MINISTRY OF EDUCATION

FIJI SEVENTH FORM EXAMINATION

2009

MATHEMATICS

COPYRIGHT: MINISTRY OF EDUCATION, REPUBLIC OF THE FIJI ISLANDS

2.

MINISTRY OF EDUCATION

FIJI SEVENTH FORM EXAMINATION 2009

MATHEMATICS

EXAMINER’S REPORT

This year 4511 candidates registered for the FSFE Mathematics Examination. Based on the raw

marks of a sample of 367 scripts, the mean score was 32.7 and the percent pass rate was 22%.

More than half of the candidates, 52%, scored fewer than 30 marks, 33% scored fewer than 20

marks and 16% scored fewer than 10 marks. It is disappointing to see such a high percentage of

candidates scoring fewer than 30 marks on an examination that was straightforward and fair. It

seems as though there may be quite a few candidates who are taking mathematics because it is

required in their school and not because they want to or need to.

This report is divided into five sections:

(1) Common Errors [page 2]

(2) Statistics [page 11]

(3) Comments and Recommendations for Selected Topics [page 13]

(4) Conclusion and General Recommendations [page 16]

(5) Marking Scheme.

Section (1), Common Errors, lists the errors that candidates made on the exam. This section is

helpful for both teachers and students as it shows how candidates approached a question and

gives examples of the types of mistakes that were made on the exam.

Section (2), Statistics, shows how this year’s candidates performed on each of the questions.

Section (3), Comments and Recommendation for Selected Topics, gives advice from the markers

to teachers and students on how to avoid making mistakes on specific questions on future exams.

Section (4), Conclusion and General Recommendations, gives general recommendations to

teachers and students on how to prepare for future exams.

Section (5), Marking Scheme, is the actual marking scheme used by the markers this year. It

contains all the different methods for answering each question and the allocation of marks for the

working.

3.

1. Common Errors

This section lists the common errors that students have made for each question in the paper.

Section A

Question 1, Mathematical Induction

Used incorrect statements, especially “Prove that n = 1”, “Assume that n=k”

Omitted brackets, k + 1(2k + 1) instead of (k + 1)(2k + 1)

Evaluated terms incorrectly, 4k – 2 instead of 4k + 1 and 2k instead of 2k + 1

In steps two (n = k) and three (n = k + 1), didn’t write out the series “1 + 5 + 9 + …”

before the term.

Question 2, Binomial Theorem

Expanded the entire binomial, but didn’t know which term was independent of x

Left out the minus sign in 3

2

x

Left out the 4 in 33

4

x or moved the 4 to the numerator

Raised the 3 to the power of 3 in 33

4

x

Stopped after finding the value of r

Question 3 (a), Limits

Differentiated incorrectly (using L’Hopital)

Substituted 0 directly, obtained 0/0 and gave the answer as 0

Cancelled the x2 in the numerator and the denominator

Question 3 (b), Limits

Used lim incorrectly: either omitted it or continued to use it after substitution

Divided all terms by x2

Substituted ½ directly, obtained 0/0 and gave the answer as 0

Factorized incorrectly

Differentiated incorrectly

4.

Question 4, Partial Fractions

Wrote the first step incorrectly, A(x2 + 1) + Bx + C(x)

Used equating coefficients incorrectly

Set up or solved the simultaneous equations incorrectly

Question 5, Trigonometry

Had no idea how to approach this question

Changed cosec 2A to 1

2sin A or ½ sin A or 22cos 1A

Substituted 3

5 and

4

5 as

1

3 42sin cos

5 5

Found incorrect value of sin A

Did not convert sin 2A to 2 sin A cos A

Found the value of angle A and thought that was the correct answer

Found the value of angle A and tried to calculate the exact answer

Question 6, Sigma Evaluation

Tried to use one of the formulas in the Eton table book

Combined the two fractions and then tried to use the Eton formula

Used plus signs instead of minus signs between terms

Used commas instead of plus signs between pairs of terms

When substituting, either stopped before 1

41 or went beyond

1

41

Question 7, Composite Functions

Wrote the domain in part (a) as x R or x=1

Wrote the range in part (b) as y R or y=0

Wrote the range in part (c) as y>3, y=3 or 3y

Wrote ranges in terms of x, instead of y

Substituted in part (d) without using brackets

5.

Question 8, Complex Numbers

Switched answers between parts (a), (b), (c) and (d)

Answered 2i for part (b)

Answered 2 for part (b)

Answered 3 2 for part (c)

Answered 3 2i for part (c)

Answered 7 for part (d)

Thought that argument meant Argand diagram and sketched one for part (e)

Answered 25.2o or 39

o for part (e)

Question 9, Differentiation

Had no idea how to approach this question

Wrote sindy

tdx

and 22 tdye

dx

Tried to use either the product or quotient rules

Wrote the chain rule incorrectly

Question 10, Probability and Statistics

Used unrelated formulas, such as x x

z

Tried to find the z score or the sample size

Divided by 1000 to find the mean

Didn’t find the square root of npq for standard deviation

Used an incorrect value for q

Section B

Question 1(a), Polynomial Functions

Did not write the answer as an equation, y = etc.

Did not include powers for (x + 2) and (x – 1)

Introduced extra terms and constants

6.

Question 1(b), Rational Functions

Wrote asymptotes as VA = and HA =

Did not give the horizontal asymptote as an equation, y = 1

Gave the horizontal asymptote as y = 0

Curves did not approach the asymptotes

Extra x intercepts were shown on the graph

Solid lines were drawn for asymptotes instead of dotted/dashed lines

Question 1(c), Piecewise Functions

Did not draw an accurate parabola with correct starting point, turning point and ending

point

Open and closed circles not correct

Question 1(d), Limits and Continuity

Included -2 in the answer for part (ii) even though the graph was not defined for -2

Question 2(a), Trigonometry

Did not find the quadrant II angle, 146.3o

Used an incorrect trig function to find the angle, 2

cos13

Question 2(b), Trigonometry

Evaluated each term to arrive at .22

Used the wrong formula, sin (A–B)

Subtracted the two angles to get sin 20o

Continued after writing the correct answer to evaluate the expression (this was ignored by

the markers)

Question 2(c), Trigonometry

Converted cos x to sin x or –sin x

Used incorrect identities

Repeated the right hand side of the proof

7.

Question 2(d), Sequences and Series

Found the first four terms of the sequence instead of the sequence of partial sums in part

(i)

Started evaluating with n=0 instead of n=1 in part (i)

Didn’t find the correct an+1 term in part (ii)

Didn’t put brackets in the numerator in the second last step while subtracting in part (ii)

Tried to prove that it was greater than 0 by substituting a value for n in part (ii) (Note that

this is not a legitimate way to prove for all values of n)

Didn’t write the denominator in all the steps of the proof in part (ii)

Did not draw the dividing line in all steps of the proof in part (ii)

Question 2(e), Series

Did not know that the 4 e of e is e ¼

Did not evaluate at least four terms (to be able to be accurate to 3 decimal places)

Did not write the final answer correct to three decimal places

Question 3(a), Complex Numbers

Had no idea how to approach this question

Multiplied the roots together

Tried to use the quadratic formula

Question 3(b), Complex Numbers

Had problems with multiplying complex numbers

Did not state the values of x and y

Wrote 4

25

iy or

4

25

iyi

Question 3(c), Complex Numbers

Wrote 7 instead of 11 for part (i)

Wrote angles of -64.8o or 64.8

o for part (i)

Didn’t write the answer in rectangular form for part (ii)

8.

Question 3(d), Complex Numbers

Used the values for r and the angle from part (c)

Used an angle of 90o instead of -90

o

Question 4(a), Statistics and Probability

Confused about what was required in this question; did not know that “either” meant

“union” in part (i)

Wrote ( ) ( ) ( )P A B P A P B in part (i)

Gave the answer to part (i) as the answer to part (ii)

Question 4(b), Statistics and Probability

Didn’t know that “not more than one” meant x = 0, 1

Question 4(c), Statistics and Probability

Expressed the final answer as 96 instead of 95

Subtracted .4772 from .5

Used the wrong formula to find the z score; it was 20 instead of 2

Stopped at .9544 instead of calculating the number of students

Question 4(d), Statistics and Probability

Calculated the wrong z value, 2.05 or 1.76 instead of 1.75

Shaded the probability curve diagram to the left (the negative side) instead of to the right

Thought it was a two-tailed test

Calculated the test statistic as -3.16 instead of 3.16

Did not properly label the regions on the probability diagram

Conclusion stated accept HA instead of reject HO

Question 5(a), Calculus

Set u = -x and dv = lnx dx

Integrated lnx to get 1/x

Didn’t include “dx” in dv = -x dx

9.

Question 5(b), Calculus

Wrote x as x1/3

Didn’t integrate x½

Didn’t use integral sign and dx properly

Left out the fraction 1/8

Didn’t show the integral as a definite integral (from 1 to 9)

Question 5(c), Calculus

Didn’t expand the function correctly or tried to integrate without expanding

Wrote dx instead of dy in the integral

Left out the dy in the integral

Used the integral sign incorrectly (for example, used the integral sign after integrating)

Integrated from -2 to 1 instead of splitting into two integrals

Substituted the values incorrectly

Did not take the absolute value of the integral from 0 to 1

Skipped the step where the integral is set up (went directly to the integration)

Question 5(d), Calculus

Had no idea how to approach this question

Tried to use the Pythagoras Theorem

Used an incorrect similar triangles ratio

Used the area of a triangle to try to solve the problem

Section C

Question 1(a), Calculus

Tried to integrate rather than differentiate

Forgot to differentiate 3x

Used the wrong formula for the derivative

Wrote 3x2 instead of (3x)

2

Question 1(b), Calculus

Integrated sec2 x to get sec x tan x

Used the integral sign incorrectly

10.

Left out the dx in the integral

Changed sec2

x to 1/cos2 x

Left out π in the formula

Used the wrong formula for volume

Question 1(c), Calculus

Incorrectly substituted after writing the growth curve equation, s = Ae-kt

Used a positive value for k as the power of e

Question 2(a), Vectors

Multiplied vectors a and b instead of subtracting b – a in part (i)

Used commas between i, j, and k instead of + signs in part (i)

Expressed the answer in vector form in part (i)

Incorrectly calculated the numerator or denominator in part (ii)

Question 2(b), Vectors

Got different answers for k due to algebraic problems

Question 2(c), Vectors

Did not write the symmetric equation in the proper form, either terms were separated by

commas instead of = signs or they were written as t = etc.

Question 2(d), Vectors

Used the incorrect ratios

Applied the ratios to the wrong vectors

Did not write the answer in coordinate form

Added -3/2 + -3/2 to get 0

Question 3(a), Computing

Didn’t know the difference between GOSUB and GOTO or didn’t know what GOSUB

was at all

11.

Question 3(b), Computing

Did not use the correct shapes

Question 3(c), Computing

Used Pascal instead of Basic

Wrote r2 as R

2 instead of R*R or R^2

Used π instead of 3.14

Asterisks (*) not used in the formula

Fraction written as 1

3 instead of 1/3

2. Statistics

The following statistics are based on samples of 80-200 scripts (only 10 scripts were in the

Computer sample). They show the percentage of candidates who scored full marks, partial

marks and zero marks for each question. Where statistics were not available, the relative

difficulty of the question is noted, from very poorly done to well done.

Section/Question Full Marks Partial Marks 0 Marks

A 1 15%

2 Poorly done

3 Poorly done

4 30% 48% 22%

5 10% 73% 17%

6 Poorly done

7 Poorly done

8 80%

9 Very poorly done

10 32% 46% 22%

B1a 33% 33% 34%

b i 73% 13% 14%

b ii 70% 5% 25%

b iii 45% 55%

b iv 25% 55% 20%

c 18% 35% 47%

d i 45% 55%

d ii 70% 30%

12.

B2a Poorly done

b Poorly done

c Well done

d i Very poorly done

d ii Well done

e Poorly done

B3a Very poorly done

b Poorly done

c Well done

d Well done

Section/Question Full Marks Partial Marks 0 Marks

B4a i 30% 33% 37%

a ii 60% 40%

b 22% 11% 67%

c 25% 26% 49%

d 10% 32% 58%

B5a 4% 9% 87%

b 15% 60% 25%

c 5% 55% 40%

d 2% 98%

C1a Very poorly done

b Poorly done

c Poorly done

C2a i Well done

a ii Well done

b Well done

c Well done

d Well done

C3a i 70% 20% 10%

a ii 30% 30% 40%

b 40% 40% 20%

c 20% 50% 30%

3. Comments and Recommendations for Selected Topics

This section contains comments and recommendations for selected topics, based on the

information received from the markers. If a phrase or sentence is written in italics, then it is a

direct quote from the markers.

13.

MATHEMATICAL INDUCTION (A1)

This question was quite well answered this year and, hopefully, gave the students

confidence since it is usually the first question attempted.

Unable to factorise or expand shows weakness in algebra.

Statements need to be correct for full marks. The correct statements are:

Prove true for n = 1

Assume true for n = k

Prove true for n = k + 1

Since it is true for n = 1 and n = k + 1, it is true by mathematical induction for all

positive integers

LIMITS (A3)

Write “lim” throughout the simplification process until the substitution is made.

This was a difficult question because some students were not able to factorise correctly.

PARTIAL FRACTIONS (A4)

This year’s question was most easily solved using equating coefficients. Most students

who tried to find the values of B and C using simultaneous equations were unable to do

so.

The major problem was the simplification of the fraction by multiplying by the

denominator of the original fraction, especially misplacing the bracket, Bx + C(x).

SIGMA NOTATION (A6)

Many students tried to answer this question without using the cancelling method. It is

important that this method be taught so it can be used for questions that can not be solved

using the formulas in the Eton table book.

COMPOSITE FUNCTIONS (A7)

Students were very weak in determining domain and range of simple expressions. They

did fairly well in finding f o g, but had difficulty simplifying the expression.

Most students substituted without using brackets [in part (d)].

14.

COMPLEX NUMBERS (A8 and B3)

Students could be taught how to use the POL key on their calculators so they can at least

check their answers when they convert from rectangular to polar form.

Very few students knew how to find a quadratic equation given the roots [B3 (a)].

Most students knew that they had to multiply by the conjugate, but could not simplify to

the correct form [B3 (b)].

Some candidates are still confused between raising a complex number to a power and

finding the roots of a complex number.

STATISTICS AND PROBABILITY (A10 and B4)

Some students totally misunderstood the mean and standard deviation question [A10].

Students need to know the meanings of terms such as “not more than” and “less than”

[B4 (b)]

Some students were not able to identify that it was a binomial distribution question [B4

(b)].

Some students used the formula for the binomial distribution [B4 (b)] and made

calculation errors. It is recommended that the students be encouraged to use the tables

instead of using the formula.

Most students were able to get the correct z value [in B4 (c)] but were not able to read

the probability value from the table and relate it to the area under the curve.

When using the normal probability curve [in B4 (d)] the critical region needs to be

clearly identified and labeled (with CR or reject H0) and the value needs to be shown on

the diagram.

Most of the students were not able to get the correct value of z [in B4 (d)]. Students need

more experience with using the distribution tables.

RATIONAL FUNCTIONS (B1)

Curves must approach the asymptotes, not pull away from them or run parallel to them.

TRIGONOMETRY (B2)

Most students did not know the formula [for changing from sum to product, B2 (b)] so I

feel they were not taught in school.

15.

SEQUENCES AND SERIES (B2)

A common mistake—the question [B2 (d) (i)] was asking for the partial sums, but the

students gave the first four terms of the sequence.

A common mistake was made in the expansion and simplification of the terms [B2 (d)

(ii)].

This was a very simple question [B2 (e), exponential series] which most students did not

answer and I think a lot of teachers miss it in the classroom.

Quite a number of students just did not know how to expand the exponential function.

CALCULUS (B5)

No one from the 100 scripts analysed scored full marks [for question B5], while more

than 75% of the scripts had zero. It was evident that either not much effort is put into the

teaching of calculus or the students did not study the topic.

Most candidates attempted only some parts of B5 or did not attempt it at all. It is as if

they had planned before the examination to ignore the calculus question. Perhaps this

calls for a change in the order of coverage of topics to cover calculus earlier in the year

rather than later. More time and room can be given to students to learn and understand

the concepts in calculus as it seems to be more challenging than other topics.

The technique of integration was not mastered well [in question B5 (a)], as candidates

obtained many different integrals.

None of the 100 scripts analysed got the correct answer [to B5 (d)]. It hasn’t been asked

in an exam in a long time so probably students (and teachers) hadn’t put in much, if any,

effort in the learning and teaching of the “shadow type” problem.

This concept needs to be thoroughly taught to students [B5 (d)].

4. Conclusion and General Recommendations

We recommend that you show this report to the form seven students so that they can

learn from others’ mistakes.

Read the Common Errors section of this report carefully to see how students approach the

questions.

Read the Marking Scheme in detail to see how the markers allocate marks to the

questions and to see how they deduct marks.

Encourage the students to cross out their work rather than using twink. Frequently, when

they use twink, they twink something out, wait for it to dry and then forget to go back and

write something over the twink. Also, if students cross out their work, the marker can at

least get an idea of what they intended and may be able to award partial marks.

16.

Introduce your students to the Eton Table Book early in the form seven year and

encourage them to refer to it throughout the year. The markers noticed many cases where

candidates were not able to use the tables correctly. They should be thoroughly familiar

with the tables and the formulas in Eton before they take the exam.

Many students are quite careless when taking the mathematics exam. They forget minus

signs, they leave out denominators in fractions, they forget to include dx or dy in the

integral, they keep writing the integral sign after integrating, they don’t bother to draw

the dividing line in fractions and they make rounding errors. Mathematics is an exact

science and students should be encouraged to solve problems carefully and accurately.

Encourage students to show their working clearly on examination papers. This will allow

markers to give partial marks if the final answer is not correct.

Many students seem to have problems with algebra: factorizing, expanding, using

powers, working with fractions and decimals. Some additional time may need to be spent

on the basics of algebra during the form seven year.

Many students do not read the questions carefully. Often key points in the questions are

in bold type to alert the students to their importance. However, it seems that the students

do not pay attention to these points. For instance, very few students found the first four

terms of the sequence of partial sums even though it was expressed in bold type in the

question. Also, this question has been asked in previous years and it should have been

pointed out to students that they needed to be ready for a question like this.

Many students do not understand which formula to use for a specific question. This

points to a lack of understanding of the topic being tested. It seems as if the students are

taught to answer questions, but are not given an understanding of the topic. It is

important to make sure that the students understand what they are doing and why.

Some students totally misunderstand what a question is asking for. It is as if they are

expecting a certain type of question and when the question is not that type, they go ahead

and try to use their method anyway.

Some students do not place their answers in the proper place in the Answer Book. For

instance, they start Question 2 in Section B at the end of Question 1 instead of turning the

page to the place where Question 2 is supposed to be answered. We recommend that

copies of the Answer Book be given to students during internal exams so that they get

used to the layout.

Since the introduction of Computer Studies in 1996, the mathematics markers have

recommended that Computing be removed from the form seven syllabus and exam. Only

about 2 or 3 percent of the exam candidates attempt the Computing question. The

Computing question also gives an unfair advantage to those students who take

Computing.

THE END