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QMI1500 Exam Pack (Question papers and Memos) Supplementary Exam 2023

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This document contains simplified and user friendly solutions to past examination question papers to help you nail QMI1500 supplementary exam once and for all. So avoid failing and get yourself this document.

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  • January 12, 2023
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  • 2022/2023
  • Exam (elaborations)
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Question 1: Example
Consider the following sequence:
𝒙𝟏 𝒚𝟏 + 𝒙𝟐 𝒚𝟐 + 𝒙𝟑 𝒚𝟑 + 𝒙𝟒 𝒚𝟒 + 𝒙𝟓 𝒚𝟓
Write it in summation notation.
Answer
Tips:
• First notice that the x and y subscripts change from 1 to 5 in the sequence.
• Also notice that the subscript is the only thing that changes in the sequence while all
else remains the same
𝒏
• Writing something in summation notation means using the summation symbol 𝒊+𝟏 𝒙𝒊 or
𝒏 𝒏
𝒊+𝟎 𝒙𝒊 or 𝒊+-𝟏 𝒙𝒊 depending on what changing number/subscript your given sequence
starts from.



ü So we already know that our “i” starts at 1 and ends at 5. Meaning our symbol will look
something like this:
𝟓
𝒊+𝟏 𝒙𝒊

ü From the summation symbol we know that the “i” represents the subscripts that are
changing meaning that the general term of our sequence is 𝒙𝒊 𝒚𝒊
ü Putting these two together, we finally get:
𝟓

𝒙𝒊 𝒚𝒊 𝒊
𝒊+𝟏


Question 2: Example
The number of girls studying at UNISA in second semester has increased in the ratio 3:2. If
the number of girls at the start of first semester were only 20, how many of girls do we have
now?
Answer
Tips:
• Another way of writing a ratio is as a fraction.

𝟓
e.g. 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐

• Sometimes they can give the ratio as words:

, 𝟓
e.g. the ratio of 5 buttons to 2 shirts → 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐

ü Convert your ratio to a fraction:
𝟑
𝟑∶𝟐→ = 𝟏. 𝟓
𝟐
ü The question says the number has increased by this ratio. When something increases by
something, we must multiply the two:
So we get: 𝟐𝟎×𝟏. 𝟓 = 𝟑𝟎

The number of girls in second semester has now increased to 30.

Question 3: Example
How many ways are there to select a committee to develop a discrete mathematics course at a
school if the committee is to consist of 3 faculty members from the mathematics department
and 4 from the computer science(CS) department, if there are 9 faculty members of the math
department and 11 of the CS department?
Answer
Tips:
• First consider if the order is important, if it is then we know to use a permutation and if
it is not then we know to use a combination.
• Consider the number of ways to choose the faculty members from the mathematics
department, and then consider the number of ways to choose the faculty members from
the CS department



ü We can pick the 3 faculty members out of the total 9 from the maths department as
follows:
𝟗!
𝒙𝟗 𝑪𝟑 = = 𝟖𝟒
𝟗 − 𝟑 !×𝟑!
ü We can pick the 4 faculty members out of the total 11 from the CS department as
follows:
𝟏𝟏!
𝒙𝟏𝟏 𝑪𝟒 = = 𝟑𝟑𝟎
𝟏𝟏 − 𝟒 !×𝟒!

Therefore, there are:
𝟗! 𝟏𝟏!
𝒙𝟗 𝑪𝟑 ×𝒙𝟏𝟏 𝑪𝟒 = × = 𝟖𝟒×𝟑𝟑𝟎 = 𝟐𝟕𝟕𝟐𝟎 𝒘𝒂𝒚𝒔
𝟗 − 𝟑 !×𝟑! 𝟏𝟏 − 𝟒 !×𝟒!

,Question 4: Example
On each delivery, a postman delivers letters to 3 of the 12 suburbs in his territory. In how
many different ways can he schedule his route?
Answer
Tips:
• The order in which he delivers to the suburbs is not mentioned, meaning that this is a
combination


ü The number of ways in which he can schedule his route is:

𝟏𝟐! 𝟏𝟐!
𝒙𝟏𝟐 𝑪𝟑 = × = 𝟐𝟐𝟎 𝒘𝒂𝒚𝒔
𝟏𝟐 − 𝟑 !×𝟑! 𝟗!×𝟑!

Question 5 & 6: Examples
Please see the hints posted on the main QMI1500 site posted by the lecturer before
proceeding to the examples below.
Example 1
Find the volume of the triangular prism shown in the diagram:




Solution:
ü 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂 𝒑𝒓𝒊𝒔𝒎 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ×𝒉𝒆𝒊𝒈𝒉𝒕
ü The base is a triangle, so we must find the area of the triangle which is:
𝟏 𝟏
𝑨= ×𝒃𝒂𝒔𝒆×𝒉𝒆𝒊𝒈𝒉𝒕 = ×𝟗×𝟏𝟐 = 𝟓𝟒𝒄𝒎𝟐
𝟐 𝟐

Hence,
𝑽 = 𝑨×𝒉 = 𝟓𝟒×𝟏𝟖 = 𝟗𝟕𝟐𝒄𝒎𝟑

, Example 2
Consider the following two similar triangles:




If 𝐴𝐵 = 6, 𝐷𝐸 = 3 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐷𝐸𝐹 = 6𝑐𝑚_ . What is the area of ABC
Solution:
ü The ratio of the given sides is:

𝑨𝑩 𝟔
= =𝟐
𝑫𝑬 𝟑
ü If you follow the hints posted by the lecturer you will find out that if ratio of the sides of
two similar triangles is 𝒙 then the ratio of the areas of the triangles is 𝒙𝟐

Therefore,
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑨𝑩𝑪 = 𝟐𝟐 ×𝑨𝒓𝒆𝒂 𝒐𝒇 𝑫𝑬𝑭
= 𝟒×𝟔 = 𝟐𝟒𝒄𝒎𝟐

Question 7: Example
There are 3 grade 12 classes at John’s high school, class one has 50 students, class two has 60
𝟏
students and class 3 has 80 students. An examiner decides to take a sample of of the entire
𝟐

grade 12 to test the new syllabus. All classes must be represented in the sample. Show how
many students will be sampled in each class
Answer
ü 𝑻𝒐𝒕𝒂𝒍 𝒈𝒓𝒂𝒅𝒆 𝟏𝟐 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 = 𝟓𝟎 + 𝟔𝟎 + 𝟖𝟎 = 𝟏𝟗𝟎
ü 𝑷𝒓𝒐𝒑𝒐𝒓𝒕𝒊𝒐𝒏𝒂𝒍 𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆 = 𝟏𝟗𝟎×𝟎. 𝟓 = 𝟗𝟓
ü Representation of each class will be:
𝟓𝟎
𝑪𝒍𝒂𝒔𝒔 𝟏 = ×𝟗𝟓 = 𝟐𝟓
𝟏𝟗𝟎

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