WMA11/01 Oct 2023, Question 6: Lines Meeting Curves: the Discriminant
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks
The question, in short
Given a rational equation in x with constant p that has two distinct real roots, derive a quadratic inequality in p and then solve it algebraically to find the possible p values.
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Part (a) · 3 marks
Rearrange the equation into a quadratic in x, then use the condition for two distinct real roots to show that 3p² − 10p − 8 > 0.
- 1Multiply by (x + p): 4(p − 2x)(x + p) = 12 + 15p
- 24(px + p² − 2x² − 2px) = 12 + 15p, so −8x² − 4px + 4p² = 12 + 15p
- 38x² + 4px + 12 + 15p − 4p² = 0
- 4Two distinct real roots: b² − 4ac > 0, so (4p)² − 4 × 8 × (12 + 15p − 4p²) > 0
- 516p² − 384 − 480p + 128p² > 0, so 144p² − 480p − 384 > 0
- 6Divide by 48: 3p² − 10p − 8 > 0
AnswerShown: 8x² + 4px + 12 + 15p − 4p² = 0; b² − 4ac = 16p² − 32(12 + 15p − 4p²) > 0 ⇒ 144p² − 480p − 384 > 0 ⇒ 3p² − 10p − 8 > 0
Part (b) · 3 marks
Solve the inequality 3p² − 10p − 8 > 0 by factorising or otherwise, stating the resulting range of p values.
- 1Critical values: 3p² − 10p − 8 = (3p + 2)(p − 4) = 0
- 2p = −2/3 or p = 4
- 3Positive quadratic is > 0 outside the roots: p < −2/3 or p > 4
Answerp < −2/3 or p > 4
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