Candidates' Performance Report

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LEMBAGA PEPERIKSAANKEMENTERIAN PELAJARAN MALAYSIA

ARAS 8, BLOK E11, KOMPLEKS KERAJAAN PARCEL E62604 PUTRAJAYA

PEPERIKSAAN SIJIL PELAJARAN MALAYSIATAHUN 2011

CANDIDATES’ PERFORMANCE REPORT

ADDITIONAL MATHEMATICS 23472 / 2

PREPARED BY:

MR MUSTAFFA BIN MAMAT

CHIEF EXAMINER ( T0100 )

5 JANUARY 2012

1. EXAMINATION : PEPERIKSAAN SIJIL PELAJARAN



MALAYSIA

2. YEAR : 2011

3. SUBJECT : ADDITIONAL MATHEMATICS 2

4. SUBJECT CODE : 3472 /2

5. CHIEF EXAMINER : MR MUSTAFFA BIN MAMAT

6. EXAMINER CODE : T0100

7. GENERAL PERFORMANCE

7.1 The performance of candidates based on 10% candidatessample.

MARKS RANGE NUMBER OF CANDIDATES0 – 10 1517

11 – 20 214921 – 30 240931 – 40 229141 – 50 2107

51 – 60 208861 – 70 183671 – 80 185381 – 90 162191 – 100 962JUMLAH 18833



7.2 From the table above,

7.21 The performance of candidates was good

For candidates in the group of low achievers, 19.47% got 0 to 20marks.

For candidates in the average achievement group, there was 56.98%who got 21 to 70 marks.

For candidates in the group of high achievers, 23.55% got 71 to 100marks.

There are 8.05% candidates who scored less than 10 marks while5.11% of the candidates scored more than 90 marks. Modal Class of marks is 21 to 30 which is 12.79% of the candidates.

7.3 The difference in performance from centres to centres is stillsignificant.

7.4 Generally, candidates performed well in topics such as SimultaneousEquation, Coordinate Geometry, Index Numbers, Linear Law, Circular Measure, Solution of Triangles, Motion On A Straight Line and Linear Programming.

8. PERFORMANCE ACCORDING TO CANDIDATES GROUP

ACHIEVEMENTGROUP

MARKSRANGE

NUMBER OFCANDIDATES

% TOTALCANDIDATES

%

LOW0-10 1517 8.05

3666 19.4711-20 2149 11.41

AVERAGE

21-30 2409 12.79

10731 56.9831-40 2291 12.1641-50 2107 11.1951-60 2088 11.0961-70 1836 9.75

HIGH71-80 1853 9.84

4436 23.5581-90 1621 8.6191-100 962 5.11

JUMLAH 18833 100 18833 100



8.1 Candidates in the High Achievement Group

8.11 Candidates in this group could answer the number of questionsas required. There are also candidates who answered morethan 4 questions in Section B and more than 2 questions inSection C.

8.12 Candidates understood the questions asked, used the formulaand the correct method, applied the concept correctly and couldsolve problems with a neat and precise working.

8.13 Candidates have good knowledge and application of inAdditional Mathematics based on response given by thecandidates.

8.14 Candidates seldom made careless mistakes during theprocess, could understand and mastered the concept in thetopics answered.

8.15 Most of the candidates were able to answer the questionsaccurately.

8.16 Solution were organized and systematic.

8.17 The final answer was correct and in the accuracy required andin accordance to the mark scheme.

8.2 Candidates in the Average Achievement Group

8.21 The performance of candidates is satisfactory. Candidates wereable to master many questions that tested on basic skill, buthad problems in the application questions.

8.22 Candidates could write the answers or solutions to some or

most of the questions, but continued to face problems with thequestions of high level of difficulty.

8.23 The answers reflected their understanding of the syllabuscontent, but they had not mastered the skills enough to applythe concepts in certain contexts.



8.24 Sometimes the calculation were irregular and inappropriate.Candidates made careless mistakes in calculation andalgebraic manipulation.

8.25 Candidates were unable to answer well in several questionssuch as 2, 3(c), 4(b), 5(c) and 6(b) from Section A and question8, 9, 10, 11, 12, 14 and 15(c)(ii).

8.26 The working shown by these candidates were notsystematically done and many crucial steps were missing or notclearly shown.

8.27 There were some candidates who were not confident and notsure of the working and this caused them to give alternatives

solutions for the same question or part of the question.

8.3 Candidates in the Low Achievement Group

8.31 Candidates did not master basic Mathematics concept, so theyhad problems involving algebra.

8.32 Candidates only have basic knowledge in the Mathematicsapplication, only abled to answer simple questions.

8.33 Majority could answer Question 1 (Simultaneous Equations), 7(Linear Law),13 (Index Numbers) and 15 (Linear Programming)only.

8.34 Most of the questions in Part A were not fully answered or noteven attempted at all.

8.35 Sometimes the symbols and formulas used in their working hadnothing to do with the question.

8.36 Some of the solutions given were meaningless and not at allconnected with the concept asked.

8.37 Candidates made mistakes when solving problems involvingfractions and the expansion of algebraic expressions.



9. INDIVIDUAL QUESTION ANALYSIS

9.1 QUESTION 1

PERCENTAGE OFCANDIDATES ANSWER 98.56MAXIMUM SCORE 5MODE 4MEAN 3.85INDEX OF DIFFICULTY 0.77LEVEL OF DIFFICULTY L

Majority of the candidates managed to answer this question well.

Strength

This question tests the skills of solving simultaneous equations. Candidatesmastered the methods of solving simultaneous equations involving linear and non-linear equations. Candidates could express one unknown in termsof the other unknown, and then eliminated one unknown. Candidates couldsolve quadratic equations using quadratic formula. They were then able tofind the values of the second unknown.

Example:



Weaknesses

Candidates used factorisation to solve the quadratic equation that could notbe factorised.

Example:

(i)

(ii)

(iii)



Few candidates were still weak in the algebraic manipulation.

Example :

Candidates were weak in algebraic expressions, especially in algebraicexpansion.

Example:

Candidates did not show the methods of solving quadratic equation. Theygave the correct answers without showing the quadratic formula. Theymight have used the scientific calculator to solve.

Example:

(i)

(ii)



Candidates gave the answers not correct to three decimal places as the

task of the question.

Example:

(i)

(ii)



(iii)

(iv)

(v)

Few Candidates were stillused the quadratics formulaincorrectly.



Examples:

(i)

(ii)

Few candidates interchanged the values of x and y .

Examples:

There were still candidates who did not find the values of the secondunknown.

Examples:



(i)

(ii)

Candidates were careless in their calculations.

Examples:



Most of the candidates in the high achievement group were careless for thevalue of y=-1.590 (3 d.p). Most of them give the answer y=-1.59 only.

Example:

QUESTION 2

PERCENTAGE OFCANDIDATES ANSWER

97.15

MAXIMUM SCORE 6MODE 2



MEAN 2.73INDEX OF DIFFICULTY 0.45LEVEL OF DIFFICULTY M

Majority of the candidates were weak in answering this question.

Strength

Candidates were able to use the laws of logarithm and the formula changeof base correctly to symplify the logaritmic expreesion. Candidates alsomanaged to solve the logaritmic equation by changing the equation inlogaritmic form to index form.

Example:

(i)

(ii)



Weaknesses

Candidates were weak in laws of logarithm.

Examples:

Candidates were weak in using the formula change of base of logarithm.

Examples:

Candidates could not differentiate between expression and equation.

Examples:

(i)

(ii)



Candidates unable to express equation in log form to index form.

Example:

QUESTION 3




91.92

MAXIMUM SCORE 7MODE 0MEAN 4.40INDEX OF DIFFICULTY 0.63LEVEL OF DIFFICULTY M

Majority candidates in the high achievement group excellently answeredthis question. The performance of candidates in the moderate group issatisfactory and the performance of candidates in the low achievementgroup is unsatisfactory. They were able to answer correctly for part (a) onlyand a few of them able to answer part (b).

Strength

Part (a)Majority candidates knew to use

y

y 5103

567=

to find the value of r .

Part (b)Candidates knew to use the formula

( ) 8471

15

=−

−

r r a

to find the values of a .

Part (c)Only a few candidates knew to use

100001 >−nar

and solve correctly to find

the smallest value of n.

Example:



Weaknesses

Part (a)Some candidates assumed that the value of r = y.

Examples:

Candidates could not solve

567

5103 y

y =

correctly.

Examples:

Part (b)

Some candidates used wrong formula to solve this part of question suchthat S n of AP or T n of GP.

Part (c)

Candidates unable to complete solving the problem due to lack of knowledge in logarithms.

Example:



Candidates used S n > 10000 instead of T n > 10000 .

Examples:

(i)

(ii)

Candidates used listing method but incomplete.

Example:



Candidates that used trial and improvement method to find the smallestvalue of n, did not show two values (before and after), to conclude n = 8.

Example:

(i)

(ii)



Candidates did not apply ‘log10’ on both side of the inequality.

Example:

Candidates used the wrong symbol to interpret the word ‘exceeds’ . Theyinterpret as

10000≤nT

or T n = 10000

Example:

Candidates made mistake in algebraic manipulation.

Example:

(i) (ii)



Some candidates mistook 567 as 2nd term, this resulting the value of the1stterm became 189

Example:

A lot of candidates don’t know how to solve the equation by using theconcept of logarithm.

Example:

QUESTION 4




93.36


Strength s

Most of the candidates know how to use the formulae of mean and

variance correctly.

Example:



Weaknesses

Majority done well in this questions especially 4(a) ,find the mean. Thecorrect way to answer 4(a) is adding all items that is: 2, y, 5, 2y+1, 10 &12, then divide by 6 because there are only 6 items. But there are somestudents divided by 7, because they think that 2y+1 is 2 different items.

Example:

Another mistake is interpret the formula wrongly when find variance. Itshould be square each item and sum up together. But the candidate willsum up together before they square it. This will lead to the wrong answer.



Most of the candidates do not remember about the impact of data changeon the standard deviation. If any data is added to a constant, the dispersionmeasure for the new data has not changed. Whereas, if all data ismultiplied by a constant c, the new standard deviation is c x the originalstandard deviation.

Example:



QUESTION 5


94.41

MAXIMUM SCORE 7

MODE 0MEAN 4.52INDEX OF DIFFICULTY 0.65LEVEL OF DIFFICULTY M

Strengths

Candidates manage to find the area of the polygon using area formula.Candidates can find the coordinate of a point and can give the equation of

the locus of a moving point.

Example:



Weaknesses



The majority of the candidates could answer part (b) of this questioncorrectly. As for part (c), most of them interchanged the ratio or mademistakes when expanding and simplifying the equation.

Not many candidates in the high achievement group could score full marksdue to part (c).

(a) Candidates knew how to find area of triangle by using the correctformula, but made careless mistakes.

Example:

(i)

Candidate did not know the meaning of modulus.



(ii)

(iii)

Candidate used the formula wrongly.

Example:

Candidate listed only two points.

some candidates used the formula wrongly by listing only 3 points.



Certain candidates didn’t know how to find area of triangle given threepoints.

Example:

(i)

(ii)

Most of the candidates know how to use the formula to find coordinates of

point C by substituting

into

However quite a number of candidates interchange the ratio of AC and CB.Many of the high achievers did not get full marks because of careless intheir working.



Example:

Candidates used the ratio formula wrongly

Examples:

(i)

ormula correctly for x coordinate, but wrongly for y coordinate.



(ii)

Candidates did not write the coordinate in formal notation.

Examples:

(i)

(ii)



Most candidates could use the distance formula correctly. Many mademistakes when equating condition of the point P, by either equate 2PA =PB or PA = 0 or PA = AB.

Example :

(i)

(ii)

(iii)



(iv)

Candidate used PA = 2 PB, but made algebra mistakes

Example:

(i)

Candidate made algebra mistake.



(ii)

QUESTION 6


92.47

MAXIMUM SCORE 7MODE 5MEAN 3.77INDEX OF DIFFICULTY 0.54

LEVEL OF DIFFICULTY M

Strengths

Most of the candidate will try to answer this question. Candidateunderstand the shape of sine graph and can sketch well the given functionwith correct maximum and minimum. Able to sketch 1½ cycles for 0≤ x ≤ 2π. Can sketch correctly the reflection of the given function.Candidate can sketch the reciprocal graph and state the number of

solutions correctly.

Example:



weaknesses

Quite a number of candidates could not get the correct amplitude becausethey assumed the optimum points are at/2 and 3/2.

Examples:

(i)

(ii)



Some of the candidates sketched cosine graph.

Example:

Some of the candidates sketch correct curve with wrong number of cycles.

Examples:



(i)

(ii)

Most of the candidates could get the correct equation for y =/x butsketch straight line instead of reciprocal curve.

Example:



QUESTION 7


99.46

MAXIMUM SCORE 10MODE 4



MEAN 5.28INDEX OF DIFFICULTY 0.53LEVEL OF DIFFICULTY M

This question is the most popular among the candidates. Most candidatesmanaged to answer very well and scored high marks for this question.

StrengthAll the candidates who answered this question could get the values of

y

1

easily.Almost all knew how to plot the points accurately from the table and drawthe graph of

y

1

against x according to the given scales. They also had no

problems drawing the line of best fit.Candidates managed to change the non-linear equation into linear form, Y = mx + c. They knew how to find the gradient of the line using the gradientformula and were also able to read the Y-intercept. They could then equatethe gradient to

n

p

and the Y - intercept to

n

1

. Only a few of the candidates

were able to answer section (b)(i) accurately. Many of the others just tookthe value of

y

1

as their answers.

Example:



Weaknesses



Candidates unable to express non-linear equation to linear equation.

Example:

Candidates were able to change to linear form but could not relate the gradient and the Y -intercept to the constants

n

p

and

n

1

respectively.

Examples:

(i)



(ii)

Quite a number of candidates contructed the line for the values of

y

1

but

they gave the value of y.

Example:

(i)

(ii)

A few of the candidatesplotted point (0.2 ,2.747) inaccurately.

Example:



Weak in algebraic manipulation.

Example:

Candidates find the value of n and of y by usingsubstitution and calculation.

Example:

Candidates substituted points into Y = mX + c to find the value of k or of c instead of reading the Y - intercept or finding the gradient from the graph.



Examples:

Once they could notwrite out the linear equation, the gradient they find willequal to p only.

Example:

QUESTION 8


55.20




This question is not popular among the candidates. Only candidates in thehigh achievement group managed to answer this question well.

StrengthCandidates could find the equation of the curve by integrating

dx x y ∫ = 2

and calculate the value of c by substituting(1, 3) into

c x y += 2

.

Question 10b, candidates managed to find the area of the shaded regionby using Area of rectangle −

∫ +

1

0

2

)2( dx x

or

∫ −

3

2

2

1

)2( dy y

.

Question 10c, candidates could find the volume generated when theshaded region is revolved through

°360

about the y -axis by integrating

dy y )2(3

2

∫ −π

.

Example:



Weaknesses

Candidates find the equation of tangent instead of the equation of thecurve.Examples:

(i)

(ii)

Candidates were weak in algebraic manipulation.



Examples:

Candidates unable to find the equation of the curve from the gradientfunction given.

Examples:

Candidates used the wrong limit for integration to find the area under thecurve.

Examples:

(i)



(ii)

(iii)

When candidates found the area of shaded region wrt x, most of themforgot use the area of rectangle minus the area under the curve.

Examples:

Candidates also used the wrong formula to find the volume wrt y.

Examples:



QUESTION 9

PERCENTAGE OFCANDIDATES ANSWER 84.04MAXIMUM SCORE 10MODE 10MEAN 6.16INDEX OF DIFFICULTY 0.62LEVEL OF DIFFICULTY M

Strength

Majority of the candidates were able to find the angle θ by usingTrigonometric ratio and knew how to use the formula for arc length anddetermine the perimeter of the sector. Almost all candidates able to use thethe formula for area of a sector.



Example:

Weaknesses

most of the candidates were able to use trigonometry ratio to find

θ

in

radians. However, some of the candidates mistook

ϑ

= 600.

Example:

some of the candidates mistook CD = 2 cm.



Example:

The candidates could find the area of sector APD / sector AOC correctlybut not the area of triangle COP. Some of the candidates were confusedthe area of the shaded region as sector of APD minus area of “sector” CAPas they mistook CP = AP.

Example:

QUESTION 10


90.73

MAXIMUM SCORE 10



MODE 7MEAN 6.00INDEX OF DIFFICULTY 0.60LEVEL OF DIFFICULTY M

Majority candidates were able to answer this question correctly except tosolve the collinear question in part (a). Most of them just skipped theproving of collinear.

Strength

Candidates were able to use the triangle law or polygon law to find

AB

and

PC

. Candidates were able to substitute u = 3i and v = 2i + 5j into

PC

.

Candidates also able to find the magnitude of vector

PC

.

Example:



Weaknesses

Candidates unable to find

PC

correctly.

Example:

Some candidates assumed that 8i + 5j = 13

Example:



Candidates unable to simplify the answer in finding the unit vector in thedirection of

PC

Example:

Candidates could not apply the ratio correctly that result in finding thevector wrongly.

Examples:

(i) (ii)

Candidates assumed that

u DC 8−=

and

v BC −=

Example s :

(i) (ii)



Candidates unable to write the vector in the form of column vector correctly.

Example:

(i) (ii)

Candidates unable to conclude the relation between the vectors

PR

and

PC

correctly.

Example:



Candidates understand the triangle of law but some of them failed todefined the vector.

Example s :



Candidates failed to understand the concept of unit vector.

Example:

.(i)

(ii)

(ii)

:



QUESTION 11

PERCENTAGE OF

CANDIDATES ANSWER 64.99MAXIMUM SCORE 10MODE 0MEAN 4.72INDEX OF DIFFICULTY 0.47LEVEL OF DIFFICULTY M

This is not a popular question. Candidates who tried this question weremostly from the high achievement group. The performance of thecandidates were only satisfactory. Only candidates in the high achievementgroup excellently answered this question. The performance of candidatesin the moderate group is unsatisfactory and candidates in the lowachievement group did not try this question at all.

Question 11(a)

Strength

Candidates in the high achievement group and average achievement groupwere able to use the correct formula for part (a)(i). Even the averagecandidates manage to get marks when using the Binomial Distributionformula and knew that p + q =1.

Example:



Weaknesses

Candidates used incorrect values of p and q and p+ q ≠ .1

Example:

Most of them did not use the calculator correctly when calculating theprobability and the answer given is round off to 3 decimal places.

Example:

(i)

(ii)

(iii)

(iv)



Candidates made calculation mistakes even though they used calculator.

Example:

Candidates used incomplete combination of P(X=10) + P(X=9) +P(X=8).

Example:

Average candidates managed to write P(X=0) + P(X=1) +P(X=3) but used p= 0.8 and q=0.2.

Example:



(i)

(ii)

(ii)

Candidates used 1 - P(X=10) + P(X=9) +P(X=8) with incorrect pairs of pand q but p + q =1.

Example:

(i)



(ii)

Some of the candidates used nPr instead of nCr .

Example:

Question 11(b)(i) and 11(b)(ii)

Only candidates in the high achievement group excellently answered thispart of the question. The candidates in the moderate group can onlyanswer part (b)(i) and the candidates in the low achievement group did nottry this question at all.



Strength

Most candidates who answered this question were able to use

in part (b)(i) and p= in part (b)(ii).

Examples:

Candidates did not find the probability P (z >1.2) in part (b)(i) after gettingthe value of z. Some make mistakes when reading the value from the table

or calculator.

Examples:



In part (b)(ii) candidates could not find the correct value of z for

. Those who could find the value of z used the positive value of z to find t.

Example:



In part 11 (b)(i) candidates found but did notproceed to find P (z >1.2). In (b)(ii) candidates used z = 1.6 instead of finding z from the table.

Example:



Candidates make mistakes when reading the table to find P(z >1.2) or zwhen Q (z) =1.6.

Example:



QUESTION 12


32.72


This question is not the main candidates’ choice for Section C. Overall, thisquestion is answerd by the high achievement group candidates and most of them performed well.

Strength

Almost all the candidates knew to find the value of time when itsacceleration is zero by substituting a = 0. Most candidates able to find v by using integration ,

∫ = dt av

. Candidates knew to find the maximum

velocity by substituting value of t from a = 0. Candidates knew to find thetime when the particle stops instantaneously by equating v = 0 .

Candidates able to find s by using integration ,

∫ = dt v s

.

Candidates knew how to find the total distance travelled in the first 6seconds correctlly

Example:



Weaknesses



Some candidates did not know how to factorise the quadratic equations.

Example:

Candidates did not know how to find the total distance travelled in the first 6seconds correctlly.

Examples:

Some candidates made error in calculation.

Example:

QUESTION 13




90.57

MAXIMUM SCORE 10MODE 10MEAN 7.56INDEX OF DIFFICULTY 0.76LEVEL OF DIFFICULTY L

A popular question among the candidates. Most of the candidates cananswer this question excellently. Even the candidates in the average andsome of the lower achievement group are able to solve the problemscompletely.The overall performance of this question is good. This is a popular question

and candidates who answer this question managed to get at least 4 marksin part (a).

Strength

Candidates were able to find the price index and prices for the specificitems as well as the composite index. The formulae has been appliedcorrectly without any major mistakes especially for candidates in the highand average achivement group.

Example:



Weaknesses

A few candidates did not apply the formula correctly to get the answer for 13(a)

Example:



candidates did not know that the percentage expenditure is actually theweight, but they try to calculate a new set of weightages by consideringthe prices of year 2005 and Price index in the year 2007 based on theyear 2005.

Examples:



Some of the candidates did wrongly in part (b). They did not divide thesum of price index plus weightage with the sum of weightage.

Example 1

Candidates made calculation mistakes especially in part (b) and part (c).

Example 1



Candidates did not know a price index number is to measure and comparechanges in the prices of goods for the current period with that of the baseperiod.

Examples:



Candidates mistakes 15% as 50% and used the composite index of 2009based on 2007 150 innstead of 115.

Example:

Candidates used instead of

Candidates did further calculation after getting required the compositeindex part (d).

Example :



QUESTION 14

PERCENTAGE OFCANDIDATES ANSWER 65.51MAXIMUM SCORE 10MODE 0MEAN 6.29INDEX OF DIFFICULTY 0.63LEVEL OF DIFFICULTY M

This question is a popular choice for Section C. Many candidates chose toanswer this question and scored good marks.

Strength Candidates were able to use the sine rule, cosine rule and the formula for the area of triangle. Candidates were able to calculate correctly and gavethe answers accurately and within the range of the answers allowed by themarking scheme.

Example:



Weaknesses

Candidates used the wrong formula although the formula is given in thequestion paper.

Example:

Candidates were careless in their calculation.



Examples:

Candidates used the formula wrongly where the lengths and the angles didnot correspond or candidates use the wrong formula.

Examples:

Candidates round off too early will affect the answer.

Examples:

Candidates using the wrong sides or the wrong angles which they apply inthe Sine Rule and Cosine Rule.



Examples:

QUESTION 15




49.93


Most of the candidates did not prefer to answer this question. However,candidates in the high achievement group were able to solve the problemcompletely. The candidates in the average performance group can only getmark for part (a) and only some of them draw the graph correctly. Thecandidates of low performance group are not able to answer this question.

Strength

Candidates in the High Achievement Group could produce a very neatworking demonstrating their understanding of the topic.

Example:

Weaknesses

Average candidates were able to form the three inequalities and draw thecorrect straight line.



Examples:

Some of the average candidates could not form the required inequalities.

Examples:



In part (c), the maximum expenditure was found using the followingexpression:

K = 5(coordinate x) + 7(coordinate y).Or the point used was outside the region.

Examples:



10. CANDIDATES PERFORMANCE COMPARED TO PREVIOUS YEAR

Generally, the candidates’ performance in this examination is better thanthe previous year’s. Not many candidates scored 0 out of all the scriptsmarked. In addition, there are 23.55% candidates who scored more than 70marks. Half of them scored 21 – 70.

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