A Level Mathematics Practice Paper E – Pure Mathematics.docx 70116q

Pearson Edexcel Level 3 GCE Mathematics Advanced Level Paper 1 or 2: Pure Mathematics Practice Paper E Time: 2 hours

Paper Reference(s)

9MA0/01 or 9MA0/02

You must have: Mathematical Formulae and Statistical Tables, calculator Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Instructions  Use black ink or ball-point pen.  If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).  Answer all questions and ensure that your answers to parts of questions are clearly labelled.  Answer the questions in the spaces provided – there may be more space than you need.  You should show sufficient working to make your methods clear. Answers without working may not gain full credit.  Inexact answers should be given to three significant figures unless otherwise stated. Information  A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.  There are 14 questions in this paper. The total mark is 100.  The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question. Advice  Read each question carefully before you start to answer it.  Try to answer every question.  Check your answers if you have time at the end.  If you change your mind about an answer, cross it out and put your new answer and any working underneath.

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Answer ALL questions. 1.

Prove by exhaustion that 1+ 2 + 3+ ...+ n º

n(n + 1) for positive integers from 1 to 6 inclusive. 2

(3 marks) ___________________________________________________________________________________ 2.

(a) When θ is small, show that the equation

1+ sinq + tan 2q 1 can be written as . 1- 3q 2cos3q -1 (4 marks)

(b) Hence write down the value of

1+ sinq + tan 2q when θ is small. 2cos3q -1

(1 mark) ___________________________________________________________________________________ 3.

A stone is thrown from the top of a building. The path of the stone can be modelled using the parametric equations x  10t , y  8t  4.9t 2  10 , t  0, where x is the horizontal distance from the building in metres and y is the vertical height of the stone above the level ground in metres. (a) Find the horizontal distance the stone travels before hitting the ground. (4 marks) (b) Find the greatest vertical height. (5 marks) ___________________________________________________________________________________

4.

Given that x  sec 4 y , find (a)

dy in of y. dx

(2 marks) (b) Show that

dy k  , where k is a constant which should be found. dx x x 2  1

(3 marks) ___________________________________________________________________________________ f  x 

5.

5

6 3   7x 2 x x2

(a) Find f  x  dx .



(3 marks) (b) Evaluate

 f  x  dx , giving your answer in the form 9

4

m  n ln p , where m, n and p are rational

numbers. (3 marks) ___________________________________________________________________________________

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6.

Figure 1 shows a sketch of part of the graph y = f(x) where f ( x)  3 x  4  5

Figure 1 (a) State the range of f. (1 mark) 1 (b) Given that f ( x)   x  k , where k is a constant has two distinct roots, state the possible values of k. 3

(7 marks) ___________________________________________________________________________________ 7.

f (x) º

Show that f (x) can be written in the form A +

9x 2 + 25x +16 9x 2 - 16

B C , where A, B and C are constants to be found. + 3x - 4 3x + 4

(7 marks) ___________________________________________________________________________________

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8.

A ball is dropped from a height of 80 cm. After each bounce it rebounds to 70% of its previous maximum height. (a) Write a recurrence relation to model the maximum height in centimetres of the ball after each subsequent bounce. (2 marks) (b) Find the height to which the ball will rebound after the fifth bounce. (2 marks) (c) Find the total vertical distance travelled by the ball before it stops bouncing. (4 marks) (d) State one limitation with the model. (1 mark) ___________________________________________________________________________________

9.

Solve 6sin(  60)  8 3 cos in the range 0 „  „ 360 . Round your answer to 1 decimal place. (4 marks) ___________________________________________________________________________________

10.

Use proof by contradiction to show that there is no greatest positive rational number. (4 marks) ___________________________________________________________________________________ 1

11.

1 The first three in the binomial expansion of (a  bx) 3 are 4  x  cx 2  8

.

(a) Find the values of a and b. (5 marks) (b) State the range of values of x for which the expansion is valid. (2 marks) (c) Find the value of c. (2 marks) ___________________________________________________________________________________

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12.

The diagram shows a cuboid whose vertices are O, A, B, C, D, E, F and G. a, b and c are the vectors OA , OB and OC respectively. The points M and N lie on OA such that OM : MN : NA  1: 2 :1 . The points K and L lie on EF such that EK : KL : LF  1: 2 :1 .

Figure 1 Prove that the diagonals KN and ML bisect each other at P. (10 marks) ___________________________________________________________________________________

13.

The value of a computer, V, decreases over time, t, measured in years. The rate of decrease of the value is proportional to the remaining value. Given that the initial value of the computer is V0 , (a) show that V  V0e kt . (4 marks) 1 After 10 years the value of the computer is V0 . 5

(b) Find the exact value of k. (3 marks) (c) How old is the computer when its value is only 5% of its original value? Give your answer to 3 significant figures. (3 marks) ___________________________________________________________________________________

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14.

p t  

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1 t 1 ln  t  1  cos    t 2  199.3,0 „ t „ 12 . 10  2  10

Figure 3 Figure 3 is a graph of the price of a stock during a 12-hour trading window. The equation of the curve is given above. (a) Show that the price reaches a local maximum in the interval 8.5 < t < 8.6 . (5 marks) Figure 3 shows that the price reaches a local minimum between 9 and 11 hours after trading begins. (b) Using the Newton–Raphson procedure once and taking t0 = 9.9 as a first approximation, find a second approximation of when the price reaches a local minimum. (6 marks)

____________________________________________________________________________ TOTAL FOR PAPER IS 100 MARKS

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