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NDA 1 2024 Maths, question 94

Topic: Integral calculus

Passage · Q93 to Q94

Let f(x)=∣x−1∣f(x)=|x-1|, g(x)=[x]g(x)=[x] and h(x)=f(x)g(x)h(x)=f(x)g(x) where [⋅][\cdot ] is greatest integer function.

Q94

What is ∫02h(x)dx\int _{0}^{2}h(x) dx equal to ?

  1. (a) −32-\frac{3}{2}
  2. (b) −1-1
  3. (c) 00
  4. (d) 12\frac{1}{2}Correct answer

Solution.

Answer: (d) 12\frac{1}{2}

  1. On 0 < x < 1, floor(x) = 0 and h(x) = 0.
  2. On 1 < x < 2, floor(x) = 1 and h(x) = x − 1.
  3. The integral is therefore ∫ from 1 to 2 of (x − 1) dx = 1/2.