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Question #40724

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{r} = x\hat{i} + y\hat{j}$ be the position vector of a point $P$ moving on a closed curve $C$ in the $xy$ -plane. Let $\vec{v}_1 = x\hat{i} + y\hat{j} + \hat{k}$ and $\vec{v}_2 = y\hat{i} - x\hat{j} + \hat{k}$ . The curve $C$ is defined by the locus of $P$ such that $|\vec{v}_1 \times \vec{v}_2 + (x^2+y^2)\hat{k}| = 2\sqrt{2}$ . Let $f(P) = [\alpha] + [\beta] + [\gamma]$ , where $\alpha, \beta, \gamma$ are the scalar components of $\vec{v}_1 \times \vec{v}_2$ along $\hat{i}, \hat{j}, \hat{k}$ respectively, and $[ \cdot ]$ denotes the greatest integer function. Find the number of points on the curve $C$ where $f(P)$ is discontinuous.

Options

A.
12
B.
14
C.
16
D.
18

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

Locus of a Moving Point|Geometric Interpretation of Cross Product|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of vector locus geometry and calculus-based discontinuity analysis.

The primary trap lies in the Greatest Integer Function (GIF) applied to parametrically varying vector components.

Students often correctly identify the locus as a circle but fail to properly analyze the simultaneous discontinuities of $[2\sqrt{2}\sin\phi]$ and $[2\sqrt{2}\cos\phi]$ .

A common mistake is assuming that at points where both inner functions are integers, the discontinuities might 'cancel out' to make the function continuous.

However, careful left-hand and right-hand limit analysis reveals that these points remain discontinuous (either jump or removable), leading to exactly 16 points of discontinuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Locus of a Moving Point
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free