4PM1/01 Jun 2024, Question 10: When a Line Meets, Touches or Misses a Curve

Part (a) · 2 marks

Use the x-intercept (5/4, 0) and the vertical asymptote x = 3 to determine the integer values of a and b.

1. 1y = 0 when ax − 5 = 0, x = 5/4, so a = 4
2. 2Vertical asymptote x = 3, so b = 3

Answera = 4, b = 3

Part (b) · 5 marks

Draw C, clearly labelling both asymptotes with equations and the coordinates where C meets the coordinate axes.

1. 1Vertical asymptote x = 3
2. 2Horizontal asymptote: y → −4 as x → ±∞
3. 3x = 0 gives y = −5/3
4. 4y = 0 gives x = 5/4
5. 5Left branch passes through (0, −5/3) and (5/4, 0); right branch lies below y = −4

AnswerTwo-branch curve y = (4x − 5)/(3 − x) with asymptotes x = 3 and y = −4; crosses the axes at (0, −5/3) and (5/4, 0); left branch rises from y = −4 through the intercepts towards x = 3, right branch below y = −4

Part (c) · 9 marks

Using algebra, show that no intersection with 4y − 7x = k occurs precisely for m < k < n, and find integers m and n.

1. 1dy/dx = [(3 − x)(4) − (4x − 5)(−1)]/(3 − x)² = 7/(3 − x)²
2. 2Gradient of l = 7/4
3. 37/(3 − x)² = 7/4 gives x = 1 or x = 5
4. 4Points (1, −1/2) and (5, −15/2)
5. 5Tangent lines: 4y − 7x = −9 and 4y − 7x = −65
6. 6No intersection for −65 < k < −9

Answer−65 < k < −9 (m = −65, n = −9)

More past-paper questions on when a line meets, touches or misses a curve

• 4PM1/01 Jan 2023 Q5 · 8 marks
• 4PM1/02R Jan 2020 Q11 · 14 marks
• 0606/11 Oct/Nov 2025 Q2 · 6 marks