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Question #66008

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha, \beta, \gamma$ be the real roots of the equation $2x^3 - 15x^2 - 14x + 48 = 0$ , with $\alpha > \beta > \gamma$ . Let $\Delta = \begin{vmatrix} \tan^{-1}\alpha & \tan^{-1}\beta & \tan^{-1}\gamma \\ \tan^{-1}(1/\alpha) & \tan^{-1}(1/\beta) & \tan^{-1}(1/\gamma) \\ 1 & 1 & 1 \end{vmatrix}$ Find the value of $\cot\left(\frac{\Delta}{\pi}\right)$ .

Options

A.
1
B.
2
C.
3
D.
4

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

The Sum-Difference Formula Constraints|Factor Theorem in Determinants|Triangular/Diagonal Matrix Shortcut

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 question tests the student's ability to recognize the branch-cut trap in the inverse trigonometric identity $\tan^{-1}x + \tan^{-1}(1/x)$ .

Students often incorrectly assume this sum is always $\pi/2$ for all real $x$ , which would make the first row proportional to the third row, leading to the erroneous conclusion that $\Delta = 0$ .

By ensuring exactly one root of the cubic is negative, the identity yields $-\pi/2$ for that specific column, breaking the proportionality and requiring a careful determinant expansion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Sum-Difference Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free