Question

If \( a \in \mathbb{Z}^+ \), \([x]\) is the greatest integer not more than \( x \) and \[ \int_0^a \left\lfloor x \right\rfloor \, dx = 127, \text{ then } a = ? \]

• A. 6
• B. 7
• C. 8
• D. 9

Hint:

When dealing with integrals involving the greatest integer function, break the range into intervals where \( \left\lfloor x \right\rfloor \) is constant and compute the integral over each interval.

Answer 1

The Correct Option is C

We are given the integral involving the greatest integer function. By solving the integral and evaluating the sum for different values of \( a \), we find that the solution for \( a \) is 8. Thus, the correct answer is option (3).