Question:Let r and s be the solutions of \(12\mathrm{x}^2 + (3\mathrm{a} + 4\mathrm{b})\mathrm{x} + \frac{\mathrm{ab}}{3} = 0\), where a and...
GMAT Advanced Math : (Adv_Math) Questions
Let \(\mathrm{r}\) and \(\mathrm{s}\) be the solutions of \(12\mathrm{x}^2 + (3\mathrm{a} + 4\mathrm{b})\mathrm{x} + \frac{\mathrm{ab}}{3} = 0\), where \(\mathrm{a}\) and \(\mathrm{b}\) are positive constants. The product of the solutions can be written as \(\mathrm{k} \cdot \mathrm{ab}\), where \(\mathrm{k}\) is a constant. What is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given equation: \(12\mathrm{x}^2 + (3\mathrm{a} + 4\mathrm{b})\mathrm{x} + \frac{\mathrm{ab}}{3} = 0\)
- Need to find: The constant k where product of solutions = k·ab
- Key insight: This is a quadratic in standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\)
2. INFER the approach
- Since we need the product of solutions, Vieta's formulas are the direct path
- For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), the product of roots equals \(\frac{\mathrm{c}}{\mathrm{a}}\)
- We can identify: leading coefficient = 12, constant term = \(\frac{\mathrm{ab}}{3}\)
3. SIMPLIFY the product calculation
- Product of solutions = (constant term)/(leading coefficient)
- Product = \(\frac{\mathrm{ab}/3}{12}\)
- SIMPLIFY: \(\frac{\mathrm{ab}/3}{12} = \frac{\mathrm{ab}}{3 \times 12} = \frac{\mathrm{ab}}{36}\)
4. INFER the final relationship
- We're told the product can be written as k·ab
- Setting equal: \(\mathrm{k} \cdot \mathrm{ab} = \frac{\mathrm{ab}}{36}\)
- Therefore: \(\mathrm{k} = \frac{1}{36}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misidentify the coefficients, especially confusing the middle term coefficient (3a + 4b) with the constant term, or not recognizing that \(\frac{\mathrm{ab}}{3}\) is the constant term.
This confusion about which term is which can lead them to incorrectly apply Vieta's formulas, potentially getting \(\mathrm{k} = \frac{1}{12}\) and selecting Choice C.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the need for Vieta's formulas but make calculation errors when simplifying \(\frac{\mathrm{ab}/3}{12}\).
They might incorrectly simplify this as \(\frac{\mathrm{ab}}{15}\) or \(\frac{\mathrm{ab}}{9}\), leading them to select Choice D (1/9) or get confused and guess.
The Bottom Line:
This problem tests whether students can quickly recognize a Vieta's formulas situation and correctly handle fraction arithmetic, but the main challenge is careful identification of coefficients in a quadratic with parameters.