JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

\int_{0}^{5} \cos \left( \pi \left( x - \left[ \frac{x}{2} \right] \right) \right) dx

Choose the correct answer:

Explanation

Solution

We break the integral based on the values of the Greatest Integer Function $\left[ \frac{x}{2} \right]$ over the interval $[0, 5]$ :

For $x \in [0, 2)$ : $\left[ \frac{x}{2} \right] = 0$ . Integral $= \int_{0}^{2} \cos(\pi x) dx = \left[ \frac{\sin(\pi x)}{\pi} \right]_{0}^{2} = 0$ .
For $x \in [2, 4)$ : $\left[ \frac{x}{2} \right] = 1$ . Integral $= \int_{2}^{4} \cos(\pi(x - 1)) dx = \int_{2}^{4} -\cos(\pi x) dx = -\left[ \frac{\sin(\pi x)}{\pi} \right]_{2}^{4} = 0$ .
For $x \in [4, 5]$ : $\left[ \frac{x}{2} \right] = 2$ . Integral $= \int_{4}^{5} \cos(\pi(x - 2)) dx = \int_{4}^{5} \cos(\pi x) dx = \left[ \frac{\sin(\pi x)}{\pi} \right]_{4}^{5} = 0$ .

Summing these up: $0 + 0 + 0 = 0$ .

Correct Option: (D)

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