MERIDIAN JUNIOR COLLEGE JC1 Promotional Examination Higher 2
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H2 Mathematics
9740/01
Paper 1
2 October 2015
3 Hours
Additional Materials: Writing paper List of Formulae (MF 15)
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READ THESE INSTRUCTIONS FIRST
Write your name and civics group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
You are expected to use a graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states
otherwise.
Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question. ___________________________________________________________________
This document consists of 7 printed pages.
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2 1
A cruise liner offers three types of cabins for its passengers to choose from, namely, interior, balcony and suite. The number of passengers on board occupying each type of cabin is as follows: No. of Passengers
Interior 350
Balcony 100
Suite 50
Total 500
It is desired to sample 10% of the passengers on board from different cabin types to find their opinions of the service quality provided. (i) Describe how a simple random sample may be obtained in this context.
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(ii) Explain a disadvantage of simple random sampling in this context.
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(iii) Suggest another sampling method that would not have this disadvantage, and describe how it can be carried out in this context.
2
2 x 2 − 9 x + 12 Without the use of a graphing calculator, solve 2 ≤ 1. x − 5x + 4 Hence solve
2 x 2 − 9 x + 12 > 1. x2 − 5 x + 4
MJC/2015 JC1 Promotional Examination/9740/01
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3 3
The diagram shows the graph of y = f ( x ) . Sketch, on separate diagrams, the graphs of (i)
⎛ x ⎞ y = f ⎜ − 1⎟ , ⎝ 2 ⎠
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(ii)
y2 = f ( x) ,
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showing in each case the equations of the asymptotes and the points of intersection with the axes. 4
Michael planted a bonsai tree in his backyard. In order to keep the tree at a certain height, he plans to trim its height by 15% at the start of each year. By the end of each year, the bonsai tree is expected to grow by 0.5 metres in height. Let un denote the height of the bonsai tree in metres at the end of the nth year. It is given that u0 =
2 , 3
where u0 represents the initial height of the bonsai tree. (i) Write down a recurrence relation for un . (ii) Show that un =
10 ⎛ 8 ⎞ − 0.85n ⎜ ⎟ for n ≥ 0 . 3 ⎝ 3 ⎠
MJC/2015 JC1 Promotional Examination/9740/01
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4
(iii) By sketching the graph of y =
10 ⎛ 8 ⎞ − 0.85 x ⎜ ⎟ , comment on the height of the 3 ⎝ 3 ⎠
bonsai tree in the long run.
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(iv) If Michael does not want the height of the bonsai tree to grow beyond two metres in the long run, what is the minimum percentage that the height of the bonsai tree should be trimmed each year?
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(i) Show that
1 2 3 r 2 + 3r − 1 and find + − = ( r − 1)! r ! ( r + 1)! ( r + 1)!
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⎛ r 2 + 3r − 1 ⎞ ⎜⎜ ⎟⎟ . ∑ r =1 ⎝ ( r + 1)! ⎠ n
⎛ r 2 + 5r + 3 ⎞ (ii) Hence find ∑ ⎜ ⎟⎟ . ⎜ r =1 ⎝ ( r + 2 )! ⎠ N
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(iii) Using your answer in part (i), give a reason why the series
⎛ r 2 + 3r − 1 ⎞ ⎜⎜ ⎟⎟ ∑ r =1 ⎝ ( r + 1)! ⎠ ∞
converges, and find the value of the sum to infinity.
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(There is no need to express your answers in parts (i) and (ii) as a single algebraic fraction.)
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Prove by mathematical induction that
n
4r − 3 4n + 5 = 5 − n , for n ∈ r 2 2 r =1
∑
+
.
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Hence evaluate the following in terms of N. (i)
(ii)
2N
4r − 3 , r r = N +1 2
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∑ N
4r
∑2 r =1
r
.
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5 7
(a)
Differentiate the following with respect to x , giving your answers as single fractions. (i)
(ii)
(b)
⎛ ⎞ x , ln ⎜ ⎜ √ (1− 2 x ) ⎟⎟ ⎝ ⎠
1 cos
−1
(x ) 2
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.
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The variables x and y are related by 2
e xy = y ( x 2 + 2e x ) . Find the value of
8
1 dy when x = 0 and y = . 2 dx
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A tea cup is designed such that the portion that is used to contain tea is an open cylinder inscribed in a sphere of radius 5 units. The cylinder has base radius r units and height h units and the upper and lower parts of the sphere are removed, as shown in the diagram below.
h r
(
(i) Show that h = 2 25 − r
1 2 2
).
Use differentiation to determine the exact
maximum volume of tea that the cup can hold as r and h vary.
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(ii) It costs $0.02 per unit2 to coat the interior of the open cylinder with a layer of thermal insulation of negligible thickness. Find the cost of the thermal insulation for the tea cup with maximum volume found in part (i).
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6 9
The function f is defined by
f :x
x 2 − 2 x − 1, − 1 ≤ x ≤ 1.
(i) Define f −1 in a similar form and sketch the graphs of y = f ( x ) and y = f −1 ( x ) on a single diagram, showing clearly the relationship between the graphs.
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The function g is defined by
g:x
⎧⎪ 9 − 3x, 0 ≤ x < 3, ⎨ 2 ⎪⎩( x − 3) , 3 ≤ x < 6,
and that g ( x ) = g ( x + 6 ) for all real values of x. (ii) Sketch the graph of y = g ( x ) for −2 ≤ x ≤ 8 .
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(iii) Give a reason why the composite function gf exists and hence state its range. [2]
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⎛ 0 ⎞ ⎛1⎞ ⎛ −1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The equations of two planes p1 and p2 are r = ⎜ 1 ⎟ + β ⎜1⎟ + γ ⎜ 0 ⎟ , β , γ ∈ ⎜ 2 ⎟ ⎜1⎟ ⎜ 4 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
r
and
⎛ 2 ⎞ ⎜ ⎟ ⎜ 0 ⎟ = −7 a respectively, where a is a constant. ⎜ 1 ⎟ ⎝ ⎠
(i) Find the equation of p1 in scalar product form.
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(ii) Calculate the acute angle between p1 and p2 .
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(iii) Find the position vector of the foot of perpendicular from point A ( a,1, a ) to
p2 in terms of a.
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Another plane p3 is defined by the equation 3a − bx − z = 0 , where b is a constant. It is also given that the distance between p2 and p3 is greater than √ 5 units. (iv) State the value of b and using your result in part (iii), find the set of possible values of a. (You must show sufficient working to justify your answer).
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7 11
(a)
The sum of the first n terms of a series is n ( 4n − 1) . Obtain an expression for the nth term of the series. Hence prove that the series is arithmetic.
(b)
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A geometric series has common ratio r, and an arithmetic series has first term a and common difference d, where a and d are non-zero. The second, fifth and seventh terms of the arithmetic series are three consecutive terms of the geometric series. (i) Show that the geometric series is convergent.
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(ii) It is given that the first term of the geometric series is the same as the first term of the arithmetic series. Let S be the sum to infinity of all the even-numbered terms of the geometric series and An be the sum of the first n terms of the arithmetic series. Given a > 0 , find the least value of n such that S + An < 0 .
END OF PAPER
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