100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
COS1501 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS } R48,72
Add to cart

Exam (elaborations)

COS1501 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS }

 1 purchase

COS1501 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS }

Preview 4 out of 124  pages

  • December 21, 2024
  • 124
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers
All documents for this subject (25)
avatar-seller
Jennifer2024
lOMoAR cPSD| 49511909




COS1501/XOS1501/MO001




1



Downloaded by Vincent master
()

, lOMoAR cPSD| 49511909




Activity 1-11:

1. Factorising: If we need to factorise an expression of the form x2 + ax + b or (x2 – ax – b or x2 +
ax – b or x2 – ax + b) we need to find some c and some d, such that a = c + d and b = (c)(d)
and (x + c)(x + d) = x2 + ax + b.

(a) x2 + 6x + 9 = x x + (3x + 3x) + (3)(3)
= (x + 3)(x + 3)


(b) x2 – x – 2 = x x + (x – 2x) + (1)(-2)
= (x + 1)(x – 2)

(c) x2 – 5x + 6 = x x – 2x –3x + (-2)(-3)
= (x – 2)(x – 3)

(d) x2 + 4x – 12 = x x – 2x + 6x + (-2)(6)
= (x – 2)(x + 6)

2. Solve x2 − 4x + 4 = 0 by factorising.

Well, at school we learnt that: the “+” in front of the last term means that our factorised version
x2 − 4x + 4 will be either of the form (x + ?)(x + ?) or of the form (x − ?)(x −? ), and that the “−” in
front of the middle term tells us that our factorised version must be of the form (x − ?)(x − ?).

Experimenting with numbers that, when multiplied, give us 4, we soon find that our factorised version has
to be (x − 2)(x − 2), or (x − 2)2 if you prefer.
Our equation x2 − 4x + 4 = 0 may therefore be rewritten as (x − 2)2 = 0.
By Property 9, at least one of the factors on the left-hand side must be zero, and both factors are
(x - 2), so we get that x − 2 = 0 ie
that x = 2.

3. Complete the square to solve x2 – 4x = 12.
If x2 − 4x = 12

then x2 − 4x + 4 − 4 = 12 (by Property 8, since 4 − 4 = 0)
ie x2 − 4x + 4 = 12 + 4 (by Property 6 with k = 4)
ie (x − 2)2 = 16 (factorise)
ie x − 2 = 4 or x − 2 = −4 (taking square roots)
ie x = 6 or x = −2 (using Property 6 again, with k = 2).

4. Is 21 a prime number?
No. Refer to the definition of prime numbers on p 16. The
numbers 3 and 7 are factors of 21 (3 × 7 = 21).

5. What is the value of 5! (5 factorial)?
5! = 5 × 4 × 3 × 2 × 1 = 120.

2



Downloaded by Vincent master ()

, lOMoAR cPSD| 49511909




COS1501/XOS1501/MO001




Study unit 2

Activity 2-8

1. Define the words “even” and “odd” for positive integers.
Definitions:
- An integer n is even if n is a multiple of 2.
(We can say a positive integer n is even if n = 2k for some positive integer k. You can think of even positive
integers as numbers n of the form n = 2k, where k is some positive integer.) - An integer n is odd if n is not
even.
(Using the general form of an even positive integer, we can now say that n is odd if n = 2k + 1 for some
positive integer k. You can think of odd positive integers as numbers n of the form n = 2k + 1, where k is
some positive integer.)


2. Is it the case that m + (n k) = (m + n)(m + k) for all positive integers m, n and k?
Substitute a few values and see whether the idea is plausible.
Take m = 1, n = 2, and k = 3, then the left-hand side is m +
(n k) = 1 + (2 3) = 7 while the right-hand side becomes
(m + n)(m + k) = (1 + 2) (1 +3) = 3 4 = 12.
This is a counterexample to show that it is not the case that m + (n k) = (m + n) (m + k) for all positive
integers m, n and k.

3. Are there any even prime numbers besides 2?
No. Any even prime number other than 2 would have three factors: 1, because 1 is a factor of every
number; 2, because the number we are talking about is supposedly even; and the number itself, because
a number is always a factor of itself. However, primes cannot have so many factors, which means that 2
is the only even prime number.

4. If m and n are even positive integers, is m + n even?
If m and n are even positive integers, then each is a multiple of 2, in other words
m = 2k for some k Z+. and n = 2j for some j Z+. So
m + n = 2k + 2j
= 2(k + j)
which means m + n is also even.


5. If m and n are odd positive integers, is m n odd?
If m and n are odd positive integers, then both m and n can be written in the following general form:
m = 2k + 1 for some k in Z+.
and n = 2j + 1 for some j in Z+.
So m n= (2k + 1)(2j + 1)
= 4kj + 2k + 2j + 1
= 2(2kj + k + j) + 1
which means that m n is odd.

3



Downloaded by Vincent master
()

, lOMoAR cPSD| 49511909




An additional exercise:
If m and n are prime, is m + n and m − n prime?
No, not usually.
It can occasionally happen that m + n is also prime: take m = 3 and n = 2 then m + n = 5.
But for other values m + n may not be prime: take m = 3 and n = 7, for instance. 3 + 7 = 10, which is not a
prime number.

What about the difference between two prime numbers? The difference m − n will sometimes be prime
and sometimes not. E.g. 5 – 3 = 2, which is a prime number, but 23 – 3 = 20, which is not prime.


5 Learning unit 3 – Sets
Study Material



Study guide
You should cover study unit 3 in the study guide. Where the study unit refers to the CAI tutorial, you should
also work through the relevant part in the CAI tutorial. The theory, examples and exercises will help you
understand the concepts. If you have not received your CAI tutorial with your study material, please see
the Orientation section for downloading instructions.

Time allocated
You will need one week to master this learning unit.



Notes
Background
The previous study units covered different number systems that you will come across in the remaining
units in this study guide. In this study unit, you are introduced to set theory. Set theory includes topics
involving sets. It is essential for your computing studies that you understand set theory concepts because
sets can be considered as fundamental building blocks for mathematics.

Applications
You should be familiar with all the notations and properties of sets. We use set notation to define
different kinds of sets. In this study unit, we build new sets from old ones. The Venn diagrams in the next
study unit will visually help you to understand the definitions in this study unit.



Activities

Do the activities provided in study unit 3 to consolidate your knowledge of the work in this learning unit.

Study unit 3

Activity 3-3

1. In each of the following cases, describe the set more concisely, firstly, using list notation and then
using set-builder notation.


4



Downloaded by Vincent master ()

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying this summary from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller Jennifer2024. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy this summary for R48,72. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

69066 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy summaries for 15 years now

Start selling
R48,72  1x  sold
  • (0)
Add to cart
Added