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Question #75712

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, a] \to [0, b]$ be a strictly convex, continuously differentiable bijective function with $f(0)=0$ . Let $I = \int_0^a f(x) \, dx$ . For a positive integer $n$ , let $x_k = \frac{ka}{n}$ for $k=0, 1, \dots, n$ . Consider the position vectors $\vec{v}_k = \lambda x_k \hat{i} + \mu f(x_k) \hat{j}$ , where $\lambda, \mu > 0$ . If $L = \lim_{n \to \infty} \sum_{k=1}^n |\vec{v}_{k-1} \times \vec{v}_k|$ , which of the following expressions is equal to $L$ ?

Options

A.
$\lambda \mu (ab - 2I)$
B.
$\frac{1}{2} \lambda \mu (ab - 2I)$
C.
$\lambda \mu (ab - I)$
D.
$\lambda \mu (2ab - I)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Scalar Multiplication Geometric Scaling|Integral as the Limit of a Sum|Axis of Integration Selection

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the conceptual understanding of Riemann sums as measures of area under different axes, combined with vector geometry scaling.

The trap lies in the vector cross product generating two distinct Riemann sums: one along the x-axis ( $\sum f \Delta x$ ) and one along the y-axis ( $\sum x \Delta f$ ).

Students often fail to recognize $\lim \sum x \Delta f$ as the integral of the inverse function $\int f^{-1}(y) dy$ (or equivalently, integrating by parts $\int x f'(x) dx$ ), leading to incorrect evaluations.

Additionally, a common lethal mistake is assuming the sum represents the area of the triangles ( $\frac{1}{2}|\vec{a} \times \vec{b}|$ ), leading to an erroneous factor of $1/2$ in the final answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Multiplication Geometric Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free