AQA MFP1 p QP JUN15

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12345678TOTAL

(JUN15MFP101)

Answer all questions.

Answer each question in the space provided for that question

1 The quadratic equation 2x2þ 6x þ 7 ¼ 0 has rootsa andb

(a) Write down the value of a þ b and the value of ab

[2 marks]

(b) Find a quadratic equation, with integer coefficients, which has roots a2 1 and

b2 1

[5 marks]

(c) Hencefind the values of a2 andb2

[2 marks]

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3 Answer space for question 1

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2 (a) Explain why

ð4

0

x  4

x1:5 dx is an improper integral

[1 mark]

(b) Either find the value of the integral

ð4

0

x  4

x1:5 dx or explain why it does not have a finite value

[4 marks]

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5 Answer space for question 2

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3 (a) Show that ð2 þ iÞ3 can be expressed in the form 2 þ bi, whereb is an integer.

[3 marks]

(b) It is given that 2 þ i is a root of the equation

z3þ pz þ q ¼ 0 wherep and qare real numbers

(i) Show that p ¼ 11 and find the value ofq

[4 marks]

(ii) Given that 2  i is also a root of z3þ pz þ q ¼ 0, find a quadratic factor of

z3þ pz þ q with real coefficients

[2 marks]

(iii) Find the real root of the equation z3þ pz þ q ¼ 0

[2 marks]

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7 Answer space for question 3

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4 (a) Find the general solution, in degrees, of the equation

2 sinð3x þ 45Þ ¼ 1

[5 marks]

(b) Use your general solution to find the solution of 2 sinð3x þ 45Þ ¼ 1 that is closest to

200

[1 mark]

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9 Answer space for question 4

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5 (a) The matrix Ais defined byA ¼ 2 c

Given that the image of the pointð5, 2Þunder the transformation represented byAis

ð2, 1Þ, find the value of cand the value of d

[4 marks]

(b) The matrixB is defined by B ¼

ffiffiffi 2

2 p

 ffiffiffi 2

2 p

(i) Show that B4 ¼ kI, wherek is an integer andI is the2  2 identity matrix

[2 marks]

(ii) Describe the transformation represented by the matrixBas a combination of two

geometrical transformations

[5 marks]

(iii) Find the matrixB17

[2 marks]

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13 Answer space for question 5

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6 A curveC1has equation

x2

9  y

2

16 ¼ 1 (a) Sketch the curveC1, stating the values of its intercepts with the coordinate axes

[2 marks]

(b) The curveC1 is translated by the vector k

0

  , where k < 0, to give a curve C2

Given thatC2passes through the origin ð0, 0Þ, find the equations of the asymptotes

ofC2

[3 marks]

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7 (a) The equation 2x3þ 5x2þ 3x  132 000 ¼ 0 has exactly one real roota.

(i) Show thata lies in the interval 39 < a < 40

[2 marks]

(ii) Taking x1¼ 40 as a first approximation to a, use the Newton–Raphson method to

find a second approximation,x2, toa Give your answer to two decimal places

[3 marks]

(b) Use the formulae for Xn

r¼1

r2 and Xn

r¼1

r to show that

Xn r¼1

2rð3r þ 2Þ ¼ nðn þ pÞð2n þ qÞ

wherep and qare integers

[5 marks]

(c) (i) Expresslog84r in the form lr, wherel is a rational number

[1 mark]

(ii) By first finding a suitable cubic inequality fork, find the greatest value ofk for which

X60

r¼kþ1

ð3r þ 2Þ log84r is greater than 106 060

[4 marks]

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8 A curveC has equation

y ¼ xðx  3Þ

x2þ 3 (a) State the equation of the asymptote ofC

[1 mark]

(b) The line y ¼ k intersects the curveC Show that 4k2 4k  3 4 0

[5 marks]

(c) Hencefind the coordinates of the stationary points of the curve C

(No credit will be given for solutions based on differentiation.)

[5 marks]

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21 Answer space for question 8

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