Question

The number of values of $x$ satisfying the equation $3\cos \left(2x\right)-10\cos \left(x\right)+7=0$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D

$4$

The explanation for the correct answer.

Solve for the value of $x$

$3\cos \left(2x\right)-10\cos \left(x\right)+7=0$

$\Rightarrow$ $3(2{\cos }^{2}x–1)–10\cos x+7=0$

$\Rightarrow$ $3{\cos }^{2}x–5\cos x+2=0$

$\Rightarrow$$3{\cos }^{2}x–3\cos x–2\cos x+2=0$

$\Rightarrow$$3\cos x(\cos x–1)–2(\cos x–1)=0$

$\Rightarrow$$(3\cos x–2)(\cos x–1)=0$

$\Rightarrow$$\cos x=\frac{2}{3},\cos x=1$

As we know,$\cos x$ is positive in quadrants $I$and $IV.$

The figure shows the solutions of $\cos x=\frac{2}{3},\cos x=1$respectively in the region $[0,2\pi ]$

Hence, the number of values of $x$in $[0,2\pi ]$ is $4.$

Hence, option (D) is the correct answer.