6B CAPE Pure Mathematics Unit 1
Term 1
HW 4
October 26, 2021.
Due: Tuesday, November 2, 2021.
1) The function 𝑔(𝑥) = 𝑥 2 + 2𝑥 + 8 has domain 𝑋 = {𝑥 ∈ 𝑹: 𝑥 ≤ −1}. Define the
inverse function of g stating clearly the domain and range of 𝑔−1 (𝑥).
(6 marks)
2) For each of the following determine whether the function is injective, surjective,
bijective or neither giving reason(s) to support you conclusion.
a) 𝑓(𝑥) = 5𝑥 − 9
b) ℎ(𝑥) = 2𝑥 2 − 8, x  0
c) 𝑝(𝑥) = 𝑥 2 + 2𝑥, 𝑥 ≤ 2
(5 marks each)
3) Show by counter example that the statement “The square root of a real number x is
always less than x.” is FALSE.
(5 marks)
n
4) For the series ∑(𝑟 2 + 2)
r=1
i. Write the expression for the nth term of the series and use this expression
to find the nth term of the series if n = 14
(3 marks)
ii. If n = 6, expand the series fully (write out the terms of the series).
(3 marks)
iii. Hence evaluate the series when n = 6 and use another method of
evaluation to check if your sum is correct
(6 marks)
n
5) Let 𝑆𝑛 = ∑ 𝑟 for 𝑛 ∈ 𝑵. Find the value of n for which 3𝑆2𝑛 = 11𝑆𝑛 .
(4 marks)
r=1
(2007 Paper 2 #2a)
n
1
r=1
3
6) Show that ∑ 𝑟(𝑟 + 1) = 𝑛(𝑛 + 1)(𝑛 + 2), 𝑛 ∈ 𝑵.
(5 marks)
50
Hence, or otherwise, evaluate ∑ 𝑟(𝑟 + 1).
(3 marks)
r = 31
(2008 Paper 2 #1c)
Total Score 50 marks
S. Kenny-Folkes
October 2021