2msu-ar-2005

8/19/2019 2msu-ar-2005

http://slidepdf.com/reader/full/2msu-ar-2005 1/7

Mathematical Studies

2 0 0 5 A S S E S S M E N T R E P O R T

Mathematics Learning Area

8/19/2019 2msu-ar-2005


1

MATHEMATICAL STUDIES

2005 ASSESSMENT REPORT

GENERAL COMMENTS

The distribution of results suggest this examination was a fair, well balanced, and well graded test

of the students, knowledge and ability to apply their knowledge as described by the learning

outcomes and content of this subject (see the curriculum statement). There were many parts of the

paper where students consistently performed well. These areas covered a broad range of the

course content. The examination provided enough questions that differentiated between student

abilities, but at the same time rewarded those students who had worked hard to gain the basics

required to pass this course.

It is pleasing to see that many students gained a good number of marks in the last few questions

of the examination. It is important that students know that the last few questions do not contain

only ‘challenging’ tasks. These questions are written so that students who have worked

adequately throughout the course can achieve well in them.

Fully correct solutions were presented by students for each of the sixteen questions with the last

two questions having the lowest number of responses that scored full marks.

As has been the case in the past two years many students were well rehearsed in the routine

procedures included in this subject. However, the examination again included questions that

required students to (among other things):

1. ‘use mathematics as a tool to analyse data and other information elicited from the study

of situations’ (for example Question 9);

2. look for reasons that explain results (for example, Questions 3 (c), 4 (b), 7 (b), 12 (a) (iii),

and 12 (b));

3. communicate mathematically.

It is believed that some improvement has been seen in these areas. However, much room for

further improvement remains.

It is critically important that all students understand that marks are awarded for documented steps

of logic that support their final answer. Answers alone will generally be awarded no marks.

It is worth noting that many students may not be aware of when the electronic technology they are

using returns an exact result as opposed to a decimal approximation (for example in Questions 5

and 9). Students should know how to ascertain when a result is an approximation. They should

also realise that, when a decimal approximation is returned, algebraic methods may be one way to

determine the associated exact value. Students must be aware that electronic technology is a

suitable tool to gain an answer in some instances and not in others, and that they will be expected

to apply ‘by head algebraic processing’ and then commit this to paper in some parts of some

questions (for example Questions 2 (b), 6 (b) (ii), and 15 (b)). It was not the intent of this

examination to make the algebra parts difficult or tricky, but rather to have students using algebra

in appropriate situations (see the comments made in Question 12 for more details).

8/19/2019 2msu-ar-2005


2

ASSESSMENT COMPONENT 1: EXAMINATION

Question 1

Parts (a) and (b) were generally done well. In part (c), integrating ln 2 x caused many students

difficulty.

Question 2

This question was also generally well done. Some students could not employ the algorithm to find

the determinant of a three by three matrix correctly in part (b). It was disappointing to note that a

considerable number of students could not use their electronic technology to find the inverse of

the matrix they required in part (d).

Question 3

Overall students performed quite well in this question. Most common errors occurred in part (b)

(ii), where students computed 1−P(Y ≤ 6) rather than 1−P(Y ≤ 5) and part (c) where it was

disappointing that many students, even though they had completed part (b) correctly, had little

idea how to deal with what was asked.

Question 4

Part (a) of this question was well done. While there were a good number of excellent responses to

part (b), far too many students were unable to provide a logical reason for their choice. A

common response was that f ( x) was a quadratic and so Graph C represented g ( x) as it was a cubic.

It was important that students realised that the form of each function was not given and that

analysis of the graphs features was required.

Successful reasoning could be developed by considering how the area under the graph of f ( x)

accumulated for increasing x or by considering the changing value of the slope of the tangents to

each of the possibilities given for g ( x). Understanding of this type can be enhanced in the teaching

and learning phase with the help of electronic technology in the form of animations.

Question 5

Students generally provided very good solutions to this question. When using electronic

technology for part (b), some students failed to realise that an approximation for 0 was returned

for the value of d and therefore wrote d in their script as a very small number written in standard

form. Such answers were penalised.

Question 6

Many students wrote very good solutions to this question as well. It was, however, disappointing

to observe the lack of care and detail students took in ensuring they earned the 3 marks for part(a). Students must do what a question requests of them, in this case ‘clearly indicate’ two things

on the graph; some did not do this. Graphs that illustrated incorrect shape (mostly due to lack ofcare) were penalised. A disappointing number of students reported that the x – axis has equation x

= 0.

In part (b) (i), what should have been a straight-forward integration caused problems for more

than a few students. In such simple cases, students might ask, ‘What do I have to differentiate to

get …’ rather than attempting to employ an algorithm.

8/19/2019 2msu-ar-2005


3

The application of the rules of logarithms was not done well by quite a few students in part (b)

(ii). Too many students who were able to apply the rules well stopped short of providing the

answer in the required form of ln a.

Question 7

While the majority of students were able to correctly compute the confidence interval sought in

part (a), many were unable to demonstrate they understood the significance of it by answering

part (b) correctly. During the teaching and learning phase, students need to be exposed to many

situations where they have to compute a confidence interval and use it to make a decision, rather

than only computing the interval.

A disturbing number of students were not able to answer part (c). An appropriate formula to have

applied is provided on the formula sheet. Some students who did use the correct formula did not

use the value of 0.25 for p*. Some students attempted to employ the idea of standard error (SE)

but unfortunately seemed not to understand the relationship between SE and the width of a

confidence interval.

Question 8

It was somewhat surprising that for many students, the idea of a chord approximating the slope of

a tangent at a point on a curve seemed unfamiliar. This is a fundamental idea that students should

be well aware of.

The large majority of students had rehearsed the routine process of first principles differentiation,

required in part (b), very well. Those students who substituted 2 for x at the beginning of the

process had a far easier route to success. Common errors included:

• starting with3 x + h

x + h +1 instead of

3( x + h)

x + h +1;

• algebraic errors such as sign errors made when expanding and incorrectly dealing with

fractions.

Question 9

The computational side of this question was handled well by the majority of students. It is worth

noting, though, that some students, who chose to compute mentally, simply squared each of the

elements in the matrices. Of those who used electronic technology to compute the matrices, some

did not pay enough attention to the electronic output and mistook -0.33 as an approximation to

−1

3 rather than −

33

100.

Students who had given answers to part (a) in fractional form were best placed to notice the

patterns that led to the correct conjecture in part (b) (i). Hence, quite a few students were only

able to correctly offer the element with value zero ( a1,2).

While a good number of students provided excellent responses to part (b) (ii), many clearly didnot understand what is required to prove a conjecture. More than a few, incorrectly considered it

appropriate to generate further arithmetic examples.

Question 10

This was a routine question in which many students scored well. There was, however, a small

group of students for whom the concepts in this question seemed to be completely unfamiliar.

8/19/2019 2msu-ar-2005


4

Common errors included:

• use of incorrect notation in part (b) (i); most notably µ = x . Students must indicate to the

marker that they are aware of the fact that the mean of the sample means is equal to the

mean of the population, i.e., µ (Y500) = µ ;

• the misconception that provided n is greater than 30 then the distribution of Yn will be

approximately normal, as too often reported as a response to part (b) (iv). Studentsneeded to observe that the distribution of Y200 was approximately normal and so

increasing the sample size to 500 should result in a distribution that more closely

approximates a normal distribution.

Relatively few students answered part (c) of this question well. Students needed to look back over

the calculations they had made and determine how they related to part (c). They needed to

appreciate that the probability of a sample with the characteristics of Shane’s collection of claims

being selected randomly from the population of claims was very small. Hence they could

conclude that either the sample was not a random selection or, if it was, its selection was a rare

event.

Question 11Many students produced excellent solutions to this question. The skill of implicit differentiation

was well implemented.

In part (c) it was obvious that some students were not aware how to find the solutions to a cubic

equation. Some students did not reduce the resulting equation to the form ax3+ bx

2+cx + d = 0.

While not necessary, had they done so, it may have been obvious that its solutions could have

been found using electronic technology.

Question 12

Most students handled parts (a) (i) and (ii) with little trouble. Marks were lost in part (a) (i) when

students did not make it clear where 16

x 2 originated from and in part (a) (ii) if they omitted to find

the y coordinate.

In part (a) (iii) many students were not able to supply a complete argument to show what was

requested. Many failed to state that the condition for two lines to be perpendicular is that the

product of the line’s slopes is -1 (or similar).

Part (b) was not handled well. The algebraic processes involved were not difficult but many

students struggled, some not even getting started. Students need to experience situations like this

question in the teaching and learning phase. First exploring a situation for a given specific case

(where electronic technology can be employed unless exact values are required) and then

exploring the general case (where algebraic methods will be required). For those students who didmade a good attempt, the common omission was the verification that the minimum value

occurred when x = b . Either a sign diagram or use of the second derivative was required.

Question 13

This was the first year that a question involving a hypothesis test had been included. It was clear

that many students had been well taught and that the support materials made available by

SSABSA had been well utilised.

8/19/2019 2msu-ar-2005


5

However, like in Question 10 there appears to be a group of students for whom the concepts in

this question seemed to be completely unfamiliar.

Common errors included:

• incorrect statement of the null hypothesis, e.g., H o = 1

2;

• failure to exclude customer 10’s result (0);• students attempting to use a distribution other than the binomial distribution;

• failure to compute the probability associated with a two tailed test.

In drawing a conclusion from a Sign Test (part (e)), students need to be particular in the language

they use. They were required to state something equivalent to ‘insufficient evidence was found to

suggest that the frequency of rounding down was different to the frequency of rounding up.’ It is

important that, during the teaching and learning phase, students come to understand the difference

between a statement like that given above and one that (incorrectly) concludes that ‘the frequency

of rounding down is the same as the frequency of rounding up’.

Question 14

Many students answered parts (a) and (b) very well. In part (b) some chose to use electronic

technology while others preferred to use an algebraic method.

The explanation required in part (c) was not handled well. Students should be aware that

explanations in the form of mathematical statements like f ( x)dx =

−4

0

∫ − f ( x)dx

0

m

∫ are acceptable.

The most successful solutions involved a discussion of the values of the integral, not areas.

Most students knew what was required in part (e). However, many students made both algebraic

and arithmetic errors, resulting in an incorrect cubic equation. As in Question 11 (c), manystudents failed to use electronic technology to find the solutions to the cubic equation that

resulted. Of the students who did, it was pleasing to see that many rejected the imaginary

solutions as possible answers.

Question 15

The differentiation required in part (a) was handled well by most students.

Part (b) requested an exact value and hence the use of electronic technology was of use for

checking purposes only. Many students were aware of what was required, but many of these

made errors in attempting the process.

A disappointing number of students failed to appreciate that, since R(t ) was a rate, a definite

integral of this function would give the amount of the substance over some time period. Of thosewho did, too many did not round their answer as requested in part (c) (ii).

In part (d), few students seemed to realise they could use electronic technology to graph R'(t )

(without entering its equation) and then proceed to determine that t = 34.5. Even fewer students

provided a correct interpretation for this, which is based on an understanding of the point of

inflection.

8/19/2019 2msu-ar-2005


6

Question 16

Many students provided good answers for parts (a) and (b).

Part (d) (i) was not handled well. Some students worked only with XB, failing to equate it to X .

Some misinterpreted the question and did not realise they had to start with X = XB and show the

working to reach the given set of equations.

In part (d) (ii) about half the students were able to provide a correct interpretation of

a + b + c = 1.

In attempting part (d) (iii) very few students realised that a + b + c = 1 could replace one of the

equations from the system introduced earlier and then a unique solution could be found.

A very small proportion of the cohort were able to provide a suitable interpretation of the

situation when X = XB. Students were required to ‘read’ this statement and appreciate that when

multiplying X by B, X did not change and hence the proportion of trees in each age group was not

changing.

Chief Examiner

Mathematical Studies

Documents

2msu-ar-2005