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Question #41296

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B, C$ be the angles of a triangle such that the determinant $\begin{vmatrix} \sec^2 A & \tan A & 1 \\ \sec^2 B & \tan B & 1 \\ \sec^2 C & \tan C & 1 \end{vmatrix} = 0$ and $\sin(A-B) = \cos C$ . Let $S$ be the sum of all possible values of $A$ . Find $S$ .

Options

A.
$\frac{9\pi}{8}$
B.
$\frac{15\pi}{8}$
C.
$\frac{11\pi}{8}$
D.
$\frac{3\pi}{4}$

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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

Factor Theorem in Determinants|Conditional Identity Constraints|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Factor Theorem in determinants with domain constraints of trigonometric functions (Quadrant Adjustment Logic).

Students easily reduce the determinant to a product of differences of tangents, concluding the triangle is isosceles.

However, the presence of $\sec^2 \theta$ and $\tan \theta$ implicitly restricts the domain to $\theta \neq \pi/2$ .

When solving the conditional identity $\sin(A-B) = \cos C$ , students find families of solutions that include right-angled triangles (e.g., $A=\pi/2$ or $C=\pi/2$ ).

Failing to reject these extraneous solutions—where the determinant is mathematically undefined—leads to the lethal trap sum of $15\pi/8$ instead of the correct $9\pi/8$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Factor Theorem in Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free