Gradient of linear graphs — Paper 3 · IGCSE Mathematics 0580

7 (a) For y Examiner's Use 3 2 1 x –3 –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 The simultaneous equations 2x − y = 3 and x + y = 2 can be solved graphically. (i) Which of these equations is shown by the line on the grid above? Answer(a)(i) [1] (ii) Find the gradient of the line on the grid. Answer(a)(ii) [2] (iii) Complete the table below for the other equation. x −1 0 1 2 3 y [2] (iv) Draw this line on the grid above. [1] (v) Use your graphs to write down the solution to the two equations. Give your values correct to 1 decimal place. Answer(a)(v) x = y = [3] (b) Use algebra to solve the following simultaneous equations exactly. For Show all your working. Examiner's Use 2x − y = 3, x + y = 2. Answer(b) x = y = [4]

13 marks

Mark scheme: 7 (a) (i) y = 2x – 3 oe 1 (ii) 2 oe 2 SC1 for gradient of other line (–1) (iii) 3 2 1 0 –1 2 1 for two correct (iv) correct line drawn 1 (v) (x =) 1.6 1.7, or 1.8 3 2 for correct answers not to 1 dp (y =) 0.2, 0.3, or 0.4 or 1 for 1 answer correct (b) eliminating one of the M1 working must be seen variables but second M1 can imply the eliminating the other M1 first variable (√) 1.66 or 5/3 only A1 0.3 or 1/3 only A1 SC1 for 1.67 and 0.333 [13]

This question in 0580/31 Oct/Nov 2005

7 For y Examiner's Use B A 6 5 4 3 2 1 x 3 2 1 0 1 2 3 4 5 6 7 8 1 2 3 4 Two straight lines labelled A and B are shown on the grid above. (a) Find the gradient of line A. Answer(a) [2] (b) The equation of line B can be written as y = mx + c. Find the values of m and c. Answer(b) m = c = [2] (c) (i) On the diagram draw the line which is parallel to B and passes through the point (1,−1). [1] (ii) Write down the equation of this line. Answer(c) (ii) [2]

7 marks

Mark scheme: 7 (a) –1 2 k SC1 for 1 SC1 for − K (b) (m =) 2 1 (c =) 3 1 (c) (i) Correct line drawn. 1 must cross both axes and line A (ii) y = 2x – 3 oe 2ft SC1 for m = 2 or c = –3. Follow through their line for 2 and SC1. 7

This question in 0580/31 Oct/Nov 2006

5 For y Examiner's Use 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 l –3 (a) Find the gradient of the line l. Answer(a) [2] (b) (i) Complete the table below for x + 2y = 6 . x 0 2 y 0 [3] (ii) On the grid, draw the line x + 2y = 6 for −4 Y x Y 6 . [2] (c) The equation of the line l is 4x + 3y = 4. Use your diagram to solve the simultaneous equations 4x + 3y = 4 and x + 2y = 6 . Answer(c) x = y = [2]

9 marks

Mark scheme: − 4 rise 5 (a) oe, –1.2 to –1.4 2 B1 for attempt at 3 run (b) (i) 3, 2, 6 3 B1 for each value (ii) Correct continuous line 2ft Minimum length (0,3) to (6,0) B1 for plotting their 3 points (c) x = −2, y = 4 2ft B1 for their x, B1 for their y from their intersections

This question in 0580/31 Oct/Nov 2011

6 For y Examiner's Use B 6 4 2 A E C x –4 –2 0 2 4 6 8 10 12 –2 –4 –6 Triangle ABC is drawn on a 1cm2 grid. E is the point (0, 0). (a) Write down the gradient of the line AB. Answer(a) [2] (b) The gradient of BC is – 0.5 . Write down the equation of the line BC in the form y = mx + c. Answer(b) y = [2] (c) Write down the ratio AE : EC. For Give your answer in its simplest form. Examiner's Use Answer(c) : [2] (d) Measure angle ABE. Answer(d) Angle ABE = [1] (e) Triangle ABE is similar to triangle BCE. Explain what the word similar tells you about the triangles ABE and BCE. Answer(e) [2] (f) Calculate the area of triangle ABC. Answer(f) cm2 [3] (g) ABCD is a rectangle. (i) Mark point D on the grid. [1] (ii) Write down the co-ordinates of D. Answer(g)(ii) ( , ) [1]

14 marks

Mark scheme: (e) 6 × 10–3 4 M1 ‘50’ × ‘120’ figs seen in area calculation A1 for 6000 seen (implied by 0.006 later) M1 for dividing by 1000², 0.05 & 0.12 seen or ×10–6 oe somewhere B1 ft from ‘their 0.006’ provided SF power is –ve Or SC1 for 0.6 × 10–2 oe 9 (a) (i) 226 to 226.224 cm³ 3 M1 π × 3² × 8 B1 for units : cm³ (ii) 8 cao www 4 B1 1500 used M1ft 3 × their (a)(i) 4 their 1500 M1ft 3 × their (a)(i) 4 16 (b) 5.09 (5.092 to 5.10) 2 M1 π (c) 148 cm² 3 M2 for 2 × 4 × 5 + 2 × 4 × 6 + 2 × 5 × 6 SC1 for 2 × 4 × 5 oe or 4 × 5 + 4 × 6 + 5 × 6 implied by 40, 48, 60 or 74, or list of 20, 20, 24, 24, 30, 30 (d) (i) mv oe 1 (ii) msv oe 1ft Ft (d)(i) × s (iii) 1000 msv oe 1ft Ft (d)(ii) × 1000

This question in 0580/31 May/June 2012

5 (a) (i) Complete the table for y = 5 + 3x – x2. For Examiner′s Use x –2 –1 0 1 2 3 4 5 y –5 5 7 5 –5 [3] (ii) On the grid, draw the graph of y = 5 + 3x – x2 for –2 Y x Y 5. y 8 7 6 5 4 3 2 1 x –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 –6 [4] (b) Use your graph to solve the equation 5 + 3x – x2 = 0 . Answer(b) x = … or x = … [2] (c) (i) On the grid, draw the line of symmetry of y = 5 + 3x – x2. [1] For Examiner′s Use (ii) Write down the equation of this line of symmetry. Answer(c)(ii) … [1] (d) (i) On the grid, draw a straight line from (–1, 1) to (3, 5). [1] (ii) Work out the gradient of this line. Answer(d)(ii) … [2] (iii) Write down the equation of this line in the form y = mx + c. Answer(d)(iii) y = … [1] _____________________________________________________________________________________

15 marks

Mark scheme: 5 (a) (i) 1, 7, 1 1, 1, 1 (ii) 8 points correctly plotted P3FT P2FT for 6 or 7 correct P1FT for 4 or 5 correct Correct smooth curve through all 8 C1 correct points IGCSE – October/November 2013 0580 33 (b) –1.1 to –1.3 and 4.1 to 4.3 1FT, 1FT (c) (i) Line x = 1.5 drawn 1 (ii) x = 1.5 oe 1FT Equation of their line in (c)(i) (d) (i) Ruled continuous line drawn 1 rise (ii) 1 2 M1 for for their line run (iii) [y =] x + 2 1FT their (d)(ii) + their 2

This question in 0580/33 Oct/Nov 2013

10 (a) (i) Write down the gradient of the line y =- 4 x + 7 . … [1] (ii) Write down the equation of a line parallel to y = 2x + 3 . y = … [1] (iii) Write down the co-ordinates of the point where the graph of y = 6x - 5 crosses the y-axis. ( … , … ) [1] (iv) The point (k, 7) lies on the line y = 4x - 3 . Find the value of k. k = … [2] (b) (i) Complete the table of values for y = x 2 - x - 5 . x - 3 - 2 - 1 0 1 2 3 4 y 7 - 3 - 5 [3] (ii) On the grid, draw the graph of y = x 2 - x - 5 for - 3 G x G 4 . y 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 4 –1 –2 –3 –4 –5 –6 [4] (iii) Write down the co-ordinates of the lowest point on the graph. ( … , … ) [1] (iv) (a) On the grid, draw the line of symmetry of the graph. [1] (b) Write down the equation of this line. … [1]

15 marks

Mark scheme: 10(a)(i) – 4 1 10(a)(ii) 2x + k k ≠ 3 1 10(a)(iii) (0, –5) 1 10(a)(iv) 2.5 2 M1 for 7 = 4k – 3 or better 10(b)(i) 1, –5, –3, 1, 7 3 B2 for 4 correct B1 for 3 correct 10(b)(ii) Correct smooth curve 4 B3FT for 8 or 7 correct plots or B2FT for 5 or 6 correct plots or B1FT for 3 or 4 correct plots 10(b)(iii) (0.5, h ) 1 where –5.5 ⩽ h < –5 10(b)(iv)(a) Correct line of symmetry drawn 1 10(b)(iv)(b) x = 0.5 oe 1

This question in 0580/31 May/June 2018

8 (a) (i) Write down the co-ordinates of the point where the line y = 6x - 3 crosses the y-axis. ( … , … ) [1] (ii) Write down the equation of the straight line that • passes through the origin and • is parallel to y = 6x - 3 . … [1] (b) y 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 (i) On the grid, draw the line through the point (- 3, - 2) that is perpendicular to the y-axis. [1] (ii) On the grid, draw the line y =- 2x . [1] (c) The equations of two straight lines are y = 3x + 13 and y = 7x - 3 . Use algebra to solve these two simultaneous equations to find the co-ordinates of the point where the lines meet. You must show all your working. ( … , … ) [3] Question 9 is printed on the next page.

7 marks

Mark scheme: 8(a)(i) (0, −3) 1 8(a)(ii) y = 6 x oe 1 8(b)(i) y = −2 drawn, ruled 1 8(b)(ii) y = −2 x drawn, ruled 1 8(c) For correct method seen to M1 3 x + 13 = 7 x − 3 oe eliminate one variable x = 4 A1 y = 25 A1 If M0 scored, SC1 for 2 values that substitute to give y – 3x rounding to 13.0, or y – 7x rounding to −3.0 or SC1 if no working shown, but 2 correct answers given

This question in 0580/32 May/June 2019

6 The line L is shown on the grid. y 25 20 L 15 10 5 x – 4 – 3 – 2 – 1 0 1 2 3 4 5 – 5 – 10 – 15 (a) Find the equation of the line L in the form y = mx + c . y = … [3] (b) The equation of a different line is y = 3x - 4 . (i) Write down the gradient of this line. … [1] (ii) Write down the co-ordinates of the point where this line crosses the y˗axis. ( … , … ) [1] (c) On the grid, draw the graph of y =- 2x + 1 for - 4 G x G 5 . [3]

8 marks

Mark scheme: 6(a) 4x + 2 3 B2 for 4x + c or B1 for mx + 2, m ≠ 0 4k and M1 for rise/run of k 6(b)(i) 3 1 6(b)(ii) (0, –4) 1 6(c) Correct ruled line 3 B2 for 2 correct points plotted from x = –4 to x = 5 or B1 for one correct point plotted soi or M1 for line with gradient –2 If B0 or M0 scored, SC1 for a correct table with a minimum of 3 correct coordinates

This question in 0580/31 Oct/Nov 2019

4 The diagram shows a line L and two points, A and B, on a grid. y 66 L 5 A 44 3 2 1 B x – 6 – 5 – 4 – 3 – 2 – 1 00 1 2 3 4 5 6 7 88 – 1 – 2 (a) Write down the coordinates of point A. ( … , … ) [1] (b) (i) Find the gradient of line L. … [1] (ii) Write down the equation of line L in the form y = mx + c . y = … [2] (c) (i) Draw a line that is perpendicular to line L and passes through the point A. [1] (ii) This line crosses the x-axis at point C. Mark point C on the grid and write down the coordinates of point C. ( … , … ) [1] (iii) Find, by measuring, the perimeter of triangle ABC. … cm [2]

8 marks

Mark scheme: 4(a) ( −2,4) 1 4(b)(i) −0.5 oe 1 4(b)(ii) [ y =] − 0.5x + 3 2 FT their (b)(i) B1FT for [ y =] − 0.5x + c or for [ y =] their (b)(i)x + c or for [ y =] mx + 3 4(c)(i) Correct ruled line drawn 1 4(c)(ii) (–4, 0) 1 FT their (c)(i) for x-coord 4(c)(iii) 23.0 to 23.8 2 FT provided their 3 lengths seen M1 for AB + AC + BC soi or B1FT for AB = 8.7 to 9.1 or BC = 10 or AC = 4.3 to 4.7

This question in 0580/31 May/June 2021

8 The grid shows a line L. y 6 5 L 4 3 2 1 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 – 5 – 6 (a) Find the equation of line L. Give your answer in the form y = mx + c . y = … [2] (b) (i) Complete the table of values for y = 2x + 5 . x −5 −3 0 y −5 5 [1] (ii) On the grid, draw the graph of y = 2x + 5 . [1] (c) Write down the coordinates of the point which lies on both line L and the graph of y = 2x + 5 . ( … , … ) [1] (d) Write down the equation of the line that is parallel to y = 2x + 5 and passes through the point (0, 18). … [1]

6 marks

Mark scheme: 8(a) 1 2 M1 for rise ÷ run [y =] – + 2 or for [y =] kx + 2 (k ≠ 0) 2x 1 or [y =] – 2x + j oe 8(b)(i) −1 1 8(b)(ii) Correct ruled line on grid 1 8(c) −1.2, 2.6 1 FT Line L and their (b)(ii) 8(d) y = 2x + 18 1

This question in 0580/31 May/June 2022

12 The equation of line L is y = 5x - 3 . (a) Write down the gradient of line L. … [1] (b) Write down the equation of a line parallel to line L. y = … [1]

2 marks

Mark scheme: 12(a) 5 1 12(b) 5x + k (k ≠ −3) 1

This question in 0580/31 May/June 2025

15 The equation of a line is y =- 5x + 7 . (a) Write down the gradient of this line. … [1] (b) Find the coordinates of the point where this line crosses the y-axis. ( … , … ) [1]

2 marks

Mark scheme: 15(a) −5 1 15(b) 0 , 7 1

This question in 0580/32 May/June 2025