Question

The number of ordered pairs $\left(\alpha ,\beta \right)$ , where $\alpha ,\beta \in \left(-\pi ,\pi \right)$ satisfying $\cos \left(\alpha -\beta \right)=1$ and $\cos \left(\alpha +\beta \right)=\frac{1}{e}$ is

• A. 0
• B. 1
• C. 2
• D. 4

Answer 1

Answer: (D)

4

$\cos (a+b)=\frac{1}{e}\cos (a-b)=1-\pi \le (a,b)\le \pi$

Graph of $\cos x$ -(i)

$\cos (a-b)=1\therefore a-b=0(-\pi \le a,b\le \pi )a=b\cos \left(2a\right)=\frac{1}{e}(-\pi \le a\le \pi ,-2\pi \le 2a\le 2\pi )$

So graph of $\cos 2a$ (ii)

$\therefore$ There are $4$ ordered pairs.