Start Learning Free

Before You Begin

Question #72192

Difficulty: 8/10 | 🟢 Free

The Question

Let r⃗=xi^+yj^\vec{r} = x\hat{i} + y\hat{j} be the position vector of a point. A family of curves F\mathcal{F} in the xyxy-plane is defined by the differential equation ∣r⃗×dr⃗∣=3(r⃗⋅dr⃗)|\vec{r} \times d\vec{r}| = \sqrt{3} (\vec{r} \cdot d\vec{r}) Let F⊥\mathcal{F}_\perp be the family of orthogonal trajectories to F\mathcal{F}. If a curve C∈F⊥C \in \mathcal{F}_\perp passes through the point (x,y)=(1,0)(x, y) = (1, 0) and intersects the ray θ=π23\theta = \frac{\pi}{2\sqrt{3}} at a distance RR from the origin, then the sum of all possible values of RR is:

Options

  • A.

    eπ/2+e−π/2e^{\pi/2} + e^{-\pi/2}

  • B.

    eπ/2e^{\pi/2}

  • C.

    eπ/6+e−π/6e^{\pi/6} + e^{-\pi/6}

  • D.

    e−π/2e^{-\pi/2}

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

Implicit to Explicit Conversion|Geometric Interpretation of Cross Product|Geometrical Interpretation of Modulus

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 the cross product and the 'modulus trap' in differential equations.

Students often compute the 2D cross product r⃗×dr⃗\vec{r} \times d\vec{r} as the scalar (xdy−ydx)(x dy - y dx) and drop the absolute value, solving only xdy−ydx=3(xdx+ydy)x dy - y dx = \sqrt{3}(x dx + y dy).

This yields only one branch of the differential equation (drdθ=r3\frac{dr}{d\theta} = \frac{r}{\sqrt{3}}), leading to a single orthogonal trajectory and an incomplete sum.

The question enforces the geometric constraint that the magnitude of the cross product is r2∣dθ∣r^2|d\theta|, requiring both positive and negative cases to find the complete family of implicit curves.

Quick Facts About This Question

Question Information

SubjectMathematics
ChapterDifferential Equations - Class 12
ConceptImplicit to Explicit Conversion
Difficulty8
Cognitive Load7
Target Solve Time170 sec
AccessFree