Inequalities — Paper 4 · IGCSE Mathematics - International 0607

7 y 10 x –4 0 4 –10 f x = 9 - x 2 ^ h (a) On the diagram, sketch the graph of y = f x for values of x between -4 and 4. ^ h, [4] (b) Solve f x = 7 . ^ h … [2] (c) The equation 9 - x 2 = k has two solutions. Find the range of values of k. … [2]

8 marks

Mark scheme: 7(a) Correct Graph 4 B1 for maximum point on or close to y-axis B1 for correct shape between their –3 and 3 10 y f(x)=abs(9-x^2) B1 for mod graph x -4 4 -10 7(b) [x =] ±4, ± 2 2 B1 for any 2 correct answers or ± 1.41 or ± 1.414... 7(c) k > 9 2 B1 for each k = 0

This question in 0607/41 May/June 2017

8 y 5 x –400 0 600 f(x) = 3sinx (a) Sketch the graph of y = f(x) for - 400 ° G x G 600 ° . [3] (b) Find the x co-ordinates of the local maximum points of f(x) for - 400° G x G 600° . x = … or x = … or x = … [3] (c) The point (30, 3) is on the graph. The point (a, 3) is also on the graph where 600° 1 a 1 900 ° . Find the two possible values of a. a = … or a = … [2] x (d) g(x) = 3 − 100 Solve the inequality g x 2 f x . ^ h ^ h … [3]

11 marks

Mark scheme: 8(a) 3 WithW correct shape with ttwo max on right of y-axxis anda one on leeft, all abovee x-axis and reasonabler qualityq oro B2 for corrrect shape anand all above x-axis oro B1 for corrrect shape Correct skettch 8(b) –270, 90, 4550 3 B1B for each SC2S for all correctc but wwith y co-ordss oro SC1 for twwo correct wwith y co-ordss 8(c) 750, 870 2 B1B for each 8(d) x < 54.7 1 54.745 to 54.775 164 < x < 2667 2 163.51 to 163.6 , 266.6... B1B for one innequality oro B1 for botth values seeen IfI 0 scored, B1B for straighht line with negativen gradientg crosssing curve thhree times beetween xx = 0 and x == 400. May bbe freehand.

This question in 0607/42 May/June 2017

6 y 6 x 0 10 f(x) = x - 5 log x (a) On the diagram, sketch the graph of y = f(x) for 0 1 x G 10 . [2] (b) Find the co-ordinates of the local minimum point. ( … , … ) [2] (c) Find the range of f(x) for the domain 1 G x G 5 . … [2] (d) Solve the equation f(x) = 2. x = … or x = … [2] (e) Solve the inequality f(x) 1 2. … [1] (f) (i) Find f(0.001), f(0.000 01) and f(0.000 000 1). f(0.001) = … , f(0.000 01) = … , f(0.000 000 1) = … [1] (ii) Complete the statement. The y-axis is … to the graph of y = f(x). [1]

11 marks

Mark scheme: 6(a) Correct sketch 2 B1 for correct shape 6666 5555 4444 3333 2222 1111 0000 0000 2222 4444 6666 8888 10101010 6(b) (2.17, 0.488) or (2.171…, 0.4877…) 2 B1 for each 6(c) 0.488 - f ( x ) - 1.51 2 FT their 0.488 or 0.4877... - f ( x ) - 1.505... B1 for 0.488 - f ( x ) oe or f ( x ) - 1.51 oe 6(d) 0.502 or 0.5015… 2 B1 for each 5.83 or 5.827… 6(e) 0.502 < x < 5.83 1 FT their (d) or 0.5015... < x < 5.827... 6(f)(i) 15.[0] or 15.00… 1 25.[0] or 25.00… 35. [0] or 35.00… 6(f)(ii) [an] asymptote oe 1

This question in 0607/43 May/June 2017

10 (a) Solve the equation 4x 2 = 12 - 3x . Give your answers correct to 2 decimal places. You must show all your working. x = … or x = … [4] (b) Solve the inequality 4x 2 2 12 - 3x . … [2] (c) Solve the inequality 4x 2 + 5 G 12 - 3x . … [4]

10 marks

Mark scheme: 10(a) appropriate sketch giving one M2 M1 for sketch of parabola or parabola and positive and one negative answer or 2 −±3 … fully correct use of formula straight line or 3 − 4(4)( −12) or 2(4) oe 1.4[0] and –2.15 final answers B2 B1 for each If 0 scored B1 for 1.397… and –2.147... or SC1 for 2.15 and –1.4[0] 10(b) x > 1.40 and x < −2.15 2 FT [ x ] > their max(a), [ x ] < their min(a) B1 for each 10(c) −1.75 - x - 1 nfww 4 B3 for 1, − 1.75 oe B2 for 1 inequality correct B1 for 1 correct value seen or M2 for appropriate sketch or correct factorising or correct use of formula or M1 for 4 x 2 + 3 x − 7 - 0

This question in 0607/43 Oct/Nov 2017

9 y 50 x 0 –3 5 –50 f ()x = x 3 - 3x 2 - 4x + 1 for - 3 G x G 5 . (a) On the diagram, sketch the graph of y = f ( x) . [2] (b) Write down the co-ordinates of the local minimum. ( … , … ) [2] (c) Find the range of values of k so that f ()x = k has only one solution. … [2] (d) g (x) = 3x 2 - 6x - 4 for - 3 G x G 5 . The graph of y = f ( x) intersects the graph of y = g (x) twice. Solve f (x) 2 g (x) . … [2]

8 marks

Mark scheme: 9(a) Correct sketch 2 B1 for cubic graph with max/min incorrect 9(b) (2.53, –12.1) 2 B1 for each co-ordinate 9(c) k < –12.1 2 B1 for each k > 2.13 FT their –12.1 9(d) –0.726 < x < 1.26 2 B1 for both critical values seen or for [k <] x < 1.26 or for –0.726 < x [ < k]

This question in 0607/41 Oct/Nov 2018

2 y 3 0 x 5 x + 1 (a) On the diagram, sketch the graph of y = log for 0 1 x G 5 . [2] b x l x + 1 (b) Write down the equations of the asymptotes to the graph of y = log b x l. … … [2] x + 1 (c) Solve the equation log = 0. 5 . b x l x = … [1] x (d) On the same diagram, sketch the graph of y = for 0 1 x G 5 . [1] 2 x + 1 x (e) Solve the equation log = . b x l 2 x = … [1] x x + 1 (f) On your diagram, shade the region where y G 0.5 , y H and y H log [1] 2 b x l.

8 marks

Mark scheme: 2(a) Correct sketch 2 Must not cross axes 1111 0.80.80.80.8 0.60.60.60.6 0.40.40.40.4 B1 for correct shape 0.20.20.20.2 1111 0000 0000 1111 2222 3333 4444 5555 2(b) y = 0, x = 0 2 B1 for each If 0 scored, SC1 for answers x-axis and y-axis 2(c) 0.462 or 0.4624 to 0.4625 1 2(d) Correct sketch 1 3333 2.52.52.52.5 2222 1.51.51.51.5 1111 0.50.50.50.5 0000 0000 1111 2222 3333 4444 5555 2(e) 0.742 or 0.7415 to 0.7416 1 2(f) Region that is below y = 0.5 and 1 above other two graphs.

This question in 0607/42 May/June 2019

9 (a) Solve the following equations. 135 (i) = 5 x x = … [1] (ii) 3x + 5 = 7x + 25 x = … [2] (iii) 8x 2 = 11 - 2x x = … or x = … [4] (b) Solve the following inequalities. (i) 6 - 2x H 10 … [2] 1 (ii) 2 3 x - 2 … [3] (c) Solve the simultaneous equations. You must show all your working. 3x + 5y =-3 5x - 2y = 26 x = … y = … [4] (d) Solve the equation. log x + 4 log 2 = log 13 x = … [3]

19 marks

Mark scheme: 9(a)(i) 27 1 9(a)(ii) –5 2 M1 for 5 − 25 = 7 x − 3x or better 9(a)(iii) 1.05 or 1.054… 4 2 −±2 2 −×4 8 ×−11 –1.3[0] or –1.304… M3 for 2 × 8 or correct sketch which would lead to solution. b 2 or M2 for correct or b − 4 ac correct 2a or M1 for 8 x 2 + 2 x − 11 or − 8 x 2 − 2 x + 11 or sketch of 8 x 2 or 11 − 2x 9(b)(i) x - − 2 oe 2 M1 for 6 − 10 . 2x or − 2 x . 10 − 6 or 3 − x . 5 or better If 0 scored SC1 for x . − 2 or x = –2 9(b)(ii) 1 3 7 2 < x < 2 oe M2 for x = 2 and x = 3 3 or correct sketch which would lead to solution. or M1 for 1 > 3( x − 2) or better or sketch of 1 y = x − 2 1 or B1 for x < 23 or for x > 2 9(c) Correctly equating one set of M1 coefficients oe Correct method to eliminate one M1 variable [x=] 4 B1 [y=] –3 B1 If 0 scored SC1 for correct substitution into one of original equations and evaluation to find other variable. 9(d) 13 3 4 or 0.8125 M1 for log2 or better 16 p M1 for correct use of log p − log q = log q or use of log p + log q = log pq

This question in 0607/43 May/June 2019

7 (a) (i) Factorise 2x 2 - 11x - 6 . … [2] (ii) Using your answer to part (i), solve 2x 2 - 11x - 6 1 0. … [2] (b) Solve the equation 3x 2 - x - 5 = 0. Give your answers correct to 2 decimal places. You must show all your working. x = … or x = … [3]

7 marks

Mark scheme: 7(a)(i) (2 x + 1)( x − 6) final answer 2 M1 for (2x + a)(x + b) where ab = –6 or a + 2b = –11 or 2x(x – 6) + x – 6 or x(2x + 1) – 6(2x + 1) or correct answer seen 7(a)(ii) − 0.5 < x < 6 2 FT their (i) only from factors giving positive x2 term B1 for each or –0.5 and 6 seen 7(b) Appropriate sketch indicating answers M1 2 Allow 61 for ( −1) − 4(3)( −5) (one positive and one negative) or correct substitution in formula or correct completion of square 1.47 B2 B1 for each –1.14 or both correct but not rounded to 2dp 1.468… , –1.135…

This question in 0607/42 Oct/Nov 2019

9 y 6 x – 0.5 0 4.5 – 6 f ( )x = x 3 - 6x 2 + 8x for - 0.5 G x G 4.5 (a) On the diagram, sketch the graph of y = f ( x) . [2] (b) Solve the inequality f ( )x 1 0 . … [3] (c) Find the positive value of k when f ( )x = k has two different solutions. k = … [2]

7 marks

This question in 0607/41 May/June 2020

7 y 9 x – 6 0 2 – 3 1 (a) f ( )x = 2 + x + 2 (i) On the diagram, sketch the graph of y = f ( x) for values of x between - 6 and 2. [2] (ii) Write down the coordinates of the points where the graph crosses the axes. ( … , … ) and ( … , … ) [2] (iii) Write down the equations of the asymptotes of the graph. … , … [2] (b) g ( x) = ( x + 4) 2 On the diagram, sketch the graph of y = g ( x) for - 6 G x G - 1 . [2] (c) Solve the equation. f ( x) = g ( x) … [3] (d) Solve the inequality. f ( x) H g ( x) … [2]

13 marks

Mark scheme: 7(a)(i) 8 y f(x)=2+1/(x+2) 2 B1 for correct ‘hyperbolic shape’ 7 6 B1 for intersects with axes correct 5 4 (approx.) 3 2 1 x -5 -4 -3 -2 -1 1 -1 -2 7(a)(ii) (–2.5, 0) 2 B1 for each (0, 2.5) 7(a)(iii) x = –2 2 B1 for each y = 2 y f(x)=2+1/(x+2)f(x)=(x+4)^2 5 7(b) 4 2 B1 for correct ‘quadratic shape’ 3 B1 for min point at (–4, 0) (approx.) 2 1 -5 -4 -3 -2 -1 1 x -1 -2 -3 -4 -5 7(c) [x =] – 5.30 3 B1 for each correct answer [x =] –3 [x =] –1.70 7(d) −5.30 ≤ x ≤−3 2 B1 for each and −<2 x ≤−1.70

This question in 0607/42 May/June 2020

7 y 5 – 1.5 0 1.5 x – 5 3 1 f ( )x = x - x (a) On the diagram, sketch the graph of y = f ( x) , for values of x between - .15 and 1.5 . [3] (b) Write down the equation of the asymptote of the graph. … [1] (c) Solve the equation f ( )x = 2 for values of x between - .15 and 0. x = … or x = … [2] (d) Solve the inequality f ( )x + x 2 G 2 for values of x between - .15 and 1.5 . … [3]

9 marks

Mark scheme: 7(a) Correct sketch 3 B1 for modulus graph B1 for correct for x > 1, or –1 < x < 0 4444 B1 for x = –1 and 1 when y = 0 plotted 2222 correctly. .5.5.5.5 -1-1-1-1 -0.5-0.5-0.5-0.5 0000 0000 0.50.50.50.5 1111 1.51.51.51.5 Maximum 2 marks if sketch not fully -2-2-2-2 correct -4-4-4-4 7(b) x = 0 1 7(c) –1.4[0] or –1.395… 2 B1 for each –0.475 or –0.4746… 7(d) –1.15 ⩽ x ⩽ –0.536 3 B2 for one fully correct inequality or ––1.154 to –1.153...⩽ x ⩽ –0.5357 or B1 for –1.15 ⩽ x ⩽ – k to –0.5356 – k ⩽ x ⩽ –0.536 or 0.536 ⩽ x ⩽ k AND k ⩽ x ⩽ 1.15 0.536 ⩽ x ⩽ 1.15 or M1 for suitable sketch, or 0.5356 to 0.5357 ⩽ x ⩽ 1.153 to e.g. f(x) + x2 ⩽ 2 1.154 or B1 for 4 correct solutions seen

This question in 0607/41 Oct/Nov 2020

8 (a) y 4 x – 2 0 2 (i) On the diagram, sketch the graph of y = 1.5 -x for - 2 G x G 2 . [2] (ii) Solve the inequality 0.5 G 1.5 -x G 1. … [3] (iii) Solve the equation 1.5 -x = x 2 for - 2 G x G 2 . … [3] (iv) On your diagram shade the regions where 1.5 -x 1 x 2 for - 2 G x G 2 . [1] (b) y O x - 2x + b The diagram shows a sketch of the graph of y = . x + a The asymptotes of the graph are x = 2 and y =- 2 . The graph passes through the point (0, 2). Find the value of a and the value of b. a = … b = … [3] (c) y y = k M x O x = a x = b f(x) is a function such that • the asymptotes of the graph are x = a , x = b and y = k • when x 1 a , the gradient of the graph is positive • when x 2 b , the gradient of the graph is negative • M is the only local maximum point • the graph does not cross any asymptote. On the diagram sketch the graph of y = f ( x) . [3]

15 marks

Mark scheme: 8(a)(i) Correct sketch4444 2 B1 for exponential shape 3333 2222 1111 2222 -1-1-1-1 0000 0000 1111 2222 8(a)(ii) 0 ≤ x ≤1.71 or 1.709 to 1.710 3 B2 for either correct or B1 for 0 and 1.71 or 1.709 to 1.710 seen 8(a)(iii) –1.3[0] or –1.302... 3 B2 for one correct 0.843 or 0.8429... or B1 for sketch4444 of y = x2 added to diagram 3333 2222 1111 2222 -1-1-1-1 0000 0000 1111 2222 8(a)(iv) Two areas shaded which are above 1 y = 1.5 − x and below y = x2 8(b) [a =] –2 3 B1 for a = –2 [b =] –4 b M1 for = 2 oe a 8(c) Correct4444 sketch 3 B1 for each branch 3333 2222 1111 1111 0000 0000 1111 2222 3333 4444 5555 6666 -1-1-1-1

This question in 0607/43 Oct/Nov 2020

3 (a) (i) Write down the coordinates of the point where the line y =- 2x + 3 crosses the y-axis. ( … , … ) [1] (ii) Write down the gradient of the line y =- 2x + 3 . … [1] (b) The line x + y = 6 crosses the line x =- 2 at point A. Find the y-coordinate of A. … [1] (c) Find the equation of the straight line that passes through the points (3, -1) and (12, 5). … [3] (d) The line L passes through the point (3, 4). Line L is perpendicular to the line 2y = 5x + 6 . Find the equation of line L. … [4] (e) y 7 6 5 4 3 2 1 – 2 – 1 0 1 2 3 4 5 6 7 x – 1 – 2 (i) On the grid, draw the lines y = 4, x + y = 3 and y = x - 1 . [3] (ii) By shading the unwanted regions, find and label the region R that satisfies these three inequalities. y G 4 x + y H 3 y H x - 1 [1]

14 marks

Mark scheme: 3(a)(i) (0, 3) 1 3(a)(ii) –2 1 3(b) 8 1 3(c) 2 3 2 y = x − 3 oe final answer B2 for answer x − 3 3 3 OR 5 −−( 1) M1 for oe 12 − 3 M1 for correct substitution of point into y = (their m)x + c or e.g. y – 5 = (their m)(x – 12) 3(d) 2 26 4 2 26 y = − x + oe final answer B3 for answer − x + oe 5 5 5 5 OR 5 M1 for gradient 2 −1 M1 for m = or better their ( 52 ) M1 for (3, 4) substituted into y = (their m)x + c or e.g. y – 4 = (their m)(x – 3) 3(e)(i) 3 correct ruled lines 3 B1 for each line correct 3(e)(ii) Clear indication of correct 1 FT if appropriate region

This question in 0607/42 Feb/March 2021

7 y 13 – 4 0 3 x – 3 1 g ( )x = , x ! 2 x - 2 (a) On the diagram, sketch the graph of y = g(x) for values of x between - 4 and 3. [3] (b) Write down the equations of the asymptotes of the graph of y = g(x). … … [2] (c) h ( x) = ( x + 1) 2 - 3 Solve the inequality g ( x) 2 h ( x) . … [4]

9 marks

Mark scheme: 7(a) Correct sketch 3 B2 for correct branches but joined or for ‘correct’ but with excessive overlap or ‘curl back’ B1 for one correct branch 7(b) y = 0 B2 B1 for each x = 2 7(c) –2.67 < x < 0.524 B2 B1 for x > −2.67 or x < 0.524 or –2.7 < x < 0.52 2 < x < 2.15 B2 B1 for either x > 2 or x < 2.145… If B0, B0 scored, then SC1 for 2 of the boundaries –2.67, 0.524, 2.15 seen

This question in 0607/42 Feb/March 2021

4 y 15 – 5 0 5 x – 15 f ( )x = 10 - x 2 (a) On the diagram, sketch the graph of y = f(x) for - 5 G x G 5 . [2] (b) Solve the equation f(x) = 6. … [2] (c) Solve f ( )x 2 6 . … [3] (d) Find the values of k for which f(x) = k has exactly two solutions. … [2]

9 marks

Mark scheme: 4(a) Correct sketch 2 B1 for correct middle section 4(b) ± 4 2 B1 for 2 correct solutions ± 2 4(c) x < –4 3 B1 for each –2 < x < 2 x > 4 4(d) 0 2 B1 for each [k ] > 10

This question in 0607/42 May/June 2021

12 (a) Solve. 2 (i) 9 = 5 - x x = … [3] 6 (ii) 2 3 x - 4 … [3] (b) (i) Solve the equation, giving your answers correct to 3 significant figures. 2x 2 - 5x + 1 = 0 x = … or x = … [3] (ii) Use your answers to part (b)(i) to solve 2 ( tan y) 2 - 5 ( tan y) + 1 = 0 for 0° G y G 180° . y = … or y = … [2]

11 marks

Mark scheme: 12(a)(i) –0.5 oe 3 x 1 M2 for = − or 4 x = − 2 2 4 2 or M1 for = 5 − 9 oe or 9 x = 5 x − 2 oe x 12(a)(ii) 4 < x < 6 3 B2 for x < 6 seen and not spoiled or B1 for [x =] 6 seen OR 6 − 3 x + M2 for 12[ > 0] x − 4 3( x − 4) or M1 for soi x − 4 OR M2 for correct graph showing answers or M1 for appropriate graph 12(b)(i) 0.219 3 B2 for 0.2192... or 0.22 and 2.280 to 2.281 2.28 or M1 for correct curve or correct use of formula 12(b)(ii) 12.4 or 12.35 to 12.36... 2 B1 for each 66.3 or 66.31 to 66.33 FT their (b)(i) 13 For all parts accept decimals or percentages with the usual rules for 3sf Do not penalise incorrect cancelling or converting Do not accept ratios or words

This question in 0607/43 Oct/Nov 2021

3 y 5 – 3 0 1 x – 5 1 f ( x) = 2 x + 4 - 2 x (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 3 and 1. [3] (b) Write down the equation of the asymptote of the graph. … [1] (c) Find the coordinates of the local maximum. ( … , … ) [1] (d) g( )x = x 3 - 5x for - 3 G x G 1. Solve f ( x) G g( x) . … [4]

9 marks

Mark scheme: 3(a) correct sketch 3 B2 for correct branches but joined or touching y-axis B1 for one correct branch 3(b) x = 0 1 3(c) (−1, 1) 1 3(d) −2.31 ⩽ x < 0 and 0 < x ⩽ 0.388 4 B3 for −2.3 ⩽ x ⩽ 0.388 or −2.31 ⩽ x ⩽ 0.39 or B2 for –2.3 ⩽ x ⩽ 0.39 or –2.31 ⩽ x or x ⩽ 0.388 or B1 for – 2.31 or 0.388 seen or for correct sketch

This question in 0607/41 May/June 2022

4 y 4 – 4 0 4 x – 4 (a) On the diagram, sketch the graph of y = f( x) , where f( )x = 4 - 2 x for values of x between - 4 and 4. [3] (b) Write down the x-coordinates of the points where the graph meets the x-axis. x = … and x = … [1] (c) On the diagram, sketch the graph of y = g( x) , where g ( x) = 0 .25 x 2 for values of x between - 4 and 4. [2] (d) Write down the equation of the line of symmetry of the graph of y = g( x) . … [1] (e) Find the value of the x-coordinate of each point of intersection of the two graphs. x = … and x = … [2] (f) On your diagram shade the region defined by f( x) H g( x). [1]

10 marks

Mark scheme: 4(a) Correct sketch 3 B1 for inverted ‘v’ B1 for symmetrical about y-axis 4 4(b) –2, 2 1 4(c) Correct sketch 2 Must touch x-axis at origin and without serious curl backs B1 for a u-shaped parabola 4(d) x = 0 1 4(e) –1.66 or –1.657 to –1.656 2 B1 for each 1.66 or 1.656 to 1.657 4(f) Correct region shaded 1 Dependent on at least B2 in (a) and at least B1 in (c)

This question in 0607/42 May/June 2022

10 (a) P = 5 Work out the value of P when x =- 18 and y = 28 . P = … [3] (b) Simplify fully. 5 y 4 x # 2 x 3 … [2] (c) Factorise fully. (i) 15ab - 25bc … [2] (ii) 6 x 2 y 5 - 16 x 3 y 3 … [2] (iii) 6cd - 3 - 9d + 2c … [2] (d) Make x the subject of the formula. 2 x 3ax = 1 - a + 2 x = … [4] (e) Solve the inequality. 3 - x 2 1 2 + x … [3]

18 marks

Mark scheme: 10(a) – 84 3 M1 for correct substitution B1 for answer 84 10(b) 10 y 1 2 20 xy 10 xy 20 y 5 y  2 or 3 3 y or 3.3 (or 3.33 or 3.333…)y B1 for or or or 3 6 x 3 x 6 3 final answer or correct answer seen 10(c)(i) 5b(3a – 5c) final answer 2 M1 for b(15a – 25c) or 5(3ab – 5bc) or correct answer seen 10(c)(ii) 2x2y3(3y2 – 8x) final answer 2 M1 for x2y3(6y2 – 16x) or 2y3(3x2y2 – 8x3) or 2x2(3y5 – 8xy3) or better i.e. answers which are correct and have only one common factor left inside brackets e.g. 2x2y(3y4 – 8xy2) or correct answer seen 10(c)(iii) (2c – 3)(3d + 1) final answer 2 M1 for 2c(3d + 1) – 3(3d + 1) or 3d(2c – 3) + 2c – 3 or correct answer seen 10(d) a  2 4 M1 for correctly eliminating fractions [ x  ] oe 2 M1 for correctly expanding brackets 3a  6 a  2 final answer M1 for correctly collecting all terms in x on one side and other terms on other side of equation M1 for correctly isolating x by factorising and dividing Max 3 marks only if final answer is incorrect 10(e) –2 < x < 0.5 final answer 3 M2 for –2 and 0.5 SOI or M1 for correct graph(s) sketched 1  2 x or M1 for  0 oe 2  x or B1 for 0.5 soi

This question in 0607/42 May/June 2022

5 y 4 x 0 –1 5 – 4 (a) On the diagram, sketch the graph of y = f ( x) , where 1 f ( x) = for values of x between - 1 and 5. [3] ( x - 1)( x - 2)( x - 3) (b) Write down the y‑coordinate of the point where the curve meets the y‑axis. y = … [1] (c) Write down the equations of all the asymptotes to the graph of y = f ( x) . … [3] (d) On the diagram, sketch the graph of y = g ( x) , where g ( x) = x - 1 , for values of x between - 1 and 5 . [1] (e) Find the x‑coordinate of each point of intersection of the two graphs. x = … or x = … [2] (f) Solve the inequality f ( x) 2 g ( x) . … [3]

13 marks

Mark scheme: 5(a) Correct sketch f(x)=1/((x-1)(x-2)(x-3)) 3 B1 for graph in 4 sections B1 for rectangular hyperbola type on outside 2 sections not crossing x-axis B1 for 2 quadratic type sections (one inverted) Max 2 marks if not fully correct 5(b) 1 1 –0.167 or –0.1667 to –0.1666 or − 6 5(c) x = 1, x = 2, x = 3, y = 0 3 B2 for 3 correct or B1 for 1 correct If 0 scored, SC1 for all four with  5(d) 1 Can be good freehand, cutting negative y-axis and positive x-axis 5(e) x = 0.487 or 0.4871… 2 B1 for each x = 3.18 or 3.178 to 3.179 5(f) [–1 < ] x < 0.487 3 B1 FT their(e) for each 1 < x < 2 3 < x < 3.18

This question in 0607/41 Oct/Nov 2022

5 (a) y 8 0 x – 6 7 – 5 1 (i) On the diagram sketch the lines y =- x + 3 , 2y = x + 5 and y = x for - 6 G x G 7 . 2 [4] (ii) Show, by shading, the region that satisfies these inequalities. 1 y 2- x + 3 2y 1 x + 5 y 2 x [2] 2 (b) y 9 0 x – 1 4.7 – 10 f ( x) = ( x - 2) 3 - 5x + 12 for - 1 G x G 4.7 (i) On the diagram, sketch the graph of y = f ( x) . [2] (ii) Write down the coordinates of the local maximum. ( … , … ) [2] (iii) The equation ( x - 2) 3 - 5x + 12 = k has exactly 2 solutions. Find the values of k. k = … or k = … [2] (iv) g ( x) =- ( x - 1) 2 for - 1 G x G 4.7 On the diagram, sketch the graph of y = g ( x) . [2] (v) Solve f ( x) = g ( x) . x = … [1]

15 marks

Mark scheme: 5(a)(i) correct sketch 4 B1 for correct sketch of 2 y = x + 5 1 B1 for correct sketch of y = − x + 3 2 B1 for correct sketch of y = x passing through (0,0) B1 for all intersections in 1st quadrant 5(a)(ii) correct region indicated 2 FT their lines B1 for region satisfying 2 inequalities or for shading shown but region not clearly indicated 5(b)(i) correct sketch 2 M1 for positive cubic curve with a maximum and minimum 5(b)(ii) (0.709, 6.3[0]) 2 B1 for one correct coordinate 5(b)(iii) –2.3[0], their 6.3[0] 2 B1 for each

This question in 0607/42 Oct/Nov 2023

4 y 40 0 x -3 4 -40 f ( x) = 2x 3 - 3x 2 - 12x + 7 for -3 G x G 4 (a) Sketch the graph of y = f ( x) . [2] (b) Solve f ( x) = 0 . … [3] (c) Find the values of k for which f ( x) = k has exactly two solutions. k = … or k = … [2] (d) Find the range of values of x for which the gradient of f ( x) is negative. … [2]

9 marks

Mark scheme: 4(a) Correct sketch 2 B1 for any cubic with positive x3 4(b) –2.12 or –2.116... 3 B1 for each 0.537 or 0.5370... 3.08 or 3.079... 4(c) 14 and –13 cao 2 B1 for each or B1 for both14 and -13 seen 4(d) –1 < x < 2 2 B1 for each

This question in 0607/43 Oct/Nov 2023

11 (a) Solve. 3x + 2 2 7x - 8 … [2] (b) Factorise fully. 75x 2 - 3 … [2] (c) Simplify. 2 1 1 (i) + - 3x 6x 5x … [2] 2 x 2 + 3 x - 2bx - 3b (ii) 2 2x - 7x - 15 … [4]

10 marks

Mark scheme: 11(a) x < 2.5 oe final answer 2 M1 for 2 + 8 > 7x – 3x oe or B1 for x * 2.5 where * is =, >, ≤ or ≥ 11(b) 3(5x + 1)(5x – 1) final answer 2 B1 for 3(25x2 – 1) or (15x + 3)(5x – 1) or (15x – 3)(5x + 1) 11(c)(i) 19 2 B1 for any equivalent cao final answer or M1 for correct use of common 30x denominator 20 + 5 − 6 20 x + 5 x − 6 x e.g. , etc. oe 30 x 30 x 2 11(c)(ii) x − b 4 B3 for (x – b)(2x + 3) and (x – 5)(2x + 3) final answer or B2 for (x – b)(2x + 3) x − 5 or for (x – 5)(2x + 3) or B1 for x(2x + 3) – b(2x + 3) or 2x(x – b) + 3(x – b) or x(2x + 3) – 5 (2x + 3) or 2x(x – 5) + 3 (x – 5) or (2x + c)(x + d) where c + 2d = –7 or cd = –15

This question in 0607/42 Feb/March 2024

1 (a) Solve the equations. (i) 3x - 2 = - 14 x = … [2] (ii) 7x + 11 = 26 - 3 x x = … [2] (b) Solve the simultaneous equations. You must show all your working. 5x + 3y = - 15 3x + 5y = - 17 x = … y = … [4] (c) Solve the inequality. 2x + 1 2 9 … [4]

12 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) –4 2 2 14 M1 for 3x = –14 + 2 or x   3 3 1(a)(ii) 15 2 M1 for 7x + 3x = 26 – 11 oe or better 1.5 or oe 10 1(b) Correctly equating coefficients M1 or sketch of one equation with negative slope and y intercept Correct method to eliminate one variable M1 or sketch of other equation with negative slope and y intercept If 0 scored, SC1 for answers that satisfy one x = –1.5 oe A1 equation. SC1 for two correct answers with no working y = –2.5 oe A1 1(c) x > 4 and x < –5 4 M2 for correct sketch showing both answers Mark final answer or M1 for appropriate sketch of y = 2 x  1 OR M2 for 2x + 1 > 9 oe and 2x + 1 < –9 oe or M1 for either correct inequality or for 2x + 1 = 9 and 2x + 1 = –9 oe B1 for x > 4 or x < –5 Mark final answer

This question in 0607/41 May/June 2024

9 y 4 x 0 -2 4 - 4 1 f ( x) = ( 2x - 3)( 2x + 1) (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 2 and 4. [3] (b) Write down the equations of the asymptotes parallel to the y-axis. … [2] (c) Write down the coordinates of the local maximum. ( … , … ) [2] (d) The line y = x - 2 intersects the curve y = f ( x) three times. Find the x-coordinate of each point of intersection. x = … or x = … or x = … [3] (e) Solve the inequality f ( )x H x - 2 . … [3]

13 marks

Mark scheme: 9(a) Correct sketch 3 B1 for correct shape with 3 branches B1 for the local maximum in correct position, not above x-axis B1 for graph with no excessive overlaps, gaps or curl backs, the upper branches not crossing the x-axis 9(b) x = –0.5 x = 1.5 2 B1 for each 9(c) (0.5, –0.25) 2 B1 for each 9(d) –0.448 1.3[0] 2.15 3 B1 for each If 0 scored, SC1 for –0.45, 1.3 and 2.1 9(e) [ −2 ] x −0.5 3 B1 for each, strict inequality on the asymptote −0.448 x 1.30 values – only penalised once 1.5 x 2.15

This question in 0607/41 Oct/Nov 2024

4 y 17 0 x -3 3 -13 3 3 2 f ( )x = x - 4x + 2 g ( )x = + x x (a) On the diagram, sketch the graph of y = f ( x) for values of x between -3 and 3. [2] (b) Find the solutions of f ( )x = 0 . x = … , x = … , x = … [3] (c) On the diagram, sketch the graph of y = g ( x) for values of x between -3 and 3. [3] (d) Write down the equation of the asymptote of the graph of y = g ( x) . … [1] (e) Solve f ( x) G g ( x) . … [4]

13 marks

Mark scheme: 4(a) correct sketch 2 M1 for positive cubic shape 4(b) –2.21 0.539 1.68 3 B1 for each correct or –2.214… 0.5391… 1.675… penalise 1 mark if y co-ordinates included if 0 scored SC1 for –2.2, 0.54 and 1.7 4(c) correct sketch 3 For full marks there must be exactly one intersection in the first quadrant B2 for both branches but joined or touching the y axis or B1 for one correct branch on either side of y axis 4(d) x = 0 1 4(e) [ −3] ⩽ x ⩽ –1.96 or –1.959…. 4 B2 for x ⩽ –1.96 0 < x ⩽ 2.48 or 2.482…to 2.483 or B1 for –1.96 seen B2 for 0 < x ⩽ 2.48 or B1 for 2.48 seen

This question in 0607/43 Oct/Nov 2024

8 (a) The amount charged for electricity in one month is $E. $E is the sum of a fixed charge $f and a cost of $d for each unit of electricity used. Find a formula for the amount charged in one month when u units of electricity are used. … [2] (b) Write as a single fraction in its simplest form. x 2 x 5x - + 2 3 18 … [2] (c) Solve 7n - 9 2 21 + 2 n . … [2] (d) Solve the simultaneous equations. You must show all your working. 2x + 15y = –57 20x + 3y = 18 x = … y = … [3] (e) y is proportional to the square of ( x - 3) . y = 5 when x = 7 . Find the value of y when x = 27 . y = … [3]

12 marks

Mark scheme: 8(a) E = du + f final answer 2 M1 for du + f 8(b) x 2 M1 for correct use of common denominator eg final answer 9 9 x 12 x 5 x − + 18 18 18 8(c) n  6 final answer 2 M1 for 7n − 2n *21 + 9 or better * can be = or any inequality 8(d) correctly equating one set of M1 coefficients Or correctly making x or y the subject of an equation and correct substitution x = 1.5 A2 A1 for each y = −4 If M0 scored SC1 for correct substitution and evaluation to find the other variable. or SC1 if no working shown, but 2 correct answers given. 8(e) 180 3 5 2 M2 for y = their ( x − 3) oe 16 OR M1 for y = k ( x − 3) 2 5 A1 for k = 16

This question in 0607/43 Oct/Nov 2024

8 y 2 – 3 0 3 x – 2 1 f ( )x = x 2 + 1 (a) On the diagram, sketch the graph of y = f ( x) for values of x between -3 and 3. [2] (b) Solve f ( )x 1 x 2 + x - 1. … [4]

6 marks

Mark scheme: 8(a) correct sketch 2 B1 for correct shape but: touching x axis or maximum not on y-axis or too high or too low or distinct curl-up at one or both ends or curvature incorrect on one side. 8(b) x < –1.73 or x < –1.725… 4 B2 for x < –1.73 or x < –1.725… x > 0.852 or x > 0.8524 to 0.8525 or B1 for –1.73 to –1.72 seen B2 for x > 0.852 or x > 0.8524 to 0.8525 or B1 for 0.85 or 0.852 to 0.853 seen If 0 scored SC1 for sketch of positive quadratic

This question in 0607/42 Feb/March 2025

11 y 20 x – 10 0 10 – 20 x 3 f ( x) = ( x + 2)( x - 3) (a) Sketch the graph of y = f ( x) for values of x between -10 and 10. [3] (b) Find the coordinates of the local minimum. ( … , … ) [2] (c) Write down the equations of the asymptotes to the graph of y = f ( x) that are parallel to the y-axis. … [2] (d) Solve f ( )x 2 x + 7 . … [4]

11 marks

Mark scheme: 11(a) Correct sketch 3 No gaps or overlaps B1 for either LH branch or RH branch correct shape (Ignore any joining with middle section for this mark) B1 for middle section correct - must pass through origin with no obvious max or min (ignore any joining with outer branches) 11(b) (5.36, 8.87) 2 B1 for each If 0 scored SC1 for 5.35 and 8.86 11(c) x = –2 2 B1 for each x = 3 11(d) –2 < x < –1.78 4 B1 for –1.78 seen 3 < x < 3.94 B1 for 3.94 seen B1 for each inequality If 0 scored SC1 for straight line with positive gradient, with positive y-intercept and cutting curve twice

This question in 0607/41 May/June 2025

5 Solve. 5 - 2x 2 3 x + 7 … [2]

2 marks

Mark scheme: 5 2 2 2 x < – oe final answer B1 for [x *] – or better where * is =, >, ⩽ 5 5 or ⩾ or M1 for 5 – 7 > 3x + 2x oe or better

This question in 0607/41 Oct/Nov 2025

8 y 4 0 x – 4 4 – 4 1 (a) f ( x) = - 1 ( 2x - 1)( x + 1) (i) On the diagram, sketch the graph of y = f ( x) for values of x between -4 and 4. [3] (ii) Write down the x-intercepts. … [2] (iii) Write down the equations of the asymptotes parallel to the y-axis. … [2] (b) g ( x) = 0. 5 ( x + 1) On the diagram, sketch the graph of y = g ( x) for values of x between -4 and 4. [1] (c) Solve the inequality f ( x) H g ( x) . … [3]

11 marks

Mark scheme: 8(a)(i) Correct sketch 3 B1 for correct outer branches both crossing x-axis B1 for middle branch in correct position B1 graph in 3 sections with no excessive overlaps (except if penalised already in second B1) or gaps or curlback If sketch not correct max of 2 marks 8(a)(ii) –1.28 –1.281 to –1.280 2 B1 for each 0.781 0.7807 to 0.7808 or for –1.3 and 0.78 8(a)(iii) x = –1 2 B1 for each x = 0.5 oe 8(b) Correct line 1 8(c) x − 2.84 –2.837 to –2.836 3 B1 for each, strict inequality on the −1.33  x −1 –1.327… asymptote values, only penalised once. If 0 scored SC1 for 2 correct intersections 0.5 x 0.664 0.6640… –2.8…, –1.3…, 0.66… seen in an inequality

This question in 0607/42 Oct/Nov 2025