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Simultaneous equations

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 219

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 48 parts, 219 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q3(a) · 2 marks
3. The first three terms of an arithmetic sequence are 6k, 10 and 2k where k is a constant. (a) Find the value of k. (2)
2. June 2025 · Paper 1 · Q6(a) · 4 marks
6. A scientist is monitoring the flight of sea birds after they leave their nests on a cliff. The height above the sea, h metres, of one of the birds is modelled by the equation 1.5 h 0 0 t T h = A – B t where t seconds is the time after the bird leaves its nest and A, B and T are positive constants. Given that • the bird was 17.6 m above the sea exactly 4 seconds after leaving its nest • the bird was 11.9 m above the sea exactly 9 seconds after leaving its nest (a) find a complete equation for the model. (4)
3. June 2025 · AS Paper 1 · Q6(b) · 5 marks
Given also that f′ (2) = 14 (b) find the value of a and the value of b. (5)
4. June 2025 · AS Paper 1 · Q14(b) · 5 marks
2 f (x) x ≠ 0 ; g (x) = a :9 x where a is a constant. Given that the first 3 terms, in ascending powers of x, of the series expansion of g (x) are – 2 x – 21 – 90x (b) find the possible pairs of values of a and k. (5)
5. June 2025 · Paper 2 · Q14(b) · 5 marks
Given also that • the point N lies on BC such that BN : NC = 1 : 4 • AN intersects BD at X (b) find BX : XD (5)
6. June 2025 · AS Paper 2 (Statistics) · Q4(b)(ii) · 2 marks
(ii) Find the value of a (2)
7. June 2025 · AS Paper 2 (Statistics) · Q5(c) · 4 marks
(c) Find (i) the value of p (ii) the value of q (iii) the value of s (iv) the value of t (4)
8. June 2025 · Paper 3 (Statistics) · Q5(b) · 5 marks
The heights of women in the tennis club are normally distributed with a mean of μ cm and a standard deviation of sigma cm. Given that 40% of the women are shorter than 170 cm and 15% are taller than 175 cm, (b) find the value of μ and the value of sigma Show your working clearly. (5)
9. June 2024 · AS Paper 1 · Q2(b) · 4 marks
Given also that (x – 2) is a factor of f (x), (b) express f (x) in the form f (x) = (2x + k) (x – 4) (x – 2) where k is a constant to be found. (4)
10. June 2024 · Paper 1 · Q10(c) · 5 marks
The line l2 has equation y = 8x The region R, shown shaded in Figure 3, is bounded by the curve, the line l1 and the line l2 (c) Use algebraic integration to find the exact area of R. (5)
11. June 2024 · Paper 1 · Q15(ii) · 4 marks
(ii) A student was asked to prove by contradiction that / / O “There are no positive integers x and y such that 3 x y x y ” The start of the student’s proof is shown below. Assume that positive integers x and y exist such that / / O x y x y If 3x + 2y = 14 and 2x – 5y = 2 x y x y , Notintegers x y Show the calculations and statements needed to complete the proof. (4)
12. June 2024 · Paper 2 · Q9(a) · 4 marks
9. H (m) A(0, 2) B(20, 0.8) x (m) Figure 3 The graph in Figure 3 shows the path of a small ball. The ball travels in a vertical plane above horizontal ground. The ball is thrown from the point represented by A and caught at the point represented by B. The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown. With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3. The ball reaches its maximum height when x = 9 A quadratic function, linking H with x, is used to model the path of the ball. (a) Find H in terms of x. (4)
13. June 2024 · Paper 2 · Q15(c) · 5 marks
(c) Prove that the normal to C at P does not meet C again. You should use algebra for your proof and make your reasoning clear. (5)
14. June 2024 · Paper 3 (Mechanics) · Q6(b) · 6 marks
The coefficient of friction between the rod and the ground is μ. Given that the rod is in limiting equilibrium, (b) find the value of μ. (6)
15. June 2024 · Paper 3 (Statistics) · Q5(d) · 5 marks
Students in the school athletics club also throw the discus. The random variable D~N μ sigma 2 represents the distance, in metres, that a student can , throw the discus. Given that P(D < 16.3) = 0.30 and P(D 29.0) = 0.10 < (d) calculate the value of mu and the value of sigma (5)
16. June 2023 · Paper 1 · Q5(b) · 3 marks
Given also that the sum of all the y values in the table is 97.2 (b) find the value of a and the value of b (3)
17. June 2023 · AS Paper 1 · Q7(a) · 4 marks
7. The distance a particular car can travel in a journey starting with a full tank of fuel was investigated. • From a full tank of fuel, 40 litres remained in the car’s fuel tank after the car had travelled 80 km • From a full tank of fuel, 25 litres remained in the car’s fuel tank after the car had travelled 200 km Using a linear model, with V litres being the volume of fuel remaining in the car’s fuel tank and d km being the distance the car had travelled, (a) find an equation linking V with d. (4)
18. June 2023 · AS Paper 1 · Q10(b) · 5 marks
Given that • lines l1 and l2 intersect at the point C • line l1 crosses the x-axis at the point A (b) find the exact area of triangle ABC, giving your answer as a fully simplified fraction in the form p q (5)
19. June 2023 · Paper 1 · Q10(b) · 6 marks
The line with equation y = 2x – 1 intersects C at 2 distinct points. (b) Find the range of possible values of k. (6)
20. June 2023 · AS Paper 1 · Q16 · 6 marks
16. A curve has equation y = f ( ), x x 0 Given that • f ( ) ∑ x x a x b , where a and b are constants / • the curve has a stationary point at 4 3 , • the curve meets the y-axis at -5 find f ( ), x giving your answer in simplest form. (6)
21. June 2022 · Paper 1 · Q5(a) · 4 marks
5. The height, h metres, of a tree, t years after being planted, is modelled by the equation 2 = at + b 0 t < 25 h where a and b are constants. Given that • the height of the tree was 2.60 m, exactly 2 years after being planted • the height of the tree was 5.10 m, exactly 10 years after being planted (a) find a complete equation for the model, giving the values of a and b to 3 significant figures. (4)
22. June 2022 · AS Paper 1 · Q11(b) · 5 marks
(b) Find the shortest distance between C and l. (5)
23. June 2022 · AS Paper 2 (Statistics) · Q5(b) · 5 marks
The random variable X = mR + nG where m and n are integers. P(X = 20) = 1 6 and P(X = 50) = 1 (b) Find the value of m and the value of n (5)
24. Oct 2021 · Paper 1 · Q7(b) · 5 marks
The line l has equation y = 3x + k where k is a constant. Given that l is a tangent to C, (b) find the possible values of k, giving your answers as simplified surds. (5)
25. Oct 2021 · Paper 1 · Q12(a) · 5 marks
12. H x Figure 3 Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit. The ball is modelled as a particle travelling in a vertical plane above horizontal ground. Given that the ball • is hit from a point on the top of a platform of vertical height 3 m above the ground • reaches its maximum vertical height after travelling a horizontal distance of 90 m • is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m Given also that H is modelled as a quadratic function in x (a) find H in terms of x (5)
26. Oct 2021 · AS Paper 1 · Q15(b) · 4 marks
(b) an equation for C. (4)
27. Oct 2021 · AS Paper 1 · Q15(c) · 3 marks
The line with equation y = 1 3 x + k, where k is a non-zero constant, is also a tangent to C. (c) Find the value of k. (3)
28. Oct 2021 · AS Paper 1 · Q16(a) · 4 marks
16. The curve C has equation y = f(x) where f(x) = ax 3 + 15x 2 – 39x + b and a and b are constants. Given • the point (2, 10) lies on C • the gradient of the curve at (2, 10) is –3 (a) (i) show that the value of a is –2 (ii) find the value of b. (4)
29. Oct 2021 · Paper 2 · Q8(b) · 5 marks
Given that • the point P (–1, – 4) lies on C • the normal to C at P has equation 19x + 26y + 123 = 0 (b) find the value of p and the value of q. (5)
30. Oct 2021 · AS Paper 2 (Mechanics) · Q3(b) · 7 marks
(b) find, in terms of h only, the height above the ground at which P first comes to instantaneous rest. (7)
31. Oct 2020 · AS Paper 1 · Q8(d) · 4 marks
μ (0, 94) (8, 50) l t Figure 2 The temperature, μ °C, of a second cup of tea t minutes after it was placed on a table in a different room, is modelled by the equation -t μ = A + Be 8 t 0 where A and B are constants. Figure 2 shows a sketch of μ against t with two data points that lie on the curve. The line l, also shown on Figure 2, is the asymptote to the curve. Using the equation of this model and the information given in Figure 2 (d) find an equation for the asymptote l. (4)
32. Oct 2020 · Paper 1 · Q11(a) · 4 marks
11. y C1 C2 A x B Figure 3 Circle C1 has equation x2 + y2 = 100 Circle C2 has equation (x - 15)2 + y2 = 40 The circles meet at points A and B as shown in Figure 3. (a) Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. (4)
33. Oct 2020 · AS Paper 1 · Q14(a) · 7 marks
14. A curve has equation y = g(x). Given that ● g(x) is a cubic expression in which the coefficient of x3 is equal to the coefficient of x ● the curve with equation y = g(x) passes through the origin ● the curve with equation y = g(x) has a stationary point at (2, 9) (a) find g(x), (7)
34. Oct 2020 · Paper 1 · Q16 · 4 marks
16. Prove by contradiction that there are no positive integers p and q such that 4 p2 - q2 = 25 (4)
35. Oct 2020 · Paper 2 · Q9(a) · 4 marks
9. A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, theta °C, at time t seconds after heating began, is modelled by the equation theta = A - Be- 0.07t where A and B are positive constants. Given that ● the initial temperature of the ethanol was 18 °C ● after 10 seconds the temperature of the ethanol was 44 °C (a) find a complete equation for the model, giving the values of A and B to 3 significant figures. (4)
36. Oct 2020 · Paper 2 · Q14(a) · 3 marks
14. A circle C with radius r ● lies only in the 1st quadrant ● touches the x-axis and touches the y-axis The line l has equation 2x + y = 12 (a) Show that the x coordinates of the points of intersection of l with C satisfy 5x 2 + (2r - 48) x + (r 2 - 24r + 144) = 0 (3)
37. Oct 2020 · Paper 3 (Mechanics) · Q4(b) · 7 marks
(b) Hence, or otherwise, find the value of μ. (7)
38. Oct 2020 · Paper 3 (Mechanics) · Q5(a) · 6 marks
5. U m s-1 45° 25 m N A 100 m Figure 2 A small ball is projected with speed U m s-1 from a point O at the top of a vertical cliff. The point O is 25 m vertically above the point N which is on horizontal ground. The ball is projected at an angle of 45° above the horizontal. The ball hits the ground at a point A, where AN = 100 m, as shown in Figure 2. The motion of the ball is modelled as that of a particle moving freely under gravity. Using this initial model, (a) show that U = 28 (6)
39. June 2019 · AS Paper 1 · Q1(b) · 2 marks
The lines l1 and l2 meet at the point P. (b) Find the x coordinate of P. (2)
40. June 2019 · AS Paper 1 · Q7(b) · 2 marks
The line l has equation y = –2x + 5 (b) Show that the x coordinate of any point of intersection of l with C is given by a solution of the equation 2x2 – 4x + k2 = 0 (2)
41. June 2019 · Paper 2 · Q7(b) · 3 marks
The bars of soap are sold for £2 each. On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of £80 Using the above information, (b) show that y = 0.84x + 428 (3)
42. June 2019 · Paper 3 (Mechanics) · Q4(b) · 9 marks
(b) Find the magnitude of the resultant force acting on the ramp at A. (9)
43. June 2019 · Paper 3 (Mechanics) · Q5(b) · 6 marks
(6) (b) Find (1) the size ofangle 0, (11) the value of w.
44. June 2018 · Paper 1 · Q3 · 4 marks
3. A r cm theta r cm B Figure 1 Figure 1 shows a sector AOB of a circle with centre O and radius r cm. The angle AOB is theta radians. The area of the sector AOB is 11 cm2 Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r. (4)
45. June 2018 · Paper 1 · Q6(b) · 4 marks
(b) Find an equation for C. (4)
46. June 2018 · Paper 1 · Q9(b) · 5 marks
The curve is used to model the shape of a cycle track with both x and y measured in km. The points P and Q represent points that are furthest west and furthest east of the origin O, as shown in Figure 4. Using part (a), (b) find the exact coordinates of the point P. (5)
47. June 2018 · Paper 1 · Q12(a) · 4 marks
12. The value, £V, of a vintage car t years after it was first valued on 1st January 2001, is modelled by the equation V = Ap t where A and p are constants Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012 (a) (i) find p to 4 decimal places, (ii) show that A is approximately 24 800 (4)
48. June 2018 · Paper 1 · Q14(c) · 5 marks
The line with equation x + y = k, where k is a constant, intersects C at two distinct points. (c) State the range of values of k, writing your answer in set notation. (5)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 1 · Q3
Mark scheme June 2025 Paper 1 Q3(a)
June 2025 · Paper 1 · Q6
Mark scheme June 2025 Paper 1 Q6(a)
June 2025 · AS Paper 1 · Q6
Mark scheme June 2025 Paper 1 Q6(b)
June 2025 · AS Paper 1 · Q14
Mark scheme June 2025 Paper 1 Q14(b)
June 2025 · Paper 2 · Q14
Mark scheme June 2025 Paper 2 Q14(b)
June 2025 · AS Paper 2 (Statistics) · Q4
Mark scheme June 2025 Paper 2 (Statistics) Q4(b)(ii)
June 2025 · AS Paper 2 (Statistics) · Q5
Mark scheme June 2025 Paper 2 (Statistics) Q5(c)
June 2025 · Paper 3 (Statistics) · Q5
Mark scheme June 2025 Paper 3 (Statistics) Q5(b)
June 2024 · AS Paper 1 · Q2
Mark scheme June 2024 Paper 1 Q2(b)
June 2024 · Paper 1 · Q10
Mark scheme June 2024 Paper 1 Q10(c)
June 2024 · Paper 1 · Q15
Mark scheme June 2024 Paper 1 Q15(ii)
June 2024 · Paper 2 · Q9
Mark scheme June 2024 Paper 2 Q9(a)
June 2024 · Paper 2 · Q15
Mark scheme June 2024 Paper 2 Q15(c)
June 2024 · Paper 3 (Mechanics) · Q6
Mark scheme June 2024 Paper 3 (Mechanics) Q6(b)
June 2024 · Paper 3 (Statistics) · Q5
Mark scheme June 2024 Paper 3 (Statistics) Q5(d)
June 2023 · Paper 1 · Q5
Mark scheme June 2023 Paper 1 Q5(b)
June 2023 · AS Paper 1 · Q7
Mark scheme June 2023 Paper 1 Q7(a)
June 2023 · AS Paper 1 · Q10
Mark scheme June 2023 Paper 1 Q10(b)
June 2023 · Paper 1 · Q10
Mark scheme June 2023 Paper 1 Q10(b)
June 2023 · AS Paper 1 · Q16
Mark scheme June 2023 Paper 1 Q16
June 2022 · Paper 1 · Q5
Mark scheme June 2022 Paper 1 Q5(a)
June 2022 · AS Paper 1 · Q11
Mark scheme June 2022 Paper 1 Q11(b)
June 2022 · AS Paper 2 (Statistics) · Q5
Mark scheme June 2022 Paper 2 (Statistics) Q5(b)
Oct 2021 · Paper 1 · Q7
Mark scheme Oct 2021 Paper 1 Q7(b)
Oct 2021 · Paper 1 · Q12
Mark scheme Oct 2021 Paper 1 Q12(a)
Oct 2021 · AS Paper 1 · Q15
Mark scheme Oct 2021 Paper 1 Q15(c)
Oct 2021 · AS Paper 1 · Q16
Mark scheme Oct 2021 Paper 1 Q16(a)
Oct 2021 · Paper 2 · Q8
Mark scheme Oct 2021 Paper 2 Q8(b)
Oct 2021 · AS Paper 2 (Mechanics) · Q3
Mark scheme Oct 2021 Paper 2 (Mechanics) Q3(b)
Oct 2020 · AS Paper 1 · Q8
Mark scheme Oct 2020 Paper 1 Q8(d)
Oct 2020 · Paper 1 · Q11
Mark scheme Oct 2020 Paper 1 Q11(a)
Oct 2020 · AS Paper 1 · Q14
Mark scheme Oct 2020 Paper 1 Q14(a)
Oct 2020 · Paper 1 · Q16
Mark scheme Oct 2020 Paper 1 Q16
Oct 2020 · Paper 2 · Q9
Mark scheme Oct 2020 Paper 2 Q9(a)
Oct 2020 · Paper 2 · Q14
Mark scheme Oct 2020 Paper 2 Q14(a)
Oct 2020 · Paper 3 (Mechanics) · Q4
Mark scheme Oct 2020 Paper 3 (Mechanics) Q4(b)
Oct 2020 · Paper 3 (Mechanics) · Q5
Mark scheme Oct 2020 Paper 3 (Mechanics) Q5(a)
June 2019 · AS Paper 1 · Q1
Mark scheme June 2019 Paper 1 Q1(b)
June 2019 · AS Paper 1 · Q7
Mark scheme June 2019 Paper 1 Q7(b)
June 2019 · Paper 2 · Q7
Mark scheme June 2019 Paper 2 Q7(b)
June 2019 · Paper 3 (Mechanics) · Q4
Mark scheme June 2019 Paper 3 (Mechanics) Q4(b)
June 2019 · Paper 3 (Mechanics) · Q5
Mark scheme June 2019 Paper 3 (Mechanics) Q5(b)
June 2018 · Paper 1 · Q3
Mark scheme June 2018 Paper 1 Q3
June 2018 · Paper 1 · Q6
Mark scheme June 2018 Paper 1 Q6(b)
June 2018 · Paper 1 · Q9
Mark scheme June 2018 Paper 1 Q9(b)
June 2018 · Paper 1 · Q12
Mark scheme June 2018 Paper 1 Q12(a)
June 2018 · Paper 1 · Q14
Mark scheme June 2018 Paper 1 Q14(c)