If m and M are the minimum and the maximum values of 4 + 1 \over 2 sin2 2x - 2cos4 x, x \in R, then M - m is equal to :

Trigonometric Ratio and Identites PYQ — JEE Main 2016 · 9 Apr · Shift 1 · Q32 with Solution

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Correct answer: $\FRAC{9}{2}$

Let $f(x)=4+\frac12\sin^2 2x-2\cos 4x.$ We need the maximum value $M$ and minimum value $m$ of $f(x)$ , then compute $M-m$ .
Use the identity $\sin^2 2x=\frac{1-\cos 4x}{2}.$ So, $\frac12\sin^2 2x=\frac12\cdot \frac{1-\cos 4x}{2}=\frac{1-\cos 4x}{4}.$

Hence, $f(x)=4+\frac{1-\cos 4x}{4}-2\cos 4x.$

=\frac{17}{4}-\frac94\cos 4x.$$ 4. Since $$-1\le \cos 4x\le 1,$$ we get the range of $f(x)$ from $$f(x)=\frac{17}{4}-\frac94\cos 4x.$$ Because the coefficient of $\cos 4x$ is negative: - Maximum occurs when $\cos 4x=-1$ - Minimum occurs when $\cos 4x=1$ So, $$M=\frac{17}{4}-\frac94(-1)=\frac{17}{4}+\frac94=\frac{26}{4}=\frac{13}{2},$$ $$m=\frac{17}{4}-\frac94(1)=\frac{8}{4}=2.$$ 5. Therefore, $$M-m=\frac{13}{2}-2=\frac{13}{2}-\frac{4}{2}=\frac{9}{2}.$$ 6. Compare with the options: $$\frac{9}{2}$$ is not among the given options. So the stored answer $\text{B} = \frac94$ is not correct.

View written solutionFree

Correct answer: $\FRAC{9}{2}$

Let $f(x)=4+\frac12\sin^2 2x-2\cos 4x.$ We need the maximum value $M$ and minimum value $m$ of $f(x)$ , then compute $M-m$ .
Use the identity $\sin^2 2x=\frac{1-\cos 4x}{2}.$ So, $\frac12\sin^2 2x=\frac12\cdot \frac{1-\cos 4x}{2}=\frac{1-\cos 4x}{4}.$

Hence, $f(x)=4+\frac{1-\cos 4x}{4}-2\cos 4x.$

=\frac{17}{4}-\frac94\cos 4x.$$ 4. Since $$-1\le \cos 4x\le 1,$$ we get the range of $f(x)$ from $$f(x)=\frac{17}{4}-\frac94\cos 4x.$$ Because the coefficient of $\cos 4x$ is negative: - Maximum occurs when $\cos 4x=-1$ - Minimum occurs when $\cos 4x=1$ So, $$M=\frac{17}{4}-\frac94(-1)=\frac{17}{4}+\frac94=\frac{26}{4}=\frac{13}{2},$$ $$m=\frac{17}{4}-\frac94(1)=\frac{8}{4}=2.$$ 5. Therefore, $$M-m=\frac{13}{2}-2=\frac{13}{2}-\frac{4}{2}=\frac{9}{2}.$$ 6. Compare with the options: $$\frac{9}{2}$$ is not among the given options. So the stored answer $\text{B} = \frac94$ is not correct.