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Question #40570

Difficulty: 8/10 | 🟢 Free

The Question

Let $k$ be the number of circular permutations of 7 distinct objects such that 4 specified objects are always together, and out of these 4, two specific objects are never adjacent to each other. A variable plane $\Pi$ makes non-zero intercepts $a, b, c$ on the $x, y, z$ axes respectively, such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{k}$ . The plane $\Pi$ always passes through a fixed point $Q$ . Let $L_1$ be the line given by the equations $x - 2y = 3$ and $y + 2z = 1$ . A line $L_2$ passes through $Q$ and is parallel to $L_1$ . Find the shortest distance between the line $L_2$ and the $z$ -axis.

Options

A.
$\frac{72\sqrt{5}}{5}$
B.
$\frac{36\sqrt{5}}{5}$
C.
$72\sqrt{2}$
D.
$\frac{12\sqrt{5}}{5}$

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

Circular Permutations Constraint|Variable plane intercepts|Symmetric vs Non-symmetric form

Why This Question Is Tricky

Common Mistakes

This question fuses combinatorics with 3D geometry to create a multi-stage trap.

First, students must correctly calculate the circular permutations, avoiding the trap of applying circular arrangement rules to the internal linear block of the 4 grouped objects.

Second, they must deduce the fixed point Q from the intercept constraint.

The lethal trap lies in the non-symmetric form of the line L1.

Students frequently misidentify the direction ratios by directly reading the coefficients (e.g., assuming (1, -2, 0) or (0, 1, 2)) instead of computing the cross product of the normal vectors of the two planes.

This cascades into an incorrect shortest distance calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Circular Permutations Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free