Question:
The number of solutions of the equation \(cos \bigg(x+\frac{π}{3}\bigg)cos\bigg(\frac{π}{3-x}\bigg)=\frac{1}{4}cos^22x,x ∈ [-3π, 3π]\) is:
The number of solutions of the equation \(cos \bigg(x+\frac{π}{3}\bigg)cos\bigg(\frac{π}{3-x}\bigg)=\frac{1}{4}cos^22x,x ∈ [-3π, 3π]\) is:
Updated On: Jun 20, 2024
- 8
- 5
- 6
- 7
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The Correct Option is D
Solution and Explanation
\(cos \bigg(x+\frac{π}{3}\bigg)cos\bigg(\frac{π}{3-x}\bigg)=\frac{1}{4}cos^22x,x ∈ [-3π, 3π]\)
\(⇒ cos2x + cos \frac{2π}{3} = \frac{1}{2} cos^22x\)
\(⇒ cos^22x - 2cos^2x - 1 = 0\)
\(⇒ cos2x = 1\)
Therefore, x can have \(-3π, -2π, -π,0,π,2π,3π\) [Seven different results]
Hence, the correct option is (D): \(7\)
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Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
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