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Question #41636

Difficulty: 8/10 | 🟢 Free

The Question

A line $L$ passes through the origin $O$ and has direction ratios $(1, -1, \sqrt{2})$ . A point $P(x, y, z)$ lies on the line $L$ at a distance $r$ ( $r > 0$ ) from $O$ . The point $P$ satisfies the condition $x \sin\left(\frac{\pi r}{2}\right) + y \cos\left(\frac{\pi r}{2}\right) + z \ge \frac{r}{\sqrt{2}}$ and its distance $r$ satisfies $\frac{r^2 - 11r + 28}{r - 3} \le 0$ . Find the sum of all possible integral values of $r$ .

Options

A.
14
B.
3
C.
25
D.
11

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

Parametric form pitfalls|Direction Cosines constraint|Wavy Curve (Method of Intervals)|General Solution Principal Values

Why This Question Is Tricky

Common Mistakes

The student uses the parametric form of a line to express coordinates in terms of the distance parameter $r$ .

The primary pitfall arises when solving the resulting trigonometric inequality: students often restrict themselves to the principal values (e.g., $k=0$ ), ignoring the general solution and the periodic nature of the constraint.

Furthermore, combining this with the Wavy Curve method requires careful intersection of discrete valid integers from disjoint intervals.

Failing to account for the periodicity leads to missing valid segments of the line (such as $r \in \{5, 6\}$ ), resulting in an incomplete set of points and an incorrect sum.

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