MATHEMATICS WITH CALCULUS - Pacific Community · 2019. 4. 3. · 1-1.8 1.9 0 1.10 For questions 1.8...
Transcript of MATHEMATICS WITH CALCULUS - Pacific Community · 2019. 4. 3. · 1-1.8 1.9 0 1.10 For questions 1.8...
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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.)
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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
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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?
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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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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