Let f :[0,2] rightarrow R be the function defined by f(x)=(3-sin (2 π x)) sin (π x-π/4 )-sin (3 π x+π/4 ) If α, β in[0,2] are such that \ x in[0,2]: f(x) geq 0\ =[α, β] , then the value of β-α is \ \ \ \

Mathematics · Trigonometry Ratios and Identities

JEE Advanced 2020 — Paper 1 — Question 42

Let $\mathrm{f}:[0,2] \rightarrow \mathrm{R}$ be the function defined by

f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)

If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$ , then the value of $\beta-\alpha$ is $\_\_\_\_$

Numerical answer — enter this value.

$(3-\sin 2 \pi x) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)$

$=(3-\sin 2 \pi x) \sin \left(\pi x-\frac{\pi}{4}\right)+\sin \left(3 \pi x-\frac{3 \pi}{4}\right)$

$=\left(3-\sin 2 \pi x+3-4 \sin ^{2}\left(\pi x-\frac{\pi}{4}\right)\right) \sin \left(\pi x-\frac{\pi}{4}\right)$

$\Rightarrow \sin \left(\pi x-\frac{\pi}{4}\right) \geq 0$

$\Rightarrow \pi \geq \pi x-\frac{\pi}{4} \geq 0 \quad(\because x \in[0,2])$

$\Rightarrow \frac{5}{4} \geq x \geq \frac{1}{4} \Rightarrow \beta-\alpha=1$

Answer key and solution verified before publishing.

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