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Question #67698

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = |x-1|(x^2 - px + 2)$ , where $p \in \mathbb{R}$ . Let $p_0$ be the unique real value of $p$ for which $f(x)$ is differentiable at $x=1$ . Let $L$ be the line of intersection of the planes $P_1: x + y + z = 1$ and $P_2: x + 2y + p_0 z = 2$ . Let $S$ be the set of all real values of $k$ such that every point on the line $L$ satisfies the equation $(k - p_0)(x + 3y + k z) = k^2 - 5k + 6$ Find the sum of all elements in $S$ .

Options

A.
5
B.
8
C.
3
D.
6

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

System of Linear Equations (Cramer/Inverse)|Line lying in a plane|Cross Multiplication Trap|Right-Hand vs Left-Hand Derivatives

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 'Cross-Multiplication Trap' (Concept 62) integrated with system dependency and differentiability.

Students must first evaluate RHD and LHD to find $p_0=3$ .

The core trap lies in the equation $(k-3)(x+3y+kz) = (k-3)(k-2)$ .

Students commonly divide both sides by $(k-3)$ to find the plane condition, yielding $k=5$ .

However, by doing so without checking the zero-denominator condition, they implicitly assume $k \neq 3$ and lose the singular solution $k=3$ (where the equation degenerates to the identity $0=0$ , which trivially contains the line).

This forces a rigorous case-by-case analysis of parameters.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	System of Linear Equations (Cramer/Inverse)
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free