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Question #41752

Difficulty: 8/10 | 🟢 Free

The Question

Consider two planes $P_1$ and $P_2$ given by the equations: $P_1: (k-5)x + \frac{1}{k-2}y + (k-5)z = 1$ $P_2: (k-5)x + \frac{k-4}{k-3}y + (5-k)z = 2$ where $k \in \mathbb{R}$ . Let $S$ be the set of all integers $k \in [0, 10]$ for which the planes $P_1$ and $P_2$ intersect at a strictly acute angle (i.e., $0 < \theta < \frac{\pi}{2}$ ). Find the sum of all elements in $S$ .

Options

A.
40
B.
45
C.
49
D.
54

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Angle between two planes|Linear Independence vs Dependence|Wavy Curve (Method of Intervals)|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of algebraic inequalities and geometric boundary conditions.

The primary trap lies in the definition of a 'strictly acute angle' ( $0 < \theta < \pi/2$ ).

Students typically use the wavy curve method on the dot product $\vec{n_1} \cdot \vec{n_2} > 0$ to find $k \in (2, 3) \cup (4, \infty)$ and blindly sum the integers in this range, yielding 45.

However, they must verify Linear Independence.

At $k=5$ , the $x$ and $z$ components of the normal vectors vanish, making the vectors parallel ( $\theta = 0$ ).

Failing to exclude $k=5$ triggers the main distractor.

The denominator restrictions naturally exclude $k=2$ and $k=3$ , acting as a secondary filter.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Angle between two planes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free