AQA MFP4 p QP JUN15

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(JUN15MFP401)

Answer all questions.

Answer each question in the space provided for that question

1 The pointsU,V andW have position vectors u,vand wrespectively relative to an

originO, where

u ¼

1 2 2

2 4

3

5, v ¼

3

4 2

2 4

3

5 and w ¼

a 7

2

2 4

3 5

(a) Findðu  vÞ:win terms of a

[2 marks]

(b) Given thatu,v and ware linearly dependent:

(i) find the value ofa;

[1 mark]

(ii) expressu as a linear combination ofv andw

[3 marks]

Answer space for question 1

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3 Answer space for question 1

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2 The vectorsa,b andc are such that c  a ¼ 2i and b  a ¼ 3j.

Simplify ða þ 2b  6cÞ  ða  b þ 3cÞ, giving your answer in the form li þ mj

[5 marks]

Answer space for question 2

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5 Answer space for question 2

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3 (a) Factorise completely the determinant

a b  c bc

b a  c ca

c a þ b ab

























[6 marks]

(b) Hence, or otherwise, find the values ofafor which the equations

ax þ y  6z ¼ 0 3x þ ða  2Þy  2az ¼ 0

2x þ ða þ 3Þy þ 3az ¼ 0

do not have a unique solution

[3 marks]

Answer space for question 3

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7 Answer space for question 3

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4 (a) Find the eigenvalues and corresponding eigenvectors of the matrix

M ¼ 1 1

[6 marks]

(b) Given that U ¼ 4 b

a 2

and U1MU ¼ 3 0

0 2

, find the values of a andb

[3 marks]

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9 Answer space for question 4

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5 A system of equations is given by

2x  11y  3z ¼ 1 5x  10y  4z ¼ 6 9x  17y  7z ¼ 11 (a) Find the solution of this system of equations, showing all your working

[5 marks]

(b) Interpret this solution geometrically

[1 mark]

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11 Answer space for question 5

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6 The lineL has equation r 

2 1 0

2 4

3 5

0

@

1

A 

3

1 2

2 4

3

5 ¼

0 0 0

2 4

3

5

The planePcontains the lineL and the point Að4, 1, 2Þ

(a) Show thatAdoes not lie on the line L

[1 mark]

(b) Find an equation of the planeP, giving your answer in the form r:n ¼ c

[5 marks]

(c) The pointD has coordinatesð8, 2, 6Þ Find the coordinates of the image of Dafter

reflection in the planeP

[5 marks]

Answer space for question 6

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13 Answer space for question 6

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Answer space for question 6

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15 Answer space for question 6

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7 The matrix A ¼ 3:4 2

1:2 1

represents a transformation that is a shearS followed by a transformationT

(a) The shearSis such that the image of the point ð1, 1Þisð5, 3Þ and the line y ¼ x

is a line of invariant points Find the matrix that representsS

[4 marks]

(b) (i) Hence find the matrix that represents the transformationT

[4 marks]

(ii) Give a full description of the transformationT

[2 marks]

Answer space for question 7

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17 Answer space for question 7

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8 The linear transformationTis represented by the matrix M ¼

1 1 1

2 4

3

5

(a) In the case whenM is a non-singular matrix:

(i) find the possible values ofk;

[3 marks]

(ii) findM1 in terms of k

[5 marks]

(b) In the case when k ¼ 1, the matrix M1 is applied to a solid shape of volume6 cm3

Find the volume of the image

[3 marks]

(c) In the case when k ¼ 5, verify that the image of every point under T lies in the plane

x  y þ z ¼ 0

[3 marks]

(d) Find the value ofk for which Thas a line of invariant points and obtain the Cartesian

equations of this line

[5 marks]

Answer space for question 8

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19 Answer space for question 8

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Answer space for question 8

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21 Answer space for question 8

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