mathematics with calculus
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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 2β21 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 4π₯ β 8 = 2(π₯ + 5) β π₯
1.2 Find the point of intersection of the lines π¦ + 3 = π₯ πππ 3π₯ + π¦ = 1
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1.3 Solve π₯+1
2 β€
4 β π₯
β3
1.4 Make π the subject of the formula in the equation: β2βπ
3= π₯ + 1
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1.5 Factorise π₯2 β 15π₯ + 26
_
1.6 Solve the quadratic equation: 2π₯2 β 7π₯ = β3
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1.7 Use the Laws of Indices to simplify (π2)
π Γ π4π Γ π
π5π
1.8 Divide π₯3 β 13π₯2 β 50π₯ β 56 by (π₯ β 7)
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1.9 3π₯2 + ππ₯ β 4 has a remainder of 5 when divided by (π₯ + 3). Find the
value of βπβ.
1.10 Use the Binomial Theorem to expand and simplify (π₯ β 1
π₯)
3
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1.11 Simplify 2β3 + 4β3 β β27
1.12 Solve (1 + 2π₯)
3 β
5π₯
2 =
(π₯ β 3)
4
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1.13 Simplify ((2π₯)3 . π¦β4
6π₯5 . π¦β7 )β2
1.14 Find the value of π that will make π₯2 β 16π₯ + π a perfect square.
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1.15 Two complex numbers are given as: π’ = 2 πππ π
3 and π£ = 8 πππ
π
2 .
Find π£
π’ (Leave your answers in rectangular form)
π£
π’=
π1
π2 πππ (π1 β π2)
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1.16 The polar form of a complex number π is given as:
π = 8 (cos βπ
2+ ππ ππ β
π
2)
Find the cube roots of π and display the roots on an Argand diagram.
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Extended Abstract
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STRAND 2: TRIGONOMETRY
2.1 Prove the following identities:
a. sec π₯ + tan π₯ = 1 + π πππ₯
πππ π₯
b. (1 + πππ‘2π)(1 β πππ 2π) = 1
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2.2 Find the exact value of πππ 2π₯ if πππ π₯ = β3
2
Use the information in the diagram. [Do not use the calculator in this
problem].
πππ2π₯ = 2 ππππ₯ cos π₯
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β3
x
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2.3 Use the grid below to sketch the graph of π¦ = 2 πΆππ 2π₯ for 0 β€ π₯ β€ 2π
π
2 π
3π
2 2π
2.4 Expand and simplify πΆππ (π
2+ π), using the Compound Angle Formula:
cos(π΄ + π΅) = πππ π΄πππ π΅ β π πππ΄π πππ΅.
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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 π 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 π₯ is β(π₯) discontinuous? π₯ = 4 πππ π₯ = ______
3.2 Find ππππ₯ β4 2β βπ₯
4βπ₯
3.3 Calculate ππππ₯ ββ (π₯+3)(4β2π₯)
(2π₯β5)2
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3.4 Use the quotient rule to differentiate:
π2π₯
π₯ + 1
3.5 The displacement of a particle at any time, π‘, is given by the equation:
π (π‘) = 6π‘3 + 2π‘ β 1
βπ‘
Find the acceleration of the particle at π‘ = 4 π ππππππ .
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3.6
A rectangle has its opposite vertices located on the graph of π₯π¦ = 8. The π₯ and π¦ axes act as an axis of symmetry to the orientation of the rectangle.
Find the minimum perimeter of this rectangle.
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Extended Abstract
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STRAND 4: INTEGRATION
4.1
Find the integral β« βπ₯ + 1 ππ₯
4.2
Find β« 2π2π₯+2ππ₯
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4.3 Find β« 3π₯2 + 2π₯ β 1 ππ₯2
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 π‘ seconds is given
by the equation:
π = (6 + 2π‘) π/π 2
a. What is the acceleration of the particle in a quarter of a minute?
b. The particle was momentarily at rest at π‘ = 12π ππ. 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:
π π΅
π π πΆ π
150 g of the substance was present initially.
Show that the amount present (N) at any time, π‘, 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
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