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Question #59922

Difficulty: 8/10 | 🟢 Free

The Question

Let $V(x_1, x_2, x_3)$ be the variance of the real numbers $x_1, x_2, x_3$ . Let $S$ be the set of all points $(x_1, x_2, x_3) \in \mathbb{R}^3$ such that $\sum_{i=1}^3 \sin x_i = 0$ and $\sum_{i=1}^3 \cos x_i = 0$ . Let $M$ be the minimum value of $V$ on $S$ . At any point in $S$ where $V$ attains this minimum $M$ , the gradient of $V$ can be expressed as $\nabla V = \lambda_1 \nabla \left(\sum_{i=1}^3 \sin x_i\right) + \lambda_2 \nabla \left(\sum_{i=1}^3 \cos x_i\right)$ . The value of $\frac{M}{\lambda_1^2 + \lambda_2^2}$ is:

Options

A.
$\frac{9}{8}$
B.
$\frac{8}{9}$
C.
$\frac{3}{4}$
D.
$\frac{4}{3}$

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

Variance Invariance under Translation|Objective Function Gradient|Conditional Identity Constraints

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 problem tests constrained optimization intersecting with trigonometric periodicity.

The primary trap is treating the trigonometric constraints purely algebraically or ignoring the periodicity, which leads students to incorrectly assume the minimum variance is zero.

Furthermore, students must recognize that variance is invariant under translation, allowing them to shift the variables by their mean to simplify the gradient equations.

Failing to apply this shift makes the Lagrange multiplier system computationally intractable.

The question elegantly combines Class 11 statistics (variance shift) with Class 12 advanced calculus (objective function gradients).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free