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Question #72133

Difficulty: 8/10 | 🟢 Free

The Question

Let $a_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1+x)^{14}$ . Consider the family of lines $L_r$ defined for $r \in \{1, 2, \dots, 13\}$ by the symmetric equations: $L_r : x - r = \frac{y - r^2}{a_r} = \frac{z - r^3}{a_r}$ Let $S$ be the set of all unordered pairs of distinct lines $\{L_i, L_j\}$ chosen from this family. Find the number of pairs in $S$ that are skew lines.

Options

A.
78
B.
72
C.
66
D.
84

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

Shortest distance between skew lines|Coefficient of x^r in Expansion|Complementary Counting

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the complete definition of skew lines (non-intersecting AND non-parallel) using a parameterized family of lines.

The mathematical trap lies in the intersection check: students will easily prove that no two distinct lines in this family intersect (since $i^3-i^2 = j^3-j^2 \implies i=j$ ).

A common mistake is to immediately conclude that all $\binom{13}{2} = 78$ pairs are skew.

However, they must perform a domain check for parallelism.

The direction ratios depend on binomial coefficients, which are symmetric ( $a_i = a_{14-i}$ ).

Complementary counting is required to subtract the 6 parallel pairs, yielding 72.

This forces validation of both conditions of skewness.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Shortest distance between skew lines
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free