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Question #41730

Difficulty: 8/10 | 🟢 Free

The Question

Let $t_0 \in (\pi, 2\pi)$ be the unique solution to the equation $\sin^{-1}(\sin t) + \cos^{-1}(\cos t) = \frac{\pi}{2}$ . A line $L$ passes through the point $P(2, -1, 3)$ and is parallel to the vector $\vec{v} = \sin^{-1}(\sin t_0) \hat{i} + \cos^{-1}(\cos t_0) \hat{j} + \tan^{-1}(\tan t_0) \hat{k}$ . If the line $L$ intersects the plane $x + y + z = 10$ at the point $Q$ , then the value of $PQ^2$ is:

Options

A.
44
B.
12
C.
108
D.
56

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

Parametric form pitfalls|The Sum-Difference Formula Constraints|Principal Value Branch Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify the principal value branches of inverse trigonometric functions.

A common lethal mistake is assuming $f^{-1}(f(x)) = x$ universally, which leads to the incorrect direction vector $\vec{v} \propto \langle 1, 1, 1 \rangle$ and a 'ghost' intersection point yielding $PQ^2 = 12$ .

By forcing the parameter $t_0$ into the third quadrant, the problem demands rigorous application of domain constraints for $\arcsin$ , $\arccos$ , and $\arctan$ to find the true parametric direction of the line.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free