Let α =(cos 85 ° sin 55 ° sin 65 ° /cos 5 ° sin 35 ° sin 25 ° ) is a root of the quadratic equation 2x2-px+q=0 , where p,q∈ Q , then the value of (p + q) is

Question

Let α =(cos 85 ° sin 55 ° sin 65 ° /cos 5 ° sin 35 ° sin 25 ° ) is a root of the quadratic equation 2x2-px+q=0 , where p,q∈ Q , then the value of (p + q) is (A) 2 (B) 5 (C) 8 (D) 10

Tardigrade Admin · Accepted Answer

α =(cos 85 ° sin 55 ° sin 65 ° /cos 5 ° sin 35 ° sin 25 ° )=(sin 5 ° sin 55 ° sin 65 ° /cos 5 ° cos 55 ° cos 65 ° ) =((1/4) sin 15 ° /(1)4 cos 15 ° )=tan15° =2-√3 Other root will be 2+√3[p , q ∈ Q] So, required equation is x2-(4 x)+1=0 2x2-8x+2=0 ∴ p+q=10

Q. Let $α = \frac{cos 8 5 ^{\circ} s in 5 5 ^{\circ} s in 6 5 ^{\circ}}{cos 5 ^{\circ} s in 3 5 ^{\circ} s in 2 5 ^{\circ}}$ is a root of the quadratic equation $2 x^{2} - p x + q = 0$ , where $p, q \in Q$ , then the value of $(p + q)$ is

$2$

$5$

$8$

$10$

Q. Let α=cos5∘sin35∘sin25∘cos85∘sin55∘sin65∘​ is a root of the quadratic equation 2x2−px+q=0 , where p,q∈Q , then the value of (p+q) is

2

5

8

10

α=cos5∘sin35∘sin25∘cos85∘sin55∘sin65∘​=cos5∘cos55∘cos65∘sin5∘sin55∘sin65∘​ =41​cos15∘41​sin15∘​=tan15∘ =2−3​ Other root will be 2+3​[p,q∈Q] So, required equation is x2−(4x)+1=0 2x2−8x+2=0 ∴p+q=10

Q. Let $α = \frac{cos 8 5 ^{\circ} s in 5 5 ^{\circ} s in 6 5 ^{\circ}}{cos 5 ^{\circ} s in 3 5 ^{\circ} s in 2 5 ^{\circ}}$ is a root of the quadratic equation $2 x^{2} - p x + q = 0$ , where $p, q \in Q$ , then the value of $(p + q)$ is

$2$

$5$

$8$

$10$