Equations — Paper 3 · IGCSE Mathematics (9-1) 0980

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 0980/32 May/June 2019

10 (a) Solve these equations. (i) 5x =- 30 x = … [1] (ii) 4x - 2 = 28 x = … [2] (iii) 3( 2x + 7) = 12 x = … [3] (b) Solve the simultaneous equations. You must show all your working. 5x - 2y = 44 2x + 3y = 10 x = … y = … [4]

10 marks

Mark scheme: 10(a)(i) −6 1 10(a)(ii) 7.5 or 7 12 2 M1 for 4x = 28 + 2 oe 10(a)(iii) −1.5 or −1 12 3 M1 for 6x + 21 [= 12] or 2x + 7 = 12 ÷ 3 or better M1 for their 6x = 12 – their 21 or 2x = their 4 – 7 10(b) Equating coefficients of one variable M1 Accept any correct method such as by multiplying the equations by substitution e.g. scalars e.g. M1 for rearranging one equation to make 15x – 6y = 132 either x or y the subject 4x + 6y = 20 M1 for substituting their rearranged equation into the other equation Adding or subtracting equations to M1 A1 A1 as in other method eliminate one variable e.g. 19x = 152 If 0 scored, SC1 for two values that satisfy one equation or SC1 for two correct answers [x =] 8 A1 without any correct working [y =] −2 A1

This question in 0980/32 May/June 2020

4 (a) Simplify. 6a - 3b + 2a - 4b … [2] (b) Expand. 5 ( x - 3) … [1] (c) Solve these equations. x (i) = 18 3 x = … [1] (ii) 5x + 18 = 8 x = … [2] (iii) 12x - 3 = 4x + 21 x = … [2] (d) 6 10 # 6 x = 6 2 Find the value of x. x = … [1] (e) The Fraser family and the Singh family go to the cinema. The Fraser family buys 6 adult tickets and 2 child tickets for $124. The Singh family buys 3 adult tickets and 5 child tickets for $100. Find the price of an adult ticket and the price of a child ticket. Adult ticket $ … Child ticket $ … [5]

14 marks

Mark scheme: 4(a) 8a – 7b 2 B1 for 8a or –7b in final answer or for 8a – 7b seen then spoilt 4(b) 5x – 15 1 4(c)(i) 54 1 4(c)(ii) −2 2 M1 for 18 8 5x = 8 – 18 or x + = or better oe 5 5 4(c)(iii) 3 2 M1 for 12x – 4x = 21 + 3 or better oe 4(d) −8 cao 1 4(e) 17.5[0] 5 B1 for 6a + 2c = 124 B1 for 3a + 5c = 100 9.5[0] oe M1FT for a correct method to eliminate one variable A1 for 17.5[0] A1 for 9.5[0] If 0 scored after B0, B1 or B2, SC1 for two values that satisfy one of the/their original equations or family

This question in 0980/31 Oct/Nov 2020

9 (a) Complete the table of values for y = . x x - 5 - 3 - 2 - 1 1 2 3 5 y - 15 15 [3] 15 (b) On the grid, draw the graph of y = for - 5 G x G - 1 and 1 G x G 5 . x y 16 14 12 10 8 6 4 2 0 x – 5 – 4 – 3 – 2 – 1 1 2 3 4 5 – 2 – 4 – 6 – 8 – 10 – 12 – 14 – 16 [4] (c) On the grid, draw the line y = 6 . [1] 15 (d) Use your graph to solve = 6 . x x = … [1]

9 marks

Mark scheme: 9(a) −3 −5 −7.5 7.5 5 3 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 9(b) Correct curve 4 B3FT for 7 or 8 points plotted correctly or B2FT for 5 or 6 points plotted correctly or B1FT for 3 or 4 points plotted correctly 9(c) Correct ruled line 1 9(d) 2.5 or 2.4 to 2.6 1 FT their line (y = k) and their curve

This question in 0980/31 Oct/Nov 2020

4 (a) Put a ring around the fraction that is equivalent to . 12 35 20 49 82 64 62 36 84 144 110 [1] (b) Write these numbers in order, starting with the smallest. 7 8 2 0.6 58% 12 13 3 … 1 … 1 … 1 … 1 … [2] smallest (c) Write 0.724 as a fraction in its simplest form. … [1] (d) The mass, m grams, of a ball is 415 g, correct to the nearest 5 grams. Complete the statement about the value of m. … G m 1 … [2] (e) Ruth uses three-quarters of a bag of flour to make one cake. Work out the number of bags of flour she needs to buy to make 7 cakes. … [3] (f) A tin of soup costs $t and a packet of biscuits costs $p. (i) 3 tins of soup and 2 packets of biscuits cost $15.50 . Complete the equation. 3t + 2p = … [1] (ii) 5 tins of soup and 4 packets of biscuits cost $28.50 . Write down another equation in terms of t and p. … [1] (iii) Solve the two simultaneous equations. You must show all your working. t = … p = … [3]

14 marks

Mark scheme: 4(a) 49 1 84 4(b) 7 8 2 2 B1 for 4 in correct order 58% 0.6 or M1 for 0.583[...], [0.6], 0.58, 0.615[…] 12 13 3 or 0.61 or 0.62, 0.66[...] or 0.67 or 0.667 or 0.7 oe 4(c) 181 1 cao 250 4(d) 412.5 417.5 2 B1 for each If 0 scored, SC1 for both correct but reversed 4(e) 6 3 3 M1 for 7 × oe 4 1 A1 for 5.25 or 5 4 4(f)(i) 15.5[0] 1 4(f)(ii) 5t + 4p = 28.5[0] 1 4(f)(iii) For correct method to eliminate one M1 FT their two linear equations variable [t =] 2.5 A1 [p =] 4 A1 If 0 scored, SC1 for 2 values satisfying one of the correct equations or their (f)(i) or (f)(ii) or SC1 if no working shown, but 2 correct answers

This question in 0980/32 May/June 2021

(c) Write down the order of rotational symmetry of the graph. … [1] (d) (i) On the grid, plot and join the points (-8, -3) and (6, 4). [2] 18 (ii) Write down the values of x where this line intersects the graph of y = . x x = … and x = … [2] (iii) Find the equation of this line in the form y = mx + c . y = … [2]

14 marks

Mark scheme: 5(a) −2.25 −4.5 −9 9 4.5 2.25 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 5(b) 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 5(c) 2 1 5(d)(i) (−8, −3) and (6, 4) plotted and joined in a 2 B1 for one point correctly plotted or both ruled line correctly plotted but not joined, or ruled 5(d)(ii) −7.3 to −6.9 and 4.9 to 5.3 2 B1FT for each 5(d)(iii) 1 2 1 [y =] x + 1 oe final answer B1 for x + c (c ≠ +1) or 2 2  1  kx + 1  k ≠ 0 or   2  or B1FT for (their m)x + c or kx + their intercept (k ≠ 0)

This question in 0980/32 May/June 2021

3 (a) a (i) Write down the mathematical name for the type of angle a. … [1] (ii) Measure angle a. … [1] (b) Kate describes a quadrilateral. • All the sides are the same length. • It has only two lines of symmetry. (i) Draw a sketch of this quadrilateral. [1] (ii) Write down the mathematical name for this quadrilateral. … [1] (iii) One of the interior angles of this quadrilateral is 70°. Work out the other three interior angles. … , … , … [2] (c) The diagrams show the angles in a triangle and two angles on a straight line. 2y° NOT TO SCALE 6y° x° x° x° (i) The triangle is used to write down an equation in terms of x and y. 2x + 2y = 180 Give the geometrical reason why this equation is correct. Reason … [1] (ii) Use the diagram with two angles on a straight line to write down another equation in terms of x and y. … [1] (iii) Solve these simultaneous equations. You must show all your working. x = … y = … [3]

11 marks

Mark scheme: 3(a)(i) Obtuse 1 3(a)(ii) 113 1 3(b)(i) Sketch of a rhombus 1 3(b)(ii) Rhombus cao 1 3(b)(iii) 70, 110, 110 2 B1 for 110 or M1 for (360 – 70 – 70 ) ÷ 2 oe or M1 for 70  70  x  x  360 oe soi 3(c)(i) Angles [in a] triangle add to 1 180 3(c)(ii) x  6 y  180 oe 1 3(c)(iii) Correctly eliminating one M1 FT their (c)(ii), if linear in x and y variable [ x  ] 72 A1 [ y  ]1 8 A1 If M0 scored, SC1 for 2 values satisfying one of the original equations or their equations in (c)(ii) SC1 if no working shown but 2 correct answers given

This question in 0980/32 May/June 2022

4 (a) Find (i) a multiple of 3 between 70 and 80, … [1] (ii) a factor of 63 between 5 and 10, … [1] (iii) a cube number between 60 and 90, … [1] (iv) the reciprocal of 7. … [1] 2 (b) Work out of 84. 7 … [1] (c) Find the value of (i) 3 3375 , … [1] (ii) 120. … [1] (d) Rana hires a car. The cost is $74 per day plus a delivery cost of $17.50 . Rana pays a total of $461.50 . Calculate the number of days that Rana hires the car. … days [2] (e) A train to town A leaves a station every 25 minutes. A train to town B leaves the same station every 45 minutes. Both trains leave at 08 00. Find the next time both trains leave together. … [3]

12 marks

Mark scheme: 4(a)(i) 72 or 75 or 78 1 4(a)(ii) 7 or 9 1 4(a)(iii) 64 1 4(a)(iv) 1 1 or 0.143 or 0.142[8..] 7 4(b) 24 1 4(c)(i) 15 1 4(c)(ii) 1 1 4(d) 6 nfww 2 ( 461.5 –17.5 ) M1 for oe 74 4(e) 11 45 3 B2 for 225 or 3 hr 45 mins or M1 for 225k or 3  3  5  5 or [25 =] 5  5 and [45 =] 3  3  5 or two correct factor trees/tables of both 25 and 45 OR M2 for listing times/multiples of both 25 and 45 to at least 11 45 or 225 or M1 for listing at least 3 consecutive times/multiples of each correctly or one full list

This question in 0980/31 Oct/Nov 2022

7 (a) Simplify. 5g - 3h - 7g + 6h … [2] (b) j = 4k + 7m Find the value of j when k =- 5 and m = 6 . j = … [2] (c) Factorise completely. 14x 3 + 49x … [2] (d) Solve. 8 ( 3t - 9) = 108 t = … [3] (e) (i) 9 24 ' 9 w = 9 5 Find the value of w. w = … [1] (ii) 4x 2 = 256 Find the value of x. x = … [1] (f) Ranjit’s age is x years. Suzi’s age is 3 times Ranjit’s age. Juan’s age is 4 years more than Suzi’s age. The total of their ages is 46 years. Use this information to write down an equation and solve it to find the value of x. x = … [4]

15 marks

Mark scheme: 7(a) –2g +3h final answer 2 B1 for –2g or 3h in final answer or –2g+ 3h seen then spoilt 7(b) 22 2 M1 for 4  –5 + 7  6 or B1 for –20 or [+]42 7(c) 7x(2x2 + 7) final answer 2 B1 for 7(2x3 +7x) or x(14x2 + 49) or correct answer seen then spoilt 7(d) 7.5 3 M1 for a first correct step 24t – 72 = 108 or 3t –9 =13.5 M1FT for a second correct step e.g. 24t =180 or 3t =22.5 7(e)(i) 19 1 7(e)(ii) 8 1 7(f) x + 3x + 3x + 4 = 46 4 M2 for a correct equation which would or 7x + 4 = 46 lead to 7x + 4 = 46 leading to x = 6 or B1 for 3x or 3x + 4 seen M1 for 7x = 42 or for rearranging their equation to ax = b B1 for [x =] 6

This question in 0980/31 Oct/Nov 2022

8 (a) T = 5P + 3Q Find the value of T when P = 6 and Q = 8 . T = … [2] (b) Simplify. 3a - 7b + 2a + 4b … [2] (c) Multiply out. 5 ( 2x - 3y) … [1] (d) Solve. 5x - 1 = 3x + 19 x = … [2] (e) Make t the subject of the formula p = t5 - 3 . t = … [2] (f) Entry to a castle costs $x for an adult and $y for a child. Entry for 2 adults and 3 children costs $15.00 . Entry for 3 adults and 5 children costs $23.50 . Write down a pair of simultaneous equations to show this information and solve them to find the value of x and the value of y. You must show all your working. x = … y = … [6]

15 marks

Mark scheme: 8(a) 54 2 M1 for 5×6 + 3×8 or 30 or 24 8(b) 5a – 3b final answer 2 B1 for 5a or – 3b in final answer or for correct answer seen and spoilt 8(c) 10x – 15y final answer 1 8(d) 10 2 M1 for 5x – 3x = 19 + 1 or better 8(e) p  3 2 M1 for p + 3 = 5t or 5p t 53 oe [t=] oe final answer 5 8(f) 2x + 3y = 15 and 3x + 5y = 23.5 B2 B1 for each correctly equating one set of M1 FT coefficients correct method to eliminate one M1 FT variable Dependent on the coefficients being the same for one of the variables Correct consistent use of addition or subtraction using their equations [x =] 4.5 A1 [y =] 2 A1 If M0 scored, SC1 for 2 values satisfying one of correct equations or their equations

This question in 0980/32 May/June 2023

7 (a) Simplify. 5a + 3b + 2a - 4b … [2] (b) P = 8x + 3y Find the value of x when P = 21 and y =-5 . x = … [2] (c) Make v the subject of the formula S = kv2 . v = … [2] (d) Multiply out and simplify. ( x - 3)( x + 5) … [2] (e) Nasser has x marbles. Selina has 15 more marbles than Nasser. Hanif has 3 times as many marbles as Selina. In total they have 150 marbles. Find the value of x. x = … [5]

13 marks

Mark scheme: 7(a) 7a – b final answer 2 B1 for 7a or –b in final answer or 7a – b seen then spoilt 7(b) 4.5 2 M1 for 21 = 8x + 3  −5 oe or better 7(c) S 2 2 S final answer M1 for v = or S = k  v k k 7(d) x2 + 2x – 15 final answer 2 B1 for three correct terms from x2 – 3x + 5x – 15 7(e) 18 5 B1 for x + 15 or 3  their (x + 15) oe M1 for x + their (x + 15) + their (3( x + 15)) = 150 or better M1 for 5x + 60 = 150 or better or their linear equation simplified to ax + b = 150 M1 for their [ax + b = c] 150 − b solved to x = a

This question in 0980/31 Oct/Nov 2023

2 (a) Simplify. (i) 5a - 6a + 3a … [1] (ii) 6x 2 - 6x - 4x 2 - x … [2] (b) Find the value of c 2 + d 2 when c = 7 and d =- 5 . … [2] (c) The time, T minutes, to cook a chicken with a mass of m kg is T = 35m + 20 . (i) Make m the subject of the formula. m = … [2] (ii) Find the mass of a chicken that takes 83 minutes to cook. … kg [2] (d) Solve these simultaneous equations. You must show all your working. 5x - 6y = 24 15x + 8y = 33 x = … y = … [3]

12 marks

Mark scheme: 2(a)(i) 2a final answer 1 2(a)(ii) 2x2 – 7x final answer 2 B1 for 2x2 or −7x in final answer or for 2x2 – 7x seen then spoilt. 2(b) 74 2 B1 for 49 or 25 2(c)(i) T  20 2 T 20 [m =] final answer M1 for T – 20 = 35m or  m  35 35 35 2(c)(ii) 1.8 2 FT their (c)(i) for 2 marks or 1 mark M1 for (83 – 20) ÷ 35 2(d) Correctly eliminates one variable M1 Making the coefficients the same for one of the variables and correct consistent use of addition or subtraction using their equations alternative substitution method. M1 for correct rearrangement of one equation to make either x or y the subject and correct substitution of their rearrangement into 2nd equation. [x =] 3 A1 If A0 scored SC1 for 2 values satisfying one of the original equations. [y =] −1.5 A1

This question in 0980/32 May/June 2024

10 (a) Solve. 4x - 7 = 3 x = … [2] (b) Simplify. 2 (i) `x6j … [1] (ii) `5x 3 y 4j # `2 x 2 y 2j … [2] (c) Expand and simplify. (i) 4a + 5 - 2 ( a - 1) … [2] (ii) ( d + 7)( d - 3) … [2]

9 marks

Mark scheme: 10(a) 2.5 oe 2 7 3 M1 for 4x = 3 + 7 or for x − = oe 4 4 10(b)(i) x12 final answer 1 10(b)(ii) 10x5y6 final answer 2 B1 for two of 10, x5, y6 correct in final answer or for 10x5y6 seen then spoilt 10(c)(i) 2a + 7 final answer 2 B1 for 2a or + 7 seen in final answer or for 2a + 7 seen then spoilt 10(c)(ii) d 2 + 4d – 21 final answer 2 B1 for d 2 + 7d – 3d – 21 with at least 3 terms correct

This question in 0980/31 Oct/Nov 2024

14 (a) P = 6 a + 5b Find the value of b when P = 25 and a = 3 . b = … [2] (b) Make T the subject of the formula W = kT + y . T = … [2]

4 marks

Mark scheme: 14(a) 1.4 2 M1 for 25 = 6 +3 5b or better 14(b) W − y W y 2 W y T =  or T =  − M1 for W − y = kT or = T + k k k k k final answer

This question in 0980/32 May/June 2025

22 In this question, all lengths are in centimetres. NOT TO 4x - 9 SCALE 2x - 5 2x + 2 The diagram shows a right-angled triangle. (a) Write down an expression, in terms of x, for the perimeter of the triangle. Give your answer in its simplest form. … [2] (b) The perimeter of the triangle is 40 cm. Work out the value of x. x = … [2] (c) Work out the area of the triangle. … cm2 [3]

7 marks

Mark scheme: 22(a) 8x − 12 final answer 2 M1 for 2 x −+5 2 x + 2 + 4 x − 9 soi or better or B1 for 8x or −12 , in final expression or correct expression seen then spoilt 22(b) 6.5 2 12 40 M1 for 8x = 40 + 12 or x − = 8 8 or a correct first step in rearranging their kx + j = 40 k ≠ 0, j ≠ 0 22(c) 60 3 FT their (b) for 3 marks if their (b) > 2.5 M2 for 1  ( 2  their (b) − 5 )  ( 2  their (b) + 2 ) oe or 2 better or M1 for substituting their (b) into 2 x − 5 and 2 x + 2 1 or M1 for  (2 x − 5)  (2 x + 2) 2

This question in 0980/32 May/June 2025

8 Solve. (a) 8x = 32 x = … [1] (b) 6x - 3 = 12 x = … [2] (c) Represent the inequality -4 G x 1 2 on the number line. x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 [2]

5 marks

Mark scheme: 8(a) 4 1 8(b) 2.5 oe 2 3 12 M1 for 6x = 12 + 3 oe or x – = oe 6 6 8(c) 2 B1 for correct line with circles at both ends or for correct circles but line missing –5 –4 –3 –2 –1 0 1 2 3 4

This question in 0980/31 Oct/Nov 2025