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Question #67442

Difficulty: 8/10 | 🟢 Free

The Question

Consider the following system of linear equations: $(2\cos\theta) x + y + z = 1$ $x + (2\cos\theta) y + z = 2\cos\theta$ $x + y + (2\cos\theta) z = 4\cos^2\theta$ Let $S$ be the set of all values of $\theta \in [0, 3\pi]$ for which the system has infinitely many solutions, and $T$ be the set of all values of $\theta \in [0, 3\pi]$ for which the system has no solution. Find the value of $\sum_{\theta \in S} \theta + \sum_{\theta \in T} \theta$ .

Options

A.
$\frac{25\pi}{3}$
B.
$4\pi$
C.
$\frac{13\pi}{3}$
D.
$\frac{22\pi}{3}$

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

Elementary Row Operations|Cramer's Rule Constraints|Even Multiplicity Roots|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Examiner Levers

Boundary Condition

Tests behavior at limits like 0, 1, infinity, or empty sets.

Difficulty Boost : +4

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' in Cramer's Rule and matrix row reduction.

The determinant $\Delta = (k-1)^2(k+2)$ has an even multiplicity root at $k=1$ .

When calculating $x = \Delta_x / \Delta$ , students often impulsively cancel the common factor $(k-1)^2$ from the numerator and denominator, yielding $x = -(k+1)/(k+2)$ .

This false algebraic reduction leads them to believe that at $k=1$ , the system has a unique solution $(-2/3, 1/3, 4/3)$ .

If they plug this point back into the equations, it works, reinforcing their false belief and causing them to completely miss the infinite solutions case (Set S).

Only by analyzing $\Delta = 0$ prior to division can the true nature of the singular matrices be revealed.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Elementary Row Operations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free