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Q.
The sum of the roots of the equation $cos 4 x+6=7cos 2 x$ in the interval $\left[0,314\right]$ is $\lambda \pi ,$ then the numerical value of $\lambda $ is
NTA AbhyasNTA Abhyas 2020
Solution:
$\left(2 \left(cos\right)^{2} 2 x - 1\right)+6=7cos 2 x$
On putting $cos 2 x=t$ , we get,
$2t^{2}-1+6=7t$
$2t^{2}-7t+5=0$
$\left(2 t - 5\right)\left(t - 1\right)=0$
$t=\frac{5}{2},1$
$t=\frac{5}{2}$ (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 $
$\Rightarrow \lambda =4950$