θ - 2 sinq )y - (cos 3θ)z = 0

and sin 3θ y + ( cos 3θ)z=0

which is homogeneous system of linear equation. But

y ≠ 0,z ≠ 0

⇒ cos 3θ = sin3θ

⇒   =  

For θ ∈ (0,π) ⇒

∴ Three such solutions are possible.

Q. 2. The number of values of q in the interval, such that  for n = 0, ±1,±2 and tanθ = cot 5θ as well as sin 2θ = cos 4θ is (2010)

Ans. Sol.  (3) 

⇒ cos θ cos 5θ - sin5θ sinθ= 0 ⇒ cos 6θ=0

⇒

⇒

Again sin 2θ= cos 4θ= 1 - 2 sin ²2θ

⇒ 2sin² 2θ + sin 2θ -1= 0 ⇒ sin 2θ = -1
 

⇒  

So common solutions are

∴ Number of solutions = 3.

Q. 3. The maximum value of the expression

Ans. Sol. Let

where g (θ) = sin ²θ+ 3 sinθ cos θ+ 5 cos²θ

Clearly f is maximum when g is minimum

Now (θ)  =

= 3 + 2cos 2θ +

Q. 4. The positive integer value of n > 3 satisfying the equation

          (2011)

Ans. (7) 

Sol.   We have,

= 2kπ where k ∈ Z

( n=not possible for any integral value of k)

As n > 3, for k = 0, we get n = 7.

Q. 5. The number of distinct solutions of the equation

 in the interval [0, 2π] is  (JEE Adv. 2015)

Ans. (8)  

Sol.   cos²2x + cos⁴x + sin⁴x + cos⁶x + sin⁶x = 2

⇒

⇒  (cos²2x - sin²2x) = 0 ⇒ cos4x = 0

⇒ 4x = (2n + 1) or            x = (2n + 1)

For x∈[0, 2π], n can take values 0 to 7

∴ 8 solutions.