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AQA Level Certificate in Further Mathematics Worksheets - Teacher Booklet Level Specification Level Certificate in Further Mathematics 8360 Worksheets - Teacher Booklet Our specification is published on our website (www.aqa.org.uk) We will let centres know in writing about any changes to the specification We will also publish changes on our website The definitive version of our specification will always be the one on our website, this may differ from printed versions You can get further copies of this Teacher Resource from: The GCSE Mathematics Department AQA Devas Street Manchester M16 6EX Or, you can download a copy from our All About Maths website (http://allaboutmaths.aqa.org.uk/) Copyright © 2012 AQA and its licensors All rights reserved AQA retains the copyright on all its publications, including the specifications However, registered centres for AQA are permitted to copy material from this specification booklet for their own internal use The Assessment and Qualifications Alliance (AQA), is a company limited by guarantee registered in England and Wales (company number 3644723), and a registered charity 1073334 Registered address: AQA Devas Street, Manchester M15 6EX Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Contents Coordinate Geometry - Circles Geometric Problems and Proof 17 Algebraic Proof 26 Trigonometry 31 Matrices - 37 Matrices - 41 Inequalities 46 Functions 53 Coordinate Geometry - Calculus 59 10 Factor Theorem 68 11 Sequences 75 12 Algebraic Problems – including ratio 84 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Glossary for Mark Schemes These examinations are marked in such a way as to award positive achievement wherever possible Thus, for these papers, marks are awarded under various categories M Method marks are awarded for a correct method which could lead to a correct answer A Accuracy marks are awarded when following on from a correct method It is not necessary to always see the method This can be implied B Marks awarded independent of method M Dep A method mark dependent on a previous method mark being awarded B Dep A mark that can only be awarded if a previous independent mark has been awarded ft Follow through marks Marks awarded following a mistake in an earlier step SC Special case Marks awarded within the scheme for a common misinterpretation which has some mathematical worth oe Or equivalent Accept answers that are equivalent eg, accept 0.5 as well as Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Coordinate Geometry - Circles Question Write down the equation of each of these circles (a) Centre (0, 3) radius (2 marks) (b) Centre (1, 5) radius (2 marks) (c) Centre (3, 4) radius (d) Centre (8, 15) radius 17 (2 marks) Does this circle pass through the origin? Show working to support your answer (4 marks) Mark Scheme (a) x + (y 3) = B2 B1 LHS, B1 RHS (b) (x 1) + (y + 5) = 16 B2 B1 LHS, B1 RHS (c) (x + 3) + (y 4) = B2 B1 LHS, B1 RHS (d) (x 8) + (y 15) = 289 B2 B1 LHS, B1 RHS (8) + (15) M1 oe 64 + 225 = 289, Yes A1 Question Write down the centre and radius of each of these circles (a) x + y = 36 (b) (x 3) + (y 4) = 100 (c) (x + 5) + y = 2 (2 marks) (2 marks) (2 marks) Mark Scheme (a) ( r) = (centre =) (0, 0) B2 B1 For each (b) (r) = 10 (centre =) (3, 4) B2 B1 For each (c) ( r) = B2 B1 For each (centre =) (5, 0) Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question (non-calculator) AB is the diameter of a circle A is (3, 6) and B is (5, 12) Work out the equation of the circle (5 marks) Mark Scheme 3 12 or 2 M1 (1, 9) A1 (5 1) (12 9) M1 oe ft Their centre A1 (x 1) + (y 9) = 25 A1 ft ft Their centre and radius Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question (non-calculator) PQ is a diameter of a circle, centre C Not drawn accurately y Q C (1, 2) P (1, 1) x O (a) Work out the coordinates of Q (b) Work out the equation of the circle (1 mark) (3 marks) Mark Scheme (a) (b) (3, 3) B1 22 12 M1 A1 (x 1) + (y 2) = B1 ft oe ft Their radius Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question (non-calculator) A (12, 6) and B (14, 4) are two points on a circle, centre C (20, 12) y Not drawn accurately C (20, 12) A (12, 6) M B (14, 4) x O (a) Work out the coordinates of the midpoint M, of AB (2 marks) (b) Show that the length CM = (3 marks) (c) Work out the radius of the circle (2 marks) Mark Scheme (a) (b) (c) 12 14 64 or 2 M1 (13, 5) A1 (20 13)2 (12 5)2 M1 ft Their M 98 A1 72 72 49 = A1 72 (1 1) = (20 12)2 (12 6)2 M1 10 10 A1 oe Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question A linear sequence starts a+b a + 3b a + 5b a + 7b ………… The 5th and 8th terms have values 35 and 59 (a) Work out a and b (4 marks) (b) Work out the nth term of the sequence (2 marks) Mark Scheme (a) a + 9b = 35 M1 a + 15b = 59 (b) 6b = 24 M1 b=4 A1 a = 1 A1 ft 11 19 …… 8n oe B1 ft B1 ft Question A sequence has nth term 3n n n ( n 1) (a) Show that the difference between the nth and (n + 1)th terms is (b) Which are the first two consecutive terms with a difference less than 0.01? (c) Write down the limiting value of the sequence as n 80 (3 marks) (2 marks) (1 mark) Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Mark Scheme (a) 3n 3( n 1) n 1 n M1 (3n 1)( n 1) n (3n 4) n ( n 1) M1 3n n 3n 3n 4n n ( n 1) A1 = Alt (a) (3 + 1 ) (3 + ) n1 n n 1 n n ( n 1) (b) eg subtracts in different order M1 oe eg subtracts in different order M1 oe A1 n ( n 1) Any substitution and evaluation for M1 n 10 (c) oe n ( n 1) 3n =3+ n n = oe eg 1 = 10 90 or 1 = 10 11 110 oe eg 0.01n + 0.01n and attempt to solve 10th and 11th A1 B1 81 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question A sequence has nth term 5n 2n Show that the limiting value of the sequence, S, as n is 2.5 (2 marks) Mark Scheme 5n 5n = + 2n 2n 2n M1 oe 1 2 n as n n S = A1 (= 2.5) Question Here is the sequence of odd numbers …… A quadratic sequence is formed by multiplying consecutive odd numbers in successive pairs 15 35 63 …… Work out the nth term of this sequence (3 marks) Mark Scheme Odd number is 2n + or 2n M1 2n and M1 2n + Sequence is (2n 1)(2n + 1) A1 (= 4n 1) Alt Using Method A or Method B giving 4n 82 marks or any other valid method eg n2 64 63 4n – 1 16 16 36 15 35 4n Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question 10 The nth term of a sequence is 2n 3n (a) Show that the difference between the first two terms is (b) Write down the limiting value of the sequence as n 10 (3 marks) (1 mark) Mark Scheme (a) T1 = B1 T2 = 14 B1 oe = 10 10 10 B1 oe B1 (= (b) ) 83 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics 12 Algebraic problems – including ratio Note x y If x : y = : 7, then If, in a problem, two numbers are in the ratio : 7, use 4x and 7x as the numbers (usually leading to a linear equation); otherwise, use x and y as the numbers (which will lead to simultaneous equations) If x : y = : 7, what is x + 2y : 3x? = Think in terms of ‘parts’, ie parts and parts, so x + 2y : 3x = + 14 : 12 = 18 : 12 = 3:2 Question Work out the possible values of 2n 3n if n = 16 Give your answers as fractions in their simplest form Mark Scheme n=4 M1 A1 n = 4 M1 10 A1 84 (4 marks) Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question x:y=6:5 (a) Express x in terms of y (b) Show that (2 marks) x + 3y : 2x y = : (2 marks) Mark Scheme (a) x = y A1 oe 15 y 12 y 6y 5y + : 5 5 M1 oe + : 21( y ) 7( y ) : (5 ) (5 ) A1 x= (b) M1 6y 85 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question A point P divides XY in the ratio : Y (6a, 11b) Not drawn accurately P X (a, b) Work out the coordinates of P, in terms of a and b (3 marks) Mark Scheme 3 of (6a a) or of (11b b) 10 10 M1 oe (2.5a, 4b) A2 oe A1 For each coordinate SC2 (1.5a, 3b) 86 Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question Here is a linear sequence a+b a + 3b a + 5b a + 7b …… Given that 2nd term : 4th term = : 1st term = Work out a and b (5 marks) Mark Scheme a 3b = a 7b M1 5a + 15b = 2a + 14b M1 Allow one error 3a + b = A1 oe a + b = 4 A1 ft 2a = a = and b = 6 A1 ft 87 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question You are given that ab + a = and a:b=4:3 Work out the possible pairs of values of a and b (7 marks) Mark Scheme M1 a = b oe b= 3a A1 a 3a +a=5 M1 4b 4b b+ =5 3 3a + 4a 20 = A1 4b + 4b 15 = (3a + 10)(a 2) = M1 (2b + 5)(2b 3) A1 ft b= A1 ft a= 10 a= 10 b= 88 a=2 b= a= 4b b= a=2 Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question The sum of the ages of two people is 90 years Six years ago, their ages were in the ratio : How old are they now? Do not use trial and improvement You must show your working (5 marks) Mark Scheme Let their ages years ago be 8x and 5x M1 8x + 5x = 90 12 M1 13x = 78 A1 Allow 90 for M1 (x = 6) Their and their (48) Alt M1 (30) 54 and 36 A1 x + y = 90 M1 x6 = y6 M1 18 = 8y 5x A1 Eliminates a letter M1 (x =) 54 and (y =) 36 A1 89 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question O is the centre of the circle Not drawn accurately y O Given that x:y=4:5 x Work out the value of y Do not use trial and improvement You must show your working (7 marks) Mark Scheme x, x and 180 2x M1 seen or on diagram M1 x = y A1 oe M1 oe M1 oe 9y = 90 M1 oe y = 50 A1 x= 4y 2y = 180 2x (or y = 90 x) y = 90 90 4y Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question A rectangular picture is surrounded by a frame of constant width All measurements are in centimetres a Not drawn accurately 7x 3x b Given that a:b=3:2 Work out x (5 marks) Mark Scheme a = 7x + 18 or b = 3x + 18 B1 their (7 x 18 ) = their (3 x 18 ) M1 14x + 36 = 9x + 54 M1 Rearranging 5x = 18 M1 Solving x = 3.6 A1 oe 91 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question If x : y = : and y : z = 10 : Find, in its simplest form (a) x:z (3 marks) (b) 10x : 7y (2 marks) (c) x+y:y (2 marks) Mark Scheme (a) (b) (c) 92 x : y = : 10 M1 x : y : z = : 10 : M1 x:z=2:3 A1 10 : M1 6:7 A1 3+5:5 M1 8:5 A1 oe oe x y x = + or +1 y y Level Certificate in Further Mathematics – 8360 – Worksheets – Teacher Booklet Question 10 A cuboid has dimensions 2n, n and n cm A diagonal has length 2n + cm Not drawn accurately n1 2n + n 2n Work out n (6 marks) Mark Scheme (2n) + n M1 (2n) + n + (n 1) = (2n + 1) M1 4n + n + n n n + M1 Allow one error 2n 6n = M1 Rearranging 2n(n 3) = M1 (allow ÷ by n ) n=3 A1 oe = 4n + 2n + 2n + ; or 2n = 6n 2n = 93 AQA Level Certificate in Further Mathematics from 2011 onwards Qualification Accreditation Number: 600/2123/8 For updates and further information on any of our specifications, to find answers or ask us a question, register with Ask AQA at: aqa.org.uk/askaqa Free launch meetings are available in 2011 followed by further support meetings through the life of the specification Further information is available at: http://events.aqa.org.uk/ebooking Copyright © 2012 AQA and its licensors All rights reserved The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales company number 3644723) and a registered charity (registered charity number 1073334) Registered address: AQA, Devas Street, Manchester M1 6EX SP/03/11 ... circle (2 marks) Mark Scheme (a) (b) (c) 12 14 64 or 2 M1 (13, 5) A1 (20 13 )2 ( 12 5 )2 M1 ft Their M 98 A1 72 72 49 = A1 72 (1 1) = (20 12) 2 ( 12 6 )2 M1 10 10 A1 oe Level Certificate. .. 42 12 M1 A1 15 Worksheets – Teacher – Booklet – 8360 – AQA Level Certificate in Further Mathematics Question 12 (non-calculator) The equation of this circle is x + y = 20 P (4, 2) is a point... or M2 M1 Allow one error in expansions 4x + 20 x + 25 and (2x + 5) A1 oe eg, 4x + 20 x + 25 and (2x + 5)(2x + 5) Explains that only solution is A1 oe 4x + 20 x + 25 (x = ) 2. 5 28 eg, explains
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