Before You Begin

Question #40426

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(\theta) = \begin{vmatrix} \cos\theta & \sin\theta & 1 \\ \sin\theta & 1 & \cos\theta \\ 1 & \cos\theta & \sin\theta \end{vmatrix}$ . Find the sum of all values of $\theta \in [0, 2\pi)$ such that $f(\theta) = 0$ .

Options

A.
$\frac{3\pi}{2}$
B.
$3\pi$
C.
$\frac{5\pi}{2}$
D.
$\pi$

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

Tangent Half-Angle Substitution|Factor Theorem in Determinants|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of trigonometric determinants and the domain pitfalls of the tangent half-angle substitution ( $t = \tan(\theta/2)$ ).

A common advanced technique is to convert trig determinants into rational functions of $t$ .

However, doing so maps $\theta = \pi$ to $t \to \infty$ .

The determinant simplifies to a rational function where the degree of the numerator (5) is less than the denominator (6), meaning it vanishes at infinity.

Students who only check the roots of the numerator polynomial will find $t = -1$ ( $\theta = 3\pi/2$ ) but completely miss the singularity at $\theta = \pi$ , leading to an incomplete solution sum.

Alternatively, recognizing the left-circulant matrix structure avoids this trap entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Tangent Half-Angle Substitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free