Question

The sum of the roots of the equation $cos4x+6=7cos2x$ in the interval $\left[0,314\right]$ is $\lambda \pi ,$ then the numerical value of $\lambda$ is

• A. $4950$
• B. $2475$
• C. $9900$
• D. $4945$

Answer 1

Answer: (A)

$\left(2{\left(cos\right)}^{2}2x-1\right)+6=7cos2x$
On putting $cos2x=t$ , we get,
$2t^2-1+6=7t$
$2t^2-7t+5=0$
$\left(2t-5\right)\left(t-1\right)=0$
$t=\frac{5}{2},1$
$t=\frac{5}{2}$ (not possible)
$t=1\Rightarrow cos2x=1\Rightarrow 2x=2n\pi$
$\Rightarrow x=n\pi$
The roots in $\left[0,314\right]$ are
$\pi ,2\pi ,3\pi ,.....,99\pi$ $\left\{100\pi >314\right\}$
Sum of roots $=\pi +2\pi +3\pi +....+99\pi =4950\pi$
$\Rightarrow \lambda =4950$