Question

If $cos(x+\pi /3)cosx=a$ has real solutions, then

• A. number of integral value of a are $3$
• B. sum of number of integral value of a is $0$
• C. when $a=1$ number of solutions for $xϵ[0,2\pi ]$ are 3
• D. when $a=1$, number of solutions for $xϵ[0,2\pi ]$ are 2

Answer 1

The correct option is B sum of number of integral value of a is $0$

$\cos (x+\frac{\pi }{3})\cos x=a$

$\Rightarrow (\cos x.\cos \frac{\pi }{3}-\sin x.\sin \frac{\pi }{3})\cos x=a$ ........ Using $\cos (a+b)=\cos a\cdot \cos b-\sin a\cdot sinb$

$\Rightarrow (\cos x-\sqrt{3}\sin x).\cos x=2a$

$\Rightarrow {\cos }^{2}x-\sqrt{3}\sin x.\cos x=2a$ .

$\Rightarrow \frac{1+\cos 2x}{2}-\frac{\sqrt{3}}{2}\sin 2x=2a$ ......... [$\because {\cos }^{2}x=\frac{1+\cos 2x}{2}$]

$\Rightarrow \cos 2x-\sqrt{3}\sin 2x=4a-1$

Thus equation is only solvable if,

$-2\le 4a-1\le 2$ ......... From concept of extrema of trigonametric function

$\Rightarrow a\in [-\frac{1}{4},\frac{3}{4}]$

Number of integral solution is 1 which is 0.

sum of number of integral value of a is 0.