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NDA 2 2024 Maths, question 77

Topic: Differential calculus

Q77

If f(x)=[x]2−30[x]+221=0f(x)=[x]^{2}-30[x]+221=0, where [x][x] is the greatest integer function, then what is the sum of all integer solutions?

  1. (a) 1313
  2. (b) 1717
  3. (c) 2727
  4. (d) 3030Correct answer

Solution.

Answer: (d) 3030

  1. Let m=⌊x⌋.
  2. Then m²−30m+221=(m−13)(m−17)=0.
  3. Thus real solutions lie in [13,14) or [17,18).
  4. The integer solutions are exactly 13 and 17, whose sum is 30.