6667/01 Edexcel GCE - Pearson qualifications | Edexcel, … · · 2018-04-20Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary Monday

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6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 30 January 2012 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) Nil

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75. There are 24 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

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P40086AW850/R6667/57570 5/4/5

*P40086A0124*

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*P40086A0224*

1. Given that z1 1= − i ,

(a) find arg( z1 ).(2)

Given also that z2 3 4= + i, find, in the form a b+ i , a b, ,∈

(b) z z1 2 ,(2)

(c) zz

2

1

.(3)

In part (b) and part (c) you must show all your working clearly.

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Question 1 continued

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___________________________________________________________________________ Q1

(Total 7 marks)

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*P40086A0424*

2. (a) Show that f ( )x x x= + −4 1 has a real root in the interval [ . , . ]0 5 1 0 .(2)

(b) Starting with the interval [ . , . ],0 5 1 0 use interval bisection twice to find an interval of width 0.125 which contains .

(3)

(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice tof ( )x to obtain an approximate value of Give your answer to 3 decimal places.

(5)

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(Total 10 marks)

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*P40086A0624*

3. A parabola C has cartesian equation C y x2 16= . The point P t4 2,( t8 ) is a general point on C.

(a) Write down the coordinates of the focus F and the equation of the directrix of F C.(3)

(b) Show that the equation of the normal to C at C P isP y tx t t+ = +8 4 3.(5)

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(Total 8 marks)

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*P40086A0824*

4. A right angled triangle T has verticesT A ( , ),1 1 B ( , )2 1 and C ( , ).2 4 When T is transformed T

by the matrix P =⎛⎝⎜

⎞⎠⎟

01

1 0

, the image is ′T .

(a) Find the coordinates of the vertices of ′T .(2)

(b) Describe fully the transformation represented by P. (2)

The matrices Q =⎛⎝⎜

−−

⎞⎠⎟

43

21

and R =⎛⎝⎜

⎞⎠⎟

13

2 4

represent two transformations. When T isT

transformed by the matrix QR, the image is ′′T .

(c) Find QR.(2)

(d) Find the determinant of QR.(2)

(e) Using your answer to part (d), find the area of ′′T .(3)

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___________________________________________________________________________ Q4

(Total 11 marks)

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*P40086A01224*

5. The roots of the equation

z z z3 28 22 20 0− + − =

are z z z1 2 3, and .

(a) Given that z1 3= + i, find z2 and z3.(4)

(b) Show, on a single Argand diagram, the points representing z z z1 2 3, . and (2)

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(Total 6 marks)

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*P40086A01424*

6. (a) Prove by induction

r n nr

n3

1

2 214

1= +=

∑ ( )(5)

(b) Using the result in part (a), show that

( ) ( )r n n n nr

n3

1

3 22 14

2 8=

∑ − = + + −(3)

(c) Calculate the exact value of ( )rr

3

20

50

2=∑ − .

(3)

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*P40086A01624*


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___________________________________________________________________________ Q6

(Total 11 marks)

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*P40086A01824*

7. A sequence can be described by the recurrence formula

u u n un n+ = + =1 12 1 1 1, ,

(a) Find u2 and u3 .(2)

(b) Prove by induction that unn= −2 1

(5)

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(Total 7 marks)

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*P40086A02024*

8. A =⎛⎝⎜

⎞⎠⎟

0 12 3

(a) Show that A is non-singular.(2)

(b) Find B such that BA2 = A.(4)

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(Total 6 marks)

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*P40086A02224*

9. The rectangular hyperbola H has cartesian equationH xy = 9

The points P pp

Q qq

3 3 3 3, ,⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

and lie on H, where HH p q≠ ± .

(a) Show that the equation of the tangent at P isP x p y p+ =2 6 .(4)

(b) Write down the equation of the tangent at Q.(1)

The tangent at the point Pt and the tangent at the point P Q intersect at R.t

(c) Find, as single fractions in their simplest form, the coordinates of R in terms of p and q.

(4)

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TOTAL FOR PAPER: 75 MARKS

END

Q9

(Total 9 marks)

Documents

6667/01 Edexcel GCE - Pearson qualifications | Edexcel, … · · 2018-04-20Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary Monday