Before You Begin

Question #41164

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u} = \sin^{-1}(\sin 10)\hat{i} + \cos^{-1}(\cos 10)\hat{j} + \tan^{-1}(\tan 10)\hat{k}$ . If the vectors $\vec{u}$ , $\vec{v} = 2\hat{i} + \hat{j} + 3\hat{k}$ , and $\vec{w} = \hat{i} + \lambda\hat{j} + 2\hat{k}$ are coplanar, then the value of $\lambda$ is:

Options

A.
$\frac{\pi - 4}{3\pi - 10}$
B.
$\frac{\pi}{10 - 3\pi}$
C.
$\frac{4}{5}$
D.
$\frac{2}{5}$

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 Quadrant Adjustment Logic|Collinearity and Coplanarity|Coplanarity Condition

Why This Question Is Tricky

Common Mistakes

This question tests the 'Quadrant Adjustment Logic' intertwined with vector coplanarity.

The primary trap lies in the evaluation of the inverse trigonometric functions.

Students frequently assume $\sin^{-1}(\sin x) = x$ or incorrectly map the angle to the wrong principal branch, failing to recognize that 10 radians falls in the 4th quadrant relative to $3\pi$ for sine and tangent, but requires referencing $4\pi$ for cosine to stay within $[0, \pi]$ .

This leads to incorrect signs or offsets in the vector components (e.g., assuming all components are identical or merely differ by a sign).

When these flawed components are substituted into the scalar triple product determinant, it yields an incorrect value for $\lambda$ , perfectly matching the distractor options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Quadrant Adjustment Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free