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← JEE Main 2022 · 27 Jul · Shift 1
MathematicsSingle correct+4 / -1Question 7 of 90

JEE Main 2022 · 27 Jul · Shift 1, Mathematics question 7

Let f:R→Rf:\mathbf R\to\mathbf R be a function defined as

f(x)=asin⁡(π[x]2)+[2−x], a∈R,f(x)=a\sin\left(\frac{\pi[x]}2\right)+[2-x],\ a\in\mathbf R,

where [t][t] is the greatest integer less than or equal to tt. If lim⁡x→−1f(x)\displaystyle\lim_{x\to-1}f(x) exists, then the value of ∫04f(x) dx\displaystyle\int_0^4f(x)\,dx is equal to

Choose one option
Answer and worked solution

Official answer: B

Limit at −1-1. From the left, [x]=−2[x]=-2 and [2−x]=3[2-x]=3: value 3. From the right, [x]=−1[x]=-1 and [2−x]=2[2-x]=2: value 2−a2-a. So a=−1a=-1.

Integrate piece by piece on (0,1),(1,2),(2,3),(3,4)(0,1),(1,2),(2,3),(3,4): f=1,−1,−1,−1f=1,-1,-1,-1.

∫04f=−2\int_0^4f=-2. (B)

Topics:Properties of definite integralsLimits
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