Question

Sum of integral values of n such that sin * (2 sin x + cos x)=n, has at least one real solution is
(A) 3
(B)
1
(C) 2
(D) 0​

Answer 1

Answer:

0

Step-by-step explanation:

sin * (2 sin x + cos x)=n

Let say 2 sin x + cos x = α

then Sinα = n

possible integral values of n

are -1 , 0 , 1

=> α = -π/2 ,

α = 0 , π

α = π/2

2 sin x + cos x  = 0

=> 2Sinx = -Cosx

=> tanx = -1/2

Hence real solution exist  => n = 0 is one value

Let say y = 2 sin x + cos x

dy/dx = 2Cosx - Sinx

2Cosx - Sinx = 0

=> Tanx = 2  => Sinx = ±2/√5  & Cosx = ±1/√5

Max & min value of

2 sin x + cos x  = ± (2 * 2/√5 + 1/√5) = ± 5/√5 = ±√5  = ± 2.23

Hence real solution will Exist for  -π/2 & π/2  (±1.57) too

hence n = - 1 , 0  , 1

Sum = -1 + 0 + 1  =0