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Question #74686

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(0, 2)$ and $B(4, 18)$ be two points in the $xy$ -plane. A point $P(x(\lambda), y(\lambda))$ divides the line segment $AB$ internally in the ratio $\lambda : 1$ ( $\lambda > 0$ ). Let $k$ be the non-zero value of $\lambda$ for which $P$ lies on the parabola $y = x^2 + x + 2$ . Evaluate the integral $I = \int_{0}^{k} \frac{x(\lambda)}{(\lambda+1)^2} \left[ y(\lambda) \right] d\lambda$ where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
$\frac{683}{64}$
B.
$\frac{685}{64}$
C.
$\frac{341}{32}$
D.
$\frac{683}{32}$

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

Section Formula Internals|Substitution (Change of Variable)|Greatest Integer Function (GIF)

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the integration of coordinate geometry (section formula) with calculus (GIF integration) via a substitution trap.

Students must first solve a recursive parameter condition to find the upper limit $k$ .

The core trap lies in recognizing that the section ratio $\lambda$ creates a rational function inside the GIF.

Instead of splitting the integral in the $\lambda$ -domain (which leads to complex algebraic boundaries and arithmetic errors), students must use the change of variable $u = y(\lambda)$ .

This elegantly maps the jump discontinuities of the GIF to integers, transforming a messy rational-GIF integral into a clean summation of piecewise polynomials.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Section Formula Internals
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free