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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

If the sum of the roots of the equation 4 x+6=7 2 x in the interval...

If the sum of the roots of the equation

\cos 4 x+6=7 \cos 2 x

in the interval

[0,314]

is

k \pi, k \in R

. Find k .

Correct answer

4950

Step-by-step solution

\begin{aligned} & \cos 4 x+6=7 \cos 2 x \\ & 2 \cos ^2 2 x-1+6=7 \cos 2 x \\ & 2 \cos ^2 2 x-7 \cos 2 x+5=0 \\ & 2 \cos ^2 2 x-2 \cos 2 x-5 \cos 2 x+5=0 \\ & 2 \cos 2 x(\cos 2 x-1)-5(\cos 2 x-1)=0 \\ & (\cos 2 x-1)(2 \cos 2 x-5)=0\end{aligned}

\begin{aligned} & \cos 2 x=\frac{5}{2} \quad(\text { not possible }) \\ & \cos 2 x=1\end{aligned}

\begin{aligned} & 2 x=2 n \pi \\ & \Rightarrow \quad x=n \pi\end{aligned}

sum in

[0,314]

is

\pi[1+2+3+\ldots \ldots+100]-100 \pi=5050 \pi-100 \pi=4950 \pi \Rightarrow k=4950

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