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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 2โ21 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.)
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STRAND 1: ALGEBRA
1.1 Solve the linear equation 4๐ฅ โ 8 = 2(๐ฅ + 5) โ ๐ฅ
1.2 Find the point of intersection of the lines ๐ฆ + 3 = ๐ฅ ๐๐๐ 3๐ฅ + ๐ฆ = 1
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1.3 Solve ๐ฅ+1
2 โค
4 โ ๐ฅ
โ3
1.4 Make ๐ the subject of the formula in the equation: โ2โ๐
3= ๐ฅ + 1
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1.5 Factorise ๐ฅ2 โ 15๐ฅ + 26
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1.6 Solve the quadratic equation: 2๐ฅ2 โ 7๐ฅ = โ3
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1.7 Use the Laws of Indices to simplify (๐2)
๐ ร ๐4๐ ร ๐
๐5๐
1.8 Divide ๐ฅ3 โ 13๐ฅ2 โ 50๐ฅ โ 56 by (๐ฅ โ 7)
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1.9 3๐ฅ2 + ๐๐ฅ โ 4 has a remainder of 5 when divided by (๐ฅ + 3). Find the
value of โ๐โ.
1.10 Use the Binomial Theorem to expand and simplify (๐ฅ โ 1
๐ฅ)
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1.11 Simplify 2โ3 + 4โ3 โ โ27
1.12 Solve (1 + 2๐ฅ)
3 โ
5๐ฅ
2 =
(๐ฅ โ 3)
4
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1.13 Simplify ((2๐ฅ)3 . ๐ฆโ4
6๐ฅ5 . ๐ฆโ7 )โ2
1.14 Find the value of ๐ that will make ๐ฅ2 โ 16๐ฅ + ๐ a perfect square.
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1.15 Two complex numbers are given as: ๐ข = 2 ๐๐๐ ๐
3 and ๐ฃ = 8 ๐๐๐
๐
2 .
Find ๐ฃ
๐ข (Leave your answers in rectangular form)
๐ฃ
๐ข=
๐1
๐2 ๐๐๐ (๐1 โ ๐2)
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1.16 The polar form of a complex number ๐ is given as:
๐ = 8 (cos โ๐
2+ ๐๐ ๐๐ โ
๐
2)
Find the cube roots of ๐ and display the roots on an Argand diagram.
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STRAND 2: TRIGONOMETRY
2.1 Prove the following identities:
a. sec ๐ฅ + tan ๐ฅ = 1 + ๐ ๐๐๐ฅ
๐๐๐ ๐ฅ
b. (1 + ๐๐๐ก2๐)(1 โ ๐๐๐ 2๐) = 1
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2.2 Find the exact value of ๐๐๐ 2๐ฅ if ๐๐๐ ๐ฅ = โ3
2
Use the information in the diagram. [Do not use the calculator in this
problem].
๐๐๐2๐ฅ = 2 ๐๐๐๐ฅ cos ๐ฅ
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โ3
x
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2.3 Use the grid below to sketch the graph of ๐ฆ = 2 ๐ถ๐๐ 2๐ฅ for 0 โค ๐ฅ โค 2๐
๐
2 ๐
3๐
2 2๐
2.4 Expand and simplify ๐ถ๐๐ (๐
2+ ๐), using the Compound Angle Formula:
cos(๐ด + ๐ต) = ๐๐๐ ๐ด๐๐๐ ๐ต โ ๐ ๐๐๐ด๐ ๐๐๐ต.
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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 ๐ in your equation.)
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STRAND 3: DIFFERENTIATION
3.1
Use the graph drawn below to answer question 3.1.
At which value(s) of ๐ฅ is โ(๐ฅ) discontinuous? ๐ฅ = 4 ๐๐๐ ๐ฅ = ______
3.2 Find ๐๐๐๐ฅ โ4 2โ โ๐ฅ
4โ๐ฅ
3.3 Calculate ๐๐๐๐ฅ โโ (๐ฅ+3)(4โ2๐ฅ)
(2๐ฅโ5)2
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3.4 Use the quotient rule to differentiate:
๐2๐ฅ
๐ฅ + 1
3.5 The displacement of a particle at any time, ๐ก, is given by the equation:
๐ (๐ก) = 6๐ก3 + 2๐ก โ 1
โ๐ก
Find the acceleration of the particle at ๐ก = 4 ๐ ๐๐๐๐๐๐ .
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3.6
A rectangle has its opposite vertices located on the graph of ๐ฅ๐ฆ = 8. The ๐ฅ and ๐ฆ axes act as an axis of symmetry to the orientation of the rectangle.
Find the minimum perimeter of this rectangle.
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STRAND 4: INTEGRATION
4.1
Find the integral โซ โ๐ฅ + 1 ๐๐ฅ
4.2
Find โซ 2๐2๐ฅ+2๐๐ฅ
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4.3 Find โซ 3๐ฅ2 + 2๐ฅ โ 1 ๐๐ฅ2
1
4.4
Find the area of the shaded region.
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4.5
A particle moves in a straight line so that its acceleration at ๐ก seconds is given
by the equation:
๐ = (6 + 2๐ก) ๐/๐ 2
a. What is the acceleration of the particle in a quarter of a minute?
b. The particle was momentarily at rest at ๐ก = 12๐ ๐๐. Find the speed of the particle after 2 seconds.
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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:
๐ ๐ต
๐ ๐ ๐ถ ๐
150 g of the substance was present initially.
Show that the amount present (N) at any time, ๐ก, is given by the expression
N = A0ekt, and hence, find the amount present after 200 years.
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Extra Blank Page If Needed
THE END