AQA Level 2 Certificate in Further Mathematics. Worksheets - Teacher Booklet

Transcription

1 AQA Level Certificate in Further Mathematics Worksheets - Teacher Booklet

2

3 Level Specification Level Certificate in Further Mathematics 860 Worksheets - Teacher Booklet

4 Our specification is published on our website ( 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 6 6EX Or, you can download a copy from our All About Maths website ( Copyright 0 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 6447), and a registered charity 074. Registered address: AQA Devas Street, Manchester 5 6EX

5 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Contents Coordinate Geometry - Circles 7 Geometric Problems and Proof 7 Algebraic Proof 6 4 Trigonometry 5 Matrices Matrices Inequalities 46 8 Functions 5 9 Coordinate Geometry - Calculus 59 0 Factor Theorem 68 Sequences 75 Algebraic Problems including ratio 84 5

6 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Glossary for s 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. BDep 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 6

7 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Coordinate Geometry - Circles Question Write down the equation of each of these circles. (a) Centre (0, ) radius (b) Centre (, 5) radius 4 (c) Centre (, 4) radius 7 (d) Centre (8, 5) radius 7 Does this circle pass through the origin? Show working to support your answer. ( marks) ( marks) ( marks) (4 marks) (a) x + (y ) = 4 B B LHS, B RHS (b) (x ) + (y + 5) = 6 B B LHS, B RHS (c) (x + ) + (y 4) = 7 B B LHS, B RHS (d) (x 8) + (y 5) = 89 B B LHS, B RHS (8) + (5) oe = 89, Yes Question Write down the centre and radius of each of these circles. (a) x + y = 6 (b) (x ) + (y 4) = 00 (c) (x + 5) + y = ( marks) ( marks) ( marks) (a) (r) = 6 (centre =) (0, 0) B B For each (b) (r) = 0 (centre =) (, 4) B B For each (c) (r) = (centre =) (5, 0) B B For each 7

8 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question (non-calculator) AB is the diameter of a circle. A is (, 6) and B is (5, ). Work out the equation of the circle. (5 marks) 5 or 6 (, 9) ( 5 ) ft Their centre 5 ) ( 9 oe (x ) + (y 9) = 5 ft ft Their centre and radius 8

9 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 4 (non-calculator) PQ is a diameter of a circle, centre C. y Not drawn accurately Q C (, ) P (, ) O x (a) Work out the coordinates of Q. ( mark) (b) Work out the equation of the circle. ( marks) (a) (, ) B (b) oe 5 (x ) + (y ) = 5 Bft ft Their radius 9

10 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 5 (non-calculator) A (, 6) and B (4, 4) are two points on a circle, centre C (0, ). y Not drawn accurately C (0, ) A (, 6) M B (4, 4) O x (a) Work out the coordinates of the midpoint M, of AB. ( marks) (b) Show that the length CM = 7 ( marks) (c) Work out the radius of the circle. ( marks) (a) 4 or 6 4 (, 5) (b) ( 0 ) ( 5) ft Their M = 7 7 ( ) = 7 (c) ( 0 ) ( 6) oe 0 0

11 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 (0, ), (0, ) and (4, ) are three points on a circle, centre C. y (4, ) Not drawn accurately C O x Work out the coordinates of C. ( marks) 0 4 C (, 5)

12 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 7 AB is a diameter of the circle ABC. C (4, 6) Not drawn accurately A (, ) B (6, k) Work out the value of k. (5 marks) Gradient AC = 6 oe 4 = 6 oe Gradient BC = 6 k = 4 6 k = Bft

13 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 8 A circle has equation (x 5) + (y 4) = 00 Show that the point (, ) lies on the circle. ( marks) ( 5) + ( 4) = 00 Question 9 The point (, ) lies on the circle (x a) + (y 4) = 00 Work out the two possible values of a. (5 marks) ( a) + ( 4) = a a + a + 6 (= 00) Allow error a 6a + 05 = 0 (a 5)(a ) = 0 a = 5 and a =

14 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 0 A circle passes through the points (0, ) and (0, ) and has centre (6, k) y Not drawn accurately (6, k) O x (a) Work out the value of k. (b) Hence find the equation of the circle. (5 marks) (a) oe eg, + 4 k = 7 (b) 6 ( 7 ) oe ft Their k 5 (x 6) + (y 7) = 5 ft ft Their k and their radius 4

15 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question (non-calculator) The equation of this circle, centre C, is (x ) + (y 5) = 7 P (4, ) is a point on the circle. y Not drawn accurately C O P (4, ) x (a) Show working to explain why OP is a tangent to the circle. (5 marks) (b) Show that the length OP is equal to the radius of the circle. ( marks) (a) C is (, 5) B Gradient CP = Gradient OP = = B So perpendicular (ie, tangent) (b) r = 7 B OP = 4 = 7 5

16 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question (non-calculator) The equation of this circle is x + y = 0 P (4, ) is a point on the circle. y Not drawn accurately P (4, ) O x Work out the equation of the tangent to the circle at P. Give your answer in the form y = mx + c ( marks) Gradient OP = 4 B Gradient of tangent = y = (x 4) y = x + 0 Bft 6

17 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Geometric Problems and Proof Question SQ is a tangent to the circle at Q. PR = QR R Not drawn accurately P S Q Prove that RQ bisects angle PQS. ( marks) Let angle SQR = x angle RPQ = x alternate segment angle RQP = x isosceles triangle Any order of angles RQS = RQP SC Correct solution without reasons 7

18 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question PQRS is a trapezium. Angle PSR = angle QRS P Q Not drawn accurately S R Prove that PQRS is a cyclic quadrilateral. ( marks) Let angle PSR = x = angle QRS SPQ = 80 x Allied angles on parallel lines PQR = 80 x SPQ + QRS = 80 PSR + PQR = 80 PQRS is a cyclic quadrilateral (converse of) opposite angles add up to 80 SC Correct solution without reasons 8

19 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question p : q : r = 4 : 6 : 5 S s r R Not drawn accurately q Q p P Work out s. (5 marks) p + r = 80 4x + 5x = 80 oe (9x = 80) x = 0 6x = 0 ft Their x s = 60 ft ft Their x 9

20 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 4 O is the centre of the circle. AOBC and EDC are straight lines. E Not drawn accurately D A y x O B x C Prove that 4x + y = 90 (4 marks) BED = x angles in same segment AEB = 90 angle in semicircle = 90 In Δ ACE y + x + x + x + 90 = 80 angle sum of a triangle = 80 y + 4x = = 90 SC Correct solution without reasons 0

21 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 5 QS bisects both of the angles PSR and PQR. P y y S Not drawn accurately Q x x R Prove that QS is a diameter of the circle. (4 marks) x + y = 80 opposite angles of a cyclic quadrilateral = 80 x + y = 90 QPS = 90 angle sum of triangle = 80 QS is diameter (converse of) angle in a semicircle = 90) SC Correct solution without reasons

22 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 6 RSX is a straight line. XT is a tangent to the circle at T. SX = ST Not drawn accurately R S X T Prove that triangle RXT is isosceles. ( marks) Let SXT = x STX = x isosceles triangle SRT = x alternate segment triangle RXT = is isosceles - base angles equal SC Correct solution without reasons

23 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 7 O is the centre of the circle. AB bisects angle OBC. Not drawn accurately A O B x Ox O y O C Prove that y = 90 + x (5 marks) OAB = x isosceles triangle BOA = 80 x angle sum of triangle = 80 Reflex BOA = 60 (80 x) (Angles at a point = 60) = 80 + x y = 90 + x Angle at centre = angle at circumference SC Correct solution without reasons

24 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 8 RTQ, RSP and PTV are all straight lines. PT = PQ V x Q Not drawn accurately T R S P Prove that PTV is a tangent to circle RST at T (5 marks) QTP = x isosceles triangle VTR = x vertically opposite angles equal TQP = x = RST exterior angle of cyclic quadrilateral = opposite interior angle VTR = RST PVT is tangent (converse of) alternate segment theorem A oe SC Correct solution without reasons 4

25 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 9 ABF is a common tangent to the two circles at A and B. CDE is a straight line. AC is parallel to BD. D E Not drawn accurately C A B x F Prove that AD is parallel to BE. (5 marks) EDB = x alternate segment DCA = x corresponding angles equal DAB = x alternate segment ie, DAB = EBF AD is parallel to BE (converse of) corresponding angles equal A SC Correct solution without reasons 5

26 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Algebraic Proof Question Prove that 4(p ) (p ) is always a negative integer. ( marks) 4p 4p + 4 terms with correct 0 Question Prove that 8(y + ) + ( y) is a multiple of 5 when y is a positive integer. ( marks) 8y y or 5y + 0 M 4 terms with correct 5y + 0 and 5(y + 6) oe eg, 5y + 0 and states both terms divisible by 5 Question a is a positive integer. Prove that 4a (a + ) (a) is a cube number. ( marks) 8a + 4a 4a or 8a M terms with correct 8a and (a) oe eg, 8a and states that 8 is a cube number 6

27 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 4 a and b are positive integers. a < b Prove that ax a < x ( marks) bx b a(x + ) or b(x + ) a( x ) b( x ) and cancelling seen a and explains that as numerator is smaller b than denominator value will be < oe Question 5 (a) Express x + 6x + in the form (x + a) + b where a and b are integers. ( marks) (b) Hence, prove that x + 6x + is always positive. ( marks) (a) a = B b = Bft ft their a (b) (x + ) 0 oe Allow their a Adding means always positive Must have a = and b = 7

28 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 6 Prove that, for all values of x, x + x + 6 > 0 (4 marks) (x + ) B (x + ) + 5 Bft ft Their (x + ) (x + ) 0 oe Allow their Adding 5 means always positive Must have (x + ) + 5 Question 7 f(x) = (x + ) + 8(x + ) for all values of x. Prove that there is exactly one value of x for which f(x) = 0 (4 marks) 4x + 6x + 6x x + 6 or 4x + 0x + 5 M Allow one error in expansions 4x + 0x + 5 and (x + 5) oe eg, 4x + 0x + 5 and (x + 5)(x + 5) Explains that only solution is (x = ).5 oe eg, explains that because the brackets are the same there is exactly one solution 8

29 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 8 The nth term of a sequence is n(n + ) (a) (b) Work out an expression for the (n )th term of the sequence. Give your answer in its simplest form. Hence, or otherwise, prove that the sum of any consecutive pair of terms of the sequence is a square number. ( marks) ( marks) (a) (n )(n + ) (b) n(n ) n(n + ) + n(n ) oe eg, n n n(n + ) + their (a) n + n + n Expands brackets n ft Their (a) n Alt oe n(n + ) + (n + )(n + + ) (b) n + n + n Expands brackets + n + n + oe eg, n + n + ft Their (n + )(n + + ) (n + ) 9

30 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 9 Prove that x 4 5x 0 0x x is always positive. (5 marks) ( x )( x ) M For either numerator or denominator 5( x ) factorised correctly At least one correct cancellation in the product x oe eg, 0x 5 Explains that > 0 and x 0 so x always positive oe eg, Explains that 0 > 0 and 5 > 0 and x 0x 0 so always positive 5 Question 0 f(n) = n n Prove that f(n) + f(n + ) = kn(5n ) where k is an integer. ( marks) (n) n + {(n + ) (n + )} oe 9n n or n + n + n + n 9n n + n + n + n + n oe eg, 0n n 0n n and n(5n ) oe eg 0n n and k = 0

31 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 4 Trigonometry Question (non-calculator) Work out the exact value of sin 60 + sin 0 + sin 70. Give your answer in its simplest form. ( marks) / + / Any values correctly stated in surd form / + / All values correctly stated in surd form Question (non-calculator) Are these statements true or false? True False sin 7 = sin 7 cos 54 = cos 06 sin 5 = cos 5 tan 6 = tan 06 (4 marks False True False True

32 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question (non-calculator) Work out the area of triangle ABC. Write your answer in its simplest form. C Not drawn accurately cm B A ( marks) Evidence that sin 45 = / Area = 5 6 sin 45 B 5

33 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 4 (calculator or non-calculator) Show that tan cos θ ( marks) tan sin θ cos θ seen sin θ cos θ cos θ cos θ tan cos θ Accurate method with clear steps is required for all marks Alt cos θ oe cos θ sin θ cos θ tan Accurate method with clear steps is required for all marks

34 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 5 (calculator) AC is a diameter of the circle. AC = 5 cm, AD = 4cm D Not drawn accurately A C B Work out angle ABD. (4 marks) Evidence that angle ADC is a right angle sin ACD = 5 4 ACD = [5., 5.005] Allow 5 with method seen Angle ABD = [5., 5.005] Bft ft From rd mark their angle ACD 4

35 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 (calculator) A hanging basket is made from a hemisphere and three chains. The radius of the hemisphere is 0cm. Each chain is 0cm long. The chains are equally spaced around the rim of the hemisphere. Work out angle AOB. O A B (5 marks) A triangle formed with A, B and the centre of the hemisphere with sides of 0 cm and an angle of 0 (AB =) cos 0 0 sin 60 (AB =) [7., 7.] oe eg, 00 (cos AOB =) 0 0 their AB sin - (0.5 their AB 0) 0 0 [.557,.6] ft ft Their AB Accept 4 with correct method seen 5

36 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 7 (calculator) Solve the following equation for tan = (4 marks) tan = + or tan = [54.7,54.74] or [5.6,5.] 80 + their [54.7,54.74] or 80 + their [5.6,5.] [54.7,54.74] and [5.6,5.] and 80 + their [54.7,54.74] and 80 + their [5.6,5.] ft All 4 solutions [54.7,54.74] and [5.6,5.] must be correct ft For other two solutions Question 8 (calculator) Solve the following equation for cos + cos = 0 (5 marks) (cos )( cos + ) M M Fully correct use of quadratic formula (acos + b)(ccos + d) where ac = and bd = or quadratic formula with one sign error cos = so = 80 cos = so = [70.5, 70.5] = ft ft 60 their [70.5, 70.5] 6

37 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 7 5 Matrices Question Work out (a) (b) (c) 6 5 (d) 0 0 (e) (f) ( marks) Each question marks. for a correct row by column multiplication. for the correct answer. (a) 6 0 (b) 0 5 (c) (d) (e) (f) 0 0 Question Work out (a) 4 0 (b) (c) (d) (e) 5 4 (f) 4 5 ( marks) Each question marks. for a correct row by column multiplication. for the correct answer. (a) 9 0 (b) (c) 0 0 (d) (e) 5 0 (f) 9

38 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics 8 Question (non-calculator) Work out, giving your answers as simply as possible. (a) 0 (b) 5 4 (c) 5 7 (d) (e) 4 4 (f) 7 (7 marks) marks per question. mark for multiplication of row by column, mark for simplified elements, for other elements correct. Part (c) marks. (a) 0 (b) 5 (c) (d) 0 5 (e) (f)

39 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 9 Question 4 Work out, giving your answers as simply as possible. (a) 0 0 p p (b) 0 0 y x (c) 0 0 m m (d) 0 0 a a 0 0 (e) t t (f) ( marks) Each question marks. for a correct row by column multiplication. for the correct answer. (f) marks. for pair correctly multiplied, for final answer. (a) p p (b) y x (c) m m (d) a a 0 0 (e) t t 0 0 (f) 0 0 =

40 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics 40 Question 5 Work out, giving your answers as simply as possible. (a) x x x x (b) a a (c) 0 x x (d) x y y 0 y (e) a a a a a a a a (f) 9 x x (4 marks) (a) to (d) marks each (e) and (f) marks each, for a correct multiplication, for two elements correct, for all correct. (a) x x x x (b) a a (c) 0 x x x (d) x y y y y 9 6 (e) 0 0 (f) x x x x x

41 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 6 Matrices Question A = 4 B = C = Work out (a) AB (b) BC (c) A (d) BA (e) C (f) B ( marks) Each question marks. for a correct row by column multiplication. for the correct answer. (a) 9 5 (b) (c) (d) 6 9 (e) 9 7 (f) 0 0 Question P = 5 0 Q = 4 C = Work out (a) P (b) QP (c) 5Q (d) PC (e) IQ (f) I ( marks) Each question marks. for a correct row by column multiplication. for the correct answer. (a) 4 0 (b) 5 (c) (d) 6 (e) 4 (f) 0 0 4

42 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question a 4 = 7 9 Work out the value of a. ( marks) 6 + 7a = a = 4 Question 4 Work out the values of a, b and c. a b 6 = c ( marks) a = 5 b = 4 c = 5 B B B Question 5 Work out the image of the point D (, ) after transformation by the matrix ( marks) 4 (4, ) B B For (4,?), (?, ) or. 4

43 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 The point A(m, n) is transformed to the point A (, 0) by the matrix Work out the values of m and n. (4 marks) m + n =, m + n =, For either For both Attempt to solve m =, n = Question 7 The matrix A represents a reflection in the line y = x. Write down the matrix A. The unit square is transformed by the matrix A and then by rotation through 90 about O. Work out the matrix representing the combined transformation. (4 marks) A = 0 0 B Rotation 0 0 B Combined = 0 Multiplication in correct order 0 0 4

44 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 8 Describe fully the transformation given by the matrix 0 0 ( marks) Reflection, in the line y = x B, B Question 9 (non-calculator) The unit square OABC is transformed by the matrix The area of OABC is 7. h 0 0 to the square OABC. h Work out the exact value of h. ( marks) Vertices of image A (h, 0) B (h, h) C(0, h) B Any one correct Area of OABC. = h h = 44

45 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 0 0 A = 0 and B = 0 0 The point P (, 7) is transformed by matrix BA to P. Show that P lies on the line 7x + y = 0 ( marks) BA = 0 0 B = 7 B Show this satisfies 7x + y = 0 45

46 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics 7 Inequalities Question 6 x 6 x is an integer Write down all the possible values for x. ( marks) < x 0 A correct with none incorrect or 4 correct with one incorrect Question Solve 6x 4 x ( marks) 6x + x > 4 oe x > Question Solve 4(x ) ( marks) 8x 4 < oe x < 4 oe 8x < + 4 oe x < 4 + oe x < 4 46

47 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 4 A rhombus and a rectangle are shown. The perimeter of the rhombus is greater than the perimeter of the rectangle. Not drawn accurately y + 4 y + 6 y + 0 Show that y > k where k is an integer. (4 marks) 4(y + 6) > y y y y + 4 M oe eg, 8y + 4 > 6y + 8 4(y + 6) or y y y y + 4 8y 6y > 8 4 oe y > or k = 47

48 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 5 p < and q > Tick the correct box for each statement. Always true Sometimes true Never true 5p < 0 p < 0 p + q > 0 < p q < 0 (4 marks) Always Never Sometimes Sometimes B4 B For each correct part 48

49 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 y B O A x y = 6 x (a) Write down the coordinates of points A and B. ( marks) (b) Hence, or otherwise, solve 6 x 0 ( marks) (a) (4, 0) B (4, 0) B SC 4 and 4 seen (b) 4 x 4 Bft ft Their 4 and their 4 Bft 4 < x < 4 Alt (b) (4 + x) (4 x) and 4 and 4 4 x 4 49

50 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 7 (a) Factorise x + x (b) Sketch y = x + x Label the x values of the points of intersection with the x-axis. ( mark) ( marks) (c) Hence, or otherwise, solve x + x 0 ( marks) (a) x(x + ) B (b) U-shaped parabola 0 and labelled on x-axis ft ft Their factors in (a) (c) x < and x > 0 Bft ft Their factors in (a) Bft x and x 0 Question 8 Solve (x 5)(x + ) 0 ( marks) 5 and B Sketch of graph y = (x 5)(x + ) x < and x > 5 Sign diagram using their 5 and their 50

51 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 9 Solve x + 4x < 0 (4 marks) (x + 6)(x ) (x + a)(x + b) where ab = ± or a + b = ± 4 6 and Sketch of graph y = (x + 6)(x ) 6 < x < Sign diagram using their 6 and their Question 0 Solve x x < 0 (4 marks) (x )(x + ) (x + a)(x + b) where ab = ± or oe and a + b = ± Sketch of graph y = (x )(x + ) < x < Sign diagram using their and their 5

52 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question Solve x > 4x 8 (4 marks) (x )(x 4) (x + a)(x + b) where ab = ± 8 or a + b = ± 4 and 4 Sketch of graph y = (x )(x 4) Sign diagram using their and their 4 x < and x > 4 Question A triangle and a square are shown. n 4n 8 n Work out the range of values of n for which area of triangle < area of square (5 marks) n > (4n 8)n oe 0 > n 4n n(n 4) Factorises their quadratic expression Sketch of graph of y = n(n 4) Sign diagram using their 0 and their 4 0 < n < 4 5

53 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 8 Functions Question (non-calculator) f(x) = x 50 Work out x when f(x) = 0 ( marks) x 50 = 0 x 50 oe = x = 5 Question f(x) = x + ax 8 f() = Work out the value of a. ( marks) () + a ( ) 8 = 9 8 = a oe Allow error a = 4 Question f(x) = x + x 0 Show that f(x + ) = x(x + 7) ( marks) (x + ) + (x + ) 0 x + x + x x oe Allow error x + 7x = x(x + 7) 5

54 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 4 Work out the range for each of these functions. (a) f(x) = x + 6 for all x ( mark) (b) f(x) = x 5 x 6 ( marks) (c) f(x) = x 4 x ( mark) (a) f(x) 6 B (b) f(x) B B For or seen (c) f(x) 48 B Question 5 (a) f(x) = x x Give a reason why x 0 is not a suitable domain for f(x). (b) Give a possible domain for f(x) = x 5 ( mark) ( mark) (a) Not defined when x = or cannot divide by 0 when x = B oe (b) x a where a 5 or x a where a 5 B eg x 5 x 6 Allow list of x values if all are 5 54

55 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 f(x) = x a x b The range of f(x) is 5 f(x) 5 Work out a and b. ( marks) Either x = 5 or x = 5 a = b = 4 SC a = 4, b = Question 7 Here is a sketch of f(x) = x + 6x + a for all x, where a is a constant y O x The range of f(x) is f(x) Work out the value of a. ( marks) Attempt to complete the square in the form (x + ) (x + ) 9 + a oe a = 0 55

56 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 8 (a) Factorise x 5x 4 (b) Sketch the function f(x) = x 5x 4 for all x. Label the points of intersection with the x and y axes. ( marks) ( marks) (a) (x + a)(x + b) ab = 4 or a + b = 5 (x 7)(x + ) (b) y B B Curve through their (7, 0) and (, 0) (from 8(a)) B Curve through (0, 4) B Smooth U shape O 7 x 4 56

57 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 9 f(x) = x 0 x 4 x x 0 x 5 Draw the graph of f(x) for values of x from 0 to 5 ( marks) y B B For each part O 4 5 x 4 57

58 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 0 Here is a sketch of the function f(x) for values of x from 0 to 7. f(x) = x 0 x x x 4 x 7 4 x 7 y A B x Show that area of triangle A : area of triangle B = : (4 marks) (, 0) and (7, 0) marked or used (, ) and (4, ) marked or used Either of their triangular areas calculated correctly and 4 = : 58

59 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet 9 Coordinate Geometry - Calculus Question For each of these straight lines, work out (i) The gradient of the line ( mark for each part) (ii) The gradient of the line that is perpendicular to the given line ( mark for each part) (iii) The y-intercept of the line ( mark for each part) (a) y = 5x 4 (b) y = 9 6x (c) y = x (d) 5x y + 5 = 0 (e) x y = 4 (a) 5 B (b) B 5 Bft ft their 5 Bft ft their 4 B B (c) B (d) 5 B Bft ft their 5 Bft ft 5 their 4 B 5 B (e) B 4 4 Bft ft their 4 6 B 59

60 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question For each of these straight line segments, AB, work out (i) The mid-point of AB ( marks for each part) (ii) The gradient of AB ( mark for each part) (iii) The length of AB, giving your answer as an integer or a surd ( marks for each part) (a) A = (, 4) B = (4, ) (b) A = (4, ) B = (, 5) (c) A = (5, ) B = (0, 0) (d) A = (, 6) B = (6, 0) (e) A = (, 9) B = (9, 6) (f) A = (7, ) B = (5, ) (a), B B For each coordinate (b) (, ) B B For each coordinate B 4 B 5 (7 + 7 ) (5 + 4 ) 98 or 7 4 (c) (, 4) B B For each coordinate (d) (4, ) B B For each coordinate B 5 B (5 + ) (4 + 6 ) 5 or (e) (5, ) B B For each coordinate 5 B 8 (f) (, ) B B For each coordinate B (8 + 5 ) ( + 4 ) 7 60 or

61 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question In each of these line segments, B lies between A and C. Work out the coordinates of C in each case. ( marks for each part) (a) A = (, ) B = (, ) and AB : BC = : (b) A = (4, ) B = (, 5) and AB : BC = : (c) A = (, 0) B = (, 5) and AB : BC = 5 : (d) A = (6, ) B = (0, 4) and AB : BC = : (e) A = (, 9) B = (, ) and AB : BC = 5 : 4 (a) (5, ) B B For each coordinate (b) (4, 6) B B For each coordinate (c) (5, 8) B B For each coordinate (d) (9, 7) B B For each coordinate (e) (7, 9) B B For each coordinate Question 4 Work out the coordinates of the points of intersection of the curve the straight line y = 5x + y = x + 7 and (4 marks) x + 7 = 5x + or x 5x + 6 = 0 (x )(x ) = 0 Attempt to factorise the quadratic (, ) or (, 6) ft ft Their factors (, ) and (, 6) 6

62 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 5 Line L has equation y + x = 7 Line N is perpendicular to line L and passes through (, ). Work out the equation of line N. Give your answer in the form y = ax + b (4 marks) Gradient of L = Gradient of N = B y () = (x ) y = x 6

63 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 dy Work out for each of the following dx y = 7x + ( mark) (b) y = 8 5x + x ( marks) (a) (c) y = x + 4x ( marks) (d) y = x 7x + 0x ( marks) (e) y = 4x(x + x ) ( marks) (f) y = (x 5)(x + 8) ( marks) (g) y = x(7 x)(6 x) ( marks) (h) y = (x + )(x )(x 6) (4 marks) (a) dy dx = 7 B (b) dy dx = x 5 B B For each term (c) dy dx = 9x + 4 B B For each term (d) dy dx = x 4x + 0 B B For two terms correct (e) y = 4x + 8x x B dy dx = x + 6x Bft B For two terms correct ft Their y =... (f) y = x + 9x 40 B dy dx = 6x + 9 Bft B For one term correct ft Their y =... (g) y = 4x 0x + x B dy dx Bft B For two terms correct = 4 40x + 6x ft Their y =. (h) y = x 4x 5x + 8 B B For four terms, three of which are correct dy dx = x 8x 5x Bft B For two terms correct ft Their y =... 6

64 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 7 A curve has equation y = x + x + x 4 Work out the equation of the tangent to this curve where x = Give your answer in the form y = ax + b (5 marks) dy dx = x + x + (when x = ) gradient tgt = 0 (when x = ) y = B y () = 0(x ()) oe y = 0x + 8 ft ft Their m and c Question 8 A curve has equation y = x + x 9x + Work out the equation of the normal to this curve at the point (, ) Give your answer in the form ax + by + c = 0, where a, b and c are integers. (5 marks) dy dx = x + 4x 9 (when x = ) gradient tgt = (when x = ) gradient nl = ft ft Their y () = (x ) oe x y 7 = 0 ft ft Their m and c 64

65 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 9 A curve has equation y = x 6x + 0 (a) Write down an expression for dy dx ( mark) (b) Work out the coordinates of the stationary points and determine whether they are maximum or minimum. Sketch the curve on the axes clearly labelling the stationary points. y (5 marks) O x (a) dy dx = x x (b) x x = 0 or x(x 4) = 0 x = 0 and x = 4 (0, 0) and (4, ) dy Testing the sign of for values of dx x either side of 0 and 4 Maximum at (0, 0) Minimum at (4, ) If previous earned (c) y B B For correct general shape 0, 0 Bft For labelling the stationary points x 4, 65

66 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 0 A curve has equation y = x x + k x (a) Write down an expression for dy dx ( mark) (b) The curve has a minimum point at the point where x = Work out the value of k. ( marks) (c) Work out the x coordinate of the maximum point on the curve. ( marks) (a) dy dx = x x + k B (b) () () + k = 0 k = 8 (c) x x 8 = 0 (x + 4)(x ) = 0 Maximum at x = 4 66

67 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question (a) Show that the line y = x 4 9 is the tangent to the curve y = 4 x x at the point A (, 4 ). ( 4 marks) (b) The point B on the curve is such that the tangent at B is perpendicular to the tangent at A, as shown in the diagram. y y = 4 x x Not drawn accurately B x A Work out the coordinates of B. (4 marks) (a) dy dx = x (when x = ) dy dx = = y ( 4 ) = (x ) y = x 4 Clearly shown since y = x 4 9 answer given (b) Gradient tangent at B = B x = x = ft ft Their tangent gradient B = (, ) 67

68 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics 0 Factor Theorem Question (a) Show that x(x + 4)(x 9) = x 5x 6x ( mark) (b) Write down the x values of the three points where the graph of y = x 5x 6x crosses the x-axis. ( marks) (a) x(x 5x 6) B (b) x = 0, x = 4, x = 9 B B For two solutions Question f(x) = x + x 5x 6 (a) Work out f() and f() (b) Work out f() and f() (c) Work out f() and f() ( marks) ( marks) ( marks) (d) Write down the three linear factors of f(x). ( mark) (a) f() = = 8 B f( ) = = 0 B (b) f() = = 0 B f( ) = = 4 B (c) f() = = 4 B f( ) = = 0 B (d) (x + ), (x ) and (x + ) B 68

69 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question (a) Show that (x + 5) is a factor of x + 7x + x 40 ( marks) (b) Work out the other two linear factors of x + 7x + x 40 ( marks) (c) Hence, solve x + 7x + x 40 = 0 ( mark) (a) ( 5) + 7( 5) + ( 5) 40 oe = 0 Clearly shown to = 0 (b) x + 7x + x 40 (x + 5)(x + kx 8) Sight of x and 8 in a quadratic factor (x ) (x + 4) Alt (b) Substitutes another value into the expression and tests for = 0 (x ) (x + 4) Alt (b) Long division of polynomials getting as far as x + x (x ) (x + 4) (c) (x =) 5, 4 and B 69

70 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 4 A sketch of y = x + 5x + 9x + k where k is an integer, is shown. y k x Work out the value of k. ( marks) ( ) + 5( ) + 9( ) + k = k = 0 k = 6 70

71 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 5 (a) (x + ) is a factor of f(x) = x + x + ax 7 where a is an integer. Work out the value of a. ( marks) (b) Work out the other linear factors of f(x). ( marks) (a) ( ) + ( ) + ( )a 7 = a 7 = 0 a = 0 (b) x + x 0x 7 (x + )(x + kx 4) Sight of x and 4 in a quadratic factor (x + 4) (x 6) Alt (b) Substitutes another value into the expression and tests for = 0 (x + 4) (x 6) Alt (b) Long division of polynomials getting as far as x x (x + 4) (x 6) 7

72 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 6 (x ) and (x + 4) are factors of f(x) = x + ax + bx + 4 where a and b are integers. (a) Work out the third linear factor of f(x). ( marks) (b) Work out the values of a and b. (4 marks) (a) (x )(x + 4)(x + k) x + ax + bx + 4 or 4 k = 4 (x ) (b) (x )(x + 4)(x ) (x )(x + x 8) oe x x 4x + 4 a = and b = 4 ft ft Their expansion Alt (b) Substitutes any two of x = 4, x = or x = into x + ax + bx + 4 to create simultaneous equations Any two of a 4b + 4 = 0 or 8 + 4a + b + 4 = 0 or 7 + 9a + b + 4 = 0 a = b = 4 ft ft Their first solution 7

73 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 7 (a) (x 5) is a factor of f(x) = x + kx + 9x 0 where k is an integer. Work out the value of k. ( marks) (b) Express f(x) as a product of (x 5) and a quadratic factor. ( marks) (c) Show that (x 5) is the only linear factor of f(x). ( marks) (a) (5) + k(5) + 9(5) 0 = k = 0 k = 6 (b) x 6x + 9x 0 (x 5)(x + kx + 4) Sight of x and 4 in a quadratic factor (x 5)(x x + 4) (c) Tests b 4ac for the quadratic ft Their quadratic or attempts to solve their quadratic = 0 Shows b 4ac = 5 (or < 0) and states no more linear factors States 'no solutions' to their quadratic = 0 7

74 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 8 Solve x 6x 5x 8 = 0 (5 marks) Substitutes a value of x into the expression and tests for = 0 Works out first linear factor (x + ), (x + ) or (x 9) x 6x 5x 8 (x + )(x + kx 8) or (x + )(x + kx 9) or (x 9)(x + kx + ) nd and rd linear factors Attempts to work out the quadratic factor Sight of x and 8 in a quadratic factor or sight of x and 9 in a quadratic factor or sight of x and in a quadratic factor, and 9 Alt Substitutes a value of x into the expression and tests for = 0 Works out first linear factor (x + ), (x + ) or (x 9) Substitutes another value into the expression and tests for = 0 nd and rd linear factors, and 9 Alt Substitutes a value of x into the expression and tests for = 0 Works out first linear factor (x + ), (x + ) or (x 9) Long division of polynomials getting as far as x 7x Depending on first linear factor or or x 8x x + x nd and rd linear factors, and 9 74

75 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Sequences Question A linear sequence starts Which term is the first to have a negative value? (4 marks) For the nth terms of quadratic sequences two methods are shown (see example ). Other valid methods may be used. 4n 54 4n 54 4n 0 oe 64th 75

76 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question Work out the nth term of this quadratic sequence (4 marks) Method A Subtract n from sequence 6 4 nth term of this sequence is 5n Giving n 5n + Alt Method B Using an + bn + c a + b + c = 8 4a + b + c = 9 9a + b + c = 4 a + b = 5a + b = 5 a = and b = 5 oe Giving n 5n + 76

77 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question (a) Show that the nth term of the quadratic sequence is n + n ( marks) (b) Hence, write down the nth term of these quadratic sequences. (b) (i) ( mark) (b) (ii) 5 ( mark) (a) Use Method A or B from Q marks or any other valid method (b)(i) n + n + B (b)(ii) n + 4n B 77

78 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 4 (non calculator) (a) Write down the nth term of the linear sequence ( mark) (b) Hence, write down the nth term of the quadratic sequence. (c) For the sequence in part 4(b), show that the 0th term is equal to the product of the nd and 4th terms ( mark) ( marks) (a) n + B (b) (n + ) B oe (c) = 7 B oe 88 0th is 9 = (7 ) oe 88 = 7 78

79 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 5 4 cm 5 cm cm 6 cm 4 cm 5 cm This pattern of rectangles continues. Show that the sequence of numbers formed by the areas of these rectangles has nth term n + 5n + 6 (4 marks) nth term of lengths is n + nth term of widths is n + Alt Area is (n + )(n + ) n + n + n + 6 = n + 5n + 6 nth term of 0 0 by Method A or Method B 4 marks or any other valid method 79

80 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 6 A linear sequence starts The 5th and 8th terms have values 5 and 59. a + b a + b a + 5b a + 7b.. (a) Work out a and b. (4 marks) (b) Work out the nth term of the sequence. ( marks) (a) a + 9b = 5 a + 5b = 59 6b = 4 oe b = 4 a = ft (b) 9 B ft 8n 5 Bft Question 7 A sequence has nth term n n (a) Show that the difference between the nth and (n + )th terms is n( n ) ( marks) (b) Which are the first two consecutive terms with a difference less than 0.0? ( marks) (c) Write down the limiting value of the sequence as n ( mark) 80

81 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet (a) n n ( n ) oe n eg subtracts in different order (n )( n ) n(n 4) oe n( n ) n n n n n( n ) 4n = n( n ) Alt (a) n oe = + n n eg subtracts in different order ( + ) ( + ) n n n n n( n ) oe = n( n ) (b) Any substitution and evaluation for oe n 0 eg 0.0n + 0.0n and attempt to solve eg 9 0 = 90 or 0 = 0 0th and th (c) B 8

82 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 8 A sequence has nth term 5n n Show that the limiting value of the sequence, S, as n is.5 ( marks) 5n n 5n = n + n oe 5 n 5 0 as n S = (=.5) n Question 9 Here is the sequence of odd numbers A quadratic sequence is formed by multiplying consecutive odd numbers in successive pairs Work out the nth term of this sequence. ( marks) Odd number is n + or n n and n + Alt Sequence is (n )(n + ) (= 4n ) Using Method A or Method B giving 4n marks or any other valid method eg n n n 8

83 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 0 The nth term of a sequence is n n (a) Show that the difference between the first two terms is 0 ( marks) (b) Write down the limiting value of the sequence as n ( mark) (a) T = 5 B T = 4 7 B oe ( = ) 5 B oe = (b) B 8

84 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Algebraic problems including ratio Note If x : y = 4 : 7, then y x = 7 4 If, in a problem, two numbers are in the ratio 4 : 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 = 4 : 7, what is x + y : x? Think in terms of parts, ie 4 parts and 7 parts, so x + y : x = : = 8 : = : Question Work out the possible values of n n if n = 6 Give your answers as fractions in their simplest form. (4 marks) n = 4 n =

85 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question x : y = 6 : 5 (a) Express x in terms of y. ( marks) (b) Show that x + y : x y = : ( marks) (a) x 6 = y 5 x = 5 6y oe (b) 6y 5y y 5y oe : : ( y) (5) 7( y) : (5) 85

86 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question A point P divides XY in the ratio : 7 Y (6a, b) Not drawn accurately P X (a, b) Work out the coordinates of P, in terms of a and b. ( marks) oe of (6a a) or of (b b) 0 0 (.5a, 4b) A oe For each coordinate SC (.5a, b) 86

87 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 4 Here is a linear sequence a + b a + b a + 5b a + 7b.. Given that nd term : 4th term = : 5 st term = 4 Work out a and b. (5 marks) a a b 7b = 5 5a + 5b = a + 4b Allow one error a + b = 0 oe a + b = 4 ft a = 4 a = and b = 6 ft 87

88 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 5 You are given that ab + a = 5 and a : b = 4 : Work out the possible pairs of values of a and b. (7 marks) a 4 oe = b a b = 4 a = 4b a a + a = 5 4 4b 4b b + = 5 a + 4a 0 = 0 4b + 4b 5 = 0 (a + 0)(a ) = 0 (b + 5)(b ) 0 a = a = ft b = 5 b = b = 5 b = ft 0 a = a = 88

89 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 6 The sum of the ages of two people is 90 years. Six years ago, their ages were in the ratio 8 : 5 How old are they now? Do not use trial and improvement. You must show your working. (5 marks) Let their ages 6 years ago be 8x and 5x 8x + 5x = 90 Allow 90 6 for x = 78 (x = 6) Their 6 8 and their 6 5 (48) (0) 54 and 6 Alt x + y = 90 x 6 y 6 = = 8y 5x Eliminates a letter (x =) 54 and (y =) 6 89

90 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 7 O is the centre of the circle. y Not drawn accurately 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) x, x and 80 x seen or on diagram x 4 = y 5 x = 5 4y oe y = 80 x (or y = 90 x) oe y = y oe 9y = 90 oe 5 y = 50 90

91 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 8 A rectangular picture is surrounded by a frame of constant width. All measurements are in centimetres. a Not drawn accurately 7x 9 x b 9 Given that a : b = : Work out x. (5 marks) a = 7x + 8 or b = x + 8 B oe their (7 their ( x x 8) 8) = 4x + 6 = 9x + 54 Rearranging 5x = 8 Solving x =.6 9

92 Worksheets Teacher Booklet 860 AQA Level Certificate in Further Mathematics Question 9 If x : y = : 5 and y : z = 0 : 9 Find, in its simplest form (a) (b) (c) x : z 0x : 7y x + y : y ( marks) ( marks) ( marks) (a) x : y = 6 : 0 oe x : y : z = 6 : 0 : 9 x : z = : (b) 0 : 7 5 oe 6 : 7 (c) + 5 : 5 x y y x = + or + y 5 8 : 5 9

93 Level Certificate in Further Mathematics 860 Worksheets Teacher Booklet Question 0 A cuboid has dimensions n, n and n cm. A diagonal has length n + cm. Not drawn accurately n + n n n Work out n. (6 marks) (n) + n oe (n) + n + (n ) = (n + ) 4n + n + n n n + = 4n + n + n + Allow one error n 6n = 0 Rearranging ; or n = 6n n(n ) = 0 (allow by n ) n = 6 n = 9

94 AQA Level Certificate in Further Mathematics from 0 onwards Qualification Accreditation Number: 600//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 0 followed by further support meetings through the life of the specification. Further information is available at: Copyright 0 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 6447) and a registered charity (registered charity number 074). Registered address: AQA, Devas Street, Manchester 5 6EX. SP/0/