CAPE Pure Mathematics Unit 1 2011 Paper 2

Question 3 (a)

Solution

(i) �� + ��. �� − ��

Substitute � = � + � ��� = �� + ��

= ��� + � � + ��� + �� �� ∙ ��� + � � − ��� + �� ��

Group terms with i and j

= �� + ��� + �� + ��� � ∙ �� − ��� + �� − ��� �

Apply the dot product formula (note difference of 2 squares)

= � + ��� − �� + �� + ����� − ���

= � − �� + �� − ���

Group terms (recognize that � + � = 13�, �ℎ��ℎ�������; ���� + ��� = 10��

= � + �� − ��� + ����

= 169 − 100

= $%

(ii) 2� − � = 11�

⟹ 2��� + �� � − �� + � � = 11�

⟹ 2�� + 2�� − � − � = 11�

⟹ �2� − �� + �2�� − �� = 11� + 0

∴ 2� − = 11 ⟹ = 2� − 11 ��2�� − � = 0 ⟹ � = 2��

CAPE Pure Mathematics Unit 1 2011 Paper 2

)����|�| = +� + �� = 13,� + �� = 169

,-��.�.-.��� = 2� − 11and� = 2��, �2� − 11�� + �2���� = 169

⟹ 4�� − 44� + 121 + 4��� = 169

⟹ 4�� + 4��� − 44� − 48 = 0

)����|�| = +�� + ��� = 10

⟹ �� + ��� = 100

⟹ � = +100 − ���

,-��.�.-.�� = +100 − �����.44�� + 4��� − 44� + 48 = 0

⟹ 4 5+100 − ���6�

+ 4��� − 44+100 − ��� − 48 = 0

⟹ 352 − 44+100 − ��� = 0

⟹ +100 − ��� = 35244

⟹ 100 − ��� = 8�

⟹ −��� = 64 − 100

⟹ ��� = 36

⟹ �� = 6��4.�.ℎ. − 689���4�����:���4;-.�4����;;�

,-��.�.-.��� = 6��.4� = +100 − ���

⟹ � = <100 − �6��

⟹ � = 8 ��4.�.ℎ. − 889���4�����:���4;-.�4����;;�

CAPE Pure Mathematics Unit 1 2011 Paper 2

,-��.�.-.�� = 8��.4 = 2� − 11

⟹ = 2�8� − 11 ⟹ = 5

,-��.�.-.��� = 6��.4� = 2��

⟹ � = 2�6�

⟹ � = 12

=����, = 5,� = 12,� = 8,�� = 6

>ℎ�:�?4:�, � = 5� + 12

� = 8� + 6