Question Simulator

O Level Additional Mathematics (4037)

Simultaneous equations Rare

2 questions · about 1% of the marks we indexed · examined in 2 of 9 sessions.

Questions by year

2025

  • 2025 Oct/Nov · Paper 22 · Q1 (a)4 marksLinear and non-linear simultaneous equations

2022

  • 2022 Oct/Nov · Paper 22 · Q15 marksLinear and non-linear simultaneous equations

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Linear and non-linear simultaneous equations · Core · Paper 1 style · 4 marks
  1. (a) Solve the simultaneous equations y = x + 1 and x² + y² = 25. [4]
Show answer
  1. (a) x = 3, y = 4 and x = −4, y = −3
    x² + (x + 1)² = 25 M1; x² + x − 12 = 0 A1; x = 3 or −4 M1 (factorise and substitute back); y = 4, −3 A1
Q2Linear and non-linear simultaneous equations · Core · Paper 1 style · 4 marks
  1. (a) Solve the simultaneous equations x + y = 7 and xy = 12. [4]
Show answer
  1. (a) x = 3, y = 4 and x = 4, y = 3
    x(7 − x) = 12 M1; x² − 7x + 12 = 0 A1; x = 3 or 4 A1; corresponding y A1
Q3Linear and non-linear simultaneous equations · Core · Paper 2 style · 4 marks
  1. (a) Find the coordinates of the points where the line y = 2x − 3 meets the curve y = x² − 4x + 5. [4]
Show answer
  1. (a) (2, 1) and (4, 5)
    x² − 4x + 5 = 2x − 3 M1; x² − 6x + 8 = 0 A1; x = 2, 4 A1; y = 1, 5 A1
Q4Linear and non-linear simultaneous equations · Core · Paper 1 style · 4 marks
  1. (a) Solve the simultaneous equations 2x + y = 5 and x² + y² = 10. [4]
Show answer
  1. (a) x = 1, y = 3 and x = 3, y = −1
    y = 5 − 2x, x² + (5 − 2x)² = 10 M1; 5x² − 20x + 15 = 0 A1; x = 1 or 3 A1; y = 3, −1 A1
Q5Linear and non-linear simultaneous equations · Core · Paper 2 style · 4 marks
  1. (a) Solve the simultaneous equations x − y = 2 and x² − 2y² = 7. [4]
Show answer
  1. (a) x = 3, y = 1 and x = 5, y = 3
    (y + 2)² − 2y² = 7 M1; y² − 4y + 3 = 0 A1; y = 1 or 3 A1; x = 3, 5 A1
Q6Linear and non-linear simultaneous equations · Core · Paper 1 style · 4 marks
  1. (a) Two positive numbers differ by 3 and their product is 40. Find the two numbers. [4]
Show answer
  1. (a) 8 and 5
    x − y = 3 and xy = 40 B1; y(y + 3) = 40 M1; y² + 3y − 40 = 0 A1; (y + 8)(y − 5) = 0, y = 5 (reject −8), x = 8 A1
Q7Linear and non-linear simultaneous equations · Core · Paper 1 style · 3 marks
  1. (a) Solve the simultaneous equations 3x + 2y = 12 and 5x − 3y = 1. [3]
Show answer
  1. (a) x = 2, y = 3
    eliminate y: 9x + 6y = 36 and 10x − 6y = 2 M1; 19x = 38, x = 2 A1; y = 3 A1
Q8Linear and non-linear simultaneous equations · Core · Paper 2 style · 4 marks
  1. (a) A cinema in Karachi sells adult tickets at Rs 800 and child tickets at Rs 500. For one show, 45 tickets were sold for a total of Rs 30 000. Find the number of adult tickets and the number of child tickets sold. [4]
Show answer
  1. (a) 25 adult tickets and 20 child tickets
    a + c = 45 B1; 800a + 500c = 30 000 B1; 300a = 7500 M1; a = 25, c = 20 A1
Q9Linear and non-linear simultaneous equations · Core · Paper 1 style · 3 marks
  1. (a) Solve the simultaneous equations 4x − 3y = 18 and 2x + 5y = −4. [3]
Show answer
  1. (a) x = 3, y = −2
    4x + 10y = −8 M1; subtracting, −13y = 26 so y = −2 A1; x = 3 A1
Q10Linear and non-linear simultaneous equations · Standard · Paper 2 style · 5 marks
  1. (a) The line y = 3 − x meets the curve x² + xy + y² = 7 at the points A and B. Find the coordinates of A and B. [4]
  2. (b) Find the exact length of AB. [1]
Show answer
  1. (a) (1, 2) and (2, 1)
    x² + x(3 − x) + (3 − x)² = 7 M1; x² − 3x + 2 = 0 A1; x = 1, 2 A1; y = 2, 1 A1
  2. (b) √2
    √(1² + 1²) = √2 B1
Q11Linear and non-linear simultaneous equations · Standard · Paper 1 style · 7 marks
  1. (a) The line 2y = x + 4 meets the curve xy = 6 at the points P and Q. Find the coordinates of P and Q. [4]
  2. (b) Find the coordinates of the midpoint of PQ and the length of PQ in the form k√5. [3]
Show answer
  1. (a) (2, 3) and (−6, −1)
    x(x + 4)/2 = 6 M1; x² + 4x − 12 = 0 A1; x = 2 or −6 A1; y = 3, −1 A1
  2. (b) midpoint (−2, 1); PQ = 4√5
    midpoint B1; √(8² + 4²) = √80 M1; 4√5 A1
Q12Linear and non-linear simultaneous equations · Standard · Paper 2 style · 5 marks
  1. (a) A rectangular photo frame has a perimeter of 46 cm and a diagonal of length 17 cm. Taking the sides as x cm and y cm, write down two equations and solve them to find the dimensions of the frame. [5]
Show answer
  1. (a) 15 cm by 8 cm
    x + y = 23 B1; x² + y² = 289 B1; x² + (23 − x)² = 289 M1; x² − 23x + 120 = 0 A1; x = 15 or 8, so 15 cm by 8 cm A1
Q13Linear and non-linear simultaneous equations · Standard · Paper 2 style · 5 marks
  1. (a) A farmer in Multan fences a rectangular plot of area 1200 m² against a straight wall. The wall forms one long side, and the other three sides use 100 m of fencing. Find the two possible pairs of dimensions of the plot. [5]
Show answer
  1. (a) width 20 m, length 60 m or width 30 m, length 40 m
    2w + l = 100 B1; wl = 1200 B1; w(100 − 2w) = 1200 M1; w² − 50w + 600 = 0 A1; w = 20, l = 60 or w = 30, l = 40 A1
Q14Linear and non-linear simultaneous equations · Standard · Paper 1 style · 5 marks
  1. (a) Solve the simultaneous equations xy = 12 and x² + y² = 25. [5]
Show answer
  1. (a) (3, 4), (4, 3), (−3, −4), (−4, −3)
    y = 12/x, x² + 144/x² = 25 M1; x⁴ − 25x² + 144 = 0 A1; x² = 9 or 16 M1; x = ±3, ±4 A1; matching y values A1
Q15Linear and non-linear simultaneous equations · Standard · Paper 1 style · 4 marks
  1. (a) Find the exact coordinates of the points where the line x + y = 5 meets the curve y = x² − x − 3. [4]
Show answer
  1. (a) (2√2, 5 − 2√2) and (−2√2, 5 + 2√2)
    x² − x − 3 = 5 − x M1; x² = 8 A1; x = ±2√2 A1; y = 5 ∓ 2√2 A1
Q16Linear and non-linear simultaneous equations · Standard · Paper 2 style · 6 marks
  1. (a) The line y = x − 4 meets the circle x² + y² − 6x + 4y = 12 at the points C and D. Find the coordinates of C and D. [4]
  2. (b) Find the length of the chord CD in the form k√2. [2]
Show answer
  1. (a) (6, 2) and (−1, −5)
    x² + (x − 4)² − 6x + 4(x − 4) = 12 M1; x² − 5x − 6 = 0 A1; x = 6, −1 A1; y = 2, −5 A1
  2. (b) 7√2
    √(7² + 7²) M1; 7√2 A1
Q17Linear and non-linear simultaneous equations · Standard · Paper 1 style · 4 marks
  1. (a) Solve the simultaneous equations 4x² − y² = 39 and 2x + y = 13. [4]
Show answer
  1. (a) x = 4, y = 5
    (2x − y)(2x + y) = 39 M1; 2x − y = 3 A1; adding: 4x = 16 M1; x = 4, y = 5 A1
Q18Linear and non-linear simultaneous equations · Standard · Paper 2 style · 5 marks
  1. (a) Two square plots of land in a village have a total area of 1300 m². The side of the larger plot is 10 m longer than the side of the smaller plot. Find the length of the side of each plot. [5]
Show answer
  1. (a) 30 m and 20 m
    x² + y² = 1300 and x − y = 10 B1 B1; (y + 10)² + y² = 1300 M1; y² + 10y − 600 = 0 A1; y = 20 (reject −30), x = 30 A1
Q19Linear and non-linear simultaneous equations · Standard · Paper 2 style · 4 marks
  1. (a) Find the coordinates of the points of intersection of the line y = 5 − 2x and the curve y = 3/x. [4]
Show answer
  1. (a) (1, 3) and (1.5, 2)
    x(5 − 2x) = 3 M1; 2x² − 5x + 3 = 0 A1; x = 1, 1.5 A1; y = 3, 2 A1
Q20Linear and non-linear simultaneous equations · Standard · Paper 2 style · 4 marks
  1. (a) In a market model for school bags, the price p rupees and the quantity q thousand bags satisfy p = 1700 − 20q² (demand) and p = 200 + 200q (supply). Find the equilibrium quantity and price. [4]
Show answer
  1. (a) q = 5 (5000 bags), p = Rs 1200
    1700 − 20q² = 200 + 200q M1; q² + 10q − 75 = 0 A1; (q − 5)(q + 15) = 0, q = 5 (reject −15) A1; p = 1200 A1
Q21Linear and non-linear simultaneous equations · Challenge · Paper 2 style · 5 marks
  1. (a) Solve the simultaneous equations lg x + lg y = 2 and x − y = 15. [5]
Show answer
  1. (a) x = 20, y = 5
    lg(xy) = 2 so xy = 100 M1 A1; y(y + 15) = 100 M1; y² + 15y − 100 = 0, y = 5 or −20 A1; reject −20 (log undefined), x = 20 A1
Q22Linear and non-linear simultaneous equations · Challenge · Paper 1 style · 5 marks
  1. (a) Solve the simultaneous equations 2ˣ × 2ʸ = 32 and 3^(xy) = 729. [5]
Show answer
  1. (a) x = 2, y = 3 and x = 3, y = 2
    x + y = 5 B1; xy = 6 B1; x(5 − x) = 6 M1; x² − 5x + 6 = 0, x = 2 or 3 A1; y = 3, 2 A1
Q23Linear and non-linear simultaneous equations · Challenge · Paper 1 style · 5 marks
  1. (a) Solve the simultaneous equations log₂ x + log₂ y = 5 and x + y = 12. [5]
Show answer
  1. (a) x = 4, y = 8 and x = 8, y = 4
    log₂(xy) = 5 M1; xy = 32 A1; x(12 − x) = 32 M1; x² − 12x + 32 = 0, x = 4 or 8 A1; y = 8, 4 A1
Q24Linear and non-linear simultaneous equations · Challenge · Paper 2 style · 7 marks
  1. (a) The line y = 2x + 1 meets the curve y² = 8x + 1 at the points A and B. Find the coordinates of A and B. [4]
  2. (b) Find the equation of the perpendicular bisector of AB, giving your answer in the form ax + by = c, where a, b and c are integers. [3]
Show answer
  1. (a) (0, 1) and (1, 3)
    (2x + 1)² = 8x + 1 M1; 4x² − 4x = 0 A1; x = 0, 1 A1; y = 1, 3 A1
  2. (b) 2x + 4y = 9
    midpoint (1/2, 2) B1; gradient −1/2 M1; y − 2 = −½(x − ½), i.e. 2x + 4y = 9 A1
Q25Linear and non-linear simultaneous equations · Challenge · Paper 1 style · 4 marks
  1. (a) Solve the simultaneous equations y = x − 1 and x² + 2xy − y² = 7. [4]
Show answer
  1. (a) x = 2, y = 1 and x = −2, y = −3
    x² + 2x(x − 1) − (x − 1)² = 7 M1; 2x² − 1 = 7 A1; x = ±2 A1; y = 1, −3 A1
Q26Linear and non-linear simultaneous equations · Challenge · Paper 2 style · 6 marks
  1. (a) A cyclist rides 60 km at a constant speed of v km/h, taking t hours. If her speed were 5 km/h greater, the journey would take 1 hour less. Write down two equations in v and t and solve them to find v and t. [6]
Show answer
  1. (a) v = 15 km/h, t = 4 hours
    vt = 60 B1; (v + 5)(t − 1) = 60 B1; −v + 5t − 5 = 0 M1; t(5t − 5) = 60 M1; t² − t − 12 = 0, t = 4 (reject −3) A1; v = 15 A1
Q27Linear and non-linear simultaneous equations · Challenge · Paper 1 style · 5 marks
  1. (a) Solve the simultaneous equations 1/x + 1/y = 9/20 and x + y = 9. [5]
Show answer
  1. (a) x = 4, y = 5 and x = 5, y = 4
    (x + y)/(xy) = 9/20 M1; xy = 20 A1; x(9 − x) = 20 M1; x² − 9x + 20 = 0, x = 4 or 5 A1; y = 5, 4 A1
Q28Linear and non-linear simultaneous equations · Challenge · Paper 1 style · 5 marks
  1. (a) Solve the simultaneous equations eˣ + eʸ = 5 and x + y = ln 6, giving your answers in exact form. [5]
Show answer
  1. (a) x = ln 2, y = ln 3 and x = ln 3, y = ln 2
    eˣeʸ = 6 M1 A1; let u = eˣ: u + 6/u = 5, u² − 5u + 6 = 0 M1; u = 2 or 3 A1; x = ln 2, y = ln 3 or x = ln 3, y = ln 2 A1
Q29Linear and non-linear simultaneous equations · Challenge · Paper 2 style · 6 marks
  1. (a) The line 2x − y = 4 meets the curve y² = 4x at the points A and B. Find the coordinates of A and B. [4]
  2. (b) Find the area of triangle OAB, where O is the origin. [2]
Show answer
  1. (a) (4, 4) and (1, −2)
    x = (y + 4)/2 so y² = 2(y + 4) M1; y² − 2y − 8 = 0 A1; y = 4, −2 A1; x = 4, 1 A1
  2. (b) 6
    ½|4 × (−2) − 4 × 1| M1; 6 A1
Q30Linear and non-linear simultaneous equations · Challenge · Paper 2 style · 5 marks
  1. (a) Solve the simultaneous equations log₃ x − log₃ y = 1 and x² − y² = 72. [5]
Show answer
  1. (a) x = 9, y = 3
    log₃(x/y) = 1 M1; x = 3y A1; 9y² − y² = 72 M1; y = 3 (reject −3 as logs undefined) A1; x = 9 A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Linear and non-linear simultaneous equations Rare2 questions · 2/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Linear and non-linear simultaneous equations Rare92/9

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