Before You Begin

Question #67402

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(r)$ be the coefficient of $t^{\lfloor r \rfloor}$ in the Maclaurin series expansion of $\frac{1+t}{(1-t)^3}$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function. A function $\Delta(x)$ is defined by the determinant: $\Delta(x) = \begin{vmatrix} f(x) & f(x+\frac{1}{2}) & f(x+1) \\ f(x+\frac{1}{2}) & f(x+1) & f(x+\frac{3}{2}) \\ f(x+1) & f(x+\frac{3}{2}) & f(x+2) \end{vmatrix}$ Find the value of the integral $\int_0^{10} \Delta(x) \, dx$ .

Options

A.
1000
B.
1050
C.
950
D.
1100

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Row/Column Exchange Parity|Coefficient of x^r in Expansion|Integral Part and Fractional Part

Why This Question Is Tricky

Examiner Levers

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Common Mistakes

This question tests the synchronization of determinant row operations with the discrete nature of the greatest integer function.

A common lethal mistake is treating $f(x)$ as a continuous algebraic function $f(x) = (x+1)^2$ and applying row operations directly on the variable $x$ .

Doing so yields a constant determinant of $-1/8$ , leading to an incorrect integral of $-1.25$ .

The trap forces students to recognize that row operations fail to preserve the boundary conditions of the floor function.

The parity shift at $x = k + 1/2$ fundamentally alters the matrix structure, requiring the integral to be split into two halves for each integer interval to correctly capture the piecewise constant values.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Row/Column Exchange Parity
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free