OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4755Further Concepts For Advanced Mathematics (FP1)

Wednesday 18 JANUARY 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.

• Answer all the questions.

• You are permitted to use a graphical calculator in this paper.

• Final answers should be given to a degree of accuracy appropriate to the context.

INFORMATION FOR CANDIDATES

• The number of marks is given in brackets [ ] at the end of each question or part question.

• You are advised that an answer may receive no marks unless you show sufficient detail of theworking to indicate that a correct method is being used.

• The total number of marks for this paper is 72.

HN/3© OCR 2006 [D/102/2663] Registered Charity 1066969 [Turn over

This question paper consists of 4 printed pages.

Section A (36 marks)

1 You are given that

(i) Calculate, where possible, 2B, CA and [5]

(ii) Show that matrix multiplication is not commutative. [2]

2 (i) Given that express and in terms of a and b. [2]

(ii) Prove that [3]

3 Find expressing your answer in a fully factorised form. [6]

4 The matrix equation represents two simultaneous linear equations in x and y.

(i) Write down the two equations. [2]

(ii) Evaluate the determinant of

What does this value tell you about the solution of the equations in part (i)? [3]

5 The cubic equation has roots

(i) Write down the values of and [2]

(ii) Find the cubic equation with roots simplifying your answer as far as possible.[4]

6 Prove by induction that [7]Sn

r�1

1r(r � 1) �

n

n � 1.

2a, 2b and 2g ,

abg .ab � bg � gaa � b � g ,

a, b and g .x3 � 3x 2 � 7x � 1 � 0

6 23 1

--

ÊË

ˆ¯ .

6 23 1

--

ÊË

ˆ¯

ÊË

ˆ¯ = Ê

ˈ¯

xy

ab

Sn

r�1 (r � 1) (r � 1),

zz* � � z �2 � 0.

z*� z �z � a � bj,

A � B .A � C,

C =-Ê

ËÁÁ

ˆ

¯˜˜

1 10 20 1

.B =-Ê

ˈ¯

2 31 4

,A = ÊË

ˆ¯

4 31 2

,

2

4755 January 2006

Section B (36 marks)

7 A curve has equation .

(i) Show that y can never be zero. [1]

(ii) Write down the equations of the two vertical asymptotes and the one horizontal asymptote.[3]

(iii) Describe the behaviour of the curve for large positive and large negative values of x, justifyingyour description. [2]

(iv) Sketch the curve. [3]

(v) Solve the inequality [4]

8 You are given that the complex number satisfies the equation ,where p and q are real constants.

(i) Find and in the form Hence show that and [6]

(ii) Find the other two roots of the equation. [3]

(iii) Represent the three roots on an Argand diagram. [2]

q � 10.p � �8a � bj .a 3a 2

z3 � 3z2 � pz � q � 0a � 1 � j

3 � x2

4 � x2 � –2.

y �3 � x2

4 � x2

3

4755 January 2006 [Turn over

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by

always lies on the line and has the same y-coordinate as P. This is illustrated in Fig. 9.

Fig. 9

(i) Write down the image of the point under transformation T. [1]

(ii) P has coordinates State the coordinates of [2]

(iii) All points on a particular line l are mapped onto the point Write down the equation ofthe line l. [1]

(iv) In part (iii), the whole of the line l was mapped by T onto a single point. There are an infinitenumber of lines which have this property under T. Describe these lines. [1]

(v) For a different set of lines, the transformation T has the same effect as translation parallel tothe x-axis. Describe this set of lines. [1]

(vi) Find the matrix which represents the transformation. [3]

(vii) Show that this matrix is singular. Relate this result to the transformation. [3]

2 � 2

(3, 6).

P�.(x, y).

(10, 50)

x

y

¢P1

P2

P1

¢P2

y � 2xP�

P�.

4

4755 January 2006

Section A

1(i)

1(ii)

4 62

2 8−⎛ ⎞

= ⎜ ⎟⎝ ⎠

B , A + C is impossible,

3 12 41 2

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

CA , A - B = 2 60 2⎛ ⎞⎜ ⎟−⎝ ⎠

4 3 2 3 11 01 2 1 4 4 5

2 3 4 3 5 01 4 1 2 8 11

−⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

−⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

≠

AB

BA

AB BA

B1 B1

M1, A1 B1

[5]

M1

E1

[2]

CA 3 2× matrix M1 Or AC impossible, or student’s own correct example. Allow M1 even if slip in multiplication Meaning of commutative

2(i)

2(ii)

( )2 2z a b= + , * jz a b= −

( )( )

( )

* 2 2

2* 2 2 2 2 0

zz a bj a bj a b

zz z a b a b

= + − = +

⇒ − = + − + =

B1 B1

[2]

M1

M1 A1

[3]

Serious attempt to find *zz , consistent with their *z ft their z in subtraction All correct

3 ( )( ) ( )

( )( )

( )( )[ ]

( )

( )( )

1

2

1

2

1 1

11 2 1

61

1 2 1 66

1

12 3 5

61

2 5 16

n

r

n

r

r r

n n n n

n n n

r

n n n

n n n

= =

+ −

= + + −

= + + −

= −

= + −

= + −

∑ ∑

M1

M1,

A1, A1

M1

A1 [6]

Condone missing brackets Attempt to use standard results Each part correct Attempt to collect terms with common denominator c.a.o.

4(i)

4(ii)

6 23x y a

x y b− =

− + =

Determinant = 0 The equations have no solutions or infinitely many solutions.

B1 B1

[2]

B1

E1 E1

[3]

No solution or infinitely many solutions Give E2 for ‘no unique solution’ s.c. 1: Determinant = 12, allow ‘unique solution’ B0 E1 E0

s.c. 2: Determinant = 10

give

maximum of B0 E1 5(i)

5(ii)

3α β γ+ + = − , 7αβ βγ γα+ + = − , 1αβγ = − Coefficients A, B and C

2 2 2 2 3 6

2 2 2 2 2 2 4 7 28

2 2 2 8 1 8

BA

CA

DA

α β γ

α β β γ γ α

α β γ

−+ + = × − = − =

× + × + × = × − = − =

−× × = × − = − =

3 26 28 8 0x x x⇒ + − + = OR

3 2

3 2

3 2

22

3 7 1 02 2 2

3 7 1 08 4 2

6 28 8 0

x x ωω

ω ω ω

ω ω ω

ω ω ω

= ⇒ =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⇒ + − + =

⇒ + − + =

B2 [2]

M1

A3 [4]

M1 A1

A1

A1 [4]

Minus 1 each error to minimum of 0 Attempt to use sums and products of roots ft their coefficients, minus one each error (including ‘= 0’ missing), to minimum of 0 Attempt at substitution Correct substitution Substitute into cubic (ft) c.a.o.

6

( )1

11 1

n

r

nr r n=

=+ +

∑

n = 1, LHS = RHS = 12

Assume true for n = k

Next term is ( )( )

11 2k k+ +

Add to both sides

( )( )( )

( )( )

( )( )( )

( )( )

2

2

1RHS1 1 2

2 11 2

2 11 2

11 2

12

kk k k

k kk k

k kk k

kk k

kk

= ++ + +

+ +=

+ +

+ +=

+ +

+=

+ +

+=

+

But this is the given result with k + 1 replacing k. Therefore if it is true for k it is true for k + 1. Since it is true for k = 1, it is true for k = 1, 2, 3

B1

E1

B1

M1

A1

E1

E1

[7]

Assuming true for k (must be explicit) (k + 1)th term seen c.a.o.

Add to 1

kk +

(ft)

c.a.o. with correct working True for k, therefore true for k + 1

(dependent on 12

kk++

seen)

Complee argument

Section A Total: 36

7(i)

7(ii)

7(iii)

7(iv)

7(v)

Section B 23 0x+ ≠ for any real x.

y = -1, 2, 2x x= = − Large positive x, 1y −→ − (e.g. consider 100x = ) Large negative x, 1y −→ − (e.g. consider 100x = − ) Curve 3 branches correct Asymptotes labelled Intercept labelled

2

2 22

2

3 2 3 8 24

11

( ) 11

x x xx

x

x

+= − ⇒ + = − +

−⇒ =

⇒ = ±

From graph, 11 2 or 2 11x x− ≤ < − < ≤

E1

[1]

B1, B1

B1 [3]

M1 B1

[2]

B1 B1

B1 [3]

M1

A1

B1 A1

[4]

Evidence of method required From below on each side c.a.o. Consistent with (i) and their (ii), (iii) Consistent with (i) and their (ii), (iii) Labels may be on axes Lose 1 mark if graph not symmetrical May be written in script Reasonable attempt to solve Accept 11

2x < − and 2 x< seen c.a.o.

8(i)

8(ii)

8(iii)

( )( )

22

3

1 j 2 j

1 j 2 j 2 2 j

α

α

= + =

= + = − +

( )( )

3 23 02 j 2 3 2 j 1 j 08 2 0

8 and 2 0 10

z z pz qp q

p j p qp p q q

+ + + =⇒ − + × + + + =

⇒ + + + − =

= − + − = ⇒ =

1 j− must also be a root. The roots must sum to –3, so the other root is z = -5

M1, A1

A1

M1

M1

A1

[6]

B1 M1 A1

[3]

B2

[2]

Substitute their 2α and 3α into cubic Equate real and imaginary parts to 0 Results obtained correctly Any valid method c.a.o. Argand diagram with all three roots clearly shown; minus 1 for each error to minimum of 0 ft their real root

Section B (continued)

9(i)

9(ii) 9(iii)

9(iv)

9(v)

9(vi)

9(vii)

( )25,50

1 ,2

y y⎛ ⎞⎜ ⎟⎝ ⎠

6y = All such lines are parallel to the x-axis. All such lines are parallel to 2y x= .

102

0 1

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

10 1det 0 1 0 02

20 1

⎛ ⎞⎜ ⎟ = × − × =⎜ ⎟⎜ ⎟⎝ ⎠

Transformation many to one.

B1 [1]

B1, B1

[2]

B1

[1]

B1 [1]

B1

[1]

B3 [3]

M1

E2 [3]

Or equivalent Or equivalent Minus 1 each error s.c. Allow 1 for reasonable attempt but incorrect working Attempt to show determinant = 0 or other valid argument May be awarded without previous M1 Allow E1 for ‘transformation has no inverse’ or other partial explanation

Section B Total: 36 Total: 72