Before You Begin

Question #40775

Difficulty: 8/10 | 🟢 Free

The Question

For each $n \in \mathbb{N}$ , let $\Delta_n$ be the area of the triangle with vertices $O(0,0)$ , $A_n\left( \sum_{k=1}^n \frac{k}{n^2}, 0 \right)$ , and $B_n\left( 0, \sum_{k=1}^n \frac{32k^3}{n^4} \right)$ . Let $\Gamma_n$ be the area of the triangle with vertices $O(0,0)$ , $C_n\left( \sum_{k=1}^n \frac{2(n-k)}{n^2}, 0 \right)$ , and $D_n\left( 0, \sum_{k=1}^n \frac{12(n-k)^2}{n^3} \right)$ . Evaluate $\lim_{n \to \infty} ([\Delta_n] + [\Gamma_n])$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
2
B.
3
C.
4
D.
5

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

Area of Triangle via Determinants|Integral as the Limit of a Sum|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous understanding of the integral as the limit of a Riemann sum and its interaction with step function discontinuities.

A common mistake is to commute the limit and the greatest integer function, incorrectly assuming $\lim [\Delta_n] = [\lim \Delta_n]$ .

Students who do this will evaluate both area limits as 2 and choose the trap answer 4.

The 'lethal collision' occurs because the coordinates are defined via Riemann sums of strictly monotonic functions.

By recognizing that right Riemann sums overestimate increasing functions and underestimate decreasing functions, students must deduce that $\Delta_n$ approaches 2 from above ( $[\Delta_n]=2$ ) while $\Gamma_n$ approaches 2 from below ( $[\Gamma_n]=1$ ), yielding the correct sum of 3.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Area of Triangle via Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free