GANSBAAI ACADEMIA

MATHEMATICS
Grade 11 / /

PROJECT

March 2014

Total: 60
Time: 2 weeks
EXAMINATOR
MODERATOR / L. Havenga
L. Mostert

TASK 1 [20]

A:

1. x2-2xy+y2=0

∴x-yx-y=0 ü

∴x=y or x=y ü

∴xy=11 or xy=11 ü

∴1=1 ü (4)

2. 5x2-13xy+8y2=0

∴5x-8yx-y=0 ü

∴5x=8y or x=y ü

∴xy=85 or xy=11 ü

∴1,6=1 ü (4)

3. 8x2-21xy+13y2=0

∴8x-13yx-y=0 ü

∴8x=13y or x=y ü

∴xy=138 or xy=11 ü

∴1,625=1 ü (4)

[12]

B:

1. x+y=13 =>y=13-x

8x-5y=0

Substitute into

8x-513-x=0 ü

∴8x-65+5x=0

∴13x=65

∴x=5 ü

y=13-5=8 ü

xy=85=1,6 ü (4)

2. x+y=21 => y=21-x

13x-8y=0

Substitute into

13x-821-x=0 ü

∴13x-168+8x=0

∴21x=168

∴x=8 ü

y=21-8=13 ü (3)

3. xy=138=1,625 ü (1)

TASK 2 [8]

1. 34 ü 55 ü 89 ü (3)

2. 11; 21; 32; 53; 85; 138; 2113; 3421; 5534; 8955 ü ü (2)

3. 1; 2; 1,5; 1,667; 1,6; 1,625; 1,615; 1,619; 1,618; 1,618 ü ü (2)

4. The numbers in the sequence get closer and closer to 1,618. ü ü (1)

TASK 3 [8]

1 and 2

T1 / T2 / T3 / T4 / T5 / T6 / T7 / T8 / T9 / T10
1 / 1,41421 / 1,55377 / 1,59805 / 1,61185 / 1,61612 / 1,6174 / 1,617851 / 1,617978 / 1,618017
1,41421 / 1,55377 / 1,89805 / 1,61185 / 1,61612 / 1,61744 / 1,61785 / 1,617978 / 1,618017 / 1,618029
1,73205 / 1,65289 / 1,62877 / 1,62135 / 1,61906 / 1,61835 / 1,61813 / 1,618064 / 1,618043 / 1,618037

Row 1 ü ü

Row 2 ü ü

Row 3 ü ü (6)

3. The terms of the sequence converge or get closer and closer to 1,618. ü ü (2)

TASK 4 [8]

1 and 2

T1 / T2 / T3 / T4 / T5 / T6 / T7 / T8 / T9 / T10
2 / 1,5 / 1,66667 / 1,6 / 1,625 / 1,61538 / 1,61905 / 1,6176471 / 1,61818 / 1,61798
1,5 / 1,66667 / 1,6 / 1,625 / 1,61538 / 1,61905 / 1,61765 / 1,6181818 / 1,61798 / 1,61806
1,3333 / 1,75 / 1,57143 / 1,63636 / 1,61111 / 1,62069 / 1,61702 / 1,6184211 / 1,61789 / 1,61809

Row 1 ü ü

Row 2 ü ü

Row 3 ü ü (6)

3. The terms of the sequence converge or get closer and closer to 1,618. ü ü (2)

TASK 5 [8]

1. 1,61803398 ü (1)

2. They are equal to one another. ü ü (2)

3. Use the quadratic formula:

φ=x=--1±-12-41-121 ü

∴1±52 ü

BUT the ratio is positive ∴φ=1±52 ü (3)

4. Irrational number ü (1)

5. 1,618033989 ü. (1)

TASK 6 [8]

1. xy=yx-y

∴xx-y=y2 ü

∴x2-xy-y2=0 ü (2)

2. If

x2-x1-12=0 ü

∴x=--1±-12-41-121 ü

∴x=1±52 ü (3)

3. The number is the Golden Ratio . ü (1)

4. This does not factorise, but from Task 1 we know that:

xy=-b±b2-4ac2a ü

and from tasks 4 and 5 we know that:

xy=1±52. ü (2)

TOTAL: 60

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