Before You Begin

Question #40390

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left( \left[ \left( \frac{180 k}{n} \right)^\circ \right] + \left[ \left( \frac{360 k}{n} \right)^\circ \right] \right)$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
$9 - \frac{33}{2\pi}$
B.
269
C.
$9 - \frac{11}{\pi}$
D.
$3\pi - \frac{33}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Radian-Degree Conversion Identity|Integral as the Limit of a Sum|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual understanding of Riemann sums combined with the degree-to-radian conversion inside a discontinuous step function.

Students commonly treat the degree symbol as a mere label or linear scalar, incorrectly evaluating the integral of $[180x] + [360x]$ to get 269.

The trap forces the realization that $x^\circ = \pi x / 180$ radians, which shifts the step function's jump discontinuities to irrational multiples of $\pi$ .

This misalignment destroys standard integer-based telescoping summation, requiring precise piecewise integration over irrational boundary points.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Radian-Degree Conversion Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free