Question Simulator

O Level Mathematics D (4024)

Coordinate geometry Banker

31 questions · about 4% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 12 · Q15 (a(i), a(ii), b)5 marksEquation of a line · Perpendicular & parallel lines
  • 2026 May/June · Paper 13 · Q194 marksPerpendicular & parallel lines
  • 2026 May/June · Paper 22 · Q5 (a, b, c)4 marksLength midpoint & gradient
  • 2026 May/June · Paper 23 · Q7 (a, b)3 marksLength midpoint & gradient

2025

  • 2025 Oct/Nov · Paper 12 · Q8 (a, b, c)5 marksLength midpoint & gradient
  • 2025 Oct/Nov · Paper 21 · Q11 (a(ii))2 marksLength midpoint & gradient
  • 2025 Oct/Nov · Paper 21 · Q16 (a, b)4 marksEquation of a line · Perpendicular & parallel lines
  • 2025 Oct/Nov · Paper 22 · Q18 (a, b(i), b(ii))9 marksEquation of a line · Perpendicular & parallel lines
  • 2025 May/June · Paper 12 · Q14 (c(ii))3 marksLength midpoint & gradient
  • 2025 May/June · Paper 12 · Q17 (a, b)5 marksEquation of a line · Perpendicular & parallel lines
  • 2025 May/June · Paper 13 · Q6 (a, b(i), b(ii))4 marksEquation of a line · Straight-line graphs
  • 2025 May/June · Paper 13 · Q214 marksPerpendicular & parallel lines

2024

  • 2024 Oct/Nov · Paper 11 · Q11 (a, b)3 marksLength midpoint & gradient
  • 2024 Oct/Nov · Paper 12 · Q24 (a, b)5 marksLength midpoint & gradient · Perpendicular & parallel lines
  • 2024 Oct/Nov · Paper 21 · Q3 (d)2 marksStraight-line graphs
  • 2024 Oct/Nov · Paper 21 · Q7 (d)4 marksPerpendicular & parallel lines
  • 2024 May/June · Paper 11 · Q23 (a, b, c)7 marksOfficial paper ↗Length midpoint & gradient · Perpendicular & parallel lines
  • 2024 May/June · Paper 22 · Q11 (a, b, c)10 marksLength midpoint & gradient · Perpendicular & parallel lines

2023

  • 2023 Oct/Nov · Paper 21 · Q2 (a(i))1 markStraight-line graphs
  • 2023 Oct/Nov · Paper 21 · Q3 (a(i), a(ii), b(i), b(ii))7 marksEquation of a line · Length midpoint & gradient
  • 2023 Oct/Nov · Paper 22 · Q7 (a(i), b)6 marksLength midpoint & gradient · Perpendicular & parallel lines
  • 2023 May/June · Paper 11 · Q26 (a, b)5 marksLength midpoint & gradient · Perpendicular & parallel lines
  • 2023 May/June · Paper 12 · Q245 marksPerpendicular & parallel lines
  • 2023 May/June · Paper 22 · Q4 (d(i), d(ii))2 marksStraight-line graphs

2022

  • 2022 Oct/Nov · Paper 11 · Q16 (a)1 markLength midpoint & gradient
  • 2022 Oct/Nov · Paper 12 · Q15 (a)1 markEquation of a line
  • 2022 Oct/Nov · Paper 12 · Q16 (a, b(i), b(ii))4 marksLength midpoint & gradient · Perpendicular & parallel lines
  • 2022 Oct/Nov · Paper 21 · Q6 (b)4 marksLength midpoint & gradient
  • 2022 May/June · Paper 21 · Q11 (a, b)7 marksLength midpoint & gradient · Perpendicular & parallel lines
  • 2022 May/June · Paper 22 · Q6 (a)2 marksStraight-line graphs
  • 2022 May/June · Paper 22 · Q10 (a, b(i), b(ii))12 marksEquation of a line · Length midpoint & gradient · Perpendicular & parallel lines

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.

Q1Perpendicular & parallel lines · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Write down the gradient of a line parallel to y = 3x − 2. [1]
  2. (b) Find the equation of the line parallel to y = 3x − 2 that passes through (1, 7). [2]
  3. (c) Write down the gradient of a line perpendicular to y = 3x − 2. [1]
Show answer
  1. (a) 3
    B1.
  2. (b) y = 3x + 4
    7 = 3 + c M1; y = 3x + 4 A1.
  3. (c) −1/3
    B1.
Q2Perpendicular & parallel lines · Core · Paper 1 style · no calculator · 4 marks

Four lines have these equations.L₁: y = 4x + 1L₂: 2y = 8x − 3L₃: y = −¼x + 5L₄: 4y = x + 2

  1. (a) Which two lines are parallel? Give a reason. [2]
  2. (b) Which line is perpendicular to L₁? Give a reason. [2]
Show answer
  1. (a) L₁ and L₂: both have gradient 4
    L₂ is y = 4x − 1.5 M1; A1
  2. (b) L₃: its gradient is −¼ and 4 × (−¼) = −1
    M1 A1
Q3Perpendicular & parallel lines · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Write down the gradient of the line 3x + y = 7. [1]
  2. (b) Find the equation of the line parallel to 3x + y = 7 that passes through (−2, 3). [2]
Show answer
  1. (a) −3
    y = −3x + 7 B1
  2. (b) y = −3x − 3
    3 = −3(−2) + c M1; c = −3 A1
Q4Perpendicular & parallel lines · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Write down the gradient of a line perpendicular to y = (2/5)x − 1. [1]
  2. (b) Find the equation of the line perpendicular to y = (2/5)x − 1 that passes through (0, 4). [1]
  3. (c) Find the coordinates of the point where this line crosses the x-axis. [2]
Show answer
  1. (a) −5/2
    B1
  2. (b) y = −(5/2)x + 4
    B1
  3. (c) (8/5, 0)
    0 = −2.5x + 4 M1; x = 1.6 A1
Q5Perpendicular & parallel lines · Standard · Paper 1 style · no calculator · 6 marks

Line L has equation 2x + 5y = 10.

  1. (a) Find the gradient of L. [2]
  2. (b) Find the equation of the line perpendicular to L that passes through (4, 1). Give your answer in the form y = mx + c. [3]
  3. (c) Show that the point (6, 6) lies on your line from part (b). [1]
Show answer
  1. (a) −2/5
    y = −2/5 x + 2 M1; −2/5 A1.
  2. (b) y = 5/2 x − 9
    Gradient 5/2 M1; 1 = 10 + c M1; y = 5/2 x − 9 A1.
  3. (c) 5/2 × 6 − 9 = 6
    B1.
Q6Perpendicular & parallel lines · Standard · Paper 1 style · no calculator · 5 marks

A is the point (−1, 6) and B is the point (5, 2).

  1. (a) Find the midpoint of AB. [1]
  2. (b) Find the gradient of AB. [1]
  3. (c) Find the equation of the perpendicular bisector of AB. Give your answer in the form ax + by = c. [3]
Show answer
  1. (a) (2, 4)
    B1
  2. (b) −2/3
    B1
  3. (c) 3x − 2y = −2 (or 2y = 3x + 2)
    Perpendicular gradient 3/2 M1; y − 4 = (3/2)(x − 2) M1; A1
Q7Perpendicular & parallel lines · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) The line through (2, k) and (6, 1) is perpendicular to the line y = 2x + 3. Find k. [3]
  2. (b) Find the equation of this line. [2]
Show answer
  1. (a) k = 3
    Gradient must be −½ B1; (1 − k)/4 = −½ M1; k = 3 A1
  2. (b) y = −½x + 4
    1 = −½(6) + c M1; c = 4 A1
Q8Perpendicular & parallel lines · Standard · Paper 1 style · no calculator · 7 marks

A is (1, 1), B is (5, 3) and C is (3, 7).

  1. (a) Find the gradients of AB and BC, and explain what they show. [3]
  2. (b) Find the area of triangle ABC. [2]
  3. (c) A circle passes through A, B and C. Find the coordinates of its centre. Give a reason. [2]
Show answer
  1. (a) AB: ½, BC: −2. ½ × (−2) = −1, so AB is perpendicular to BC (angle ABC = 90°).
    B1 B1; product −1 B1
  2. (b) 10 units²
    AB = BC = √20 M1; ½ × √20 × √20 A1
  3. (c) (2, 4)
    AC is a diameter (angle in a semicircle) B1; midpoint of AC B1
Q9Perpendicular & parallel lines · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) The line through A(3, −1) and B(7, k) is parallel to the line 2y − 3x = 5. Find k. [3]
  2. (b) Find the equation of AB in the form ax + by = c. [2]
Show answer
  1. (a) k = 5
    Gradient 3/2 B1; (k + 1)/4 = 3/2 M1; A1
  2. (b) 3x − 2y = 11
    y + 1 = (3/2)(x − 3) M1; A1
Q10Perpendicular & parallel lines · Challenge · Paper 1 style · no calculator · 6 marks

A is (−2, 3), B is (4, −1) and C is (6, 2).

  1. (a) Show that AB is perpendicular to BC. [2]
  2. (b) ABCD is a rectangle. Find the coordinates of D. [2]
  3. (c) Find the area of the rectangle ABCD. [2]
Show answer
  1. (a) Gradients −2/3 and 3/2; product −1
    Gradient AB = −4/6 = −2/3 and gradient BC = 3/2 M1; product = −1 A1.
  2. (b) (0, 6)
    D = A + (C − B) = (−2 + 2, 3 + 3) M1; (0, 6) A1.
  3. (c) 26 square units
    AB = √52, BC = √13 M1; √676 = 26 A1.
Q11Perpendicular & parallel lines · Challenge · Paper 1 style · no calculator · 6 marks

P is the point (7, 1) and L is the line y = 2x − 3.

  1. (a) Find the equation of the line through P that is perpendicular to L. [2]
  2. (b) Find the coordinates of the foot of the perpendicular from P to L. [2]
  3. (c) Find the shortest distance from P to L, giving your answer as a surd in its simplest form. [2]
Show answer
  1. (a) y = −½x + 9/2
    Gradient −½ M1; 1 = −½(7) + c, c = 4.5 A1
  2. (b) (3, 3)
    2x − 3 = −½x + 4.5 M1; x = 3, y = 3 A1
  3. (c) 2√5
    √(4² + 2²) = √20 M1; 2√5 A1
Q12Perpendicular & parallel lines · Challenge · Paper 1 style · no calculator · 6 marks

ABCD is a parallelogram. A is (−2, 1), B is (4, 3) and C is (6, 8).

  1. (a) Find the coordinates of D. [2]
  2. (b) Show that ABCD is not a rectangle. [2]
  3. (c) Find the equation of the line through D that is perpendicular to AB. [2]
Show answer
  1. (a) (0, 6)
    D = A + C − B M1 A1
  2. (b) Gradient AB = 1/3, gradient BC = 5/2; product 5/6 ≠ −1, so AB is not perpendicular to BC.
    Both gradients M1; conclusion A1
  3. (c) y = −3x + 6
    Gradient −3 M1; A1
Q13Perpendicular & parallel lines · Challenge · Paper 2 style · calculator · 8 marks

O is the origin. The line PQ has equation 3x + 4y = 24.

  1. (a) Find the coordinates of P and Q, where the line meets the x-axis and y-axis. [2]
  2. (b) Find the equation of the line through O perpendicular to PQ. [2]
  3. (c) Find the coordinates of the point N where this line meets PQ. [2]
  4. (d) Find ON and check your answer using the area of triangle OPQ. [2]
Show answer
  1. (a) P(8, 0), Q(0, 6)
    B1 B1
  2. (b) y = (4/3)x
    Gradient PQ = −3/4 M1; A1
  3. (c) (2.88, 3.84)
    3x + 4(4x/3) = 24 M1; A1
  4. (d) 4.8 units; area ½ × 8 × 6 = 24 = ½ × 10 × 4.8
    √(2.88² + 3.84²) M1; check with PQ = 10 A1
Q14Length midpoint & gradient · Core · Paper 1 style · no calculator · 4 marks

P is the point (1, 2) and Q is the point (7, 10).

  1. (a) Find the midpoint of PQ. [1]
  2. (b) Find the gradient of PQ. [1]
  3. (c) Find the length of PQ. [2]
Show answer
  1. (a) (4, 6)
    B1.
  2. (b) 4/3
    8/6 = 4/3 B1.
  3. (c) 10 units
    √(6² + 8²) M1; 10 A1.
Q15Length midpoint & gradient · Core · Paper 1 style · no calculator · 4 marks

A is the point (−4, −1) and B is the point (2, 7).

  1. (a) Find the midpoint of AB. [1]
  2. (b) Find the gradient of AB. [1]
  3. (c) Find the length of AB. [2]
Show answer
  1. (a) (−1, 3)
    B1
  2. (b) 4/3
    8/6 B1
  3. (c) 10
    √(36 + 64) M1 A1
Q16Length midpoint & gradient · Core · Paper 2 style · calculator · 5 marks

C is the point (1.5, −2.4) and D is the point (6.3, 4.1).

  1. (a) Find the length of CD, correct to 3 significant figures. [2]
  2. (b) Find the midpoint of CD. [2]
  3. (c) Find the gradient of CD, correct to 3 significant figures. [1]
Show answer
  1. (a) 8.08
    √(4.8² + 6.5²) M1 A1
  2. (b) (3.9, 0.85)
    B1 B1
  3. (c) 1.35
    6.5 ÷ 4.8 B1
Q17Length midpoint & gradient · Core · Paper 1 style · no calculator · 3 marks
  1. (a) The midpoint of PQ is (3, −2). P is the point (−1, 5). Find the coordinates of Q. [2]
  2. (b) Find the gradient of PQ. [1]
Show answer
  1. (a) (7, −9)
    2 × 3 − (−1) M1; A1
  2. (b) −7/4
    −14/8 B1
Q18Length midpoint & gradient · Standard · Paper 2 style · calculator · 6 marks

A is the point (−3, 2) and B is the point (4, 7).

  1. (a) Find the length of AB, correct to 3 significant figures. [2]
  2. (b) Find the midpoint of AB. [1]
  3. (c) Find the gradient of AB. [1]
  4. (d) The midpoint of AC is (1, −1). Find the coordinates of C. [2]
Show answer
  1. (a) 8.60 units
    √(7² + 5²) = √74 M1; 8.60 A1.
  2. (b) (0.5, 4.5)
    B1.
  3. (c) 5/7
    B1.
  4. (d) (5, −4)
    (−3 + x)/2 = 1 and (2 + y)/2 = −1 M1; (5, −4) A1.
Q19Length midpoint & gradient · Standard · Paper 1 style · no calculator · 5 marks

A is (1, 2), B is (4, 6) and C is (8, 3).

  1. (a) Show that triangle ABC is isosceles. [3]
  2. (b) Show that angle ABC is a right angle. [1]
  3. (c) Find the area of triangle ABC. [1]
Show answer
  1. (a) AB = 5, BC = 5, AC = √50
    Two lengths M1; all three A1; AB = BC stated A1
  2. (b) AB² + BC² = 25 + 25 = 50 = AC²
    B1 (converse of Pythagoras or gradients 4/3 and −3/4)
  3. (c) 12.5 units²
    ½ × 5 × 5 B1
Q20Length midpoint & gradient · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) The point (0, k) on the y-axis is the same distance from A(2, 5) as from B(6, 1). Find k. [3]
  2. (b) Find this distance, giving your answer as a surd in its simplest form. [2]
Show answer
  1. (a) k = −1
    2² + (5 − k)² = 6² + (1 − k)² M1; 29 − 10k = 37 − 2k M1; A1
  2. (b) 2√10
    √(4 + 36) = √40 M1; 2√10 A1
Q21Length midpoint & gradient · Challenge · Paper 1 style · no calculator · 7 marks

A is the point (2, k) and B is the point (8, 2k + 1).

  1. (a) The gradient of AB is 3/2. Find k. [2]
  2. (b) Find the length of AB, giving your answer as a surd in its simplest form. [2]
  3. (c) C is the point (p, 0) and AC = 10. Find the two possible values of p. [3]
Show answer
  1. (a) k = 8
    ((2k + 1) − k)/6 = 3/2 M1; k + 1 = 9, k = 8 A1.
  2. (b) 3√13
    √(6² + 9²) = √117 M1; 3√13 A1.
  3. (c) p = 8 or p = −4
    (p − 2)² + 8² = 100 M1; (p − 2)² = 36 A1; p = 8 or −4 A1.
Q22Length midpoint & gradient · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) The points A(1, 3), B(4, 9) and C(k, 15) lie on a straight line. Find k. [3]
  2. (b) Find the length AC as a surd in its simplest form. [2]
  3. (c) Find the ratio AB : BC. [1]
Show answer
  1. (a) k = 7
    Gradient AB = 2 B1; (15 − 3)/(k − 1) = 2 M1; A1
  2. (b) 6√5
    √(36 + 144) = √180 M1; 6√5 A1
  3. (c) 1 : 1
    B is the midpoint of AC B1
Q23Length midpoint & gradient · Challenge · Paper 1 style · no calculator · 7 marks

A is (−2, −1), B is (3, 1), C is (5, 6) and D is (0, 4).

  1. (a) Show that all four sides of ABCD are equal in length. [3]
  2. (b) Show that the diagonals AC and BD are perpendicular. [2]
  3. (c) Find the area of ABCD. [2]
Show answer
  1. (a) AB = BC = CD = DA = √29
    Method for one length M1; all four A2 (A1 for two)
  2. (b) Gradient AC = 1, gradient BD = −1, product −1
    B1 B1
  3. (c) 21 units²
    AC = √98, BD = √18 M1; ½ × √98 × √18 = ½ × 42 A1
Q24Equation of a line · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Write down the gradient and the y-intercept of the line y = 5 − 2x. [2]
  2. (b) Write down the equation of the line with gradient 4 that passes through (0, −3). [1]
Show answer
  1. (a) Gradient −2, y-intercept 5
    B1 each.
  2. (b) y = 4x − 3
    B1.
Q25Equation of a line · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Find the gradient of the line through (−3, 4) and (3, −8). [1]
  2. (b) Find the equation of the line. [2]
  3. (c) Find the coordinates of the point where the line crosses the x-axis. [1]
Show answer
  1. (a) −2
    −12/6 B1
  2. (b) y = −2x − 2
    4 = −2(−3) + c M1; A1
  3. (c) (−1, 0)
    B1
Q26Equation of a line · Standard · Paper 2 style · calculator · 6 marks
  1. (a) Find the equation of the line through (2, 5) and (6, 13). [3]
  2. (b) Find where the line crosses the x-axis. [1]
  3. (c) Find the acute angle the line makes with the x-axis, correct to 1 decimal place. [2]
Show answer
  1. (a) y = 2x + 1
    Gradient 8/4 = 2 M1; 5 = 4 + c M1; y = 2x + 1 A1.
  2. (b) (−0.5, 0)
    2x + 1 = 0 B1.
  3. (c) 63.4°
    tan θ = 2 M1; θ = 63.4 A1.
Q27Equation of a line · Standard · Paper 2 style · calculator · 5 marks

Temperature in degrees Fahrenheit, F, is a linear function of temperature in degrees Celsius, C. Water freezes at 0 °C = 32 °F and boils at 100 °C = 212 °F.

  1. (a) Find the equation of the line in the form F = mC + c. [2]
  2. (b) Convert 37 °C to °F. [1]
  3. (c) Find the temperature at which the Fahrenheit and Celsius readings are equal. [2]
Show answer
  1. (a) F = 1.8C + 32
    Gradient 180/100 M1; A1
  2. (b) 98.6 °F
    B1
  3. (c) −40°
    C = 1.8C + 32 M1; C = −40 A1
Q28Equation of a line · Challenge · Paper 1 style · no calculator · 7 marks

A is the point (−1, 7) and B is the point (5, −2).

  1. (a) Show that the equation of AB is 3x + 2y = 11. [3]
  2. (b) The line AB meets the x-axis at P and the y-axis at Q. Find the area of triangle OPQ, where O is the origin. [2]
  3. (c) Find the coordinates of the point where AB meets the line y = x + 1. [2]
Show answer
  1. (a) 3x + 2y = 11
    Gradient −9/6 = −3/2 M1; y − 7 = −3/2(x + 1) M1; 2y − 14 = −3x − 3 ⇒ 3x + 2y = 11 A1.
  2. (b) 121/12 (= 10 1/12) square units
    P(11/3, 0), Q(0, 11/2) M1; ½ × 11/3 × 11/2 = 121/12 A1.
  3. (c) (9/5, 14/5)
    3x + 2(x + 1) = 11 M1; x = 9/5, y = 14/5 A1.
Q29Straight-line graphs · Core · Paper 1 style · no calculator · 3 marks

A table of values for y = 2x − 1 has x = −1, 0, 1, 2, 3 and y = …, −1, 1, …, 5.

  1. (a) Complete the table: find y when x = −1 and when x = 2. [2]
  2. (b) Use the equation (or your graph) to find x when y = 4. [1]
Show answer
  1. (a) −3 and 3
    B1 each.
  2. (b) 2.5
    2x − 1 = 4 B1.
Q30Straight-line graphs · Standard · Paper 1 style · no calculator · 6 marks

Two lines have equations y = x + 2 and y = 8 − 2x.

  1. (a) Find the coordinates of the point where the two lines meet. [2]
  2. (b) Which of the points (1, 1), (3, 5) and (4, −1) satisfy both y < x + 2 and y < 8 − 2x? [2]
  3. (c) Find the area of the triangle formed by the two lines and the x-axis. [2]
Show answer
  1. (a) (2, 4)
    x + 2 = 8 − 2x M1; (2, 4) A1.
  2. (b) (1, 1) and (4, −1)
    Test each point M1; (3, 5) fails as 5 < 5 is false; answer A1.
  3. (c) 12 square units
    x-intercepts −2 and 4, base 6, height 4 M1; 12 A1.
Q31Straight-line graphs · Challenge · Paper 2 style · calculator · 6 marks

Taxi Company A charges C = 3.5 + 1.8d dollars and Company B charges C = 6 + 1.5d dollars for a journey of d km.

  1. (a) For Company A, write down what the gradient of the graph of C against d represents. [1]
  2. (b) Find the cost of a 12 km journey with each company. [2]
  3. (c) Find the distance for which both companies charge the same. Give your answer correct to 1 decimal place. [2]
  4. (d) State which company is cheaper for journeys longer than this distance. [1]
Show answer
  1. (a) The cost per kilometre ($1.80)
    B1.
  2. (b) A: $25.10; B: $24.00
    3.5 + 1.8 × 12 = 25.10 B1; 6 + 1.5 × 12 = 24 B1.
  3. (c) 8.3 km
    3.5 + 1.8d = 6 + 1.5d M1; d = 8.33… ≈ 8.3 A1.
  4. (d) Company B
    B has the smaller gradient B1.

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).

Perpendicular & parallel lines Banker16 questions · 9/9 sessions
Length midpoint & gradient Banker17 questions · 9/9 sessions
Equation of a line 8 questions · 6/9 sessions
Straight-line graphs 5 questions · 5/9 sessions

Subtopics

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

SubtopicMarks indexedSessions
Perpendicular & parallel lines Banker629/9
Length midpoint & gradient Banker499/9
Equation of a line 216/9
Straight-line graphs 85/9

← All Mathematics D topics