Can u solve it in 10 sec

Algebra Level 3

If $a,b\epsilon(0,2)$ and the equation $\frac{x^{2}+5}{2}=x-2cos(ax+b)$ has atleast one solution then $a+b=$

\pi

\frac{\pi}{2}

e

1 solution

Tanishq Varshney
Feb 12, 2015

Just put $x=1$

wrong option selected.

We can make conclusion from this.

$\dfrac{x^2 - 2x + 5}{2} = -2\cos (ax + b)$

$(x - 1)^2 + 4 = -4\cos (ax + b)$

$(x - 1)^2 = -4( 1 + \cos (ax + b)$

$(x - 1)^2 = -8 \cos^2 \dfrac{ax + b}{2}$

Thus for $x \in \mathbb R$ , the only posibble solution is $x =1 , \implies RHS = 0 \implies \dfrac{a + b}{2} = \dfrac{\pi}{2}$

U Z - 6 years, 4 months ago

$\frac{a+b}{2}=\frac{\pi}{2}$

so $a+b=\pi$

Tanishq Varshney - 6 years, 4 months ago

That mistake only I did while solving , and too writing it , thanks edited

U Z - 6 years, 4 months ago

Actually the answer can be any odd multiple of pi.

Raghav Vaidyanathan - 6 years, 3 months ago