Gradient of linear graphs — Paper 2 · IGCSE Mathematics 0580

15 The points (2, 5), (3, 3) and (k, 1) all lie in a straight line. (a) Find the value of k. Answer(a) k = [1] (b) Find the equation of the line. Answer(b) [3]

4 marks

Mark scheme: 15 (a) 4 1 5 − 3 (b) y = –2x + 9 oe 3 M1 oe 2 − 3 M1 substitution of a point into their equation If M1 only then A1ft for y = “m”x + “c” used correctly with their numeric values p 3

This question in 0580/21 May/June 2010

16 A, B and C are points on a circle, centre O. TCD is a tangent to the circle. Angle BAC = 54°. NOT TO SCALE A O 54° D B C T (a) Find angle BOC, giving a reason for your answer. Answer(a) Angle BOC = … because … … [2] 3 (b) When O is the origin, the position vector of point C is e-4 o. (i) Work out the gradient of the radius OC. Answer(b)(i) … [1] (ii) D is the point (7, k). Find the value of k. Answer(b)(ii) k = … [1] __________________________________________________________________________________________

4 marks

Mark scheme: 16 (a) 108 1 Angle at centre is twice angle at 1 circumference oe 4 (b) (i) − oe 1 3 (ii) −1 1  2

This question in 0580/22 Oct/Nov 2014

18 y 7 6 L 5 4 3 2 1 x –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 (a) Work out the gradient of the line L. … [2] (b) Write down the equation of the line parallel to the line L that passes through the point (0, 6). … [2]

4 marks

Mark scheme: 18 (a) 2 cao 2 M1 for rise/run attempted e.g. 4/2 or other correct method for finding gradient or SC1 for y = 2x – 1 as answer (b) y = 2x + 6 oe 2FT FT for y = their(a)x + 6 B1 for y = mx + 6 (m ≠ 0 or 2) or y = 2x [+ k] or y = their(a)x [+ k] (k ≠ 6) or for answer 2x + 6 or answer their(a)x + 6 30 4

This question in 0580/23 May/June 2016

20 y 7 A 6 5 4 3 2 B 1 x 0 1 2 3 4 5 6 7 8 Point A has co-ordinates (3, 6). (a) Write down the co-ordinates of point B. ( … , … ) [1] (b) Find the gradient of the line AB. … [2] (c) Find the equation of the line that • is perpendicular to the line AB and • passes through the point (0, 2). … [3]

6 marks

Mark scheme: 20 (a) ( 7 , 1 ) 1 5 1 (b) ‒1.25 or − or −14 2 M1 for rise/run 4 4 4 −1 (c) y = x + 2 oe 3 B2 for x + 2 or y = x + 2 oe 5 5 their(b) 1 or M1 for −their ( b ) oe 4 or B1 for x seen or [ y = ] mx + 2 (m ≠ 0) 5

This question in 0580/22 Oct/Nov 2016

12 A line has gradient 5. M and N are two points on this line. M is the point (x, 8) and N is the point (k, 23). Find an expression for x in terms of k. x = … [3]

3 marks

Mark scheme: 12 k – 3 or −+3 kk 3 23 − 8 M1 forr 5 = ooe k − x M1 forr 5(k – x) = 223 – 8 or bettter 23 − 88 e.g. [x =]= k − 5

This question in 0580/21 May/June 2017

24 (a) Point A has co-ordinates (1, 0) and point B has co-ordinates (2, 5). Calculate the angle between the line AB and the x-axis. … [3] (b) The line PQ has equation y = 3x - 8 and point P has co-ordinates (6, 10). Find the equation of the line that passes through P and is perpendicular to PQ. Give your answer in the form y = mx + c. y = … [3]

6 marks

Mark scheme: 24(a) 78.7 or 78.69… 3 5 M2 for tan = oe 2 − 1 or M1 for use of tangent oe 24(b) 1 3 1 [ y = ] − x + 12 final answer M1 for gradient = − 3 3 M1 for substituting (6, 10) into y = their mx + c

This question in 0580/21 May/June 2018

5 (a) Find the co-ordinates of the point where the line y = 3x - 8 crosses the y-axis. ( … , … ) [1] (b) Write down the gradient of the line y = 3x - 8 . … [1]

2 marks

Mark scheme: 5(a) (0, –8) 1 5(b) 3 1

This question in 0580/23 May/June 2019

7 Find the gradient of the line that is perpendicular to the line 2y = 3 + 5x. … [2]

2 marks

Mark scheme: 7 2 2 5 − or – 0.4 M1 for gradient = oe soi 5 2

This question in 0580/23 Oct/Nov 2019

13 A straight line joins the points (3k, 6) and (k, -5). The line has a gradient of 2. Find the value of k. k = … [3]

3 marks

Mark scheme: 13 2.75 oe 3 M2 for 6 −−=5 2 ( 3k − k ) oe or better 6 −− 5 or M1 for oe 3k − k If 0 scored, SC1 for −2.75 oe as answer

This question in 0580/23 Oct/Nov 2019

14 Find the gradient of a line that is perpendicular to 8y + 4x = 5 . … [2]

2 marks

Mark scheme: 14 2 2 5 – 4 x M1 for y = oe or better 8

This question in 0580/21 Oct/Nov 2020

12 A straight line, l, has equation y = 5 x + 12 . (a) Write down the gradient of line l. … [1] (b) Find the coordinates of the point where line l crosses the x-axis. ( … , … ) [2] (c) A line perpendicular to line l has gradient k. Find the value of k. k = … [1]

4 marks

Mark scheme: 12(a) 5 1 12(b) 12 2 M1 for 5x + 12 = 0 ( − oe, 0) 5 12(c) 1 1 1 − oe FT 5 −their ( a )

This question in 0580/23 Oct/Nov 2020

16 y 5 4 l 3 2 1 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 (a) Find the gradient of line l. … [2] (b) Find the equation of line l in the form y = mx + c . y = … [2] (c) Find the equation of the line that is perpendicular to line l and passes through the point (12, - 7 ). Give your answer in the form y = m x + c . y = … [3]

7 marks

Mark scheme: 16(a) 3 2 M1 for correct rise over run – or − 0.75 3 4 or B1 for answer oe 4 16(b) 3 2 FT [ y = ] their (a) x + 2 oe [ y = ] − x + 2 oe 4 B1 for [ y = ] their (a) x + c or [ y = ] mx + 2 . 16(c) 4 3 −1 [ y = ] x − 23 oe M1 for gradient 3 their (a) M1 for (12, − 7) substituted into y = their mx + c

This question in 0580/21 May/June 2021

17 Find the gradient of the line that is perpendicular to the line 3y = 4x - 5 . … [2]

2 marks

Mark scheme: 17 3 2 4 x − 5 – or – 0.75 M1 for y = or better 4 3 −1 or for their gradient

This question in 0580/22 May/June 2021

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

2 marks

Mark scheme: 5(a) 5 1 5(b) (0, 7) 1

This question in 0580/22 Feb/March 2022

15 C is the point ( 5, - 1) and D is the point (13, 15). (a) Find the midpoint of CD. ( … , … ) [2] (b) Find the gradient of CD. … [2] (c) Find the equation of the perpendicular bisector of CD. Give your answer in the form y = mx + c . y = … [3]

7 marks

Mark scheme: 15(a) (9, 7) 2 B1 for each 15(b) 2 2 15 – –1 M1 for oe 13 – 5 15(c) 1 23 3 [y =] – x + oe 2 2 final answer 1 M1 for gradient = their (b) oe M1 for correct substitution of their (a) into y = (their m)x + c oe

This question in 0580/22 May/June 2023

18 y 18 16 14 12 10 8 6 4 2 0 x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 The graph of y = f ( x) is drawn on the grid. (a) Draw the tangent to the graph at the point x = 3 . [1] (b) Use your tangent to find an estimate for the gradient of the curve at the point x = 3 . … [2]

3 marks

Mark scheme: 18(a) tangent ruled at x = 3 1 18(b) 4.8 to 5.8 2 dep on a close attempt at a tangent rise M1 for also dep on close attempt at tangent run

This question in 0580/22 May/June 2024

10 Find the gradient of the line joining the points ( - 2 , 7) and (3, 1). … [2]

2 marks

Mark scheme: 10 6 2 1  7  oe M1 for oe 5 3 2

This question in 0580/23 May/June 2024

24 (a) A is the point (a, 12) and B is the point (b, 27). (i) Find the y-coordinate of the midpoint of AB. … [1] (ii) The line AB has gradient 3. Find an expression for a in terms of b. a = … [3] (b) D is the point (22, 34) and E is the point (23, 39). D is the point on CE such that 2CE = 5DE. Find the coordinates of C. ( … , … ) [3]

7 marks

Mark scheme: 24(a)(i) 19.5 1 24(a)(ii) [a =] b – 5 oe 3 27 − 1 2 M1 for 3 = oe b − a M1 for 3(b – a) = 27 – 12 or better 27 − 1 2 e.g. [a =] b – 3 24(b) 41 53 3 SC2 for answer (25.5, 51.5) or (20.5, 26.5) or ( , ) 2 2 51 103 ( , ) 2 2 OR 5 M2 for 23 − ( 23 − 22 ) 2 5 or 39 − ( 39 − 34 ) oe 2 or sketch 12.5 2.5  2.5   −2.5  or vector e.g.   or    12.5   −12.5  or M1 for 5 1 5 5 ( 23 − 22 ) or ( 39 − 34 ) 2 2 1  −1  or vector e.g.  or   5  −5 

This question in 0580/23 May/June 2025