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Question #40558

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all planes intersecting the coordinate axes at points $A(a, 0, 0)$ , $B(0, b, 0)$ , and $C(0, 0, c)$ such that $a, b, c \in \\{1, 2, 3, \\dots, 30\\}$ . Let $L_k$ be the line passing through the point $(k, k, k)$ with direction ratios $(1, -2, 1)$ . Find the number of planes in $S$ that contain the line $L_k$ for some integer $k \\ge 1$ .

Options

A.
30
B.
74
C.
45
D.
28

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

Fundamental Principle of Counting (FPC)|Line lying in a plane|Line-Plane intersection logic|Variable plane intercepts

Why This Question Is Tricky

Common Mistakes

This question bridges 3D geometry (line-plane intersection) with discrete counting (FPC and number theory).

The primary trap lies in the Diophantine equation $(2a-3k)(2c-3k) = 9k^2$ .

Students often mechanically apply the divisor function $d(N)$ to find the number of solutions, which yields a total of 74.

However, the constraint $a, c \\le 30$ (variable plane intercepts bounded by a set) actively truncates the valid solution space.

Failing to manually verify the upper bound for each divisor pair leads to massive overcounting.

This tests the student's ability to transition from continuous coordinate conditions to bounded discrete combinatorics without losing track of the initial domain constraints.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Fundamental Principle of Counting (FPC)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free