Inequalities — Paper 4 · IGCSE Mathematics 0580

9 Bernie buys x packets of seeds and y plants for his garden. He wants to buy more packets of seeds than plants. The inequality x 2 y shows this information. He also wants to buy • less than 10 packets of seeds • at least 2 plants. (a) Write down two more inequalities in x or y to show this information. … … [2] (b) Each packet of seeds costs $1 and each plant costs $3. The maximum amount Bernie can spend is $21. Write down another inequality in x and y to show this information. … [1] (c) The line x = y is drawn on the grid. Draw three more lines to show your inequalities and shade the unwanted regions. y 10 9 8 7 6 5 4 3 2 1 x 0 2 4 6 8 10 12 14 16 18 20 22 [5] (d) Bernie buys 8 packets of seeds. (i) Find the maximum number of plants he can buy. … [1] (ii) Find the total cost of these packets of seeds and plants. $ … [1]

10 marks

This question in 0580/42 Feb/March 2017

9 (a) Find the equation of the straight line that is perpendicular to the line y = x + 1 and passes through 2 the point (1, 3). … [3] (b) y 8 7 6 5 4 R 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 11 12 (i) Find the three inequalities that define the region R. … … … [4] (ii) Find the point (x, y), with integer co-ordinates, inside the region R such that 3x + 5y = 35 . ( … , … ) [2]

9 marks

Mark scheme: 9(a) y = −2 x + 5 oe 3 B2 for –2x + 5 or 1 M1 for gradient = −÷1 or better 2 M1 for substituting (1, 3) into y = (their m)x + c oe If 0 scored SC1 for (1, 3) satisfying their wrong 1 equation (c ≠ 0) with gradient ≠ 2 9(b)(i) x . 2 oe 1 SC3 for x > 2 and y < 5 and y > x 2 y - 5 oe OR B1 for x ⩾ 2 1 B1 for y ⩽ 5 y . x oe 4 1 2 B2 for y ⩾ x 2 or M1 for y ⩾ kx (k > 0) OR SC2 for all three boundary lines identified but with incorrect sign(s) If 0 scored SC1 for one or two correct boundary lines with incorrect sign(s) 9(b)(ii) (5, 4) 2 M1 for one trial of an integer point inside region or for 3 x + 5 y = 35 drawn

This question in 0580/43 May/June 2018

10 (a) Solve the simultaneous equations. You must show all your working. 6x + 5y = 27 5x - 3y = 44 x = … y = … [4] (b) y is inversely proportional to (x + 3) 2 . When x = 2, y = 8. Find y when x = 7. y = … [3] (c) Solve the inequality. 3 (x - 2) 1 7 (x + 2) … [3]

10 marks

Mark scheme: 10(a) correctly equating one set of coefficients M1 or making x or y the subject of one equation correctly correct method to eliminate one variable M1 or substitution for x or y for their rearranged formula x = 7 A2 A1 for one correct value If A0 scored, SC1 for 2 values satisfying y = −3 one of the original equations or if no working shown, but 2 correct answers given 10(b) 2 3 k M1 for y = oe ( x + 3) 2 their k M1 for y = oe (7 + 3) 2 OR M2 for 8 ( 2 + 3 ) 2 = y ( 7 + 3 ) 2 oe 10(c) x > −5 final answer 3 M1 for 3 x − 6 < 7 x + 14 M1 for their ( −6) − their14 < 7 x − 3 x oe

This question in 0580/42 Feb/March 2019

2 (a) Solve. 5x - 17 = 7x + 3 x = … [2] (b) Find the integer values of n that satisfy this inequality. - 7 1 4n G 8 … [3] (c) Simplify. (i) a 3 # a 6 … [1] (ii) (5xy 2 ) 3 … [2] 1 12 - 3 27x (iii) 3 f 64y p … [3]

11 marks

Mark scheme: 2(a) –10 2 M1 for –17 – 3 = 7x – 5x oe or better 2(b) −1, 0, 1, 2 final answer 3 B2 for 3 correct values and no incorrect values or 4 correct values and one incorrect value 7 or M2 for − < n - 2 oe 4 7 or M1 for − < n - k or k < n- 2 oe 4 2(c)(i) a9 1 2(c)(ii) 125x3y6 final answer 2 B1 for 2 correct elements if in form kxnym 2(c)(iii) []1 3 4 [ −1] 4 y  3 x  4 final answer B2 for  [1]  oe seen 3 x  4 y  OR B1 for 3x4 or 4y[1] and 13 3  64 y  M1 for  12  oe  27 x  64 y [1] 0.333 x − 4 If 0 scored, SC1 for or seen 27 x 4 0.25 y − 1

This question in 0580/43 May/June 2019

9 A car hire company has x small cars and y large cars. The company has at least 6 cars in total. The number of large cars is less than or equal to the number of small cars. The largest number of small cars is 8. (a) Write down three inequalities, in terms of x and/or y, to show this information. … , … , … [3] (b) A small car can carry 4 people and a large car can carry 6 people. One day, the largest number of people to be carried is 60. Show that 2x + 3y G 30 . [1] (c) y 10 9 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 By shading the unwanted regions on the grid, show and label the region R that satisfies all four inequalities. [6] (d) (i) Find the number of small cars and the number of large cars needed to carry exactly 60 people. … small cars, … large cars [1] (ii) When the company uses 7 cars, find the largest number of people that can be carried. … [2] Question 10 is printed on the next page.

13 marks

Mark scheme: 9(a) x + y ⩾ 6 oe 3 B1 for each y ⩽ x oe x ⩽ 8 9(b) 4x + 6y ⩽ 60 1 9(c) Correct region indicated cao 6 B1 for x + y = 6 ruled and long enough B1 for x = y ruled and long enough B1 for x = 8 ruled and long enough B2 for 2x + 3y = 30 ruled and long enough or B1 for ruled line through (0, 10) or (15, 0) but not y = 10 or x = 15 9(d)(i) 6, 6 1 9(d)(ii) 34 2 M1 for trying 4x + 6y with (4, 3) or (5, 2) or (6, 1) or (7, 0)

This question in 0580/41 Oct/Nov 2019

7 (a) Naga has n marbles. Panav has three times as many marbles as Naga. Naga loses 5 marbles and Panav buys 10 marbles. Together they now have more than 105 marbles. Write down and solve an inequality in n. … [3] (b) y is inversely proportional to x2. When x = 4 , y = 7.5 . Find y when x = 5 . y = … [3] (c) Find the nth term of each sequence. (i) 4 2 0 - 2 - 4 … … [2] (ii) 1 7 17 31 49 … … [2]

10 marks

Mark scheme: 7(a) n − 5 + 3n + 10 > 105 or better B1 n > 25 final answer B2 M1 for 4n > 100 7(b) 4.8 3 k M1 for y = or better x 2 their k M1 for [ y = ] 5 2 OR M2 for y × 5 2 = 7.5 × 4 2 7(c)(i) 6 − 2n oe final answer 2 B1 for answer 6 − kn (k ≠ 0) oe or answer j−2n oe or for correct expression shown in working and then spoilt 7(c)(ii) 2 n 2 − 1 oe final answer 2 B1 for 2nd diff = 4 or a quadratic expression or for correct expression shown in working and then spoilt

This question in 0580/42 Feb/March 2020

6 Raheem makes baskets and mats. Each week he makes x baskets and y mats. He makes fewer than 10 mats. The number of mats he makes is greater than or equal to the number of baskets he makes. (a) One of the inequalities that shows this information is y 1 10 . Write down the other inequality. … [1] (b) He takes 2 14 hours to make a basket and 1 12 hours to make a mat. Each week he works for a maximum of 22.5 hours. Show that 3x + 2y G 30 . [2] (c) On the grid, draw three straight lines and shade the unwanted regions to show these inequalities. y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x [5] (d) He makes $40 profit on each basket he sells and $28 profit on each mat he sells. Calculate the maximum profit he can make each week. $ … [2]

10 marks

Mark scheme: 6(a) y ⩾ x oe 1 6(b) 2.25x + 1.5y ⩽ 22.5 oe M1 One step shown to A1 3x + 2y ⩽ 30 6(c) y = 10 ruled 1 Broken line 3x + 2y = 30 ruled B2 Solid line B1 for line passing through (0, 15) or (10, 0) y = x ruled B1 Solid line Correct region indicated B1 6(d) 412 2 M1 for (4, 9) identified or for evaluation 40x + 28y for an integer point in the region (x > 0 and y > 0)

This question in 0580/41 May/June 2020

4 (a) Solve the inequality. 3m + 12 G 8m - 5 … [2] (b) Solve the equation. 2x + 5 14 = 3 - x 15 x = … [3] (c) Solve the simultaneous equations. You must show all your working. y = 4 - x x 2 + 2y 2 = 67 x = … , y = … x = … , y = … [6]

11 marks

Mark scheme: 4(a) m ≥ 3.4 oe final answer 2 M1 for 12 + 5 ≤ 8m – 3m or better or 3m – 8m ≤ –5 – 12 or better 4(b) x = − 0.75 oe 3 M1 for 15 ( 2 x + 5 ) = 14 ( 3 − x ) B1 for 30 x + 75 = 42 − 14 x or better 4(c) 3 x 2 − 16 x − 35[ = 0] or M3 M1 for x 2 + 2 ( 4 − x ) 2 = 67 3 y 2 − 8 y − 51[ = 0] 2 2 or ( 4 − y ) + 2 y = 67 seen B1 for 16 − 8x + x 2 or 16 − 8y + y 2 (3x + 5)(x – 7) [= 0] M1 or for correct factors for their equation or (3y – 17)(y + 3)[= 0] or for correct use of quadratic formula or completing the square for their equation x = 7, y = −3 B2 5 B1 for x = 7, x = − 3 5 2 2 x = − , y = 5 or for y = −3, y = 5 3 3 3 or for a correct pair of x and y values

This question in 0580/43 May/June 2020

8 (a) Factorise completely. 3a 2 b - ab 2 … [2] (b) Solve the inequality. 3x + 12 1 5x - 3 … [2] (c) Simplify. 3 3x 2 y 4 ` j … [2] (d) Solve. 2 6 = x 2 - x x = … [3] (e) Expand and simplify. ( x - 2)( x + 5)( 2x - 1) … [3] (f) Alan invests $200 at a rate of r% per year compound interest. After 2 years the value of his investment is $206.46 . (i) Show that r 2 + 200r - 323 = 0 . [3] (ii) Solve the equation r 2 + 200r - 323 = 0 to find the rate of interest. Show all your working and give your answer correct to 2 decimal places. r = … [3]

18 marks

Mark scheme: 8(a) ab(3a – b) final answer 2 B1 for a(3ab – b2) or b(3a2 – ab) or ab(3a – b) seen 8(b) x > 7.5 final answer 2 B1 for 12+3 < 5x – 3x oe 8(c) 27x6y12 2 B1 for two of 27, x6 and y12 correct 8(d) 1 3 M2 for 4 = 6x + 2x or better 0.5 or 2 or M1 for 2(2 – x) = 6x oe 8(e) 2x3 + 5x2 – 23x + 10 final answer 3 B2 for correct expansion of three brackets unsimplified B1 for correct expansion of two brackets with at least 3 terms correct 8(f)(i) 2 M1  r  200  1 +  = 206.46 oe  100  2r r 2 M1 1 + + oe 100 100 2 r2 + 200r – 323 = 0 A1 Correct solution reached with no errors or omissions seen If 0 scored, SC1 for 200( n ) 2 = 206.46 8(f)(ii) 2 B2 2 −200 + 200 − 4(1)( −323) B1 for 200 − 4(1)( − 323) or (r + 100)2 2 × 1 −200 + q 2 B1 for or r = 323 + 100 – 100 2 × 1 OR  206.46  B2 for 100  − 1     200  206.46 or B1 for 200 1.60 cao final answer B1

This question in 0580/41 Oct/Nov 2020

9 (a) Find the integer values that satisfy the inequality 2 1 2 x G 10 . … [2] (b) Factorise completely. (i) 6y 2 - 15 xy … [2] (ii) y 2 - 9x 2 … [2] (c) Simplify. 3 2 - x - 1 2x + 1 … [3] (d) The straight line y = 3x + 2 intersects the curve y = 2x 2 + 7x - 11 at two points. Find the coordinates of these two points. Give your answers correct to 2 decimal places. ( … , … ) ( … , … ) [6]

15 marks

Mark scheme: 9(a) 2, 3, 4, 5 2 B1 for 3 correct and no extra or 4 correct and one extra or M1 for 1 < x ⩽ 5 9(b)(i) 3y (2y – 5x) 2 B1 for 3(2y2 – 5xy) or y (6y – 15x) or for the correct answer seen and then spoiled 9(b)(ii) (y – 3x) (y + 3x) 2 B1 for (y + 3) (y – 3) 9(c) 4 x + 5 3 M1 for 3(2x + 1) – 2(x – 1) oe isw ( x − 1)(2 x + 1) M1 for (x – 1)(2x + 1) oe isw 4 x + 5 or final answer 2 x 2 − x − 1 9(d) (1.74 , 7.21 to 7.24) 6 For the y values accept any value rounded and to 2 decimal places in the given range (–3.74 , –9.20 to –9.22) cao B5 for (1.74 , 7.21 to 7.24) or (–3.74 , –9.20 to –9.22) or x = 1.74 and x = – 3.74 OR M2 for 2x2 + 4x – 13 = 0 or 2y2 + 4y – 133 = 0 or M1 for 2x2 + 7x – 11 = 3x + 2  y − 2  2  y − 2  or y = 2   + 7   – 11  3   3  AND FT their quadratic expression (not 2x2 + 7x – 11) −±4 4 2 −×4 2 × −13 M2FT for 2 × 2 15 or −±1 oe 2 or M1FT for 4 2 − 4 × 2 × −13 oe −+4 k −−4 k or for or 2 × 2 2 × 2 or (x + 1)2 [ – 13/2 – 1 = 0]

This question in 0580/43 Oct/Nov 2020

7 (a) x –2 1 Write down the inequality in x shown by the number line. … [2] (b) (i) Write x 2 + 4x + 1 in the form ( x + p) 2 + q . … [2] (ii) Use your answer to part (b)(i) to solve the equation x 2 + 4x + 1 = 0 . x = … or x = … [2] (iii) Use your answer to part (b)(i) to write down the coordinates of the minimum point on the graph of y = x 2 + 4x + 1. ( … , … ) [2] (iv) On the diagram, sketch the graph of y = x 2 + 4x + 1. y O x [2]

10 marks

Mark scheme: 7(a) –2 < x ⩽ 1 2 B1 for –2 < x or x ⩽ 1 7(b)(i) ( x + 2 ) 2 − 3 2 M1 for ( x + 2 ) 2 + k 7(b)(ii) ( x + 2 ) 2 = 3 M1 FTdep their (b)(i) for k < 0 –3.73 or –3.732... and B1 –0.268 or –0.2679... 7(b)(iii) (–2, –3) 2 2 FT their ( x + 2 ) − 3 B1 for each coordinate 7(b)(iv) Correct sketch 2 Parabola with minimum point in correct 2222 quadrant and both x-intercepts negative and positive y-intercept 1111 4444 -2-2-2-2 0000 0000 2222 4444 B1 for parabola with minimum point. -1-1-1-1 -2-2-2-2 -3-3-3-3 -4-4-4-4

This question in 0580/43 May/June 2021

6 (a) Solve. (i) 4 ( 2x - 3) = 24 x = … [3] (ii) 6x + 14 2 6 … [2] (b) Rearrange the formula V = 2x 3 - 3y 3 to make y the subject. y = … [3] 2 (c) Show that 2n - 5 - 13 is a multiple of 4 for all integer values of n. ` j [3] 2 2(d) The expression 5 + 12x - 2x can be written in the form q - 2 x + p . ` j (i) Find the value of p and the value of q. p = … , q = … [3] (ii) Write down the coordinates of the maximum point of the curve y = 5 + 12x - 2x 2 . ( … , … ) [1] (e) The energy of a moving object is directly proportional to the square of its speed. The speed of the object is increased by 30%. Calculate the percentage increase in the energy of the object. … % [2]

17 marks

Mark scheme: 6(a)(i) 1 9 3 M1 for 8x – 12 = 24 or 2x – 3 = 6 4.5, 4 or M1 for reaching ax = b correctly FT their 2 2 first step 6(a)(ii) 4 2 14 x > − or x > –11 final answer M1 for 6x > 6 – 14 or x + > 1 3 3 6 6(b) 3 3 M1 for isolating term in y 2 x − V [y =] 3 oe final answer M1 for division by 3 or FT their first step 3 M1 for cube root or FT their previous step to the final answer 6(c) 4n2 – 20n + 12 M2 B1 for 4n2 – 10n – 10n + 25 4(n2 – 5n + 3) A1 with no errors seen or e.g. 4, [–]20 and 12 are all multiples of 4 or correct explanation linked to divides each term or each coefficient by 4 expression 6(d)(i) p = –3 and q = 23 3 B2 for 23 – 2(x –3)2 OR M1 for [q] – 2x2 – 4px – 2p2 or –2(x – 3)2 seen B1 for either p = –3 or q = 23 or FT q = 5 + 2(their p)2 6(d)(ii) (3, 23) 1 FT their (d)(i) 6(e) 69 2 M1 for figs 132 oe

This question in 0580/42 Oct/Nov 2021

4 (a) Solve the simultaneous equations. You must show all your working. 2p - q = 7 3p + 2q = 7 p = … q = … [3] (b) Solve the equation. x 2x + = 1 4 3 x = … [2] (c) - 8 1 3x - 2 G 7 (i) Solve the inequality. … [3] (ii) Find the integer values of x that satisfy the inequality. … [1] (d) Factorise completely. 16a - 4 a 2 … [2] (e) Write each of the following as a single fraction, in its simplest form. 1 3 (i) ' 2a 4b … [2] x (ii) 2 - x - 1 … [2]

15 marks

Mark scheme: 4(a) Correctly eliminate one variable M1 p = 3 A2 A1 for each q = –1 If M0, SC1 for 2 values satisfying one of original equations If 0 scored SC1 for correct answers with no working 4(b) 1 12 2 3 x 8 x 111 or 11 1.09 or 1.090 to 1.091 M1 for 12 + 12 = 1 or better 4(c)(i) –2 < x ⩽ 3 3 B2 for –2 < x or x ⩽ 3 or M1 for –8 + 2 < 3x or 3x ⩽ 7 + 2 4(c)(ii) –1, 0, 1, 2, 3 1 FT dep on –ve and +ve values in their (c)(i) 4(d) 4 a (4 − a ) final answer 2 B1 for any correct partial factorisation 4(e)(i) 2b 2 1 4b final answer M1 for × or better 3a 2 a 3 4(e)(ii) x − 2 2 B1 for 2(x – 1) – x oe seen. final answer nfww x − 1

This question in 0580/43 Oct/Nov 2021

10 (a) Find all the positive integers which satisfy the inequality. 3n - 8 2 5n - 15 … [2] (b) y x = 4 9 8 7 6 5 4 10y + 8 x = 80 2y = x - 4 3 R 2 1 0 x 0 1 2 3 4 5 6 7 8 9 10 11 – 1 – 2 – 3 The region marked R is defined by three inequalities. (i) Find these three inequalities. … … … [3] (ii) Write down the largest value of 3x + y in the region R for integers x and y. … [2]

7 marks

Mark scheme: 10(a) 1, 2, 3 2 M1 for 15  8  5 n  3n oe If 0 scored, B1 for 2 correct answers and no others or 3 correct answers with one extra value 10(b)(i) 10y + 8x ⩽ 80 oe final answer 3 B1 for each x > 4 oe final answer 2y > x – 4 oe final answer If 0 scored, SC1 for 10y + 8x < 80 oe final answer and x ⩾ 4 oe final answer and 2y ⩾ x – 4 oe final answer 10(b)(ii) 23 final answer 2 M1 for 7 and 2 selected soi

This question in 0580/42 May/June 2022

3 (a) x – 2 4 Write down the inequality shown by the number line. … [1] (b) - 3 G 2x + 3 1 9 (i) Solve the inequality. … [3] (ii) Write down all the integer values of x that satisfy the inequality. … [2] (c) Solve the equations. 2 ( x + 2) (i) 3 ( 3 - x) - = 1 5 x = … [4] 5 3 (ii) = x + 3 x + 5 x = … [3]

13 marks

Mark scheme: 3(a) –2 < x ⩽ 4 oe 1 3(b)(i) –3 ⩽ x < 3 final answer 3 M2 for –3 < x < k or for k ⩽ x < 3 or for –6 ⩽ 2x < 6 3 3 9 3 or for − − ⩽ x < − 2 2 2 2 or M1 for – 3 – 3 ⩽ 2x < 9 – 3 3 3 9 or for − ⩽ x +  2 2 2 After 0 scored SC1 for -3 ⩽ x or for x < 3 3(b)(ii) –3, –2, –1, 0, 1, 2 final answer 2 FT their (i) as long as negative and positive values B1FT for one error or omission 3(c)(i) 36 4 B3 for –15x–2x = 5 + 4 – 45 or better oe OR 17 B2 for 45 - 15x – 2x – 4 = 5 oe OR M1 for correct removal of fraction or M1 for correct removal of brackets 3(c)(ii) –8 3 B2 for 5x – 3x = 9 – 25 or better or M1 for 5(x + 5) = 3(x + 3) oe or better

This question in 0580/41 Oct/Nov 2022

6 (a) P = 5k 2 - 7 (i) Find the value of P when k = 3 . P = … [2] (ii) Rearrange the formula to make k the subject. k = … [3] (b) (i) Solve. x - 3 G 5x + 7 … [2] (ii) Show your answer to part (b)(i) on the number line. x – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 [1] (c) The line y = 16 is drawn on the grid. y 40 30 20 10 0 10 20 30 40 x The region R satisfies the following inequalities. y H 16 x 2 2 2x + 3y H 72 y G 32 - x (i) By drawing three more lines and shading the region not required, find and label region R. [6] (ii) Find the integer coordinates (x, y) in the region R that give the maximum value of 2x + y . ( … , … ) [2]

16 marks

Mark scheme: 6(a)(i) 38 2 M1 for 5 × 32 – 7 oe 6(a)(ii) P + 7 3 P 2 7  oe final answer M1 for P + 7 = 5k2 or = k − 5 5 5 M1 for k2 = ……. FT their first step M1 for square root to final answer Max M2 for incorrect answer 6(b)(i) x ⩾ – 2.5 final answer 2 M1 for –4x ⩽ 7 + 3 or better 6(b)(ii) 1 FT their inequality in (b)(i) –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 6(c)(i) x = 2 broken line B1 y = 32 – x solid line B1 2x + 3y = 72 solid line B2 B1 for line passing through (0, 24) or (36, 0) Correct region indicated cao B2 B1 for region satisfying 3 of the inequalities 1 1 R 1 1 6(c)(ii) (16, 16) 2 M1 for substitution into 2x + y for any integer point in their region

This question in 0580/43 Oct/Nov 2022

8 A tailor makes x dresses and y shirts in one week. In one week • he makes at least 4 dresses • he makes no more than 7 shirts • he makes less than 14 dresses and shirts altogether 2 • the number of shirts he makes is more than of the number of dresses. 3 One of the inequalities that shows this information is x H 4 . (a) Write down the other three inequalities in x and/or y. … … … [3] (b) y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 On the grid, draw 4 straight lines and shade the unwanted regions to show these inequalities. Label the region R that satisfies the 4 inequalities. [6] (c) Use your diagram to find the smallest number of dresses and the smallest number of shirts the tailor makes in one week. … dresses and … shirts [1] (d) The profit the tailor makes on one dress is $10 and the profit on one shirt is $6. Use your diagram to find the largest profit the tailor can make in one week. $ … [2]

12 marks

Mark scheme: 8(a) y  7 oe 3 B1 for each x  y  14 oe 2 y  x oe 3 8(b) x  4 solid M4 B1 for each y  7 solid x  y  14 dashed 2 y  x dashed 3 correct shading everywhere but A2 M1dep (dependent on M4 or B1B1B1B0 where region R the only error is wrong use of solid/dashed lines) for shading the correct side of 3 of the 4 lines. R 8(c) 4 dresses and 3 shirts 1 8(d) 106 2 M1 for 10 x  6 y evaluated for (x, y) in their region R or B1 for (7, 6) After 0 scored, SC1 for answer 112 or 116

This question in 0580/41 May/June 2023

8 A baker decorates x small cakes and y large cakes. In one day, he decorates: • not more than 16 small cakes • less than 10 large cakes • more small cakes than large cakes • a total of not more than 24 cakes. One of the inequalities that shows this information is x G 16 . (a) Write down the other three inequalities in x and/or y. … … … [3] (b) On the grid, draw four straight lines and shade the unwanted regions to show these inequalities. Label the region, R, which satisfies the four inequalities. y 26 24 22 20 18 16 14 12 10 8 6 4 2 x 0 2 4 6 8 10 12 14 16 18 20 22 24 26 [6] (c) The baker earns $8 for decorating a small cake and $12 for decorating a large cake. Use your diagram to find the largest amount the baker can earn in one day by decorating cakes. $ … [2]

11 marks

Mark scheme: 8(a) y < 10 3 B1 for each y  x oe x + y ⩽ 24 oe If 0 scored, SC1 for y ⩽ 10 and y ⩽ x and x + y < 24 8(b) Correct lines and region indicated 6 B1 for each correct line and c c c R B2 for R in correct region for all 4 correct c lines or B1 for R in any one of the regions marked c or B1 for R that satisfies 3 of the correct inequalities 8(c) 228 nfww 2 M1 for 8x + 12y for any (x, y) in their R, x, y both integer or x = 15, y = 9

This question in 0580/42 May/June 2024

7 A company makes scientific calculators and graphic calculators. Each day they make x scientific calculators and y graphic calculators. These inequalities describe the number of scientific and graphic calculators they make each day. x 1 180 y G 90 x + y G 240 (a) Complete these two statements. The company makes fewer than … scientific calculators each day. The company can make a maximum of … calculators each day. [2] (b) Scientific calculators cost $12 to make. Graphic calculators cost $18 to make. Each day the company spends at least $2700 making calculators. Show that 2x + 3y H 450 . [1] (c) The region R satisfies these four inequalities. x 1 180 y G 90 x + y G 240 2x + 3y H 450 By drawing four suitable lines and shading unwanted regions, find and label the region R. y 250 200 150 100 50 0 x 50 100 150 200 250 [7] (d) Scientific calculators are sold for a profit of $10. Graphic calculators are sold for a profit of $30. Calculate the maximum profit made by the company in one day. $ … [2]

12 marks

Mark scheme: 7(a) 180 and 240 2 B1 for 180 or for 240 7(b) 12x + 18y ≥ 2700 1 and completion to 2x + 3y ≥ 450 with no errors seen 7(c) x = 180 broken straight line B5 and y = 90 solid ruled line and x + y = 240 solid ruled line B1 for x = 180 broken straight line and B1 for y = 90 solid ruled line 2x + 3y = 450 solid ruled line B1 for x + y = 240 solid ruled line B2 for 2x + 3y = 450 solid ruled line or B1 for line with a negative gradient passing through (0, 150) or (225, 0) Correct region indicated B2 1 B1 for region satisfying 3 of the inequalities 2 1 1 7(d) 4200 2 B1 for 150 and 90 or M1 for their 150 × 10 + their 90 × 30

This question in 0580/41 Oct/Nov 2024