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Q1. The difference between the squares of two consecutive even numbers is twice the sum of the numbers. For example 2 2 8 – 6 = 28 2 × (8 + 6) = 28 Prove this result algebraically. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. (Total 4 marks) Q2. The value of a vintage car rises from £36 000 to £63 000. Work out the percentage increase in the price of the car. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ............................................% (Total 3 marks) Q3. (a) Expand and simplify 2 (x + 4) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ....................................................................... (2) Page 1 of 56 (b) Hence or otherwise, show that 2 (x + 4) – 4(x + 4) ≡ x (x + 4) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (2) (Total 4 marks) Q4. Simplify fully .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ....................................................................... (Total 4 marks) Q5. Greg thinks of a positive whole number smaller than 15. He subtracts 4 from the number and then doubles his result. He subtracts 4 from the new number and then doubles this result. He repeats this process several times. He stops when he gets an answer of 40. (a) Show that when Greg starts with 12 he gets an answer of 40. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (2) Page 2 of 56 (b) Find another number that Greg could have started with to get an answer of 40. ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) (Total 4 marks) Q6. Two families go to a pantomime. The Khan family of two adults and three children pay £69. The Lewis family of three adults and five children pay £109. Work out the cost of an adult ticket and the cost of a child ticket. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer Adult ticket £ .......................... Child ticket £ .......................... (Total 5 marks) Q7. (a) A sequence starts 2 7 17 ....... The rule for finding the next term in this sequence is to multiply the previous term by 2 and then add on 3 Work out the next term. ......................................................................................................................... ......................................................................................................................... Answer .................................................. (1) Page 3 of 56 (b) The rule for finding the next term in a different sequence is to multiply the previous term by 2 and then add on a, where a is an integer. The first term is 8 and the fourth term is 127 8 ........ ......... 127 Work out the value of a. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer a = .......................................... (4) (Total 5 marks) Q8. (a) Simplify 3 m ×m 5 Answer .................................................. (1) (b) Simplify Answer .................................................. (1) (c) Simplify fully ......................................................................................................................... ......................................................................................................................... Answer .................................................. (2) (Total 4 marks) Page 4 of 56 (a) Q9. Simplify (9 + √7)(9 + √7) Give your answer in the form a + b√7 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................... (2) (b) Prove that ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (4) (Total 6 marks) Q10. (a) A magazine contains adverts, photographs and features. of the pages are adverts and are photographs. What fraction are features? ......................................................................................................................... ......................................................................................................................... Answer .................................................. (3) Page 5 of 56 (b) There are 24 photographs in the magazine. The ratio of sports photographs to other photographs is 5 : 3 How many sports photographs are there in the magazine? ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer .................................................. (2) (Total 5 marks) Q11. (a) Solve the inequality 3x + 7 > x + 8 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) (b) Make a the subject of the formula ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) (Total 4 marks) Page 6 of 56 Q12. Solve .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer .................................................. (Total 6 marks) Q13. (a) Factorise fully 3 12x – 8xyz ......................................................................................................................... Answer ................................................. (2) (b) Factorise 2 x + 3x + 2 ......................................................................................................................... Answer ................................................. (2) (c) Simplify ......................................................................................................................... Answer ................................................. (1) Page 7 of 56 (d) 2 Factorise fully 10x – 40y 2 ......................................................................................................................... ......................................................................................................................... Answer ................................................. (3) (Total 8 marks) Q14. Evaluate –2 x 10 Give your answer as a fraction in its simplest form. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ................................................ (Total 3 marks) Q15. Work out the Highest Common Factor (HCF) of 63 and 105. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ................................................ (Total 2 marks) Q16. Solve the simultaneous equations 2x + 5y = 16 4x + 3y = 11 You must show your working. Page 8 of 56 Do not use trial and improvement. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ....................................................................... (Total 3 marks) Q17. (a) Work out ......................................................................................................................... ......................................................................................................................... Answer ................................................. (3) (b) Write as an improper fraction. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) (Total 5 marks) Page 9 of 56 Q18. Here are the equations of two straight lines. y = 7x – 6 y = mx + c (a) Write down the value of m if the lines are perpendicular. ......................................................................................................................... Answer ....................................................................... (1) (b) Write down the values of m and c if the lines are reflections of each other in the y axis. ......................................................................................................................... Answer m = ............................................................... c = ............................................................... (2) (Total 3 marks) Q19. For each calculation circle the answer that is correct and is in standard form. (a) 5 7 (3 × 10 ) × (4 × 10 ) Answer 12 × 10 12 1.2 × 10 36 35 12 × 10 13 1.2 × 10 (1) (b) –8 –2 (4 × 10 ) ÷ (8 × 10 ) Answer –6 0.5 × 10 5 × 10 4 –7 5 × 10 5 × 10 –5 (1) (Total 2 marks) Page 10 of 56 Q20. (a) Tom finds the value of Sam finds the value of (2n – 1)(n + 1) (2n – 1)(n + 1) when n = 1 when n = 2 Work out the difference between Tom’s value and Sam’s value. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ....................................................................... (3) (b) Expand and simplify (2n – 1)(n + 1) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ....................................................................... (2) (Total 5 marks) Q21. x and y are integers. x<4 y<3 x+y>4 Page 11 of 56 Use a graphical method to show that there is only one possible pair of values of x and y. .................................................................................................................................. .................................................................................................................................. (Total 4 marks) Q22. You are given the identity 2 2 x – 12x + 10 ≡ (x + a) + b Work out the values of a and b. .................................................................................................................................. .................................................................................................................................. Answer a = .......................... b = .......................... (Total 2 marks) Page 12 of 56 Q23. Work out the values of a and b. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer a = ..................., b = .................. (Total 3 marks) Q24. (a) (i) Factorise 2 x –x–2 ............................................................................................................... ............................................................................................................... Answer .............................................. (2) (ii) 2 Hence, solve x – x – 2 = 0 ............................................................................................................... Answer .............................................. (1) Page 13 of 56 (b) Simplify ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer .............................................. (3) (Total 6 marks) Q25. (a) Three friends share a pizza. One eighth of it falls on the floor and is thrown away. The remainder is shared equally. What fraction of the whole pizza does each person get? ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ Answer ................................................ (3) (b) The pizza costs £6.40 This is 20% less than the original price. What was the original price of the pizza? ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ Answer £ .............................................. (3) (Total 6 marks) Page 14 of 56 Q26. (a) Multiply out x(x + 7) ......................................................................................................................... Answer ............................................ (1) (b) Find both solutions of the quadratic equation x(x + 7) = 8 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ............................................ (3) (Total 4 marks) Q27. Solve the simultaneous equations x + 3y = 11 2x – y = 1 You must show your working. Do not use trial and improvement. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer x = ............. , y = ............... (Total 3 marks) Page 15 of 56 Q28. (a) Simplify (4y 3)2 ......................................................................................................................... ......................................................................................................................... Answer ............................................ (1) (b) Expand and simplify (a + 2b)(3a – 4b) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ............................................ (3) (Total 4 marks) Q29. Prove that .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. (Total 4 marks) Page 16 of 56 Q30. (a) Show that (x – 2)(x + 2) ≡ x 2 – 4 ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ (1) (b) Hence, simplify ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ Answer ............................................. (1) (Total 2 marks) Q31. Make x the subject of the formula .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ............................................ (Total 5 marks) Page 17 of 56 Q32. (a) Work out ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ Answer ................................................. (3) (b) Write down the value of (i) Answer ................................................. (1) (ii) Answer ................................................. (1) (Total 5 marks) Q33. (a) Work out the Highest Common Factor (HCF) of 42 and 98. ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ Answer ................................................ (2) (b) Write + in the form where a and b are integers. ........................................................................................................................ ........................................................................................................................ Answer ................................................ (2) (Total 4 marks) Page 18 of 56 Q34. (a) Expand and simplify 2x 2(x + 6) + 3x(x – 5) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer .................................................. (3) (b) Factorise fully 3mh2 – 15m 2h ......................................................................................................................... ......................................................................................................................... Answer .................................................. (2) (c) Simplify fully 4rs2 × 5r3s 4 ......................................................................................................................... ......................................................................................................................... Answer .................................................. (2) (d) Solve 9x 2 + 29x – 28 = 0 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer .................................................. (3) (Total 10 marks) Q35. (a) A straight line has gradient 2 and passes through the point (0, 8) Write down the equation of the line. Answer .................................................. (1) Page 19 of 56 (b) Write down the gradient of a line that is perpendicular to the line y = 4x Answer .................................................. (1) (c) Write down the equation of a line that is perpendicular to the line y = 4x Answer .................................................. (1) (Total 3 marks) Q36. (a) Solve the equation 2(4y – 1) = 18 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer y = ............................................ (3) Page 20 of 56 (b) The diagram shows a rectangle and a square. The rectangle has length 3x – 5 and width x + 1 The rectangle and the square have the same perimeter. Not drawn accurately Work out the length of the side of the square. Give your answer as an expression in terms of x in its simplest form. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................ (4) (Total 7 marks) Q37. (a) Expand 3(x + 5) ......................................................................................................................... Answer ................................................ (1) (b) Factorise 6a – 9 ......................................................................................................................... ......................................................................................................................... Answer ................................................ (1) (Total 2 marks) Page 21 of 56 Q38. The diagram shows a sequence of triangle patterns using shaded and unshaded triangles. The number of unshaded triangles in Pattern n is The number of shaded triangles in Pattern n is Work out the difference between the number of unshaded triangles and shaded triangles in Pattern 100. You must show your working. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ................................................ (Total 3 marks) Q39. (a) Simplify (5y 5)2 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) Page 22 of 56 (b) Expand and simplify (x – 3)(x + 5) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ................................................. (2) (c) Rearrange the formula w = 3(2x + y) – 5 to make x the subject. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer x = .......................................... (3) (Total 7 marks) Q40. Solve the equation (2x – 3)2 = (x – 1)(x + 1) Give your solutions to 2 decimal places. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ................................................ (Total 5 marks) Page 23 of 56 Q41. Andy and Ben both have a collection of marbles. If Andy gives Ben 15 marbles, they will have an equal number. If Ben gives Andy 15 marbles, Andy will have four times as many as Ben. How many marbles does Andy have? You must show your working. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer .............................................. (Total 4 marks) Q42. (a) Show that (p + q)2 ≡ p2 + 2pq + q2 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (1) (b) p and q are two numbers. The sum of p and q is 10 The product of p and q is 18 Work out the value of p2 + q2 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... Answer ............................................... (3) (Total 4 marks) Page 24 of 56 Q43. The function denoted by INT(x) is defined as the integer part of the positive number x. For example, (a) INT(2.7) = 2, INT(0.68) = 0, INT (5.432) = 5 Write down the value of INT (9.44) ......................................................................................................................... Answer ................................................ (1) (b) Write down a value of x for which INT(x) = x ......................................................................................................................... Answer ................................................ (1) (c) y is a positive number less than 0.5 Explain why INT(2y) = INT(y) ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (2) (Total 4 marks) Q44. The diagram shows the cross-section of a river bed, ABCDE. BD is the surface of the river. C is the midpoint of the river. A and E are points on the river bank that are 1 metre above the river surface. Not drawn accurately Page 25 of 56 Taking the origin at B, 1 metre as 1 unit on both axes and the river surface as the x-axis, the curve ABCDE can be represented by the equation (a) Show that the depth of the river at the mid-point C is 2 metres. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (3) (b) Show that the distance AE is approximately 12.25 metres. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (5) (Total 8 marks) Q45. (a) Write down the next odd number after 2n + 1 Answer ............................................ (1) (b) Show that (2n + 1)2 ≡ 4n2 + 4n + 1 ......................................................................................................................... ......................................................................................................................... (1) Page 26 of 56 (c) Prove that the difference between the squares of consecutive odd numbers is a multiple of 8 ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... (3) (Total 5 marks) Q46. James invests £700 for 2 years at 10% per year compound interest. How much interest does he earn? ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... Answer £ ................................................. (Total 2 marks) Q47. A sketch of the line 2y – x = 4 is shown. The line crosses the axes at A and B. Page 27 of 56 (a) Calculate the coordinates of A and B. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer A(..........., .............), B (............., ............) (2) (b) Calculate the gradient of the line AB. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ................................................. (2) (Total 4 marks) Q48. (a) Factorise 7x + 14 .......................................................................................................................... Answer ................................................. (1) (b) Expand and simplify 4(m + 3) + 3(2m – 5) .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ................................................. (2) Page 28 of 56 (c) Solve the simultaneous equations: 2x + 3y = 9 3x + 2y = 1 You must show all your working. Do not use trial and improvement. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer x = ......................., y = ......................... (4) (d) Factorise x 2 + 6x – 16 .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ................................................. (2) (Total 9 marks) Q49. (a) Write down the value of 110 .......................................................................................................................... Answer .................................................... (1) (b) Find the value of 8 .......................................................................................................................... .......................................................................................................................... Answer .................................................... (2) Page 29 of 56 (c) Simplify 6–2 × 1440.5 .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ...................................................... (3) (Total 6 marks) Q50. The line l on the graph passes through the points A (0, 3) and B (–4, 11). (a) Calculate the gradient of the line l. .......................................................................................................................... .......................................................................................................................... Answer ...................................................... (2) Page 30 of 56 (b) Write down the equation of the line l. .......................................................................................................................... Answer ...................................................... (1) (c) Write down the equation of the line which also passes through the point (0, 3) but is perpendicular to line l. .......................................................................................................................... .......................................................................................................................... Answer ...................................................... (2) (Total 5 marks) Q51. Simplify fully .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Answer ....................................................................... (Total 4 marks) Page 31 of 56 (2n + 2)2 – (2n)2 M1. M1 4n2 + 8n + 4 – 4n2) M1 dep 8n + 4 A1 8n + 4 = 2(2n + 2 + 2n) or 2(2n + 2 + 2n) = 8n + 4 A1 Alternate method Let n be even (n + 2)2 – n2 M1 n2 + 4n + 4 – n2 M1 dep 4n + 4 A1 2(n + n + 2) = 2(2n + 2) = 4n + 4 or 4n + 4 = 2(2n + 2) = 2(n + n + 2) A1 [4] M2. M1 × 100 M2 complete and correct build up method. If any numerical errors calculations must be shown to give M2. M1dep 75 A1 [3] Page 32 of 56 M3. (a) x 2 + 4x + 4x + 16 At least three terms correct M1 x 2 + 8x + 16 A1 (b) – 4x – 16 or (x + 4) (x + 4 – 4) M1 x 2 + 4x = x(x + 4) or (x + 4) x A1 [4] M4. 6x(3x – 2) or 2x(9x – 6) or B1 (2)(3x + 2)(3x – 2) (6x + 4)(3x – 2) or (3x + 2)(6x – 4) B1 Partial simplification or or M1 A1 [4] M5. (a) (12 – 4) × 2 (= 16) oe M1 (16 – 4) × 2 (= 24) and (24 – 4) × 2 (= 40) oe A1 Page 33 of 56 (b) 12 ÷ 2 + 4 or (40,) 24, 16, 12, 10(, 9) M1 10 or 9 A1 [4] M6. 2a + 3c = 69 3a + 5c = 109 B1 one equation correct Any letters may be used but need to be consistent for B2 B2 × 1st by 3 or 5 × 2nd by 2 or 3 oe (to obtain consistent coefficients) M1 Two equations (max one error) and subtraction eg, 6a + 9c = 207 6a + 10c = 218 and subtraction M1 dep Adult (a =) 18 Child (c =) 11 A1 [5] M7. (a) 37 B1 (b) 16 + a (127 – a) ÷ 2 B1 2 × their (16 + a) + a 32 + 3a, 2(16 + a) + a M1 2 × their (32 + 3a) + a = 127 oe 64 + 7a = 127 M1 (a =) 9 A1 Page 34 of 56 Alternate method Evidence of multiplying 8 by 2 and adding any number Evidence of subtracting a number from 127 and dividing by 2 M1 Evidence of multiplying their answer by 2 and adding the same number Evidence of subtracting the same number from their answer and dividing by 2 M1 Refined attempt M1 (a =) 9 A1 [5] M8. (a) m8 B1 (b) m –2 or B1 (c) B1 seen or implied by cancelling common factors B2 [4] M9. (a) 81 + 9 +9 + or better 4 terms and any 3 correct M1 88 + 18 a = 88 b = 18 A1 Page 35 of 56 (b) M1 A1 =2+2 M1 = 2(1 + ) Strand (ii) Correct answer with a logical argument showing key steps Q1 Alternate method 1 M1 A1 =2+2 M1 = 2(1 + ) Strand (ii) Correct answer with a logical argument showing key steps Q1 Page 36 of 56 Alternate method 2 +6=2 (1 + ) M1 =2 +2×3 A1 = +6 M1 12 + Note: This is not a full proof Q0 [6] M10. (a) Either or oe M1 oe A1 A1 ft (b) × 24 oe M1 15 A1 [5] M11. (a) 3x – x > 8 – 7 M1 x> oe A1 Page 37 of 56 (b) a + 3 = b2 M1 a = b2 – 3 A1 [4] M12. Sight of 10x or –3(2x – 1) or 3x(2x – 1) M1 –6x + 3 or 6x 2 – 3x M1 dep 6x 2 – 7x – 3 (= 0) A1 (2x – 3)(3x + 1) (= 0) M1 x = 1.5 or – A1 Full answer with stages clearly shown Strand (ii) Q1 [6] M13. (a) 4x(3x 2 – 2yz) B1 one correct factor eg, 4(3x3 – 2xyz) or x(12x2 – 8yz) B2 (b) (x ± 1)(x ± 2) M1 (x + 1)(x + 2) A1 (c) B1 Page 38 of 56 (d) 10(x 2 – 4y 2) M1 10(x + 2y)(x – 2y) A1 for both ± 2y or 10(x + 4y)(x – y) A2 [8] M14. 5 or –5 or ±5 B1 oe B1 SC2 0.05 or equivalent fraction B1 [3] M15. Prime factorisation of either number correct (any form) 63 = 32 × 7 105 = 3 × 5 × 7 M1 21 A1 Alt List factors of at least one number correctly (excl 1 and itself) 63 = (1,) 3, 7, 9, 21 (, 63) 105 = (1,) 3, 5, 7, 15, 21, 35 (, 105) M1 21 A1 [2] Page 39 of 56 M16. 4x + 10y = 32 (4x + 3y = 11) 6x + 15y = 48 20x + 15y = 55 oe Allow one error M1 7y = 21 or 14x = 7 oe A1 ft y = 3 and x = SC1 for no working or T&I A1 [3] M17. (a) 72 and / or –72 B1 Sight of 2 (and / or –2) also B1 Sight of 36 or (but not –36 or – ) B3 (b) B1 Sight of 1 or SC1 B2 [5] M18. (a) –0.14(…) B1 (b) –7 B1 –6 B1 [3] Page 40 of 56 M19. (a) Circles 1.2 × 1013 B1 (b) Circles 5 × 10–7 B1 [2] M20. (a) (2 × 1 – 1) × (1 + 1) or (2 × 2 – 1) × (2 + 1) or 2 × 12 + 1 – 1 or 2 × 22 + 2 – 1 M1 2 or 9 A1 (±) 7 A1 (b) 2n2 + n – 1 B1 2n2 + 2n – n – 1 any three out of four terms correct B2 [5] M21. x = 4, y = 3 and x + y = 4 drawn B1 One line correct B2 x = 3 and y = 2 identified eg, as circled or marked point on graph (no lines or regions necessary) ft x = 2 and y = 3 for lines x = 3 and y = 4 interchanged B1ft Correct region identified from (their) three lines Correct region from two correct lines or (Their) x = 3 and y = 2 from (their) two lines B2ft [4] Page 41 of 56 M22. (a =) –6 B1 (b =) –26 B1 [2] M23. a+b=7 M1 ab = 10 M1 a = 2, b = 5 a = 5, b = 2 B1 [3] M24. (a) (i) (x ± a)(x ± b) ab = ± 2 M1 (x + 1)(x – 2) A1 (ii) (x = ) –1 and 2 ft Their brackets if M1 awarded B1ft (b) ft Their factorisation if M1 awarded in (a) M1 Numerator 3x + 2 + x – 2 ft Their factorisation A1ft or No ft A1 [6] Page 42 of 56 M25. (a) 1– oe M1 Their oe M1 dep oe A1 (b) Sight of 80% or 0.8 or 1.25 or B1 × 10 oe M1 8(.00) A1 [6] M26. (a) x 2 + 7x B1 (b) x 2 + 7x – 8 (= 0) M1 (x + 8)(x – 1) (x + a)(x + b) where ab = ± 8 Use of formula, allow one error M1 –8 and 1 A1 [4] Page 43 of 56 M27. 2x + 6y = 22 6x – 3y = 3 M1 7y = 21 7x = 14 A1 y = 3 and x = 2 A1 [3] M28. (a) 16y 6 B1 (b) 3a2 – 4ab + 6ab – 8b2 3 out of 4 terms correct M1 3a2 – 4ab + 6ab – 8b2 A1 3a2 + 2ab – 8b2 A1 [4] M29. One fraction correct or n(n + 6) or (n + 5)(n – 4) M1 n2 + 6n and n2 + 5n – 4n – 20 n2+ 6n or n2 + n – 20 Condone – n2 + 5n – 4n – 20 M1dep Page 44 of 56 n2 + 6n – n2 – 5n + 4n + 20 n2 + 6n – n2 – n + 20 A1 5n + 20 and Answer given Must see all working for final mark Must see use of denominators for final mark A1 [4] M30. (a) x 2 – 2x + 2x – 4 B1 (b) x+2 B1 [2] M31. y(x – 4) = 2x + 3 M1 xy – 4y = 2x + 3 M1dep xy – 2x = 4y + 3 Allow one error M1 x(y – 2) = 4y + 3 M1 (x =) oe A1 [5] Page 45 of 56 M32. (a) At least one correct M1 ft Their attempts at improper fractions M1 oe eg, A1 Alt 1+3+ oe M1 (4 +) oe M1 oe eg, A1 (b) (i) 1 B1 (ii) 5 or –5 Either of 5 or –5 scores the B1 B1 [5] M33. (a) (42 =) 2 (×) 3 (×) 7 or (98 =) 2 (×) 7 (×) 7 Accept on factor trees / repeated division / list or similar Condone redundant (× 1) M1 14 A1 Page 46 of 56 (b) √99 = 3√11 or √44 = 2√11 M1 5√11 or a = 5, b = 11 A1 [4] M34. (a) 2x 3 + 12x 2 + 3x 2 – 15x Allow 1 error M1 2x 3 + 12x 2 + 3x 2 – 15x Fully correct A1 2x 3 + 15x 2 – 15x Ignore factorising after final answer ft From 4 terms where simplification is possible A0 For fw eg, incorrect attempt to collect terms A1 ft (b) 3mh(h – 5m) B1 For partial factorisation with two factors 3m(h2 – 5mh) 3h(mh – 5m2) mh(3h – 15m) B2 (c) 20r 4s 6 B1 For 2 terms correct B2 Page 47 of 56 (d) Attempt to factorise or attempt to use formula (allow one error) (ax + b)(cx + d) where ac = 9 and bd = ± 28 M1 (9x – 7)(x + 4) or formula fully correct A1 7 / 9 and –4 Accept 0.77 ... or 0.78 A1 [10] M35. (a) y = 2x + 8 B1 (b) – oe –0.25 B1 (c) y=– (+ c) or x = –4y (+ c) B1 ft [3] M36. (a) 8y – 2 (= 18) (4y – 1 =) 18 ÷ 2 M1 8y = 18 + 2 4y = 9 + 1 M1 2.5 oe A1 Page 48 of 56 (b) 3x – 5 + 3x – 5 + x + 1 + x + 1 oe Allow one error M1 8x – 8 oe A1 (their) (8x – 8) ÷ 4 oe Condone missing brackets M1 2x – 2 oe SC2 x – 1 from 3x – 5 + x + 1 or 4x – 4 A1 ft [7] M37. (a) 3x + 15 Allow 3 × x + 15 B1 (b) 3(2a – 3) 3 × (2a – 3) B1 [2] M38. 100 × (100 + 1) ÷ 2 (or 5050) or 100 × (100 – 1) ÷ 2 (or 4950) 1 3–1 6–3 10 – 6 … M1 Page 49 of 56 (their) 5050 – (their) 4950 Pattern 1 → 1 Pattern 2 → 2 … Pattern n → n n M1 dep 100 A1 [3] M39. (a) 25y 10 B1 ?y 10 or 25y ? B2 x 2 + 2x – 15 (b) B1 x2 – 3x + 5x – 15 3 correct terms out of 4 terms seen B2 (c) (w =) 6x + 3y – 5 or w + 5 = 3(2x + y) M1 (6x) = w – 3y + 5 (w + 5)/3 = 2x + y M1 (x) = (w – 3y + 5)/6 oe eg, (x) = [(w + 5)/3 – y]/2 A1ft [7] M40. 4x 2 – 12x + 9 or 4x2 – 6x – 6x + 9 3 correct terms out of 4 seen B1 x2 – 1 or x2 – x + x – 1 3 correct terms out of 4 seen B1 (their) 3x 2 – 12x + 10 (= 0) ft (their) 4x2 – 12x + 9 and (their) x2 – 1 oe M1 Page 50 of 56 ft (their) 3x2 – 12x + 10 Must be in the form ax2 + bx + c (= 0) M1 2.82 and 1.18 A1 [5] M41. a – 15 = b + 15 T&I using two numbers with a difference of 30 a – b = ± 30 a and b have a difference of 30 M1 4(b – 15) = a + 15 Trial and improvement: Checking ratio of 2 numbers used M1 4(a – 30 – 15) = a + 15 Dependent on both Ms 4(b – 15) – (b + 15) = 30 ⇒ b = 35 M1dep (a =) 65 A1 [4] M42. (a) Evidence of 4 terms: 2 p + pq + pq (qp) + q2 B1 (b) p2 + q2 = (p + q)2 –2pq 100 = p2 + q2 + 2 × 18 oe M1 (p2 + q2 =) 102 – 2 × 18 A1 64 A1 [4] Page 51 of 56 M43. (a) 9 B1 (b) Any integer value (accept 0) B1 (c) If y < then INT (y) = 0 and 2y < 1 so INT (2y) = 0 E1 For a partial explanation, eg 1 numerical example E2 [4] M44. (a) 2x(x – 10) = 0 M1 River ( x = ) 10 ( m wide) A1 When x = 5, y = (50 – 100) ÷ 25 = –2 Working must be seen A1 (b) 2x 2 – 20x = 25 M1 2x 2 – 20x – 25 = 0 M1 Use quadratic formula, completing the square to solve equation Allow one error but not wrong formula M1dep x = –1.123 or 11.123 A1 AE = 1.123 + 11.123 ≈ 12.25 m NB error with – b gives 1.123 and – 11.123 Allow last A1 A1 [8] M45. (a) 2n + 3 oe 2n + 1 + 2 B1 (b) 4n2 + 2n + 2n + 1 B1 Page 52 of 56 (c) (2n + 3)2 – (2n + 1)2 oe vice versa M1 4n2 + 12n + 9 – (4n2 + 4n + 1) Allow lack of brackets A1 8n + 8 (= 8(n + 1)) Accept 8n + 8 as sufficient justification – 8n – 8 OK A1 [5] 700 × 1.12 – 700 or 700 × 0.1 or 70 or 700 × 1.1 or 770 or 700 × 1.12 or 847 or 140 M46. M1 147(.00) A1 [2] M47. (a) A(–4, 0) or 2y = 4 and – x = 4 seen B1 B(0,2) SC1 reversed answers B1 (b) (their difference in y’s) ÷ (± their difference in x’s) or attempt to rearrange 2y – x = 4 to y = 0.5x + 2 and 2y = 4 + x or y – 0.5x = 2 seen ft condone A(0, ?) and/or B(?, 0) from (a) M1 0.5 or ft for 0 < gradient < 1 only A1 ft [4] Page 53 of 56 M48. (a) 7(x + 2) allow one error B1 (b) 4m + 12 + 6m – 15 M1 10m – 3 allow 10m + –3 A1 (c) 6x + 9y = 27 and or 6x + 4y = 2 4x + 6y = 18 and 9x + 6y = 3 5y = 25 or 5x = –15 y=5 or x=–3 x=–3 and y=5 M1 M1 dep A1 A1 SC1 correct answer with no working or using T&I (d) (x + 8)(x – 2) B1 (x ± 8)(x ± 2) B2 [9] M49. (a) 1 B1 (b) (3√8)2 or 3√(82) Cube root and square attempted M1 4 A1 (c) 62 = 1/62 = or 1/36 M1 1440.5 = √144 or 12 M1 or 0.33 ... Allow 12/36 or equivalent 1/36 × 12 is not fully simplified ... A0 A1 [6] Page 54 of 56 M50. (a) Any correct attempt at (y-step) ÷ (x-step) Might be marked on diagram M1 –2 A1 (b) y = -2x + 3 ft. their gradient B1 (c) Gradient = Attempt at gradient of perpendicular line, ft. from their gradient in part (a) using (m1 × m2 = –1) as long as there is no contradiction between parts (a) and (b) M1 y = ½x + 3 or equivalent A1 ft [5] M51. 6x(3x – 2) or 2x(9x – 6) or B1 (2)(3x + 2)(3x – 2) (6x + 4)(3x – 2) or (3x + 2)(6x – 4) B1 Partial simplification or or M1 A1 [4] Page 55 of 56 Page 56 of 56
