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Question #41679

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $P(x,y,z)$ in $\mathbb{R}^3$ such that the distance of $P$ from the origin $O$ is $2\sqrt{3}$ , and the sum of the direction cosines of the position vector $\vec{OP}$ is $1$ . Let $\vec{v} = 3\hat{i} + 3\hat{j}$ . If $M$ and $m$ are the absolute maximum and minimum values of the dot product $\vec{OP} \cdot \vec{v}$ for all $P \in S$ , then the value of $M^2 + m^2$ is equal to:

Options

A.
192
B.
144
C.
216
D.
256

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

Objective Function Gradient|Locus via Distance Constraints|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests optimization over a 1D manifold (a circle) embedded in 3D space, defined by distance and direction cosine constraints.

The primary trap lies in the dimensionality failure: students often attempt to maximize the objective function by aligning the position vector directly with the gradient of the objective function (the vector $\vec{v} = 3\hat{i} + 3\hat{j}$ ).

However, because the feasible region is constrained to a specific plane, the gradient must be projected onto this constraint plane ( $x+y+z=0$ ) to find the true direction of maximum increase along the locus.

Failing to project the gradient vector leads to incorrect bounds, as it ignores the geometric reality that $\vec{OP}$ cannot freely point in the direction of $\vec{v}$ without violating the direction cosines sum constraint.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Objective Function Gradient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free