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South Pacific Form Seven Certificate

MATHEMATICS WITH CALCULUS

2018

INSTRUCTIONS Write your Student Personal Identification Number (SPIN) in the space provided on the top right-hand

corner of this page.

Answer ALL QUESTIONS. Write your answers in the spaces provided in this booklet. Show all working. Unless otherwise stated, numerical answers correct to three significant figures will

be adequate.

If you need more space for answers, ask the Supervisor for extra paper. Write your SPIN on all extra

sheets used and clearly number the questions. Attach the extra sheets at the appropriate places in this

booklet.

Major Learning Outcomes (Achievement Standards)

Skill Level & Number of Questions Weight/

Time

Level 1 Uni-

structural

Level 2 Multi-

structural

Level 3 Relational

Level 4 Extended Abstract

Strand 1: Algebra Apply algebraic techniques to real and complex numbers.

17 - 1 - 20%

52 min

Strand 2: Trigonometry Use and manipulate trigonometric functions and expressions.

- 2 2 - 10%

24 min

Strand 3: Differentiation Demonstrate knowledge of advanced concepts and techniques of differentiation.

- 3 2 2 20%

52 min

Strand 4: Integration Demonstrate knowledge of advanced concepts and techniques of integration.

- 3 2 2 20%

52 min

TOTAL 17 8 7 4 70%

180 min

Check that this booklet contains pages 2-22 in the correct order and that none of these pages are blank. A 4-page booklet (No.108/2) containing mathematical formulae and tables is provided.

HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

108/1

QUESTION and ANSWER BOOKLET

Time allowed: Three hours

(An extra 10 minutes is allowed for reading this paper.)

2

STRAND 1: ALGEBRA

1.1 Solve the linear equation: 𝑥

2−

𝑥+2

3= 1

1.2 Find the point of intersection of the lines 2𝑦 − 𝑥 = 2 and 𝑦 + 𝑥 = 4

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1.3 Factorize the quadratic expression: 5𝑥2 − 12𝑥 + 7

1.4 Use the Factor Theorem to factorize the function 𝑓(𝑥) = 𝑥3 + 5𝑥2 − 𝑥 − 5

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1.5 If 4𝑥+1 = 16𝑥 find the value of ′𝑥′

1.6 Find the simplest expression for 2 log2

𝑥 − log2

𝑥3

1.7 If 𝑓(𝑥) = 𝑥3 + 𝑥2 + 3𝑥 + 1 find the remainder when f(x) is divided by (𝑥 + 2)

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1.8

1.9 1.10

For questions 1.8 to1.10 use the information in the diagram below:

Write the complex number u in the polar form.

Plot the complex number 𝒗 = 2 − 𝑖 in the Argand diagram above.

If 𝒘 = 4 − 3𝑖 and 𝒗 = 2 − 𝑖, find the complex number 𝒛 where 𝒛 = 𝒗𝒘

2

iy

1

0

-4 -3 -2 -1

-1

1 2 3 4 x

-2

-3

• u

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1.11 Each Sunday a newspaper agency sells one newspaper for $1.10 while the

cost of production is $0.40 per copy. If the fixed cost of storage on Sunday is

$68, write an algebraic expression for the profit P on the sale of 𝑥 copies of

the newspaper on a Sunday.

1.12 Simplify the surd expression: √12 − (√3 − 1)

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1.13 Find the value of ′𝒌′ that will make the expression 𝑥2 − 8𝑥 + 𝑘 a perfect

square.

1.14 Simplify the expression (8𝑎−3)−1

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1.15 Solve the equation 3𝑥2 − 6𝑥 + 2 = 0 by using the quadratic formula.

1.16 Use the Binomial Theorem to expand and simplify the expression (2𝑏 − 3)3

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1.17 Make 𝑟 the subject of the formula in 𝑎𝑟2 + 𝑏 = 𝑐

1.18 Use de Moivre’s Theorem to find the two distinct roots of the equation:

𝐮2 = 2(cos𝜋

4+ 𝑖 sin

𝜋

4)

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STRAND 2: TRIGONOMETRY

2.1 The height h(𝑡) above ground level of a seat in a Ferris Wheel after a time of

′𝑡′ seconds is given by the expression h(𝑡) = −20 cos 0.349 𝑡 + 17. After

how many seconds from the start will the seat be at a height of 27 m?

2.2

Use the right-angled triangle drawn on the right

to find the value of ′𝑥′ if sec 𝛼 = √10.

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√1 + 𝑥2

1

𝛼

𝑥

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2.3

Use the compound angle formula: Sin (𝐴 − 𝐵) = Sin 𝐴 Cos 𝐵 − Cos 𝐴 Sin 𝐵 , and the data given below to show that

sin 15𝑜 =√3 − 1

2√2

[ Data: sin 30 = cos 60 =1

2; sin 45 = cos 45 =

1

√2 ; sin 60 = cos 30 =

√3

2 ]

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2.4

A mass ‘P’ at the end of a spring is pulled

down to the point ‘L’, 25 cm above a bench,

and then released. It moves vertically up

and down between ‘H’ and ‘L’ in continuous

periodic motion. The mass moves to ‘H’ and

arrives back at the lowest point ‘L’, 1.6 s

after release.

Any effects of friction may be neglected in

this problem.

The height h of the mass above the bench at

various times t can be described by

ℎ(𝑡) = 𝐴 sin 𝐵(𝑡 − 𝐶) + 𝐷.

Evaluate the constants A, B, C and D, and rewrite the equation for ℎ(𝑡).

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25 cm

highest point H

midpoint M

lowest point

L bench top

L

75 cm

P

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STRAND 3: DIFFERENTIATION

3.1 If 𝑦 = 4 ln 𝑥 find 𝑑2𝑦

𝑑𝑥2

3.2

3.3

Questions 3.2 and 3.3 refer to the graph of the function 𝑦 = 𝑔(𝑥) shown

below.

At what value of 𝑥0 is the following true?

Lim 𝑥 → 𝑥𝑜

+𝑔(𝑥) = Lim

𝑥 → 𝑥𝑜−

𝑔(𝑥) but Lim 𝑥→𝑥0𝑔(𝑥) ≠ 𝑔(𝑥0)

At which value(s) of 𝑥 is 𝑔(𝑥) not differentiable?

2

y

1

0

-4 -3 -2 -1

-1

1 2 3 x

-2

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𝑦 = 𝑔(𝑥)

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3.4 Sketch in the grid below the graph of the function 𝑓(𝑥) =1+ 𝑥

𝑥(1− 𝑥) by

considering its behaviour near the asymptotes and at infinity.

3.5 Use the definition 𝑑𝑦

𝑑𝑥= Limℎ→0

𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ to find the derivative of

𝑓(𝑥) = 𝑥2 − 3.

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𝑦 = 𝑓(𝑥)

0 1 𝑥

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3.6 Use implicit differentiation to find 𝑑𝑦

𝑑𝑥 if 𝜋sin 𝑦 = 5𝑥𝑦

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R

• C

C

ℎ

𝑟

𝑅

3.7 A cone is made from a circular sheet of radius R by cutting out a sector

(Fig.1) and then gluing together the cut edges of the remaining piece (Fig.2).

Fig.1 Fig.2

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Show that the volume of the cone formed is maximum when 𝑟 = √2

3 . 𝑅

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STRAND 4: INTEGRATION

4.1

Use 𝑉 = 𝜋 ∫ 𝑦2𝑏

𝑎𝑑𝑥.

4.2 Solve the first order differential equation 𝑥𝑑𝑦

𝑑𝑥− 𝑦 = 0 by separating the

variables.

Find the volume of the solid formed when the

shaded region enclosed between 𝑦 = 3𝑥, 𝑥 = 1,

𝑥 = 3, and the x-axis is rotated about the x-axis.

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1 3 0

𝑦

𝑥

𝑦 = 3𝑥

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4.3 Show that the differential equation: 𝑑𝑦

𝑑𝑥= −

4𝑥

𝑦 can be formed from the

parametric equations: 𝑦 = 2 sin 𝜃 and 𝑥 = cos 𝜃.

4.4 The motion of a toy car on a long track is given by the differential equation:

𝑑2𝑠

𝑑𝑡2 = 6 𝑚/𝑠2, where s is the distance travelled after t seconds of motion. If

the speed of the car is 8 m/s after 1 second, and the car is 20 meters from the

start after 2 seconds, give an expression for the distance travelled 𝑠 in terms

of the time 𝑡.

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4.5 The diagram at right shows a shaded region

between the functions 𝑦 = 𝑥 and 𝑦 = 𝑥2.

Calculate the volume of the solid formed

when this shaded region is rotated 360o

about the x-axis.

Use the formula 𝑉 = 𝜋 ∫ 𝑦2𝑏

𝑎𝑑𝑥

_

1

𝑦 = 𝑥2

0

𝑦 = 𝑥

y

x

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4.6 The diagram in Fig.1 shows the graph of 𝑦 = 𝑥2 and the shaded region

bounded by the line 𝑥 = 2, the x-axis, and the graph of 𝑦 = 𝑓(𝑥) = 𝑥2.

Fig.1 Fig.2

Calculate the volume of the solid formed (Fig.2) when the shaded region is

rotated 360o about the line 𝑥 = 2. Use the formula 𝑉 = 𝜋 ∫ 𝑥2𝑏

𝑎𝑑𝑦

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3

2

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𝑥 = 2

4

𝑓(𝑥) = 𝑥2

𝑦

𝑥 2

𝑦

𝑥

4

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4.7

It is known that the population of a colony of bacteria increases at a rate

directly proportional to the number of bacteria present i.e. 𝑑𝑁

𝑑𝑡= 𝑘𝑁 where 𝑘 is

a constant of proportionality and N is the number of bacteria. If the number of

bacteria doubles after 5 hours, how long will it take for the bacteria to triple?

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Extra Blank Page If Needed

THE END