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Question #71547

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the surface defined by the position vector $\vec{R}(u,v) = u\hat{i} + v\hat{j} + (u^2+v^2)\hat{k}$ . A family of curves $C$ is generated by projecting the intersections of $S$ with the family of planes $P_c: \vec{r} \cdot (\hat{i} - \frac{1}{2c}\hat{k}) = 0$ onto the $xy$ -plane, where $c > 0$ is a parameter.

Let $\Gamma$ be the family of orthogonal trajectories of $C$ in the $xy$ -plane. Let $\Gamma_0$ be the specific curve in $\Gamma$ that passes through the point $(7, 7)$ .

Evaluate the line integral $I = \frac{1}{4\pi} \oint_{\Gamma_0} \left( -y^2 dx + 2xy dy \right)$ where $\Gamma_0$ is traversed in the counter-clockwise direction.

Options

A.
0
B.
341
C.
343
D.
686

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

Orthogonal Trajectories|Geometric Interpretation of Cross Product|Line lying in a plane|Parametric form pitfalls

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 geometric interpretation of vector-defined surfaces and the derivation of orthogonal trajectories.

The primary trap lies in the 'Parametric form pitfall' and loss of degrees of freedom: students might attempt to parameterize the 3D intersection curve $\vec{r}(t)$ and find orthogonal vectors in 3D space before projecting, which conflates the parameter of the curve with the parameter of the family, leading to an incorrect differential equation.

Furthermore, a secondary trap is embedded in the line integral: if a student fails to find the orthogonal trajectory and mistakenly uses the original curve $C_0$ (centered at $(7,0)$ ), Green's theorem yields $\iint 4y dA = 0$ .

The value 341 is included as a distractor for arithmetic errors near the $7^3$ calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Orthogonal Trajectories
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free