JEE Advanced (Fill in the Blanks): Trigonometric Functions & Equations

Q.1. Suppose sin3x.sin 3x =cos mx is an identity in x, where C₀, C₁, ___C_n are constants, and C_n ≠ 0. Then the value of n is _________. (1981 - 2 Marks)
Ans. 6
Sol. sin³x . sin 3x =
cos mx

sin³x.sin 3x =[3 sinx - sin3x] sin 3x

=

=

= [ cos 6x + 3 cos 2x - 3 cosx - 1]
We observe that on LHS 6 is the max value of m.
∴n = 6

Q.2. The solution set of the system of equations x + y =, cosx + cosy =   where x and y are real, is _________. (1987 - 2 Marks)
Ans. φ
Sol. The equations are x + y = 2π /3 ... (i)

cos x + cos y = 3/2 ...(ii)

From eq.
(ii)  

⇒  [Using eq. (i)]

⇒  
⇒

Which has no solution.
∴ The solution of given equations is φ.

Q.3. The set of all x in the interval [0,π] for which 2sin² x - 3 sin x + 1 ≥ 0, is _________. (1987 - 2 Marks)
Ans.

Sol. We have  2 sin² x - 3 sin x + 1 ≥ 0

⇒ (2 sin  x - 1)  (sin x - 1) ≥ 0

⇒

But we know that sin x ≤ 1 and sin x ≥  0 for x ∈ [0, π]

⇒ either sin x = 1  or

⇒ either  x = π/2 or

Combining, we get


Q.4. The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the arithmetic mean of   andis equal to _________. (1987 - 2 Marks)
Ans. 
Sol. We know that A.M. ≥ G.M.
 ⇒ Min value of AM. is obtained when AM = GM
 ⇒ The quantities whose AM is being taken are equal.

i.e., Cos

 ⇒ sin α = sin β = sinγ

Also α +β + γ= 360° ⇒ α = β = γ=120° = 2π/3

∴ Min value of A.M.

Q.5. The value of  is equal to _________. (1991 - 2 Marks)
Ans. 

Sol.

   = 

Q.6. If K = sin(π/18) sin(5π/18) sin(7π/18), then the numerical value of K is _________ . (1993 - 2 Marks)
Ans. 

Sol. 

[Using

Q.7. If A > 0,B>0 and A + B = π/3, then the maximum value of tan A tan B is_________ . (1993 - 2 Marks)
Ans. 
Sol. A + B = π/3 ⇒ tan (A +B ) =

[where y = tan A tan B]

 ⇒ tan² A + (y - 1) tan A + y = 0

For real value of tan A, 3(y - 1)² - 4y ≥ 0

 ⇒

But A, B > 0 and A + B = π/3
 ⇒ A, B  < π/3

 ⇒ tan A tan B < 3

i.e.,  max. value of y is 1/3.

Q.8. General value of θ satisfying the equation tan²θ + sec 2θ = 1 is_________ . (1996 - 1 Mark)
Ans.
Sol. tan²θ + sec 2θ= 1
 where t = tanθ

 
which means θ = nπ and θ =nπ±π/3

Q.9. The real roots of the equation cos⁷ x + sin⁴ x=1 in the interval (-π, π) are ______, ______, and _________ . (1997 - 2 Marks)
Ans.
Sol. cos⁷ x  = 1- sin⁴ x = (1- sin² x) (1 + sin² x) = cos² x (1 + sin² x)
∴ cos x = 0
or x = π/2, - π/2
or  cos⁵ x = 1 + sin² x   or  cos⁵ x - sin² x = 1
Now maximum value of each cos x or  sin x is 1.
Hence the above equation will hold  when cos x = 1 and sin x = 0. Both these imply x = 0
Hence