MATHS PAST PAPERS FOR CAPE
TEST CODE 02134020/SPEC
FORM TP 02134020/SPEC
CARI B B E AN
E XAM I NAT I O NS
CO UNCI L
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1
ALGERBRA, GEOMETRY AND CALCULUS
SPECIMEN PAPER
PAPER 02
2 hours 30 minutes
The examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to THREE significant figures.
Examination Materials
Mathematical formulae and tables
Electronic calculator
Ruler and graph paper
Copyright © 2011 Caribbean Examinations Council ®
All rights reserved
02134020/CAPE/SPEC
2
SECTION A (MODULE 1)
Answer BOTH questions.
1.
(a)
Let p and q be given propositions.
(i)
Copy and complete the table below to show the truth tables of p → q and ~ p ∨ q.
[3 marks]
p
(ii)
(iii)
q
~p
p→q
~p ∨ q
Hence, state whether the compound propositions p → q and ~p ∨ q are logically equivalent, stating reasons for your answer.
[2 marks]
Use the algebra of propositions to show that p ∧ (p → q) = p ∧ q.
[3 marks]
(b)
The binary operation * is defined on the set of real numbers,
∗ = + −1
For all x, y in
, as follows:
Prove that
(i)
[3 marks]
(ii)
* is commutative in ,
[2 marks]
(iii)
(c)
* is closed in ,
*is associative in .
[4 marks]
Let y =
(i)
(ii)
.
Show that for all real values of x, −
≤
≤
Hence, sketch the graph of y for all x such that 2 ≤
≤2
[5 marks]
[3 marks]
Total 25 marks
02134020/CAPE/SPEC
GO ON TO NEXT PAGE
3
2. (a)
Two of the roots of the
FORM TP 02134020/SPEC
CARI B B E AN
E XAM I NAT I O NS
CO UNCI L
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1
ALGERBRA, GEOMETRY AND CALCULUS
SPECIMEN PAPER
PAPER 02
2 hours 30 minutes
The examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to THREE significant figures.
Examination Materials
Mathematical formulae and tables
Electronic calculator
Ruler and graph paper
Copyright © 2011 Caribbean Examinations Council ®
All rights reserved
02134020/CAPE/SPEC
2
SECTION A (MODULE 1)
Answer BOTH questions.
1.
(a)
Let p and q be given propositions.
(i)
Copy and complete the table below to show the truth tables of p → q and ~ p ∨ q.
[3 marks]
p
(ii)
(iii)
q
~p
p→q
~p ∨ q
Hence, state whether the compound propositions p → q and ~p ∨ q are logically equivalent, stating reasons for your answer.
[2 marks]
Use the algebra of propositions to show that p ∧ (p → q) = p ∧ q.
[3 marks]
(b)
The binary operation * is defined on the set of real numbers,
∗ = + −1
For all x, y in
, as follows:
Prove that
(i)
[3 marks]
(ii)
* is commutative in ,
[2 marks]
(iii)
(c)
* is closed in ,
*is associative in .
[4 marks]
Let y =
(i)
(ii)
.
Show that for all real values of x, −
≤
≤
Hence, sketch the graph of y for all x such that 2 ≤
≤2
[5 marks]
[3 marks]
Total 25 marks
02134020/CAPE/SPEC
GO ON TO NEXT PAGE
3
2. (a)
Two of the roots of the