MINISTRY OF EDUCATION

FIJI SEVENTH FORM EXAMINATION

2008

MATHEMATICS

COPYRIGHT: MINISTRY OF EDUCATION, REPUBLIC OF THE FIJI ISLANDS

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MINISTRY OF EDUCATION

FIJI SEVENTH FORM EXAMINATION 2008

MATHEMATICS

EXAMINER’S REPORT

This year the number of candidates appearing for the Mathematics paper was 4528. The paper was

based on the form seven prescription and its objectives. The questions were straight forward with

helpful hints given so as to test specific objectives. They were phrased with the intention to avoid

confusion and allow for reasonable mark allocation.

This serves as a timely reminder to the teachers to adhere to the prescription and its specific

objectives. Two obvious facts that stood out are the concepts of differential equation and point of

inflexion. It seems that only modeling type differential equations are concentrated on and the concept

of finding the point of inflexion of a given equation is totally ignored. These were evident from the

responses by the candidates.

On the brighter side a sample of size 390 revealed that 10% more candidates passed this year‟s

examination compared to last year‟s with a sample mean 5 points more than that of last year‟s.

IMPORTANT NOTES FOR TEACHERS AND MARKERS

The following is a guide for form seven mathematics teachers, markers and candidates. It provides

insight of the marking criteria. It is intended that the teachers take note of the guide and make students

aware of the marking criteria so that they know what is required of them when preparing for the

examination.

GENERAL

Unless required to show or prove, full marks are awarded for the correct answer

Unless required to simplify, full marks are awarded if the answer is not simplified

If the marker feel that a candidate deserves partial/full marks even if the approach shown is not

in the marking scheme the marker should indicate this

with letter D ( Discretion)

The answers requiring : to obtain angles, sketch trigonometric graphs or to transform into

cosR etc can be either in degrees or radians irrespective of the given interval

If the angles are rounded off to whole number, full marks are awarded

No marks are awarded for the correct conclusion if constructing of the hypothesis test is

incorrect

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Half 12

mark is deducted if :

extra answers are given,

division line is not drawn,

incorrect or overuse of “ lim ”,

drawing solid lines to represent asymptotes,

using round brackets to list x values,

continuous use of > or < in proving increasing/decreasing sequence etc, denominator missing

in proving increasing/decreasing sequence etc,

brackets are missing when giving answer in polar form,

“ dx ” notation missing,

incorrect use of “ ”,

missing from volume expression and

final answer not given as coordinates when applying ratio formula.

OVERVIEW OF SOME SUBTOPICS

The following is a guide for future candidates which intents to provide some insight to the expectancy

level, and which may be helpful in preparation for the examination.

MATHEMATICAL INDUCTION

Correct statements should be written such as :

Prove it is true for 1n ( or k = 1 )

Assume it is true for n k

Prove it is true for 1n k

Since it is true for 1n , 1n k ( and assumed true for n k ) by mathematical induction it

must be true for 2,3,4,...n ( all natural numbers )

Complete statements must be written for both ( )S k and ( 1)S k statements

Either S(1) or ( )S k can be the first step

Mark is awarded for the conclusion only if all the steps are correct

LIMITS

Write “lim” throughout the simplification process to the point before substitution is made

When asked to evaluate limit of an expression which you are unfamiliar with, use the sandwich

method (table)

POLYNOMIAL

Clearly draw any point of inflexion, labeling is not sufficient

Polynomials which can be tested have total power of 3 or more

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RATIONAL GRAPH

Graphs must approach the asymptotes as x ( and y ) approaches infinity

Graphs can have total power of 4 or less, equation such as 2

( )( 1)( 1)( 2)

xf x

x x x

can be

tested since the power is 4

PARTIAL FRACTIONS

Always multiply by the denominator of the fraction given on the left hand side to avoid

confusion

COMPOSITE FUNCTIONS

Always simplify f g or g f etc

TRIGONOMETRY

When proving an identity, write the left hand side as the first step and avoid writing “=” sign

throughout the process

It is not necessary to give the angles in radians

Only horizontal ( y axis ) shifting of graph will be tested

PROBABILITY AND STATISTICS

Familiarize with phases like “at least 3” ( 3 or more ) and “at most 3” ( 3 or less )

Z – value can be positive even if it falls on the left hand side of Z = 0 line

Clearly mark the critical region ( rejection and acceptance regions ) when constructing a

hypothesis test

SERIES

Summation of some of the series can be reduced to an arithmetic series

Formula can be applied only if r starts at 1 and do not forget to multiply by the constant, for

example: expression containing r - use the formula ( 1)

2

n n , if expression is 5r – use

( 1)5

2

n n

COMPLEX NUMBERS

A complex number is a point, not a line segment

Calculators can be used to transform a complex number to rectangular/polar form

Use brackets when raising a number to a power, for example: 1

38 use 18 ^3

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CALCULUS

Practice writing chain rule involving three variables

Write “ ” and “ dx ” at the appropriate places

When evaluating a bounded area, where possible divide the area into parts and evaluate

separately. Apply knowledge gained in form six.

Practice solving first order differential equations before attempting to solve modeling type

questions

VECTORS AND GEOMETRY

Vector AB is b – a not a multiplied by b

Write a single parametric equation of the form p = a + t d

COMMON ERRORS FOUND IN CANDIDATES’ RESPONSES

SECTION A

QUESTION 1

Incorrect statements such as prove that n =1, assume that n = k , prove that

n = k +1, n = 1 , n = k and n = k+1.

Incomplete statements for S( k) and S( k + 1 )

Not able to simplify the left hand side of S( k + 1) to a single fraction.

Writing equal sign throughout the proving process for the above.

Expanding the right hand side of S( k + 1) incorrectly.

Not able to correctly factorise the left hand side of S( k + 1) in the final step to show it is equal

to the right hand side.

QUESTION 2

Not simplifying the final fraction as required

Not using the Binomial Theorem as required.

Using r = 4 instead of r = 3

Writing 3

58

3 12

xC

instead of

538

3 12

xC

or

538

5 12

xC

Writing 2

x as 2 x , -2 x , 2x or 12x

Simplifying 3

1 as 1

Finding the first four terms instead of just the fourth term

Finding all the terms

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QUESTION 3

Part (a)

Substituting x = 0 and writing the answer as zero , undefined or

„limit does not exist‟

Not able to factorise correctly

Differentiating incorrectly while applying L‟Hopital‟s rule

Writing derivative of 2 1xe as 22 1xe or 22 xe

Not writing limit notation at appropriate places

Writing limit notation at/after the substitution step

Part (b)

Substituting x = 0

Obtaining incorrect single fraction

Dividing 4 by x instead of

3

1x

x

x x

Not writing limit notation at appropriate places

Writing limit notation at/after the substitution step

QUESTION 4

Multiplying by 3 1x x instead of 2 1x x

Writing 2 2 26 2 1 1x x A x x B x x C x x

instead of 2 26 2 1 1x x Ax x B x C x

Obtaining C = -5 instead of C = 5

Applying coefficient method incorrectly

Obtaining three equations and were unsuccessful with solving

QUESTION 5

Not starting from the left hand side or the right hand side

Wring „=‟ sign throughout the proving process

Writing 1 1

1 sin 1 sinx x

as

1 1

cos cosx x in the next step

Omitting important steps such as 1 1

1 sin 1 sinx x

=

2

1 sin 1 sin

1 sin

x x

x

and writing 2

2

cos x directly

Writing denominator as 2

1 sin x instead of 21 sin x

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QUESTION 6

Part (a)

Not simplifying the expression

Simplifying 2

2 1x as ( 2) 1x or 2x or 1x

Writing 2 1 2x instead of 2

2 1x

Part (b)

Writing x R , 1x , 1x etc

Part (c)

Writing 2y , 2x , 1x etc

Note: consistency was followed in this part from parts (a) and (b)

QUESTION 7

Changing lower and upper limits to 1 and 97 instead of 1 and 98

Writing 5 3 2r instead of 5 2 3r

Not multiplying by 5 to the appropriate formula

Writing 98 99

5 3 1002

or

98 995 3

2

instead of

98 995 3 100

2

Writing 100 3

1 1

5 3 5 3r r

r r

QUESTION 8

Writing 3 4U i or 3 4i

Interpreting „ ‟ as conjugate

Proving that both sides are equal to 33 – 56i

Obtaining incorrect expression upon expansion

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QUESTION 9

Writing derivative of ln y as 1

x

Leaving out dy

dx upon differentiating

Not able to correct make dy

dx the subject

QUESTION 10

Incorrect formula

Writing Z = 0.97 or 0.485 instead of Z = 2.17 or 2.1701

Not squaring to obtain the sample size n

Writing

2Z

nE

, n

2Z

E

or

2Z

nE

instead of

2Z

nE

Giving the answer as 117.73n , 117.73n or 118n instead of 118

SECTION B

QUESTION 1

(a)

More than two x intercepts

Not recognizing the turning point and point of inflexion

Incorrect shape

Sketching a cubic graph

More than one y intercept

(b)

Incorrect intercepts/ more than the required number of intercepts

Incorrect factorization of the numerator

Not writing the equation of the vertical asymptote; writing x -1 or (1, 0) instead of

x - 1 = 0 or x = 1

Not writing the equation of the oblique asymptote, writing x + 5 or OA = x +5 instead of

5y x

Representing asymptotes as solid lines

Graph moving away from the asymptote as x and y get larger and larger

Incorrectly drawing the oblique asymptote

Note: consistency was followed from parts (i), (ii) and (iii) while marking part (iv)

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(c)

Writing values as coordinates, or using round brackets

Listing more than the required number of values

Calculating the limit instead of reading from the graph

QUESTION 2

(a)

Writing 1

sin2

instead of 1

sin2

Obtaining 30 instead of 30 or 330

Writing answer as 2cos(30 )

(b)

Not indicating the x and the y intercepts

Incorrect horizontal shift

Incorrect shape

Indicating maximum and minimum values as 2.5 and -2.5

(c)

Not able to obtain the equation 3

sin( 45 )2

x

Obtaining only one x value

Writing solution as 75 and 105 instead of 105 and 165

Giving more than two values of x

(d)

Finding the first four terms instead of the third and fourth terms only

Writing 1

2 4

3 1n

na

n

instead of 1

2 5

3 3n

na

n

Substituting values of n to prove instead of proving in general

Writing „< 0‟ or „ > 0‟ throughout the proving process

Finding the limit to prove the lower bound

Transferring 2

3 to the right hand side to prove the lower bound

QUESTION 3

(a)

Incorrect labeling of the imaginary axis

Labeling the point as 5 i instead of as 5

Multiplying the arguments ( 90 -30 ) = - 2700

Adding their modulli

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(b)

Confusion with i i and i - i

Incorrect simplification

Writing i - i = -1 instead of 1

(c)

Writing argument as 60 instead of -60 or 300

Finding the fifth roots instead of 5Z

Dividing the argument by 5 instead of multiplying

(d)

Finding 3

8 , 38 , 8

3 or 8 3 instead of

1

38

Writing 1

38 = 23

2 instead of 2

Giving arguments as 90 , 180 and 270

Not leaving the answer in the polar form

Note: consistency was followed in the above part from part (c)

QUESTION 4

(a)

Confusion with independent and mutually exclusive events

Not able to fully explain why the events are independent

Using P(AB) = P(A) + P(B) – P(A B)

Writing the probability of neither events as 0.73 instead of 0.27

Using 1 – (0.6 +.25 + 0.12) = 0.03

(b)

Giving extra values

Using cumulative binomial distribution values ( 0.0024 + 0.0308 + 0.1631 )

Confusion with at least 3

Giving only one value corresponding to r = 3

(c)

Giving Z – value as 1.9954 instead of 2

Using standard deviation, 12 in the working to show Z = 2

Leaving the answer as 0.1915

Subtracting 0.1915 from 0.5 instead of adding

(d)

Critical Z – value given as 1.6449 instead of 1.96

Critical region not labeled

Carrying out one- tailed test

Using and instead of < and >

Incorrect conclusion

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QUESTION 5

(a)

Writing derivative of cos x as sin x instead of sin x

Writing quotient rule as ' '

2

f g fg

g

Differentiating the numerator and the denominator

(b)

Equating the original equation to zero

Equating the first derivative to zero

Writing derivative of 12x as 22x instead of 22x

(c)

Not using the given substitution

Incorrect use of “ dx ” and “ du ”

Using 1x u substitution and complicating the integrand

Not canceling 2x to simplify to 9u du

(d)

Using incorrect limits of integration

Finding the point of intersection

Not using “ ” and “ dx ” at the appropriate places

Variety of methods used that resulted in incorrect answer

(e)

Incorrect chain rule

Using 10dV

dt instead of 10

dA

dt

Using dh

dt instead of

dr

dt as required in the question

Writing 8dA

rdt

instead of 8dA

rdr

Writing the final answer as 25/

16

drcm s

dt

instead of

5/

16

drcm s

dt

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SECTION C

QUESTION 1

(a)

Writing secdx

d

instead of 2sec

dx

d

Missing out d when writing the integration

Simplifying 24 x to 24 1 tan x instead of 24 1 tan x

Replacing 4

3

4

1

5

1 by 21 tan x

Incorrect use of “ ”

Incorrect simplification

(b)

Incorrect formula

Writing “ dx ” instead of “ dy ”

Incorrect limits of integration

Integrating 2 ye as 2 1

2 1

ye

y

or 22 ye

Not rounding off the final answer to the nearest whole number

(c)

Using integration by parts to solve the equation

Using product rule to solve the equation

Writing the solution as kty Ae and finding the values for A and k

Writing dy xydx and trying to integrate

Solving using integration by parts

QUESTION 2

(a)

Multiplying vectors a and b instead of b – a

Finding a – b

Writing 5 , ,5i j k instead of 5 5i j k

Finding

5

1

5

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Incorrect calculation of a b

Not find the inverse cosine to obtain the angle

(b)

Not equating to zero

Incorrectly solving -3 +2y - 8 = 0

Trying to prove that the vectors are orthogonal

(c)

Incorrect position and direction vectors

Writing a + t d instead of p = a + t d or

x

y a td

z

Writing three equations instead of one

Writing

4

x

y

z

instead of

x

y

z

(d)

Using the ratios of 1

5 and

4

5 instead of

1

4 and

3

4

Writing

5 33 1

0 44 4

1 1

instead of

3 53 1

4 04 4

1 1

Incorrect simplification

Not giving the final answer as coordinates

QUESTION 3

(a)

Not writing the statements fully

Writing $5.00 * 0.05 instead of 5 * .05

(b)

Writing „takes take sum‟ only

Not writing „prints the sum‟ or something similar

Incorrect shapes and statements in the flowchart

Incorrect loop

Using straight lines instead of arrows

(c)

Writing sentences instead of BASIC commands

Not defining “A”

Incorrect use of IF THEN command

Writing “if E is greater than 500” instead of “IF E>500

Writing “add 50 dollars” instead of “THEN A = A + 50

THE END