Question

If $a^2-2a\cos x+1=674$ and $\tan \left(\frac{x}{2}\right)=7$ then the integral value of a is

• A. $25$
• B. $49$
• C. $67$
• D. $74$

Answer 1

The correct option is A $25$

$\cos \left(x\right)=\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$.
Hence
$\cos \left(x\right)=\frac{1-49}{1+49}$

$=\frac{-48}{50}$

$=\frac{-24}{25}$.
Hence
$a^2-2a.\left(\frac{-24}{25}\right)+1=674$
Or
$a^2+\frac{48a}{25}+1=674$
Or
$25a^2+48a+=673\times 25$

$(a-25)(25a+673)=0$
Hence integral value of $a=25.$