Subject:
Pure Mathematics
Date: April , 2019
Time: 1:30p.m.
Duration: 1 Hour & 20 Minutes
Candidates are permitted to bring the following item to their
desk: Pen, Pencil, Ruler, Calculator
Instructions to Candidates: This paper has
4
pages and 8
questions.
Candidates are reminded that the examiners shall take into
account the proper use of the English Language in determining
the mark for each response.
1. Answer all questions.
2. Show all necessary working to obtain full marks.
3. Marks allocations are shown in brackets.
4. Begin your answer to each question on a separate sheet.
Question 1 (11 marks)
Define the simple statements p, q and r as follows:
p: I do not have Covid-19
q: I go to work
r: I spread the Covid-19
a) Using proposition connectives, symbolize the following
statements:
i.
If I have Covid-19 and I go to work, then I will
spread the Covid-19.
ii.
[2]
If I do not have Covid-19, then I will go to work. [2]
b) Write a sentence in standard English corresponding to each
of the following propositions:
iii.
iv.
𝑟 ⇔ 𝑞 ∧ (∼ 𝑝)
[2]
∼ (𝑟 ∨ 𝑞)
[2]
c) Write, in words, the contrapositive of the statement in
a)ii.
[3]
Question 2 (8 marks)
Let 𝑆 be the set of rationals which differ from −1. 𝐴 binary
operation ⊗ is defined on 𝑆 by
𝑥 ⨂ 𝑦 = 𝑥 + 𝑦 + 𝑥𝑦 for all 𝑥, 𝑦 ∈ 𝑆.
a) Determine whether ⊗ is commutative or associative.
b) Show that 𝑆 has an identity with respect to ⊗.
c) Determine which elements of 𝑆 have inverses with
respect to ⊗.
[3]
[2]
[3]
Question 3 (10 marks)
a) Prove that the square of every even integer is even.
[3]
b) Prove by induction that for all 𝑛 ∈ ℤ+ ,
𝑛
∑
𝑟=1
1
𝑛
=
(3𝑟 + 1)(3𝑟 − 2) 3𝑛 + 1
[7]
Question 4 (8 marks)
𝑓(𝑥) ≡ 3𝑥 3 + 𝑝𝑥 2 + 8𝑥 + 𝑞
When 𝑓(𝑥) is divided by (𝑥 + 1) there is a remainder of −4 and when
𝑓(𝑥) is divided by (𝑥 − 2) there is a remainder of 80.
a) Find the values of the constants 𝑝 and 𝑞.
[4]
b) Show that (𝑥 + 2) is a factor of 𝑓(𝑥).
[1]
c) Solve the equation 𝑓(𝑥) = 0.
[3]
Question 5 (9 marks)
The functions 𝑓 and 𝑔 are defined by
𝑓: 𝑥 → 5𝑥 − 7, 𝑥 ∈ ℝ
𝑔: 𝑥 → 2𝑥 + 3, 𝑥 ∈ ℝ
a) Show that 𝑔 is one-to-one.
[3]
b) Express the function 𝑔𝑓 in terms of 𝑥 and state
its domain and range.
c) Solve the equation 𝑔𝑓(𝑥) = 10.
[3]
[3]
Question 6 (13 marks)
a) In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 2√3 − 1, 𝐵𝐶 = √3 + 2 and ∠𝐴𝐵𝐶 = 90𝑂 .
i.
ii.
iii.
Find the exact value of triangle 𝐴𝐵𝐶 in its
simplest form.
[3]
Show that 𝐴𝐶 = 2√5
[3]
Show that 𝑡𝑎𝑛(∠𝐴𝐶𝐵) = 5√3 − 8
[4]
b) Solve the inequality |2𝑥 − 3| < 9, and sketch the solution on
the real line.
[3]
Question 7 (14 marks)
a) Solve the following simultaneous equations, giving your
answers to two decimal places (𝟐dp).
log 6 (3𝑥 2 + 𝑦 2 ) = log 6 28
log 6 (𝑥 + 𝑦) = 1
[5]
b) Find, in exact form, the solutions to the equation
2𝑒 2𝑥 + 12 = 11𝑒 𝑥
[4]
c) A radioactive substance is decaying such that its mass, 𝑚
grams, at a time 𝑡 years after initial observation is given
by
𝑚 = 240𝑒 𝑘𝑡
Where 𝑘 is a constant.
Given that when t=180, m=160, find,
i.
ii.
the value of k.
[2]
the time it takes for the mass of substance
to be halved.
[3]
Question 8 (4 marks)
The cubic equation 2𝑥 3 – 3𝑥 2 + 4𝑥 + 6 = 0has roots 𝛼, 𝛽 and 𝛾.
Find the new equation whose roots are
2
𝛼
,
2
𝛽
EASY LIKE SUNDAY MORNING...!!!
and
2
𝛾
.
[4]