Test: JEE Main 35 Year PYQs- Trigonometric Functions & Equations - Airforce X Y / Indian Navy SSR MCQ

 ;  Period

The given equation is tanx + secx = 2 cos x;
⇒ sin x + 1 = 2cos² x ⇒ sin x + 1 = 2(1 – sin² x);
⇒ 2sin²x + sin x – 1= 0;
⇒ (2sin x – 1)(sin x + 1) = 0 ⇒ sin x =   , –1.;
⇒ x = 30°, 150°, 270°

∵ cos is non periodic

∴ cos + cos ²x can not be periodic.

.....(1)

.....(2)

Square and add (1) and (2)

u² = a² + b² 

Now (a ⁴ + b⁴ ) cos²θ sin ²θ + a ²b ² (cos ⁴θ+ sin⁴θ)

= (a⁴ + b⁴ ) cos ²θ sin²θ + a ² b ² (1 - 2 cos ²θ sin²θ)

= (a⁴ + b⁴ - 2a ² b²) cos²θ sin ²θ+ a ²  b²

…(2)

∴ from (1) , (2) and (3)

Minimum value of u² = a ² + b²

Maximum value of u²

= a ² + b²

= a ² + b² = 2(a ²+b²)

∴ Max value - Min value

= 2(a² + b² ) - (a +b² ) = (a-b)²

The direction cosines of the line are cosθ, cosβ, cosθ

∴ cos ²θ + cos ²β + cos ²θ = 1

⇒ 2 cos ²θ = sin ²β = 3 sin ²θ (given)

⇒ 2 cos ²θ = 3 - 3 cos ²θ

2 sin ² x + 5 sinx - 3=0

⇒ (sin x + 3)(2 sinx - 1) = 0            

⇒ sin x =   and sin x ≠-3

∴ In [0, 3π] , x has 4 values.

⇒ 3 tan 2 x + 8 tanx + 3= 0

⇒ 3 tan 2 x + 8 tanx + 3= 0

as  tan x <0  

We have

cos (β – γ) + cos (γ – α) + cos (α – β) =

⇒ 2 [cos (β – γ) + cos (γ – α) + cos (α – β)] + 3 = 0

⇒ 2 [cos (β – γ) + cos (γ – α) + cos (α – β)]

+ sin²α + cos²α + sin²β + cos²β + sin²γ + cos²γ = 0

⇒ [sin²α + sin²β + sin²γ + 2 sinα  sinβ + 2 sinβ sinγ

+ 2 sinγ sinα ] + [cos²α + cos²β + cos²γ + 2cosα cosβ

+ 2 cosβ cosγ + 2cosγ cosα] = 0

⇒ [sinα + sinβ + sinγ]² + (cos2α + cosβ + cos2γ )²= 0

⇒ sinα + sinβ + sinγ = 0 and cosα + cosβ + cosγ =0

∴ A and B both are true.

A = sin ² x+ cos⁴x = sin ² x + cos ² x(1- sin²x)

Now 

Given 3 sin P + 4cos Q = 6 ...(i)

4 sin Q + 3cos P = 1 ...(ii)

Squaring and adding (i) & (ii) we get

9 sin² P + 16cos²Q + 24 sin P cos Q

+ 16 sin²Q + 9cos² P + 24 sin Q cos P = 36 + 1 = 37

⇒ 9 (sin²P + cos²P) + 16 (sin² Q + cos² Q) + 24 (sinP cosQ + cosP sinQ) = 37

⇒ 9 + 16 + 24 sin ( P + Q) = 37

[∵ sin²θ + cos²θ = 1 and sinα cos B + cos2α sinβ = sin (A + B)]

⇒ (∵P + Q + R = p)

If  R =then 0 < P,Q <

⇒ cos Q <1 and sin P < 

⇒ 3 sin P + 4 cos Q < which is not true.

So R  = 

From Sine Rule

Given expression can be written as

=  1 + sec A cosec AA

Let  Consider

cos x + cos 2x + cos 3x + cos 4x = 0

⇒ 2 cos 2x cos x + 2 cos 3x cos x = 0