ce r t i f i ca t e subject code: mtm315109 of education · ce r t i f i ca t e subject code:...

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T A S M A N I A N Mathematics - Methods

C E R T I F I C A T E Subject Code: MTM315109 O F E D U C A T I O N 2010 Assessment Report

2010 Assessment Report

The examination consisted of two parts in separate booklets. Calculators were not allowed to be used during the first part of the examination. After 80 minutes of working time, candidates were stopped and the Part 1 booklet was collected. During the second 100 minutes of working time candidates were allowed to use their calculators. This is the first year in which the candidates were able to use a CAS calculator in the examination. The examination scripts were marked by two teams of markers. A team of 6 markers in the North-West marked Part 1, the non-calculator section. Part 2 was marked by a team of 10 markers working in pairs, each marking a section relating to one of the five assessment criteria. This team operated in the South of the State. After the marking was completed the Assessment Panel met and considered the distribution of results and looked at all candidate results that were identified as borderline or an anomaly. PART 1 Section A – Functions Question 1 Generally well done. Most common errors were to reflect over the x-axis rather than the y-axis, or to reflect over the line x = -1. Translation handled quite well. Some candidates translated before reflecting. The outcome is the same in this case but this process will can lead to errors in problems where multiple, progressive transformations required. Question 2 Common error was in simplifying as equal to 10.

Many candidates multiplied 5 and 4 together before transforming Some candidates ignored the logs altogether or didn’t take solution to a final value. Question 3 (a) Generally very well done. (b) Most candidates either had trouble simplifying or did not bother to try.

Candidates seemed unable to make the link between 0.1 = 10-1. Smaller number of candidates simplified but forgot to find the absolute value of the answer.

Question 4 (a) Generally very well done.

Mathematics Methods 2

Subject Code: MTM315109


(b) Explanations were often confused or included incorrect information. Looking for terms such as ‘one to many’. Confusion as to what constitutes a function (one to one or many to one). Could refer to the ‘vertical line test’ as part of explanation. Some mentioned ‘horizontal line test’ on the inverse or ‘straight line test’.

(c) Candidates began well swapping x and y routine. Many candidates were confused with algebraic manipulation required and expanded . Many missed the second solution obtained when taking square root, even though equation(s) and answer(s) terms were included as prompts in the question. Notation was often confused. Very few candidates accurately identified the domain for each inverse but markers decided not to penalise as not specifically called for although a domain was implied in original sketch.

Section B – Circular Functions Question 5 (a) Common to get answers with candidates not recognising period differences

between sin, cos and tan. (b) Most candidates recognised need to consider when x = 0. Many could not evaluate

, showing a lack of understanding of symmetry properties and possibly too much reliance upon calculators. About half the candidates got this question correct. The challenge was recognizing that tan is negative in the 2nd quadrant and being able to correctly convert from a second quadrant angle to a first quadrant angle with a negative sign.

Question 6 Most candidates interpreted the table correctly and were able to write the correct equation. As in the previous question many candidates weren’t able to convert from a 2nd quadrant angle to a 1st quadrant in order to use the exact value from the exact value table on their information sheet. Many candidates used an incorrect value when substituting from the exact value table on their information sheet. Many candidates were unable to simplify the surd expression to give the correct answer. Question 7 (a) Few problems with scale labelling although many candidates ignored y scale. (b) Poorly handled. Few candidates demonstrated an understanding of why an asymptote

exists:




Greater depth of understanding required when introducing asymptotes

for , hyperbolic and truncus

Some confusion with and

A significant number of candidates found the correct turning points for the new graph but that was as far as they could proceed.

Question 8 It was pleasing that so many candidates were able to solve this trig equation correctly and achieve full marks. For those that weren’t able to complete the question correctly, most either were unable to find angles in the 2nd and 3rd quadrant in the first step or made errors in the algebraic/numerical manipulation needed to reach the answer. Again, many candidates struggled with basic angle and symmetry concepts, with many

defining

Those using the general formula often had problems dealing with simplification. Handling of fractions sometimes weak. Issues with restricted domain and often little evidence of checking outside these bounds. Setting out of working often poor. Section C – Differential Calculus Question 9 Most candidates identified the turning point is at x = 3. However a significant number stated the y value of the turning point as y = 0. Some candidates stated that the nature of the turning point was a point of inflection. For full marks, rising point of inflection (or positive), or a diagram indicating this was required. Question 10 If candidates divided through by x before differentiating this question became much easier. Most candidates used quotient rule and if this was completed correctly and simplified full marks were awarded. Common mistakes were in simplifying the result obtained from quotient rule and forgetting the derivative of a constant is zero.




Question 11 Most candidates were able to find the zeroes of the given expression. However many candidates could not identify the general shape of a quartic and often found the x value of each turning point, but not the y value. Without the y value of each turning point and without an understanding of the general shape, many candidates were unable to sketch the function. Question 12 Once a candidate was able to see that the slope of the tangent (4) must equal to the derivative of the function of the curve they were generally able to complete the rest of this question successfully. However if candidates were unable to make this connection, they generally made a very poor attempt at part (a). A number of candidates were still able to complete part (b) by using incorrect values obtained in part (a) and if working was correct, full marks could still be obtained for part (b). A common mistake across this whole question was the incorrect addition, subtraction and multiplication of fractions. Section D – Integral Calculus In general this section was done quite well. Question 13 Many candidates gained full marks for this question. Those who didn’t tended to overlook the negative introduced when was integrated or they didn’t put a constant into the answer. Question 14 Most candidates were able to shade the right hand section correctly, but, many shaded the area under or between and on the left hand side. It was necessary to have both sections correct for full marks ensuring that the small area under the x axis on the right hand side was also shaded clearly. Question 15 For full marks on this question candidates needed to integrate correctly, find c and state the full solution for . Common errors amongst those who did not gain full marks were: Thinking that and consequently getting the wrong value for c Not finding a value for c Not stating the solution for , as the question asked.




Question 16 Only a small percentage of candidates were able to get full marks for this question.

The most common error was to integrate to , or .

Many of these candidates were able to complete the entire question with no further errors, at times displaying a good understanding of log laws and algebra. These candidates were not penalised in part (b). If, however, the candidate did not display an ability to simplify their answers and/or made further errors during the question but obviously understood the process of finding the area they tended to gain half marks. A very common error in the manipulation of this question (when the above error had been made) was to say and/or to make the following error:

32

32 disappeared or

Some candidates did not know what to do with this question or integrated such that they did not use the natural logarithm, these candidates tended to struggle to gain many marks. Section E – Probability Question 17 (a) Generally well done with few errors. Some candidates attempted to link with by

multiplying (b) Reasonably well done. Many candidates unaware of MODE term (Foundation Syll.).

Some identified highest probability but didn’t link back to score. Question 18 Many candidates didn’t recognise as hypergeometric, even with ‘without replacement’ prompt. Significant number of candidates completed parts (a) and (b) as different distributions; i.e part a as (X~H) and part b as (X~Bi). (a) No major issues with set up of fraction. (b) Generally good but some candidates made substitution errors and subtraction errors.

Many had poor formation of numbers, making it difficult or impossible for markers to determine subscripts and superscripts in




Question 19 Number of candidates found lack of direction (‘interpret’), difficult. Many took note of 3 marks and attempted to write 3 different facts about distribution. Some candidates wrote the same thing in only slightly different ways. Some wrote imprecise or incorrect statements: ‘All women were between 130 and 190 cm.’ Question 20 Some candidates had difficulty answering in terms of p, instead allocating a value which was usually a

Many candidates did not know that , and to a lesser extent, Significant number did not understand what ‘at least 2 successes’ meant. PART 2 Part 2 of the examination was marked in Hobart by a team of 10 markers with a pair of markers marking the questions relating to each of the assessment criteria. General discussion of the paper, both parts 1 and 2, took place prior to the marking. While there was general acceptance of the standard and scope of part 1 there were a number of issues raised in relation to aspects of part 2. Some discussion and guidance was given by the group to the separate marking pairs where it was perceived that difficulties may be encountered by both the candidates and the marking teams. The major discussion points were as follows: • Concern was raised that this paper seemed to be a departure from the style of previous

maths Methods papers and at all similar to the Sample Paper that was circulated. Teachers expected to be informed ir there was such a change in style and emphasis.

• Specific concerns were raised in relation to questions that some teachers felt were not on the syllabus: o Question 7 - reciprocal functions. This was not a graph of an inverse not an

application of one of the transformations specifically listed in the course document

o Question 24 - despite the second last dot point in Functions and Graphs section the syllabus document:

recognition of the general form of possible models for data presented in graphical or tabular form,using polynomial, power, circular (trigonometric), exponential and logarithmic functions

• The set-out on the page of Question 31 lacked clarity for candidates. It should have been labelled as b) i) and ii) rather than b) and c). It was not clear that the value of k = 0.2 was expected to be used for part c)

• Teachers questioned why, in Question 38, candidates should have to express a probability as a ratio.




• Concern was expressed at the poor definition of the statistical variable X in Question 40 would impact on candidates ability to interpret and thus approach a solution to this question.

Section A – Functions In general, this section was poorly attempted. Qns 21 and 24 were often not correctly interpreted. Candidates should have been able to do these questions easily if they understood what was asked. Qn22 highlighted the shortcomings of candidates using their calculator to produce a graph without fully understanding the theory of the function being investigated. Question 21 Generally poorly done. Some candidates simply stated domain and range of (-∞, ∞) without relating the function to the real life situation. Many candidates stated t≥0, h(t)≥0 rather than visualising the function and working out the appropriate values. Others stated the width/length value of 0.6 as part of the domain. Question 22 a) For 3 marks candidates were expected to label two x-intercepts, the vertical asymptote

at x=0 and draw a graph with an appropriate shape. Common errors were to draw y = |ln(x)| or simply y = ln(x). Many candidates appeared to copy directly from the calculator which showed little understanding of the function(s) involved. They often drew in a horizontal asymptote and a y-intercept.

b) For one mark candidates were expected to correctly identify the number of points of intersection between the function provided and their own graph. Many stated that there were only 2 POI’s as their calculator did not draw the ‘tails’ of the function as it approached the vertical asymptote. Interestingly, the calculator only recognised three POI’s (instead of 4) when asked to ‘G-Solv’

Question 23 a) Well done. Some candidates, however, reversed the values of ‘h’ and ‘k’. Others used

‘h=3’ instead of ‘h=-3’. b) Generally well done. Part marks were given for y = 130e-0.175x. A lot of candidates failed

to find the value of ‘h’ or made mistakes here.




Question 24 This ‘Maths Applied type’ question was very poorly done and few candidates received the full 6 marks allocated. a) This should have been done in the statistics mode in the calculator. However, credit

was given to candidates who solved simultaneous equations with two sets of values from the table. One mark was allocated. Some candidates fitted an exponential model instead of the power model being asked for.

b) For three marks candidates were expected to identify the values for a and b and show appropriate working. Common errors included transforming the equation to make L the subject or saying that a = 6.28 (or 2π).

c) Inability to complete part a) or b) or errors in the values obtained made part c) difficult to complete successfully. There were many statements about reliability that simply repeated the question. For the full two marks candidates needed to state the percentage error and make a statement on the reliability of both a and b. No credit was given for considering the correlation coefficient (r) or coefficient of determination (r2).

Section B – Circular Functions Question 25 • This question could have been completed using a CAS calculator. • Many candidates attempted simplification of the fractions unsuccessfully. Question 26 • Few candidates gained full marks in this question. • Dilating by a factor of ½ in the x-axis also caused problems for many candidates. • Many candidates did not list the stages in the transformation (as was requested). Question 27 • Part a was generally answered well. • Once candidates moved beyond part a of the question, many forgot the domain of p,

namely p>0. This caused a great deal of confusion. • Candidates tried to express the conditions in words rather than using a simple

mathematical statement. Question 28 a) Many did not find A correctly, but simply used the amplitude of the function rather

than -60. Too many candidates failed to rewrite the equation as requested.




b) Many candidates did not use their function calculated in part a to draw the graph. Rather, they re-interpreted the original word description given at the beginning of the question. Shape, scale and axis-labelling generally were done poorly. Since a grid was given, a fairly accurate graph (including the equilibrium points) was expected. Only about 5% of candidates produced a reasonably accurate graph.

c) Most candidates used their calculators to find the intersection points between the function found in part a and y=95. The question was generally well done, but some candidates could not progress past the finding of the intersection points.

Section C – Differential Calculus In this section in the presentation of work was generally sloppy and candidates did not present answers in the requested format. Candidates had a poor understanding of proof. e.g Q31(a) and Q32(a). Question 29 a) Well done. b) Very rarely done correctly. Many candidates did not answer the question in terms of

rate of change of height. Few candidates used correct units and sign. Some candidates substituted 4 into the h’(t) equation which was not the question asked.

Question 30 Candidates who knew how to differentiate a composite function did well in both parts. Many candidates did not know how to approach this question correctly. Question 31

a) Many candidates correctly differentiated but did not show that .

b) Many candidates approached this the difficult way by calculating t and then substituting into the derivative. Often units were not given.

c) Some candidates used the original equation rather than the derivative. Reasonably well done by most candidates. Many candidates gave a decimal approximation which was not required.




Question 32 Generally answered well by most candidates. a) Proof was often sloppy. b) Well done. c) A lot of candidates embarked on a successful solution but did not supply answers to all

parts of the question. Justification of the minimum was often less than rigorous. Section D – Integral Calculus Question 33 Generally well done. Most candidates evaluated using the CAS calculator, some used algebraic methods. Those who used algebra often made errors which prevented them from getting full marks. Of those who used the CAS, a very few had their calculator set in degrees instead of radians and a couple were in gradians. These gave very convoluted answers which candidates should have recognised as being inappropriate. Some candidates gave answers as decimals instead of exact values. Question 34 Many candidates did not understand how to find the co-ordinates of A using simultaneous equations and therefore could not give exact answers for part (a). This meant they could not get an exact answer for part (b), but many tried to get around this by giving answers in ridiculously large fractions. It was also very obvious for those who rounded their answers to part (a) that many did not bother to evaluate their stated integral as it did not give the answer of 10.3354 although they wrote that it did. Question 35 Very poorly done. About half the candidates mis-read the question and interpreted the graph as if it were a graph of f(x) instead of f’(x). 1 mark was awarded if all 4 answers were correct for this major mis-interpretation error. Of those who read the question correctly, some had difficulty determining the significance of points B and D. Question 36 Many candidates did not seem to understand the relationship between velocity, acceleration and displacement as it relates to calculus. A large number used formulas such as v=u+at and s=ut+1/2at2 for (a) and (b) and very few of these used them correctly. Another common mistake for part (a) was to use the approximated point (1,40) to gain the linear equation




instead of the given point (5, 180). Relatively few candidates specifically stated the constant acceleration. In part (c), many candidates added their answer from (b) to their calculated integral giving the total distance travelled rather than the distance travelled under deceleration. A large number found the equation for velocity correctly but did not complete the question. Section E – Probability Question 37 This question was reasonably well done but a surprising number of candidates were not able to determine the correct proportion. Quite a few did not use the proportion to find the actual number of bags, and a significant number of candidates did not pay attention to the instruction ‘correct to the nearest whole number’. Question 38 The majority of candidates recognised that the problem involved a hypergoemetric distribution. A good number of candidates were able to solve the problem and set up the answer, as required, as both a proper fraction and a ratio. Reliance on the calculator functions to solve the problem often led to a decimal answer and many had difficulty trying to get a fraction and thus a ratio thereafter. Question 39 a) A surprisingly large number of candidates were not able to obtain the correct

probabilities. Some candidates used incorrect parameters and others obviously used the incorrect calculator function.

b) Most candidates did well here and interpreted their values from a) consistently and/or correctly. A variety of approaches were used, including z – scores.

Question 40 a) The interpretation of the information given was fairly poorly done. There was a general

inability to follow what the question was asking. Credit was given for any display of correct knowledge of a binomial distribution.

b) Apart from some confusion over 0.65 being the probability of at least one success after n trials, this question was reasonably well attempted. The better answers showed appropriate manipulation and presented the equation in simplified form.

c) Those candidates who did something appropriate in part b) followed through well to a solution. A small number failed to recognise that the value, n, was discrete, hence the answer was 35 not 34.466.




All correspondence should be addressed to:

Tasmanian Qualifications Authority PO Box 147, Sandy Bay 7006

Ph: (03) 6233 6364 Fax: (03) 6224 0175 Email: [email protected]

Internet: http://www.tqa.tas.gov.au

ce r t i f i ca t e subject code: mtm315109 of education · ce r t i f i ca t e subject code:...

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