Equations 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

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

8 (a) Complete the table of values for y = x2 – 2x + 5 . For Examiner's Use x –3 –2 –1 0 1 2 3 4 5 y 20 8 8 20 [3] (b) On the grid, draw the graph of y = x2 – 2x + 5 for −3 Y x Y 5 . y 22 20 18 16 14 12 10 8 6 4 2 x –4 –3 –2 –1 0 1 2 3 4 5 6 [4] (c) (i) On the grid, draw the line of symmetry of the graph. [1] (ii) Write down the equation of the line of symmetry. Answer(c)(ii) [1] (d) (i) On the grid, draw the line y = 12 . [1] For Examiner's (ii) Use your graph to solve the equation x2 – 2x + 5 = 12 . Use Answer(d)(ii) x = or x = [2] (e) The equation of a straight line is y = 6 – 3x . (i) Write down the gradient of this line. Answer(e)(i) [1] (ii) Write down the co-ordinates of the point where this line crosses the y-axis. Answer(e)(ii) ( , ) [1] (iii) Write down the equation of a line parallel to y = 6 – 3x . Answer(e)(iii) [1] (f) Simplify 3(2x + 1) O=2(6 – 3x) . Answer(f) [2]

17 marks

Mark scheme: 8 (a) (20) 13 (8) 5 4 5 (8) 13 (20) 3 B2 for 4 correct B1 for 2 or 3 correct or a correct substitution seen (b) correctly plotting 9 points and 4 P3 for correctly plotting 9 points, P2 for correctly connecting with a smooth curved line plotting 7 or 8 points and P1 for 5 or 6 points C1 for a smooth curve (c) (i) correct line of symmetry cao 1 (ii) x = 1 1ft ft their line (d) (i) correct line 1 (ii) –1.9 to –1.7 and 3.7 to 3.9 1ft,1ft SC1 for correct co-ordinates (e) (i) –3 cao 1 (ii) (0,6) cao 1 (iii) y = c – 3x 1 c can be any number except 6 (f) 12x – 9 or 3(4x – 3) 2 B1 for 6x + 3, –12 + 6x, 12x or –9 IGCSE – May/June 2012 0580 33

This question in 0580/33 May/June 2012

4 (a) The table shows some values of y = x2 – 2x – 1. For Examiner′s Use x –3 –2 –1 0 1 2 3 4 y 14 2 –1 –2 7 (i) Complete the table. [2] (ii) On the grid, draw the graph of y = x2 – 2x – 1 for –3 Y x Y 4. y 16 14 12 10 8 6 4 2 x –3 –2 –1 0 1 2 3 4 –2 –4 [4] (b) Write down the equation of the line of symmetry of the graph. For Examiner′s Use Answer(b) … [1] (c) The point with co-ordinates (–3, 7) lies on the line y = –x + 4 . (i) Write down the co-ordinates of two other points on this line. Use x co-ordinates so that –3 < x Y 4 . Answer(c)(i) ( … , … ) and ( … , … ) [2] (ii) On the grid, draw the line y = –x + 4 for –3 Y x Y 4 . [1] (iii) Use both graphs to fi nd the solutions of the equation x2 – 2x – 1 = –x + 4 . Answer(c)(iii) x = … or x = … [2] _____________________________________________________________________________________

12 marks

Mark scheme: 4 (a) (i) 7, –1, 2 2 B1 for any 2 correct (ii) 8 points plotted 3ft P2ft for 6 or 7 correct P1ft for 4 or 5 correct Correct smooth curve 1 (b) x = 1 1 (c) (i) Two correct points 1,1 x –2 –1 –0 –1 –2 –3– 4 y –6 –5 –4– 3– 2 –1 –0 (ii) Correct line drawn 1 Must be ruled and continuous (iii) –1.9 to –1.7, 2.7 to 2.9 2ft 1 for each correct

This question in 0580/33 May/June 2013

5 (a) (i) Complete the table of values for y = x2 + x – 4. x –4 –3 –2 –1 0 1 2 3 y –2 –4 –2 8 [2] (ii) On the grid, draw the graph of y = x2 + x – 4 for - 4 G x G 3 . y 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 [4] (b) (i) Write down the co-ordinates of the lowest point of the graph. Answer(b)(i) ( … , … ) [1] (ii) Write down the equation of the line of symmetry of the graph. Answer(b)(ii) … [1] (c) Use your graph to solve the equation x2 + x – 4 = –3. Answer(c) x = … or x = … [2] (d) y 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 –7 L (i) In the diagram, a line L has been drawn on a 1 cm2 grid. Write down the equation of the line L. Give your answer in the form y = mx + c. Answer(d)(i) y = … [2] (ii) Find the area of the shaded triangle. Answer(d)(ii) … cm2 [1] __________________________________________________________________________________________

13 marks

Mark scheme: 5 (a) (i) 8, 2, –4, 2 2 B1 for 3 correct values (ii) Correctly plotted points and smooth 4 B3FT for 8 correct correct curve B2FT for 6 or 7 correct B1FT for 4 or 5 correct C1 for correct smooth curve passing below y = –4 (b) (i) (–0.5, k) where –4.5 ≤ k < –4 1 (ii) x = –0.5 1 (c) –1.8 ≤ x ≤ –1.4, 0.4 ≤ x ≤ 0.8 2FT B1FT, B1FT for values from their graph rise (d) (i) 2x – 3 2 M1 for or better run If zero scored, SC1 for kx – 3 (ii) 9 1

This question in 0580/32 Feb/March 2015

7 (a) The grid below shows the straight line L. y 15 10 5 x –3 –2 –1 0 1 2 3 4 5 6 –5 L The equation of the line L is y = 2x + c. Find the value of c. Answer(a) c = … [1] (b) (i) Complete the table of values for y = x 2 – 3x – 2. x –3 –2 –1 0 1 2 3 4 5 6 y 8 2 –4 –4 2 8 [2] (ii) On the grid, draw the graph of y = x 2 – 3x – 2 for –3 x 6. [4] (iii) Write down the values of x where the line L intersects the curve y = x 2 – 3x – 2. Answer(b)(iii) x = … and x = … [2]

9 marks

Mark scheme: 7 (a) –1 1 (b) (i) 16.…–2.…–2….16 2 B1 for 2 correct (ii) 10 points correctly plotted 4 B3FT for 9 or 10 points correctly plotted Correct smooth curve B2FT for 7 or 8 points correctly plotted B1FT for 5 or 6 points correctly plotted (iii) Strict FT their intersection 2FT B1 for one correct value

This question in 0580/33 May/June 2015

9 y l 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 4 –1 –2 –3 –4 –5 –6 –7 –8 (a) Write down the equation of the line l in the form y = mx + c. Answer(a) y = … [3] 2 (b) Complete the table of values for y = . x x −4 −3 −2 −1 −0.5 −0.25 0.25 0.5 1 2 3 4 y −0.7 −4 4 0.7 [3] 2 (c) On the grid, draw the graph of y = for –4  x  –0.25 and 0.25  x  4. [4] x

10 marks

Mark scheme: 9 (a) [ y = ] 2 x + 4 3 B2 for 2 x + c or kx + 4 k ≠ 0 or 2 k rise M1 for gradient = ± or attempt at k run using a triangle or co-ordinates allowing one slip (b) –0.5, –1, –2, –8, 8, 2, 1, 0.5 3 B2 for any 6 or 7 correct or B1 for any 4 or 5 correct (c) Correct curve 4 B3FT for 11 or 12 points correctly plotted B2FT for 9 or 10 points correctly plotted B1FT for 7 or 8 points correctly plotted

This question in 0580/32 Oct/Nov 2015

8 y L 18 16 14 12 10 8 6 4 2 x –2 –1 0 1 2 3 4 5 –2 –4 –6 –8 (a) The line L is drawn on the grid. Find the equation of the line in the form y = mx + c. Answer(a) y = … [3] (b) (i) Complete the table of values for y = x2 – 4x – 2. x –2 –1 0 1 2 3 4 5 y –2 –6 –5 –2 3 [3] (ii) On the grid above, draw the graph of y = x2 – 4x – 2 for –2  x  5. [4] (iii) Use your graph to solve the equation x2 – 4x – 2 = 0. Answer(b)(iii) x = … or x = … [2] __________________________________________________________________________________________

12 marks

Mark scheme: 8 (a) 5x + 3 3 B2 for 5x + c or kx + 3 k not equal 0 Rise or M1 for attempt at Run (b) (i) 10, 3, −5 3 B1 for each correct (ii) Correct curve 4 B3FT for 7 or 8 points correctly plotted B2FT for 5 or 6 points correctly plotted B1FT for 3 or 4 points correctly plotted (iii) −0.5 to – 0.4 and 4.4 to 4.5 2FT B1FT for each correct

This question in 0580/33 Oct/Nov 2015

6 y 6 4 2 x 0 –5 –4 –3 –2 –1 1 2 –2 –4 –6 –8 –10 (a) On the grid, (i) draw the line y = 3, [1] (ii) draw the line that is perpendicular to the line y = 3 that passes through the point (1, −4). [2] (b) Complete the table of values for y = 2 − 3x − x2. x −5 −4 −3 −2 −1 0 1 2 y −2 2 2 −2 [2] (c) On the grid, draw the graph of y = 2 − 3x − x2 for - 5 G x G 2 . [4] (d) Write down the co-ordinates of the highest point of the graph of y = 2 − 3x − x2. ( … , … ) [1] (e) Use your graphs to solve the equation 2 − 3x − x2 = 3. x = … or x = … [2]

12 marks

Mark scheme: 6 (a) (i) Ruled continuous line y = 3 1 (ii) Ruled continuous line x = 1 2 B1 for (1, –4) plotted or B1 for any line perpendicular to their y = 3 drawn (b) –8, 4, 4, –8 2 B1 for 3 correct (c) Completely correct curve 4 B3FT for 7 or 8 points correctly plotted B2FT for 5 or 6 points correctly plotted B1FT for 3 or 4 points correctly plotted (d) (–1.5, 4.1 to 4.4) 1 (e) –2.5 to –2.7 and –0.3 to –0.5 2FT FT intersection of their (a)(i) with their curve B1FT for one correct

This question in 0580/32 May/June 2016

7 y 20 18 L 16 14 12 10 8 6 4 2 x –2 –1 0 1 2 3 –2 –4 –6 –8 (a) The line L is drawn on the grid. Find the equation of the line in the form y = mx + c. y = … [3] (b) (i) Complete the table of values for y = x2 + 2x + 4. x −2 −1 0 1 2 3 y 4 4 7 19 [2] (ii) On the grid above, draw the graph of y = x2 + 2x + 4 for –2 G x G 3 . [4] (c) For –2 G x G 3 , write down the x co-ordinate of the point of intersection of the curve y = x2 + 2x + 4 with the line L.

10 marks

Mark scheme: 7 (a) −5x + 6 3 B2 for –5x (oe) + 6 or –5x + k or rise B1 for kx + 6 k ≠ 0 or [gradient = ] run k with correct values or [gradient =] ±5 k (b) (i) 3 12 1 , 1 (ii) Correct curve 4 B3FT for 5 or 6 correctly plotted points or B2FT for 3 or 4 correctly plotted points or B1FT for 1 or 2 correctly plotted points (c) 0.2 to 0.35 1 FT

This question in 0580/31 Oct/Nov 2016

5 (a) Complete the table of values for y = x 2 + 2x - 1. x -5 -4 -3 -2 -1 0 1 2 3 y 14 2 -1 -1 2 [3] (b) On the grid, draw the graph of y = x 2 + 2x - 1 for -5 G x G 3 . y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x –5–5 –4–4 –3–3 –2–2 –1–1 00 11 22 33 –1 –2 –3 –4 (c) (i) On the grid, draw the line of symmetry. [1] (ii) Write down the equation of the line of symmetry. … [1] (d) (i) On the grid, plot the points (- 5 , 7) and (0, - 3) and join them with a straight line, L. [2] (ii) Write down the x co-ordinate of each point where the line L crosses the graph of y = x 2 + 2x - 1. x = … and x = … [2] (iii) Work out the gradient of the line L. … [2]

15 marks

Mark scheme: 5(a) 7 –2 7 14 3 B2 for 3 correct B1 for 2 correct 5(b) Correct smooth curve 4 B3FT for 8 or 9 correct plots or B2FT for 6 or 7 correct plots or B1FT for 4 or 5 correct plots 5(c)(i) Ruled line, x = –1, drawn 1 5(c)(ii) x = –1 oe 1 5(d)(i) Ruled line L drawn, joining 2 B1 for one of the points correct and line drawn, or (–5, 7) and (0, −3) both points correct and no or wrong line. 5(d)(ii) −3.3 to −3.5, −0.5 to −0.7 2FT B1FT for one correct. 5(d)(iii) −2 2 Rise y 2 − y1 M1FT for their from part (d)(i) or their Run x2 − x1 If zero scored, SC1 for answer 2

This question in 0580/31 May/June 2017

8 (a) Complete the table of values for y = 8x - x 2 . x 0 1 2 3 4 5 6 7 8 y 0 12 15 15 12 [3] (b) On the grid, draw the graph of y = 8x - x 2 for 0 G x G 8 . y 18 16 14 12 10 8 6 4 2 0 x 1 2 3 4 5 6 7 8 [4] (c) Write down the equation of the line of symmetry of this graph. … [1] (d) Use the graph to solve 8x - x 2 = 10 . x = … or x = … [2]

10 marks

Mark scheme: 8(a) 7 16 7 0 3 B2 for 2 or 3 correct B1 for 1 correct 8(b) Correct curve 4 B3FT for 8 or 9 points plotted correctly or B2FT for 6 or 7 points plotted correctly or B1FT for 4 or 5 points plotted correctly 8(c) x = 4 1 8(d) 1.45 to 1.65 and 6.35 to 6.55 2 B1 for each or both correct as co-ordinates

This question in 0580/32 Oct/Nov 2018

8 (a) y 4 L 3 2 1 –2 –1 0 1 2 3 4 5 6 7 8 x –1 Line L is drawn on the grid. Find the equation of line L. Give your answer in the form y = mx + c. y = … [3] - 7. (b) The points (9, a) and (b, 3) lie on the line y = 23 x Work out the value of (i) a, a = … [2] (ii) b. b = … [2] (c) (i) Complete the table of values for y = x (3 - x). x -4 -2 -1 0 1 2 4 y -10 0 2 -4 [3] (ii) On the grid, draw the graph of y = x (3 - x) for - 4 G x G 4. y 5 -4 -2 0 2 4 x -5 -10 -15 -20 -25 -30 [4] (iii) Write down the co-ordinates of the highest point of the graph for - 4 G x G 4. ( … , … ) [1]

15 marks

Mark scheme: 8(a) 1 3 1 [ y = ] − x + 3 B2 for [ y = ] − x + c 2 2 or rise 1 M1 for or m = ± oe run 2 and B1 for [ y = ] kx + 3 , k ≠ 0 or c = 3 8(b)(i) –1 2 2 M1 for [ a = ] × 9 − 7 or better 3 8(b)(ii) 15 2 2 M1 for 3 = b − 7 or better 3 8(c)(i) –28, –4, 2 3 B1 for each 8(c)(ii) correct smooth curve 4 B3FT for 6 or 7 correct plots or B2FT for 4 or 5 correct plots or B1FT for 2 or 3 correct plots 8(c)(iii) (1.5 , 2.25) 1 accept (x, y) where 1 < x < 2 and 2 < y < 4

This question in 0580/32 Feb/March 2019

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

4 (a) Complete the table of values for y = 5 + 2 x - x 2 . x -2 -1 0 1 2 3 4 y 2 5 6 -3 [2] (b) On the grid, draw the graph of y = 5 + 2x - x 2 for - 2 G x G 4 . y 7 6 5 4 3 2 1 0 –2 –1 1 2 3 4 x –1 –2 –3 [4] (c) (i) On the grid, draw the line of symmetry. [1] (ii) Write down the equation of the line of symmetry. … [1] (d) Use your graph to find the solutions of the equation 5 + 2x - x 2 = 4 . x = … or x = … [2] (e) (i) On the grid, draw a line from (- 1, 2) to (1, 6) . [1] (ii) Find the equation of this line in the form y = mx + c . y = … [3]

14 marks

Mark scheme: 4(a) −3 5 2 2 B1 for 2 correct 4(b) Correct curve 4 B3FT for 6 or 7 points correct B2FT for 4 or 5 points correct B1FT for 2 or 3 points correct 4(c)(i) Ruled line x = 1 drawn 1 4(c)(ii) x = 1 1 4(d) −0.5 to −0.3 and 2.3 to 2.5 2 B1 for each If 0 scored, B1 for y = 4 drawn 4(e)(i) Correct ruled continuous line 1 4(e)(ii) [y =] 2x + 4 3 B2 for [y =] 2x + k rise or M1 for run B1 for kx + 4 , k ≠ 0, or c = 4

This question in 0580/33 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

3 (a) Complete the table of values for y = 1 + 5 x - x 2 . x - 1 0 1 2 3 4 5 y 1 5 7 1 [2] (b) On the grid, draw the graph of y = 1 + 5x - x 2 for - 1 G x G 5 . y 8 6 4 2 0 x – 1 1 2 3 4 5 – 2 – 4 – 6 [4] (c) (i) On the grid, draw the line y = 3 . [1] (ii) Use your line to solve the equation 1 + 5x - x 2 = 3 . x = … or x = … [2]

9 marks

Mark scheme: 3(a) –5, 7, 5 2 B1 for 2 correct 3(b) Correct curve 4 B3FT for 6 or 7 points correctly plotted or B2FT for 4 or 5 points correctly plotted or B1FT for 2 or 3 points correctly plotted 3(c)(i) Ruled line y = 3 1 3(c)(ii) 0.3 to 0.6 4.4 to 4.7 2 FT their y = k and their (b) B1 for one correct or B1 for both correct answers as coordinates

This question in 0580/33 Oct/Nov 2020

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 (a) Line L is shown on the grid. y 25 20 L 15 10 5 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 x – 5 – 10 Find the equation of line L in the form y = mx + c . y = … [3] (b) (i) Complete the table of values for y = x 2 + 4x . x -6 -5 -4 -3 -2 -1 0 1 2 3 y 12 5 0 -3 -3 0 5 12 [2] (ii) On the grid, draw the graph of y = x 2 + 4x for - 6 G x G 3 . y 25 20 15 10 5 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 x – 5 – 10 [4] (iii) Use your graph to solve the equation x 2 + 4x = 10 . x = … or x = … [2]

11 marks

Mark scheme: 8(a) 2.5x + 10 final answer 3 B2 for 2.5x + c OR M1 for a correct rise over run or for a right-angled triangle marked on grid with rise =10 and run = 4 oe B1 for [y =] kx +10 (k ≠ 0) 8(b)(i) –4 21 2 B1 for each 8(b)(ii) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 8(b)(iii) –5.8 to –5.6 and 1.6 to 1.8 2 FT their curve B1 for each

This question in 0580/33 Oct/Nov 2021

6 y 10 L 9 8 7 6 5 4 3 2 1 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 x – 1 – 2 – 3 – 4 (a) Find the equation of line L in the form y = mx + c. y = … [2] (b) Write down the coordinates of the point where line L crosses the x-axis. ( … , … ) [1] (c) (i) Complete the table of values for y = x 2 + 5x + 3 . x −6 −5 −4 −3 −2 −1 0 1 y 9 −1 −1 [3] (ii) On the grid, draw the graph of y = x 2 + 5x + 3 for -6 G x G 1. [4] (d) (i) On the grid, draw the line y = 6 . [1] (ii) Use your graphs to solve the equation x 2 + 5x + 3 = 6 . x = … or x = … [2]

13 marks

Mark scheme: 6(a)  y = 2 x + 7 2 B1 for 2 x + c, c  7 or B1 for mx + 7 where m is their gradient and m  2 6(b) ( −3.5, 0 ) 1 6(c)(i) 3, −3, −,3 3, 9 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 6(c)(ii) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 6(d)(i) Correct ruled line drawn 1 6(d)(ii) 0.4 to 0.7, −5.7 to −5.4 2 FT their graph and their line B1FT for each

This question in 0580/32 Feb/March 2023

6 (a) Complete the table of values for y = 5 + 3 x - x 2 . x - 2 - 1 0 1 2 3 4 5 y 1 7 –5 [3] (b) On the grid, draw the graph of y = 5 + 3x - x 2 for - 2 G x G 5 . y 12 11 10 9 8 7 6 5 4 3 2 1 – 2 – 1 0 1 2 3 4 5 x – 1 – 2 – 3 – 4 – 5 – 6 [4] (c) Write down the equation of the line of symmetry of the graph. … [1] (d) (i) Complete the table of values for y = 2x + 1. x - 1 0 2 y [2] (ii) On the grid, draw the graph of y = 2x + 1 for - 2 G x G 5 . [1] (e) Write down the coordinates of the two points where the two graphs intersect. ( … , … ) and ( … , … ) [3]

14 marks

Mark scheme: 6(a) −5 5 7 5 1 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 6(b) Correct and accurate curve 4 B3FT for 7 or 8 points correctly plotted or B2FT for 5 or 6 points correctly plotted or B1FT for 3 or 4 points correctly plotted 6(c) x = 1.5 oe 1 6(d)(i) −1 1 5 2 B1 for 2 correct 6(d)(ii) Correct ruled line 1 6(e) (−1.7 to −1.4, −2.4 to −1.8) 3 FT their curve and their line (2.4 to 2.7, 5.8 to 6.4) B2FT for 3 values correct or B1FT for 2 values correct

This question in 0580/31 May/June 2023

7 (a) y 6 L 5 4 3 2 1 – 3 – 2 – 1 0 1 2 3 4 5 x – 1 – 2 – 3 – 4 – 5 – 6 (i) Find the equation of line L. Give your answer in the form y = mx + c . y = … [2] (ii) On the grid, draw the line y = 1. [1] (iii) Write down the coordinates of the point where the two lines intersect. ( … , … ) [1] (b) (i) Complete the table of values for y = x 2 + x - 8 . x - 4 - 3 - 2 - 1 0 1 2 3 4 y 4 - 2 - 8 - 8 - 2 4 [2] (ii) On the grid, draw the graph of y = x 2 + x - 8 for - 4 G x G 4 . y 12 10 8 6 4 2 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 2 – 4 – 6 – 8 – 10 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] (iv) Use your graph to solve the equation x 2 + x - 8 = 0 . x = … or x = … [2]

13 marks

Mark scheme: 7(a)(i) [y =] 1.5x – 2 final answer 2 B1 for 1.5x + c as final answer or B1 for mx − 2, m ≠ 0, as final answer 7(a)(ii) Correct ruled line 1 7(a)(iii) (2, 1) 1 FT their (a)(ii) 7(b)(i) −6 −6 12 2 B1 for one or two correct 7(b)(ii) Correct and accurate curve 4 B3FT for 8 or 9 points accurately plotted or B2FT for 6 or 7 points accurately plotted or B1FT for 4 or 5 points accurately plotted 7(b)(iii) x = −12 oe 1 7(b)(iv) −3.5 to −3.3 2.3 to 2.5 2 B1FT for each

This question in 0580/32 May/June 2023

6 y 8 L 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 (a) Write down the equation of line L in the form y = mx + c . y = … [2] (b) (i) Complete the table of values for y = x 2 - 3x - 3 . x - 2 - 1 0 1 2 3 4 5 y 1 - 5 - 5 1 [2] (ii) On the grid, draw the graph of y = x 2 - 3x - 3 for - 2 G x G 5 . [4] (c) (i) Write down the coordinates of the lowest point of the graph of y = x 2 - 3x - 3 . ( … , … ) [1] (ii) On the grid, draw the line of symmetry of the graph of y = x 2 - 3x - 3 . [1] (iii) Write down the equation of the line of symmetry. … [1] (d) Write down the coordinates of the point where line L intersects the graph of y = x 2 - 3x - 3 for x 2 0 . ( … , … ) [1]

12 marks

Mark scheme: 6(a)  y  2 x  3 final answer 2 B1 for –2x + c as final answer or B1 for mx + 3, m  0, as final answer 6(b)(i) 7, 3, ,3 7 2 B1 for 2 or 3 correct 6(b)(ii) Correct and accurate curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 6(c)(i) 1.5, k  1 where –5.5 ⩽ k < –5 6(c)(ii) Line x  1.5 drawn accurately 1 6(c)(iii) x  1.5 oe 1 6(d)  3,  3  1 FT their curve with line L

This question in 0580/33 May/June 2023

8 y L 6 5 4 3 2 1 x - 3 - 2 - 1 0 1 2 3 4 5 6 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 (a) Find the equation of line L in the form y = mx + c . y = … [2] (b) (i) On the grid, draw the line y = x . [1] (ii) Write down the coordinates of the point where the line y = x intersects line L. ( … , … ) [1] 8(c) (i) Complete the table of values for y = . x x −5 −4 −3 −2 −1 1 2 3 4 5 y −1.6 −2.7 2.7 1.6 [3] 8 (ii) On the grid, draw the graph of y = for - 5 G x G - 1 and 1 G x G 5 . x y 8 7 6 5 4 3 2 1 x - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 [4]

11 marks

Mark scheme: 8(a) [y =] 2x − 5 final answer 2 B1 for answer of 2x + c or mx – 5 (m ≠ 0) 8(b)(i) Correct ruled line 1 8(b)(ii) (5, 5) 1 FT their ruled y = x 8(c)(i) −2 −4 −8 8 4 2 3 B2 for 3, 4 or 5 correct or B1 for 1 or 2 correct 8(c)(ii) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted

This question in 0580/31 Oct/Nov 2023

5 (a) (i) Complete the table of values for y =- x 2 + 5x + 7 . x -1 0 1 2 3 4 5 6 y 11 11 1 [3] (ii) On the grid, draw the graph of y =- x 2 + 5x + 7 for - 1 G x G 6 . y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 – 1 0 1 2 3 4 5 6 x [4] (iii) (a) Write down the equation of the line of symmetry of the graph. … [1] (b) The points ( -8, -97) and ( t, -97) also lie on the graph of y =- x 2 + 5 x + 7 . Use symmetry to find the value of t. t = … [1] (b) Write down the gradient of the line y = 9x - 4 . … [1] (c) Write down the equation of a line parallel to y =-5x + 19 . y = … [1] (d) y 7 6 L 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 Find the equation of line L in the form y = mx + c . y = … [2] (e) Make x the subject of the formula y = mx + c . x = … [2]

15 marks

Mark scheme: 5(a)(i) 1, 7, 13, 13, 7 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 5(a)(ii) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 5(a)(iii)(a) x = 2.5 oe 1 5(a)(iii)(b) 13 1 5(b) 9 1 5(c) y = −5 x + k where k  19 1 5(d)  y =  − x + 2 final answer 2 B1 for  y =  − x + c or  y =  mx + 2 where m is their gradient 5(e) − c 2 M1 for a correct first step  x = y oe final answer m y c y −=c mx or = x + m m

This question in 0580/32 Feb/March 2024

9 (a) Line L has a gradient of 4 and passes through the point (0, 3). Write down the equation of line L in the form y = mx + c . y = … [1] (b) Line G has the equation y = 2 - 6x . Line G passes through the point (a, 5). Find the value of a. a = … [3] (c) (i) Complete the table of values for y = x 2 - 6 . x -4 -3 -2 -1 0 1 2 3 4 y 10 -2 -5 -5 -2 10 [2] (ii) On the grid, draw the graph of y = x 2 - 6 for - 4 G x G 4 . y 10 9 8 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] (iv) Use your graph to solve the equation x 2 - 6 = 0 for x 2 0 . x = … [1]

12 marks

Mark scheme: 9(a) [y =] 4x + 3 1 9(b) – 12 oe 3 B1 for 5 = 2 – 6a M1 for rearranging their linear equation to ra = s 9(c)(i) 3 –6 3 2 B1 for 2 correct 9(c)(ii) correct curve 4 B3FT for 8 or 9 points plotted accurately or B2FT for 6 or 7 points plotted accurately or B1FT for 4 or 5 points plotted accurately 9(c)(iii) x = 0 1 9(c)(iv) 2.3 to 2.6 1 FT their point of intersection

This question in 0580/33 May/June 2024

8 (a) y 8 7 L 6 5 4 3 2 1 0 x 0 1 2 3 4 5 6 7 8 (i) Find the equation of line L in the form y = mx + c . y = … [2] (ii) (a) Complete the table of values for y = 8 - 2x . x 0 2 4 y 4 [2] (b) On the grid, draw the graph of y = 8 - 2x for 0 G x G 4 . [1] (iii) Find the coordinates of the point where line L intersects the graph of y = 8 - 2x . ( … , … ) [1] (b) (i) Complete the table of values for y = x 2 - 4x - 4 . x -2 -1 0 1 2 3 4 5 6 y 8 -4 -8 -4 8 [2] (ii) On the grid, draw the graph of y = x 2 - 4x - 4 for - 2 G x G 6 . y 8 7 6 5 4 3 2 1 -2 -1 0 1 2 3 4 5 6 x -1 -2 -3 -4 -5 -6 -7 -8 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] (iv) Use your graph to solve the equation x 2 - 4x - 4 = 0 . x = … or x = … [2]

15 marks

Mark scheme: 8(a)(i) [y =] 12 x + 2 final answer 2 1 B1 for x + c or y = m x + 2 2 where m is their gradient and m ≠ 0 8(a)(ii)(a) 8 [4] 0 2 B1 for each 8(a)(ii)(b) Correct graph 1 8(a)(iii) 2.4 3.2 1 FT their graph 8(b)(i) 1 −7 −7 1 2 B1 for 2 or 3 correct 8(b)(ii) Correct curve 4 B3FT for 8 or 9 points plotted correctly OR B2FT for 6 or 7 points plotted correctly OR B1FT for 4 or 5 points plotted correctly 8(b)(iii) x = 2 oe 1 8(b)(iv) −0.7 to −0.9 2 FT their graph B1 for each 4.7 to 4.9

This question in 0580/32 Oct/Nov 2024

21 The line L is shown on the grid. y 18 L 16 14 12 10 8 6 4 2 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 2 – 4 Find the equation of the line L. Give your answer in the form y = mx + c . y = … [3]

3 marks

Mark scheme: 21 3x + 5 3 B2 for 3x + c or (2.7 to 3.3)x + (4.8 to 5.2) or for a correct unsimplified answer or M1 for a correct rise/run or for a right-angled triangle with run = a and rise = b marked on graph oe where b ÷ a = 2.7 to 3.3 or B1 for mx + 5 or mx + (4.8 to 5.2) where m is their gradient

This question in 0580/31 Oct/Nov 2025

4 y 4 P 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 (a) Write down the coordinates of point P. ( … , … ) [1] (b) On the grid, draw the line y = x . [1] (c) On the grid, draw the line that goes through point P and is perpendicular to the line y = x . [1]

3 marks

Mark scheme: 4(a) –2, 3 1 4(b) correct ruled line y = x 1 4(c) correct ruled line y =−+x 1 1 FT their straight y = x

This question in 0580/32 Oct/Nov 2025

25 (a) The grid shows line L. y 5 L 4 3 2 1 0 x 0 1 2 3 4 5 6 7 8 9 10 Find the equation of line L in the form y = mx + c . y = … [3] (b) Another line, R, has equation y =- 3x + 5 . Find the equation of the line which is parallel to R and goes through the point (1, 5). Give your answer in the form y = mx + c . y = … [2]

5 marks

Mark scheme: 25(a) 1 3 1 [ y = ] x + 2 final answer B2 for x + c 4 4 or for a correct unsimplified answer or M1 for a correct rise/run or for a right-angled triangle with run = a and rise = b labelled with b 1 numerical values where = oe a 4 or B1 for mx + 2 where m is their gradient 25(b) [ y = ] −3x + 8 final answer 2 B1 for −3x + c as answer, c ≠ 5 or M1 for 5 = −3[  1] + c

This question in 0580/32 Oct/Nov 2025

26 Line L is drawn on the grid. y 22 L 20 18 16 14 12 10 8 6 4 2 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 2 – 4 – 6 Find the equation of line L in the form y = mx + c . y = … [3]

3 marks

Mark scheme: 26 [y=] 3x + 9 cao final answer 3 B2 for 3x + c or for a correct unsimplified answer or M1 for a correct rise/run or for a right-angled triangle with run = 1 and rise = 3 marked on graph oe or B1 for mx + 9 where m is their gradient

This question in 0580/33 Oct/Nov 2025