MATHEMATICS WITH CALCULUS

1

South Pacific Form Seven Certificate

MATHEMATICS WITH CALCULUS

2020

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%

60 min

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

3 2 1 - 10%

30 min

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

1 3 - 2 15%

45 min

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

2 3 1 1 15%

45 min

TOTAL 20 9 2 4 60%

180 min

Check that this booklet contains pages 2–21 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.

108/1

QUESTION and ANSWER BOOKLET (1)

Time allowed: Three hours

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

2

STRAND 1: ALGEBRA

1.1 Solve the linear equation 4𝑥 − 8 = 2(𝑥 + 5) − 𝑥

1.2 Find the point of intersection of the lines 𝑦 + 3 = 𝑥 𝑎𝑛𝑑 3𝑥 + 𝑦 = 1

Unistructural

1

0

NR

Assessor’s use only

Unistructural

1

0

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3

1.3 Solve 𝑥+1

2 ≤

4 − 𝑥

−3

1.4 Make 𝑝 the subject of the formula in the equation: √2−𝑝

3= 𝑥 + 1

Unistructural

1

0

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Unistructural

1

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4

1.5 Factorise 𝑥2 − 15𝑥 + 26

_

1.6 Solve the quadratic equation: 2𝑥2 − 7𝑥 = −3

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Unistructural

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5

1.7 Use the Laws of Indices to simplify (𝑝2)

𝑛 × 𝑝4𝑛 × 𝑝

𝑝5𝑛

1.8 Divide 𝑥3 − 13𝑥2 − 50𝑥 − 56 by (𝑥 − 7)

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Unistructural

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6

1.9 3𝑥2 + 𝒂𝑥 − 4 has a remainder of 5 when divided by (𝑥 + 3). Find the

value of “𝒂”.

1.10 Use the Binomial Theorem to expand and simplify (𝑥 − 1

𝑥)

3

Unistructural

1

0

NR

Unistructural

1

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7

1.11 Simplify 2√3 + 4√3 − √27

1.12 Solve (1 + 2𝑥)

3 −

5𝑥

2 =

(𝑥 − 3)

4


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1

0

NR

Unistructural

1

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8

1.13 Simplify ((2𝑥)3 . 𝑦−4

6𝑥5 . 𝑦−7 )−2

1.14 Find the value of 𝑘 that will make 𝑥2 − 16𝑥 + 𝑘 a perfect square.

Unistructural

1

0

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Unistructural

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9

1.15 Two complex numbers are given as: 𝑢 = 2 𝑐𝑖𝑠 𝜋

3 and 𝑣 = 8 𝑐𝑖𝑠

𝜋

2 .

Find 𝑣

𝑢 (Leave your answers in rectangular form)

𝑣

𝑢=

𝑟1

𝑟2 𝑐𝑖𝑠 (𝜃1 − 𝜃2)

Multistructural

2

1

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10

1.16 The polar form of a complex number 𝑊 is given as:

𝑊 = 8 (cos −𝜋

2+ 𝑖𝑠𝑖𝑛 −

𝜋

2)

Find the cube roots of 𝑊 and display the roots on an Argand diagram.


Extended Abstract

4

3

2

1

0

NR

11

Unistructural

1

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STRAND 2: TRIGONOMETRY

2.1 Prove the following identities:

a. sec 𝑥 + tan 𝑥 = 1 + 𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥

b. (1 + 𝑐𝑜𝑡2𝜃)(1 − 𝑐𝑜𝑠2𝜃) = 1

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2.2 Find the exact value of 𝑆𝑖𝑛 2𝑥 if 𝑐𝑜𝑠𝑥 = √3

2

Use the information in the diagram. [Do not use the calculator in this

problem].

𝑆𝑖𝑛2𝑥 = 2 𝑆𝑖𝑛𝑥 cos 𝑥


Multi-structural

2

1

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Unistructural

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2

√3

x

12

2.3 Use the grid below to sketch the graph of 𝑦 = 2 𝐶𝑜𝑠2𝑥 for 0 ≤ 𝑥 ≤ 2𝜋

𝜋

2 𝜋

3𝜋

2 2𝜋

2.4 Expand and simplify 𝐶𝑜𝑠 (𝜋

2+ 𝑃), using the Compound Angle Formula:

cos(𝐴 + 𝐵) = 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 − 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵.

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Unistructural

1

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Multistructural

2

1

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13

2.5 Sound produced from tuning forks can be modelled by trig graphs. The intensity (loudness) is measured in decibels and the pitch (how high or low the sound) is related to the frequency, which is measured in Hertz (hz) or cycles per second. Given below is the trig graph of sound produced from a tuning fork.

Use the information from the graph to find its equation. (Use 𝜋 in your equation.)

Relational

3

2

1

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14

STRAND 3: DIFFERENTIATION

3.1

Use the graph drawn below to answer question 3.1.

At which value(s) of 𝑥 is ℎ(𝑥) discontinuous? 𝑥 = 4 𝑎𝑛𝑑 𝑥 = ______

3.2 Find 𝑙𝑖𝑚𝑥 →4 2− √𝑥

4−𝑥

3.3 Calculate 𝑙𝑖𝑚𝑥 →∞ (𝑥+3)(4−2𝑥)

(2𝑥−5)2

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Uni Lateral

Level

1

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Multi-structural

2

1

0

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Unistructural

1

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Multi-structural

2

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15

3.4 Use the quotient rule to differentiate:

𝑒2𝑥

𝑥 + 1

3.5 The displacement of a particle at any time, 𝑡, is given by the equation:

𝑠(𝑡) = 6𝑡3 + 2𝑡 − 1

√𝑡

Find the acceleration of the particle at 𝑡 = 4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠.

Multistructural

2

1

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Extended Abstract

4

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16

3.6

A rectangle has its opposite vertices located on the graph of 𝑥𝑦 = 8. The 𝑥 and 𝑦 axes act as an axis of symmetry to the orientation of the rectangle.

Find the minimum perimeter of this rectangle.


Extended Abstract

4

3

2

1

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17

STRAND 4: INTEGRATION

4.1

Find the integral ∫ √𝑥 + 1 𝑑𝑥

4.2

Find ∫ 2𝑒2𝑥+2𝑑𝑥


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Unistructural

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18

4.3 Find ∫ 3𝑥2 + 2𝑥 − 1 𝑑𝑥2

1

4.4

Find the area of the shaded region.


Multistructural

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1

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Multistructural

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19

4.5

A particle moves in a straight line so that its acceleration at 𝑡 seconds is given

by the equation:

𝑎 = (6 + 2𝑡) 𝑚/𝑠2

a. What is the acceleration of the particle in a quarter of a minute?

b. The particle was momentarily at rest at 𝑡 = 12𝑠𝑒𝑐. Find the speed of the particle after 2 seconds.


Multistructural

2

1

0

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Relational

3

2

1

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20

4.6 A radioactive substance, having a half-life of 435 years, is known to decay at

a rate proportional to the amount present, which is mathematically shown as:

𝒅𝑵

𝒅𝒕 𝜶 𝒏

150 g of the substance was present initially.

Show that the amount present (N) at any time, 𝑡, is given by the expression

N = A0ekt, and hence, find the amount present after 200 years.


Extended Abstract

4

3

2

1

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Extra Blank Page If Needed

THE END

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MATHEMATICS WITH CALCULUS