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Question #40731

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be a directed line in the $xy$ -plane passing through the origin, having a slope of $-\frac{5}{12}$ and directed such that its direction vector has a positive $x$ -component. Let $\vec{v} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ . Let $p$ be the algebraic projection of $\vec{v}$ onto the directed line $L$ . If $\vec{w}$ is the reflection of the vector $\vec{v}$ in the plane passing through the origin and perpendicular to $L$ , find the value of $\vec{w} \cdot \vec{v} + 5p^3$ .

Options

A.
177
B.
257
C.
193
D.
289

Check Your Answer

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

Slope Sensitivity|Dot Product Projection Sign|Geometric Interpretation of Cross Product

Why This Question Is Tricky

Common Mistakes

This question tests the algebraic sign of the scalar projection of a vector onto a directed line.

The trap lies in the geometric interpretation: students often calculate the projection magnitude using the Pythagorean relation with the cross product ( $p = \sqrt{|\vec{v}|^2 - |\vec{v} \times \hat{u}|^2}$ ) or simply ignore the obtuse angle between the vector and the line's direction, incorrectly assuming $p = 2$ .

This sign inversion error propagates into the cubic term $5p^3$ , leading directly to the distractor 257.

Furthermore, it tests the vector reflection formula across a plane, requiring the correct identification of the plane's normal as the line's direction vector.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Slope Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free