MATHEMATICS WITH CALCULUS

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1 South Pacific Form Seven Certificate

MATHEMATICS WITH CALCULUS

2019

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.

14 1 - 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.

1 3 3 1 20%

52 min Strand 4: Integration

Demonstrate knowledge of advanced concepts and techniques of integration.

2 2 2 2 20%

52 min

TOTAL 17 8 7 4 70%

180 min Check that this booklet contains pages 2–20 in the correct order and that none of these pages are blank.

A four-page booklet (No. 108/2) containing mathematical formulae and tables is provided.

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

QUESTION and ANSWER BOOKLET

Time allowed: Three hours

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

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STRAND 1: ALGEBRA

1.1 Solve the linear inequation ^𝑥

2− 𝑥 < 1

1.2 Find the point of intersection of the lines 𝑦 − 3𝑥 = 2 𝑎𝑛𝑑 𝑦 + 𝑥 = 4

Unistructural

1 0 NR

Assessor’s use only

Unistructural

1 0 NR

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1.3 Simplify the expression (2𝑎²𝑏)³ ÷ ( _𝑏^4𝑎₋₃⁵ )

1.4 Solve the equation 𝑥²− 7𝑥 − 78 = 0 by the method of factorisation

.

Unistructural 1

0 NR

Unistructural 1

0 NR

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1.5 Divide 2𝑥³+ 4𝑥²− 2𝑥 + 3 by (𝑥 − 2)

_

1.6 Make 𝑟 the subject of the formula (^𝑎

√𝑟 − 𝑏)² = 𝑐

1.7 Find the value of 𝑥 if:

log

₈𝑥³ = 2

Unistructural 1

0 NR

Unistructural 1

0 NR

Unistructural

1 0 NR

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1.8 Write the complex number 𝒗 = √8𝑐𝑖𝑠 ^3𝜋₄ in the rectangular form.

1.9 Plot the complex number 𝒗 from question 1.8 in the Argand diagram below.

1.10 If 𝒘 = 4 + 3𝑖 and 𝒖 = 2 − 𝑖, find the complex number 𝒛 where 𝒛 = 𝒖 − 𝒘

Unistructural 1

0 NR

Unistructural 1

0 NR

Unistructural

1 0 NR

2

𝑖𝑦

1

0 -4 -3 -2 -1

-1

1 2 3 4

𝑥

-2

-3

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1.11 Simplify the surd product (3√2 − √3)(3√2 + √3)

1.12 Use the Binomial Theorem to expand and simplify the terms in the expansion of (2𝑥 − 3)³

1.13 Find the value of 𝑐 that will make 𝑓(𝑥) = 𝑥²− 10𝑥 + 𝑐 a perfect square.

Unistructural 1

0 NR

Unistructural 1

0 NR

Unistructural 1

0 NR

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1.14 Find the remainder when 𝑓(𝑥) = −2𝑥³+ 3𝑥²− 5𝑥 − 4 is divided by (𝑥 + 2)

1.15 Use the Factor Theorem to find the factors of 𝑓(𝑥) = 𝑥³− 2𝑥²− 5𝑥 + 6

Unistructural 1

0 NR

Multistructural

2 1 0 NR

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1.16 A complex number equation is given as 𝑧⁴− 1 − 𝑖 = 0. Use de Moivre’s Theorem to find the four roots of the above equation, and represent these roots on the Argand diagram.

Extended Abstract 4 3 2 1 0 NR

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STRAND 2: TRIGONOMETRY

2.1 Use the information in the triangle shown on the right to express

cot

𝜃 in terms of 𝑥.

2.2 If

tan 𝑥 =

^{sin 2𝑥}

1+cos 2𝑥 show that tan 15^o= ¹

2+√3. Use the information in the diagram on the right. [Do not use the calculator in this problem].

1

𝜃

𝑥

Multi-structural

2 1 0 NR

Multistructural

2 1 0 NR 2 1

√3 30^𝑜

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2.3

The periodic motion of a seat on a Ferris wheel is described by the graph above, where ℎ is the height above ground level and t is the time after the seat starts moving. Write the equation for this motion of the seat.

2.4 A boat on wavy water moves a total vertical distance of 2.6 m from its highest point H to its lowest point L in a time of 10 s.

Determine the height of the boat 8 s after it has started moving down from the top.

Use the formula: ℎ(𝑡) = 𝐴 cos 𝐵(𝑡 − 𝐶) + 𝐷

Relational

3 2 1 0 NR

Relational

3 2 1 0 NR

boat

2.6 m H

L 9

3 0 6

ℎ

−2𝜋 −𝜋 𝜋 2𝜋 3𝜋 4𝜋

𝑡

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11 STRAND 3: DIFFERENTIATION

3.1

Use the piecewise function drawn below

to answer

question 3.1.

At which value of 𝑥 is 𝑓(𝑥) continuous but not differentiable? 𝑥 = ___________

2

1

0 -4 -3 -2 -1

-1 1 2 3 4

-2

3.2 What are the two 𝑥-values for which 𝑓(𝑥) =_{(𝑥−2)(𝑥+7)}^3𝑥+5 is discontinuous?

3.3 Evaluate the following limit: Lim

𝑥 → −∞

3𝑥⁴+2 𝑥⁴

Assessor’s use only Lateral Level

1 0 NR

Multi-structural 2

1 0 NR

𝑥

y=𝑓(𝑥)

Unistructural

1 0 NR

Multi-structural

2 1 0 NR

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3.4 A car travelling at a uniform acceleration covers a distance of 𝑑 metres given by the equation 𝑑 = 6𝑡 + 5𝑡² where 𝑡 is the time in seconds. After what time is the speed of the car 42 m/s?

3.5 The volume 𝑉 of water in a cylinder of constant radius 5 cm is increasing at a rate of 20 cm³/s. Use the formula for the volume of a cylinder (𝑉 = 𝜋𝑟²ℎ) to find ^𝑑𝑉

𝑑ℎ and hence determine the rate of increase ( ^𝑑ℎ_𝑑𝑡 ) in the height ℎ of the water.

ℎ

5 cm

Multistructural 2

1 0 NR

Relational

3 2 1 0 NR

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3.6 The equation of a curve is given implicitly as 𝑒^𝑦+ 𝑥²= 4. Find the equation of the tangent to the curve at the point (-3, 0).

3.7 Sketch below the graph of 𝑦 = ^−6𝑥

(𝑥+1)(𝑥−1) by considering the limiting behaviour of the function at infinity and near the asymptotes (dotted lines).

Relational

3 2 1 0 NR

Relational

3 2 1 0 NR

𝑥

𝑦

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3.8 A rectangular box with a square base and open at the top is to have a volume of 512 cm³. What should be the dimensions of the box in order to minimize the materials used in making it?

open at top

ℎ

Use the formula:

𝑥

𝑉 = 𝑥²ℎ 𝑆 = 4𝑥ℎ + 𝑥²

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STRAND 4: INTEGRATION

4.1 Find the integral ∫ √𝑥³ 𝑑𝑥

4.2 Find ∫ 5𝑒^2𝑥−3𝑑𝑥

Unistructural 1

0 NR

Unistructural

1 0 NR

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4.3 Find the value of the definite integral

∫ sin 𝑥 𝑑𝑥

𝜋 2 0

4.4 Results of a research show that the rate of growth of an insect population is directly proportional to a constant rate of 10% of the number of insects 𝑃 in the population. Translate this into mathematical form and describe any symbols that you use.

Multistructural 2

1 0 NR

Multistructural

2 1 0 NR

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4.5

Solve the differential equation ^𝑑𝑦

𝑑𝑥= 𝑥²𝑦 given that 𝑦(0) = 3

4.6 The graph of 𝑦 = 𝑒^2𝑥 is rotated 360^o around the 𝑥-axis. Use the formula 𝑉 = 𝜋 ∫ 𝑦_𝑎^𝑏 ²𝑑𝑥 to calculate the volume of the solid that is formed between 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 3.

_

_______________________

_____________________________________________________________

Relational

3 2 1 0 NR

0 𝑦 = 𝑒

^2𝑥 𝑦

𝑥 1

3

Relational 3 2 1 0 NR

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4.7 The diagram shows a shaded area enclosed by the functions 𝑦 = √𝑥 and 𝑦 = 𝑥. When this area is revolved around the x-axis, the solid formed will have a cross-sectional area as shaded with a conical space as shown in the diagram.

Calculate the volume of the solid of revolution

.

𝑦 = 𝑥

𝑦 𝑦 = √𝑥

𝑥

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4.8 A new politician in an island campaigns to inform people about himself and his vision for the 1 million people living on the island. His work results in the assumption that the rate in the number of people 𝑝 who have heard of him and his policies is directly proportional to the number of people (1 − 𝑝) who have not heard of him and his policies. At the start of the campaign no one has heard of him, but after one year on the trail 0.5 million people have heard of him.

How many would have heard of the politician after two years of the campaign?

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MATHEMATICS WITH CALCULUS

1

South Pacific Form Seven Certificate