Question

The sum of all the solution of equation $\sin \pi x+\cos \pi x=0$ if $x\in [0,100],$ is

• A. $5025$
• B. $5050$
• C. $5020$
• D. $5250$

Answer 1

The correct option is A $5025$

$\sin \pi x+\cos \pi x=0$
$\Rightarrow \tan \left(\pi x\right)=-1=\tan (-\frac{\pi }{4})$
Thus general solution is,
$\pi x=n\pi -\frac{\pi }{4}$, where $n\in Z$
$\Rightarrow x=n-\frac{1}{4}$
Hence sum of solution in the given interval is
$=\sum _{k=1}^{100}k-\frac{1}{4}=\frac{100\cdot 101}{2}-\frac{1}{4}\cdot 100=5050-25=5025$