Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0)

Transcription

1 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) First teaching from September 2017 First certification from June 2019

2 2

3 Contents About this booklet 5 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 7 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 15 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 22 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 29 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 37 A level Further Mathematics Paper 4E (Further Statistics 2) 44 A level Further Mathematics Paper 3C/4C (Further Mechanics 1) 51 A level Further Mathematics Paper 4F (Further Mechanics 2) 59 A level Further Mathematics Paper 3D/4D (Decision Mathematics 1) 67 A level Further Mathematics Paper 4G (Decision Mathematics 2) 78 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 3

4 4

5 About this booklet This booklet has been produced to support mathematics teachers delivering the new Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) (first assessment summer 2019). The booklet provides additional information on all the questions in the Sample Assessment Materials, accredited by Ofqual in It details the content references and Assessment Objectives being assessed in each question or question part. How to use this booklet Callouts have been added to each question in the accredited Sample Assessment Materials. In the callouts, the following information has been presented, as relevant to the question: Specification References; Assessment Objectives. Where content references or Assessment Objectives are being assessed across all the parts of a question, these are referred to by a single callout at the end of the question rather than by a callout for each question part. Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 5

6 6

7 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 1. Prove that n 1 r 1 ( r 1)( r 3) where a and b are constants to be found. = n( an b), 12( n 2)( n 3) Specification reference (4.4): Understand and use the method of differences for summation of series including use of partial fractions. AO1.1b: Correctly carry out routine procedures (2 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2a: Make deductions (1 mark) AO3.1: Translate problems into mathematical and non-mathematical contexts into mathematical processes (1 mark) (Total for Question 1 is 5 marks) 2. Prove by induction that, for all positive integers n, is divisible by 17. f(n) = 2 3n (5 2n + 1 ) Specification reference (1.1): Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices. AO1.1b: Correctly carry out routine procedures (2 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2a: Make deductions (1 mark) AO2.4: Use mathematical models (2 marks) (Total for Question 2 is 6 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 7

8 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 3. f(z) = z 4 + az 3 + 6z 2 + bz + 65, where a and b are real constants. Given that z 3 + 2i is a root of the equation f(z) = 0, show the roots of f(z) = 0 on a single Argand diagram. Specification references (2.1, 2.3, 2.4): Solve any quadratic equation with real coefficients. Solve cubic or quartic equations with real coefficients Understand and use the complex conjugate. Know that nonreal roots of polynomial equations with real coefficients occur in conjugate pairs. Use and interpret Argand diagrams. AO1.1a: Select routine procedures (1 mark) AO1.1b: Correctly carry out routine procedures (5 marks) AO1.2: Accurately recall facts, terminology and definitions (1 mark) AO3.1a: Translate problems into mathematical and non-mathematical contexts into mathematical processes (2 marks) (Total for Question 3 is 9 marks) 8

9 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 4. Figure 1 The curve C shown in Figure 1 has polar equation r = 4 + cos At the point A on C, the value of r is 2 9 The point N lies on the initial line and AN is perpendicular to the initial line. The finite region R, shown shaded in Figure 1, is bounded by the curve C, the initial line and the line AN. Find the exact area of the shaded region R, giving your answer in the form p + q 3, where p and q are rational numbers to be found. Specification references (7.1, 7.3): Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates. Find the area enclosed by a polar curve. AO1.1b: Correctly carry out routine procedures (5 marks) AO3.1: Translate problems into mathematical contexts into mathematical processes (4 marks) (Total for Question 4 is 9 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 9

10 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 5. A pond initially contains 1000 litres of unpolluted water. The pond is leaking at a constant rate of 20 litres per day. It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water. It is assumed that the pollutant instantly dissolves throughout the pond upon entry. Given that there are x grams of the pollutant in the pond after t days, (a) show that the situation can be modelled by the differential equation, d x 4x = 50 dt 200 t (b) Hence find the number of grams of pollutant in the pond after 8 days. (c) Explain how the model could be refined. Specification references (9.1, 9.2, 9.3): dy Find and use an integrating factor to solve differential equations of form P( x) y Q( x) and dx recognise when it is appropriate to do so. Find both general and particular solutions to differential equations. Use differential equations in modelling in kinematics and in other contexts. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.2a: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2b: Make inferences (1 mark) AO3.1b: Translate problems in non-mathematical contexts into mathematical problems (1 mark) AO3.3: Translate situations in contexts into mathematical models (2 marks) AO3.4: Use mathematical models (1 mark) AO3.5c: Where appropriate, explain how to refine models (1 mark) (Total for Question 5 is 10 marks) 10

11 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 6. f(x) = x x (a) Show that f ( x) dx = A ln (x 2 x + 9) + B arctan + c, 3 where c is an arbitrary constant and A and B are constants to be found. (b) Hence show that the mean value of f(x) over the interval [0, 3] is 1 1 ln (c) Use your answer to part (b) to find the mean value, over the interval [0, 3], of f(x) + ln k, 1 where k is a positive constant, giving your answer in the form p + ln q, where p and q are constants and q is in terms of k. 6 Specification references (5.3, 5.6, A level Mathematics Pure Mathematics 6.4): Understand and evaluate the mean value of a function. 1 2 Integrate functions of the form a 2 x 2 and x 2 substitutions to integrate associated functions. loga x + loga y = loga (xy) loga x loga y = loga(x/y) k loga x = loga x k (including, for example, k = 1 and k = ½ ) a and be able to choose trigonometric AO1.1b: Correctly carry out procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (2 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) (Total for Question 6 is 9 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 11

12 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 7. Figure 2 Figure 2 shows the image of a gold pendant which has height 2 cm. The pendant is modelled by a solid of revolution of a curve C about the y-axis. The curve C has parametric equations x = cos + 1 sin 2, y = (1 + sin ) (a) Show that a Cartesian equation of the curve C is x 3 = (y 4 + 2y 3 ) (b) Hence, using the model, find, in cm 3, the volume of the pendant. Specification reference (5.1): Derive formulae for and calculate volumes of revolution. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (3 marks) AO3.4: Use mathematical models (2 marks) (Total for Question 7 is 8 marks) 12

13 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 8. The line l1 has equation x 2 y 4 z The plane Π has equation x 2y + z = 6 The line l2 is the reflection of the line l1 in the plane Π. Find a vector equation of the line l2 Specification references (6.1, 6.2, 6.3): Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D. Understand and use the vector and Cartesian forms of the equation of a plane. Find the intersection of a line and a plane. Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.5: Use mathematical language and notation correctly (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (3 marks) (Total for Question 8 is 7 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 13

14 A level Further Mathematics Paper 1 (Core Pure Mathematics 1) 9. A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line. The vertical displacement, x metres, of the top of the capsule below its initial position at time t seconds is modelled by the differential equation, 2 d x m 2 d t dx 4 + x = 200 cos t, t 0 d t where m is the mass of the capsule including its passengers, in thousands of kilograms. The maximum permissible weight for the capsule, including its passengers, is N. Taking the value of g to be 10 m s 2 and assuming the capsule is at its maximum permissible weight, (a) (i) explain why the value of m is 3 (ii) show that a particular solution to the differential equation is x = 40 sin t 20 cos t (iii) hence find the general solution of the differential equation. (b) Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released. Specification references (9.2, 9.3, 9.5, 9.6): Find both general and particular solutions to differential equations. Use differential equations in modelling in kinematics and in other contexts. Solve differential equations of form y + a y + b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function (in cases where f(x) is a polynomial, exponential or trigonometric function) Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation. AO1.1b: Correctly carry out routine procedures (7 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) AO3.4: Use mathematical models (3 marks) (Total for Question 9 is 12 marks) 14

15 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 1. The roots of the equation x 3 8x x 32 = 0 are α, β and γ Without solving the equation, find the value of (i) (ii) (α + 2)(β + 2)(γ + 2) (iii) α 2 + β 2 + γ 2 Specification reference (4.1): Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations. AO1.1b: Correctly carry out routine procedures (6 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (2 marks) (Total for Question 1 is 8 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 15

16 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 2. The plane Π1 has vector equation r.(3i 4j + 2k) = 5 (a) Find the perpendicular distance from the point (6, 2, 12) to the plane Π1 The plane Π2 has vector equation r = λ(2i + j + 5k) + μ(i j 2k) where λ and μ are scalar parameters. (b) Show that the vector i 3j + k is perpendicular to Π2 (c) Show that the acute angle between Π1 and Π2 is 52 to the nearest degree. Specification references (6.2, 6.3, 6.4, 6.5): Understand and use the vector and Cartesian forms of the equation of a plane. Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, the angle between two planes and the angle between a line and a plane. Check whether vectors are perpendicular by using the scalar product. Find the intersection of a line and a plane. Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO2.2a: Make deductions (1 mark) AO2.4: Explain their reason (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) (Total for Question 2 is 8 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 16

17 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 3. (i) M = 2 a where a is a constant. (a) For which values of a does the matrix M have an inverse? Given that M is non-singular, (b) find M 1 in terms of a (ii) Prove by induction that for all positive integers n, æ ç è ö ø n æ = ç è 3 n 0 3(3 n -1) 1 ö ø Specification references (1.1, 3.1, 3.5, 3.6): Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices. Add, subtract and multiply conformable matrices. Multiply a matrix by a scalar. Calculate determinants of 2 2 and 3 3 matrices and interpret as scale factors, including the effect on orientation. Understand and use singular and non-singular matrices. Properties of inverse matrices. Calculate and use the inverse of non-singular 2 2 matrices and 3 3 matrices. AO1.1b: Correctly carry out routine procedures (7 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) AO2.3: Assess the validity of mathematical arguments (1 mark) AO2.4: Explain their reasoning (2 marks) (Total for Question 3 is 12 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 17

18 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 4. A complex number z has modulus 1 and argument θ. (a) Show that z n + 1 = 2cos nq, n Z+ n z (b) Hence, show that cos 4 θ = 1 (cos 4θ + 4cos 2θ + 3) 8 Specification references (2.2, 2.8): Add, subtract, multiply and divide complex numbers in the form x + iy with x and y real. Understand and use the terms real part and imaginary part. Understand de Moivre s theorem and use it to find multiple angle formulae and sums of series. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (4 marks) (Total for Question 4 is 7 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 18

19 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 5. (a) Show that 4 d y d 4 x 4y y = sin x sinh x (b) Hence find the first three non-zero terms of the Maclaurin series for y, giving each coefficient in its simplest form. (c) Find an expression for the nth non-zero term of the Maclaurin series for y. Specification references (4.5, 8.2): Find the Maclaurin series of a function including the general term. Differentiate and integrate hyperbolic functions. AO1.1a: Select routine procedures (1 mark) AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (2 marks) (Total for Question 5 is 10 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 19

20 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 6. (a) (i) Show on an Argand diagram the locus of points given by the values of z satisfying z 4 3i = 5 Taking the initial line as the positive real axis with the pole at the origin and given that θ [α, α + π], where α = arctan 4 æ 4ö è ç 3ø, (ii) show that this locus of points can be represented by the polar curve with equation r = 8 cos θ + 6 sin θ The set of points A is defined by A = z : z: 0 arg z Ç { z : z 4 3i 5} 3 (b) (i) Show, by shading on your Argand diagram, the set of points A. (ii) Find the exact area of the region defined by A, giving your answer in simplest form. Specification references (2.4, 2.7, 7.1, 7.3, A level Mathematics Pure Mathematics 5.3, 5.6): Use and interpret Argand diagrams. Construct and interpret simple loci in the argand diagram such as z a > r and arg (z a) = θ. Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates. Find the area enclosed by a polar curve. Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Know and use exact values of sin and cos for 0, 6, 3, 2 and π and multiples thereof, and exact values of tan for 0, 6, 3, 2, π and multiples thereof. Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B), and tan (A ± B), understand geometrical proofs of these formulae. Understand and use expressions for a cosθ + b sinθ in the equivalent forms of r cos (θ ± α) or r sin (θ ± α) AO1.1b: Correctly carry out routine procedures (8 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (3 marks) AO2.2a: Make deductions (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) (Total for Question 6 is 13 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 20

21 A level Further Mathematics Paper 2 (Core Pure Mathematics 2) 7. At the start of the year 2000, a survey began of the number of foxes and rabbits on an island. At time t years after the survey began, the number of foxes, f, and the number of rabbits, r, on the island are modelled by the differential equations df = 0.2 f + 0.1r dt dr = -0.2 f + 0.4r dt (a) Show that 2 d f 2 dt df f 0 dt (b) Find a general solution for the number of foxes on the island at time t years. (c) Hence find a general solution for the number of rabbits on the island at time t years. At the start of the year 2000 there were 6 foxes and 20 rabbits on the island. (d) (i) According to this model, in which year are the rabbits predicted to die out? (ii) According to this model, how many foxes will be on the island when the rabbits die out? (iii) Use your answers to parts (i) and (ii) to comment on the model. Specification references (9.4, 9.6, 9.9): Solve differential equations of form y + ay + by = 0 where a and b are constants by using the auxiliary equation. Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation. Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled first order simultaneous equations and be able to solve them, for example predator-prey models AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 mark) AO3.1b: Translate problems in non-mathematical contexts into mathematical processes (2 marks) AO3.2a: Interpret solutions to problems in their original context (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) AO3.4: Use mathematical models (5 marks) AO3.5a: Evaluate the outcomes of modelling in context (1 mark) (Total for Question 7 is 17 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 21

22 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 1. Use Simpson s Rule with 6 intervals to estimate ò x 3 dx Specification reference (6.2): Simpson s Rule. AO1.1b: Correctly carry out routine procedures (5 marks) (Total for Question 1 is 5 marks) 2. Given k is a constant and that use Leibnitz theorem to show that y = x 3 e k x n d y n dx = k n 3 e k x (k 3 x 3 + 3nk 2 x 2 + 3n(n 1)k x + n(n 1)(n 2)) Specification reference (2.3): Leibnitz s theorem. AO1.1b: Correctly carry out routine procedures (2 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) (Total for Question 2 is 4 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 22

23 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 3. A vibrating spring, fixed at one end, has an external force acting on it such that the centre of the spring moves in a straight line. At time t seconds, t 0, the displacement of the centre C of the spring from a fixed point O is x micrometres. The displacement of C from O is modelled by the differential equation 2 2 d x dx 2 4 t 2 t (2 t ) x t (I) 2 dt dt (a) Show that the transformation x = t v transforms equation (I) into the equation d 2 v dt 2 + v = t (II) (b) Hence find the general equation for the displacement of C from O at time t seconds. (c) (i) State what happens to the displacement of C from O as t becomes large. (ii) Comment on the model with reference to this long term behaviour. Specification reference (3.2): Differential equations reducible by means of a given substitution. AO1.1b: Correctly carry out routine procedures (7 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO2.2a: Make deductions (2 marks) AO3.2a: Interpret solutions to problems in their original context (1 mark) AO3.4: Use mathematical models (1 mark) AO3.5a: Evaluate the outcomes of modelling in context (1 mark) (Total for Question 3 is 14 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 23

24 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 4. (a) Show that 2 d y dy 2x y 0 (I) 2 dx dx d y d d ax y b y dx dx dx where a and b are integers to be found. (b) Hence find a series solution, in ascending powers of x, as far as the term in x 5, of the differential equation (I) where y = 0 and dy dx = 1 at x = 0 Specification reference (3.1): Use of Taylor series method for series solution of differential equations. AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO2.2a: Make deductions (1 mark) AO2.5: Use mathematical language and notation correctly (1 mark) (Total for Question 4 is 9 marks) 5. The normal to the parabola y 2 = 4ax at the point P(ap 2, 2ap) passes through the parabola again at the point Q(aq 2, 2aq). The line OP is perpendicular to the line OQ, where O is the origin. Prove that p 2 = 2 Specification references (4.1, 4.3): Cartesian and parametric equations for the parabola, ellipse and hyperbola Tangents and normal to these curves. AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (3 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) (Total for Question 5 is 9 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 24

25 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 6. A tetrahedron has vertices A(1, 2, 1), B(0, 1, 0), C(2, 1, 3) and D(10, 5, 5). Find (a) a Cartesian equation of the plane ABC. (b) the volume of the tetrahedron ABCD. The plane П has equation 2x 3y + 3 = 0 The point E lies on the line AC and the point F lies on the line AD. Given that П contains the point B, the point E and the point F, (c) find the value of k such that AE Given that AF = 1 9 AD = kac. (d) show that the volume of the tetrahedron ABCD is 45 times the volume of the tetrahedron ABEF. Specification references (5.1, 5.2, 5.3): The vector product a b of two vectors. Applications of the vector product. The scalar triple product a.b c Applications of vectors to three dimensional geometry involving points, lines and planes. AO1.1b: Correctly carry out routine procedures (6 marks) AO2.2a: Make deductions (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (4 marks) (Total for Question 6 is 11 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 25

26 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) 7. P and Q are two distinct points on the ellipse described by the equation x 2 + 4y 2 = 4 The line l passes through the point P and the point Q. The tangent to the ellipse at P and the tangent to the ellipse at Q intersect at the point (r, s). Show that an equation of the line l is 4sy + rx = 4 Specification references (4.1, 4.3, 4.4): Cartesian and parametric equations for the parabola, ellipse and hyperbola. Tangents and normals to these curves Loci problems AO1.1b: Correctly carry out routine procedures (1 mark) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO2.2a: Make deductions (2 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (3 marks) (Total for Question 7 is 8 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 26

27 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA A level Further Mathematics Paper 3A (Further Pure Mathematics 1) h(x) x Figure 1 Figure 1 shows the graph of the function h(x) with equation (a) ( ) Show that x where t tan æ ö where t = tan è ç 4 4 ø. æ ö h(x) 1 x 21 è ç 2ø + x 25 æ ö x h(x) = sin x + 21sin 25cos è ç 2 ø x x [0, [0, 40] 40] 2 2 ( ) dh dx 2 2 d h ( t 6t 17)(9t 4t 3) t - 6t t 2 2t = dx 2(1 t ) t 2 ( ) : :00 00:00 04:00 08: :00 00:00 Tue 3 Jan Wed 4 Jan Source: Figure 2 Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 08:00 on the Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 3rd January 2017 to the end of the 4th January The graph of k h(x), where k is a constant and x is the number of hours after 08:00 on 3rd of January, The graph can of be h(x) used to model k is a the constant predicted and tide x is the heights, number in metres, of hours for after this 08:00 period on of time. 3rd of January, can be used to model the predicted tide heights, in metres, for this period of time. (b) (i) Suggest a value of k that could be used for the graph of k h(x) to form a suitable model. ( ) ( ) k that could be used for the graph of k h(x) (ii) Why may such a model be suitable to predict the times when the tide heights are ( ) at their peaks, but not to predict the heights of these peaks? at their peaks, but not to predict the heights of these peaks? (c) Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of (3) the highest tide height on the 4th January ( ) 2 ( ) 2 1 (6) (6) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Pearson 16 Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 86 Pearson Edexcel Level 3 Advanced GCE in Mathematics *S54440A01618* Sample Assessment Materials Issue 1 June 2017 Pearson Education Limited

28 A level Further Mathematics Paper 3A (Further Pure Mathematics 1) Specification references (1.1, 1.2, A level Mathematics Pure Mathematics 2.9, 5.6, 7.3): The t-formulae Applications of the t-formulae to trigonometric identities Understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax) and combinations of these transformations Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B), and tan (A ± B), understand geometrical proofs of these formulae. Understand and use expressions for a cosθ + b sinθ in the equivalent forms of r cos (θ ± α) or r sin (θ ± α) Apply differentiation to find gradients, tangents and normals. Maxima and minima and stationary points. Points on inflection. Identify where functions are increasing or decreasing. AO1.1a: Select routine procedures (1 mark) AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) AO3.2a: Translate problems in non-mathematical contexts into mathematical processes (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) AO3.4: Use mathematical models (3 marks) AO3.5b: Recognise the limitations of models (1 mark) (Total for Question 8 is 15 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 28

29 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 1. (i) Use the Euclidean algorithm to find the highest common factor of 602 and 161. Show each step of the algorithm. (ii) The digits which can be used in a security code are the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. Originally the code used consisted of two distinct odd digits, followed by three distinct even digits. To enable more codes to be generated, a new system is devised. This uses two distinct even digits, followed by any three other distinct digits. No digits are repeated. Find the increase in the number of possible codes which results from using the new system. Specification references (5.1, 5.7): An understanding of the division theorem and its application to the Euclidian Algorithm and congruences. Combinatorics; counting problems, permutations and combinations. AO1.1b: Correctly carry out routine procedures (5 marks) AO3.1b: Translate problems in non-mathematical contexts into mathematical processes (2 marks) (Total for Question 1 is 7 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 29

30 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 2. A transformation from the z plane to the w plane is given by w = z 2 (a) Show that the line with equation Im(z) = 1 in the z plane is mapped to a parabola in the w plane, giving an equation for this parabola. (b) Sketch the parabola on an Argand diagram. Specification reference (4.2): Elementary transformations from the z-plane to the w-plane. AO1.1b: Correctly carry out routine procedures (3 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 mark) AO2.2a: Make deductions (1 mark) (Total for Question 2 is 6 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 30

31 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 3. The matrix M is given by M = æ ç ç è ö ø (a) Show that 4 is an eigenvalue of M, and find the other two eigenvalues. (b) For each of the eigenvalues find a corresponding eigenvector. (c) Find a matrix P such that P 1 MP is a diagonal matrix. Specification references (3.1, 3.2): Eigenvalues and eigenvectors of 2 2 and 3 3 matrices. Reduction of matrices to diagonal form. AO1.1b: Correctly carry out routine procedures (7 marks) AO1.2: Accurately recall facts, terminology and definitions (1 mark) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) (Total for Question 3 is 10 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 31

32 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 4. (i) A group G contains distinct elements a, b and e where e is the identity element and the group operation is multiplication. Given a 2 b = ba, prove ab ba (ii) The set H = {1, 2, 4, 7, 8, 11, 13, 14} forms a group under the operation of multiplication modulo 15. (a) Find the order of each element of H. (b) Find three subgroups of H each of order 4, and describe each of these subgroups. æ ç ç The elements of another group J are the matrices ç ç è ç æ cos kp è ç 4 æ -sin kp è ç 4 ö ø sin æ kp è ç 4 ö ø cos æ kp è ç 4 ö ø ö ø ö ø where k =1, 2, 3, 4, 5, 6, 7, 8 and the group operation is matrix multiplication. (c) Determine whether H and J are isomorphic, giving a reason for your answer. Specification references (1.2, 1.3, 1.5, A level Further Mathematics Core Pure Mathematics 3.3): Examples of groups. Cayley tables. Cyclic groups. The order of a group and the order of an element Isomorphism Use matrices to represent linear transformations in 2-D. Successive transformations. Single transformations in 3-D. AO1.1b: Correctly carry out routine procedures (4 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (2 marks) AO2.2a: Make deductions (3 marks) AO2.4: Explain their reasoning (3 marks) AO2.5: Use mathematical language and notation properly (1 mark) (Total for Question 4 is 13 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 32

33 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA A level Further Mathematics Paper 4A (Further Pure Mathematics 2) y O x Figure 1 An An engineering student makes a a miniature arch arch as as part part of of the the design for for a a piece of of coursework. coursework. The The cross-section cross section of of this this arch arch is is modelled by by the the curve curve with with equation y = y A = A 1-1 lna x lna 2 2 cosh2x, lna x lna where where a > a 1 >1 and and A A is is a a positive positive constant. constant. The The curve curve begins begins and and ends ends on on the the x axis, x-axis, as as shown in shown Figure in 1. Figure 1. æ (a) Show that the length of this curve is æk a ö ö (a) Show that the length of this curve is k a - 2 è ç è ç a a ø 2, ø, stating the value of the constant k. stating the value of the constant k. (5) The length of the curved cross section of the miniature arch is required to be 2 m long. The length of the curved cross-section of the miniature arch is required to be 2 m long. (b) Find the height of the arch, according to this model, giving your answer to (b) Find 2 significant the height of figures. the arch, according to this model, giving your answer to 2 significant figures. (4) (c) Find also the width of the base of the arch giving your answer to 2 significant figures. (c) Find also the width of the base of the arch giving your answer to 2 significant figures. (d) Give the equation of another curve that could be used as a suitable model for the (1) (d) Give cross section the equation of of an another arch, with curve approximately that could be the used same as a height suitable and model width for as you the found cross-section using the first of an model. arch, with approximately the same height and width as you found using (You the do first not model. need to consider the arc length of your curve) (You do not need to consider the arc length of your curve) Specification reference (2.2): (2) Calculation of arc length and the area of a surface of revolution. AO1.1b: Correctly carry out routine procedures (5 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (3 marks) AO3.3: Translate situations in context into mathematical models (2 marks) AO3.4: Use mathematical models (2 marks) (Total for Question 5 is 12 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) Pearson Edexcel Level 3 Advanced GCE in Mathematics Sample Assessment Materials Issue 1 June 2017 Pearson Education Limited 2017 *S54444A01429*

34 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 6. A curve has equation z + 6 = 2 z 6 z C (a) Show that the curve is a circle with equation x 2 + y 2 20 x + 36 = 0 (b) Sketch the curve on an Argand diagram. The line l has equation az* + a*z = 0, where a C and z C Given that the line l is a tangent to the curve and that arg a = θ (c) find the possible values of tan θ Specification reference (4.1): Further loci and regions in the Argand diagram. AO1.1b: Correctly carry out routine procedures (4 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) AO3.1a: Translate problems in mathematical contexts into mathematical processes (3 marks) (Total for Question 6 is 9 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 34

35 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 7. A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 7. (a) Prove that, for n 2, (a) Prove that, for n 2, In = 2 sin xdx, n 0 0sin n xx d, n 0 0 I n = 2 (b) (b) y ni nin n = = (n (n 1)I 1)In 2 n 2 (4) O x Figure 2 A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder A designer which is has asked diameter to produce 1 m and a poster height to 0.7 completely m. cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m. He He uses uses a large a large sheet sheet of of paper paper with with height height m m and and width width of of π m. m. Figure Figure 2 2 shows shows the the first first stage stage of of the the design, design, where where the the poster poster is divided is divided into into two two sections sections by a curve. by a curve. The curve is given by the equation The curve is given by the equation y = sin 2 (4x) sin 10 (4x) y = sin 2 (4x) sin 10 (4x) relative to axes taken along the bottom and left hand edge of the paper. relative to axes taken along the bottom and left hand edge of the paper. The The region region of of the the poster poster below below the the curve curve is shaded is shaded and and the the region region above above the the curve curve remains unshaded, remains unshaded, as shown in as Figure shown 2. in Figure 2. Find Find the the exact exact area area of of the the poster poster which which is shaded. is (5) Specification reference (2.1): Further Integration Reduction formulae AO1.1b: Correctly carry out routine procedures (4 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (3 marks) AO3.1b: Translate problems in non-mathematical contexts into mathematical processes (2 marks) (Total for Question 7 is 9 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 22 Pearson Education Ltd Pearson Edexcel Level 3 Advanced GCE in Mathematics 35 Sample Assessment Materials Issue 1 June 2017 Pearson Education Limited 2017 *S54444A02229*

36 A level Further Mathematics Paper 4A (Further Pure Mathematics 2) 8. A staircase has n steps. A tourist moves from the bottom (step zero) to the top (step n). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves. If un is the number of ways that the tourist can climb up a staircase with n steps, (a) explain why un satisfies the recurrence relation un = un 1 + un 2, with u1 = 1 and u2 = 2 (b) Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps. (c) Show that the number of ways in which she could climb up to the top of this staircase is given by é 1 æ 1+ 5 ö ê ç 5 ê è 2 ø ë 401 æ ö ç è 2 ø 401 ù ú ú û Specification references (6.1, 6.2): First and second order recurrence relations. The solution of recurrence relations to obtain closed forms. AO1.1b: Correctly carry out routine procedures (4 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) AO2.4: Explain their reasoning (3 marks) (Total for Question 8 is 9 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 36

37 A level Further Mathematics Paper 3B/4B (Further Statistics 1) A level Further Mathematics Paper 3B/4B (Further Statistics 1) 1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the 5% level of significance, whether there is evidence that the level of pollution has increased. Specification reference (4.1): Extend ideas of hypothesis tests to test for the mean of a Poisson distribution. AO1b: Correctly carry out routine procedures (2 marks) AO2.2b: Make inferences (1 mark) AO2.5: Use mathematical language and notation correctly (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) (Total for Question 1 is 5 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 37

38 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 2. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p. The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is The call centre receives 1000 calls each day. (a) Find the mean and variance of the number of wrongly connected calls a day. (b) Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent. (c) Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is to 4 decimal places. (d) Comment on the accuracy of your answer in part (b). Specification references (2.1, 2.2, 2.3): The Poisson distribution The mean and variance of the binomial distribution and the Poisson distribution. The use of the Poisson distribution as an approximation to the binomial distribution. AO1.1b: Correctly carry out routine procedures (5 marks) AO2.4: Explain their reasoning (2 marks) AO3.1b: Translate problems in mathematical contexts into mathematical processes (2 marks) AO3.2b: Translate problems in non-mathematical contexts into mathematical processes (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) AO3.4: Use mathematical models (1 mark) (Total for Question 2 is 12 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 38

39 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 3. Bags of 1 coins are paid into a bank. Each bag contains 20 coins. The bank manager believes that 5% of the 1 coins paid into the bank are fakes. He decides to use the distribution X ~ B (20, 0.05) to model the random variable X, the number of fake 1 coins in each bag. The bank manager checks a random sample of 150 bags of 1 coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place. Number of fake coins in each bag or more Observed frequency Expected frequency Table 1 (a) Carry out a hypothesis test, at the 5% significance level, to see if the data supports the bank manager s statistical model. State your hypotheses clearly. The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than 5%. She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. Number of fake coins in each bag or more Observed frequency Expected frequency Table 2 (b) Explain why there are 2 degrees of freedom in this case. (c) Given that she obtains a χ 2 test statistic of 2.67, test the assistant manager s hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a 5% level of significance and state your hypotheses clearly. Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 39

40 A level Further Mathematics Paper 3B/4B (Further Statistics 1) Specification reference (6.1): Goodness of fit tests and Contingency Tables. The null and alternative hypotheses. The use of n i 1 O E i E Degrees of freedom. i i 2 as an approximate statistic. AO1.1a: Select routine procedures (1 mark) AO1.1b: Correctly carry out routine procedures (2 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.4: Explain their reasoning (2 marks) AO2.5: Use mathematical language and notation correctly (1 mark) AO3.4: Use mathematical models (2 marks) AO3.5a: Evaluate the outcomes of modelling in context (2 marks) (Total for Question 3 is 14 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 40

41 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 4. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3. Estimate the probability that the mean of the sample is greater than 2.5. Specification references (2.2, 5.1): The mean and variance of the binomial distribution and the Poisson distribution. Applications of the Central Limit Theorem to other distributions. AO1.1b: Correctly carry out routine procedures (2 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) AO3.4: Use mathematical models (1 mark) (Total for Question 4 is 4 marks) 5. The probability of Richard winning a prize in a game at the fair is Richard plays a number of games. (a) Find the probability of Richard winning his second prize on his 8th game, (b) State two assumptions that have to be made, for the model used in part (a) to be valid. Mary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable G represents the total number of games Mary plays. (c) Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game. Specification references (3.1, 3.3): Geometric and negative binomial distributions. x 1 p r 1 p r 1 Mean and variance of negative binomial distribution with P(X = x) = x r AO1.1b: Correctly carry out routine procedures (3 marks) AO3.1b: Translate problems in non-mathematical contexts into mathematical processes (1 mark) AO3.2a: Interpret solutions to problems in their original context (1 mark) AO3.3: Translate situations in context into mathematical models (1 mark) AO3.5b: Recognise the limitations of models (2 marks) (Total for Question 5 is 8 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 41

42 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 6. The probability generating function of the discrete random variable X is given by (a) Show that k = 1 36 GX (t) = k (3 + t + 2t 2 ) 2 (b) Find P(X = 3) (c) Show that Var(X) = (d) Find the probability generating function of 2X + 1 Specification references (1.1, 7.1, 7.2): Calculation of the mean and variance of discrete probability distributions. Extension of expected value function to include E(g(X)). Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson distributions. Use to find the mean and variance. AO1.1b: Correctly carry out routine procedures (9 marks) AO2.1: Construct rigorous mathematical arguments (including proofs) (4 marks) AO3.1a: Translate problems in mathematical contexts into mathematical processes (1 mark) (Total for Question 6 is 14 marks) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8FM0) 42

43 A level Further Mathematics Paper 3B/4B (Further Statistics 1) 7. Sam and Tessa are testing a spinner to see if the probability, p, of it landing on red is less than 1 5. They both use a 10% significance level. Sam decides to spin the spinner 20 times and record the number of times it lands on red. (a) Find the critical region for Sam s test. (b) Write down the size of Sam s test. Tessa decides to spin the spinner until it lands on red and she records the number of spins. (c) Find the critical region for Tessa s test. (d) Find the size of Tessa s test. (e) (i) Show that the power function for Sam s test is given by (ii) Find the power function for Tessa s test. (1 p) 19 (1 + 19p) (f) With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam s test or Tessa s test when p = 0.15 Specification references (3.1, 4.2, 8.1): Geometric and negative binomial distributions. Extend hypothesis testing to test for the parameter p of a geometric distribution. Type I and Type II errors. Size and Power of Test. The power function. AO1.1b: Correctly carry out routine procedures (9 marks) AO1.2: Accurately recall facts, terminology and definitions (1 mark) AO2.1: Construct rigorous mathematical arguments (including proofs) (1 mark) AO2.2a: Make deductions (1 mark) AO2.2b: Make inferences (2 marks) AO2.4: Explain their reasoning (1 mark) AO3.3: Translate situations in context into mathematical models (2 marks) AO3.4: Use mathematical models (1 mark) (Total for Question 7 is 18 marks) Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) 43