Before You Begin

Question #41180

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be the line in 3D space given by the symmetric equations: $\frac{x-2}{2} = \frac{y-1}{-1} = \frac{z-1}{1}$ Let $S$ be the set of all points $(x,y,z)$ on the line $L$ that satisfy the equation: $\sin^{-1}\left( \frac{(x-2)^2 + (y-1)^2 + (z-1)^2}{6} - 2 \right) + \cos^{-1}(z-1) = \frac{\pi}{2}$ Find the sum of the coordinates of all points in the set $S$ .

Options

A.
2
B.
8
C.
10
D.
0

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

Domain Constraint Intersections|Parametric form pitfalls|Projection vs. Coordinate

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D parametric geometry with the strict domain constraints of inverse trigonometric functions.

The core trap lies in solving the algebraic equation $A = B$ derived from $\sin^{-1}(A) + \cos^{-1}(B) = \pi/2$ without verifying if the resulting parameter $t$ keeps the coordinates within the valid domain $[-1, 1]$ .

Students frequently find both $t=2$ and $t=-1$ , failing to reject $t=2$ , which leads to the incorrect inclusion of the extraneous point $(6, -1, 3)$ and a wrong coordinate sum of 10.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Domain Constraint Intersections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free