MATHEMATICS Higher 2 Paper 1
ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT
9740 29 September 2015
JC 1 PROMOTIONAL EXAMINATION Time allowed: 3 hours Additional Materials: List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Index number, Form Class, graphic and/or scientific calculator model/s on the cover page. Write your Index number and full name on all the work you hand in. Write in dark blue or black pen on your answer scripts. You may use a soft pencil for any diagrams or graphs. Do not use 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 graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in the 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. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together This document consists of 7 printed pages.
Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 1 of 7
ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT JC 1 Promotional Examination 2015
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MATHEMATICS 9740 Higher 2 Paper 1
Index No: Form Class: ___________ Name: _________________________ Calculator model: _____________________ Arrange your answers in the same numerical order. Place this cover sheet on top of them and tie them together with the string provided. Question no.
Marks
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5
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6
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7
/ 2
8
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GT:
I:
Summary of Areas for Improvement Knowledge (K)
Formula (F)
Careless Mistakes (C)
Read/Interpret Presentation (P) Qn wrongly (R)
Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 2 of 7
1 Find (i)
∫
(ii) ∫ 2
ln x dx , x 2x −1
[1]
2 x2 + 4 x + 6
dx .
[4]
Given that x is a sufficiently small angle measured in radians, show that
⎛ π ⎞ sin ⎜ − x ⎟ ⎝ 4 ⎠ ≈ 1 1 + ax + bx 2 , where a and b are constants to be determined. 5 − cos x 2 2
(
)
[4]
B
3 A
x
y
C
The diagram above shows triangle ABC which has fixed points A and C and a variable point B such that AB = kBC , where k is a constant such that 0 < k < 1 . The angles BAC and BCA, measured in radians, are x and y respectively. (i)
Show that sin y = k sin x .
(ii)
By successively differentiating the equation in part (i), show that
[1]
2
cos y (iii)
d2 y ⎛ dy ⎞ − sin y ⎜ ⎟ = −k sin x . 2 dx ⎝ dx ⎠
[1]
Hence find the Maclaurin series for y , up to and including the term x3 .
[3]
4 (a) In a training exercise, an athlete runs around a circular path. He takes 20 seconds to complete his first lap of running. Subsequently, he takes 15% more time to complete each lap than his previous lap, that is he takes 23 seconds to complete his second lap. (i)
Find the time taken by the athlete to complete his 10th lap.
(ii)
Write down an expression in terms of n, for the total time taken by the athlete
[2]
to complete n laps. Hence, without the use of a graphing calculator, find the number of laps he had completed if he had been running for 32 minutes.
[4]
(b) An arithmetic progression has first term 1 and common difference 2. The terms of this arithmetic progression are grouped into sets containing 1, 2, 3, 4, … terms, as indicated below. {1}, {3, 5}, {7, 9, 11}, {13, 15, 17, 19}, {21, 23, 25, 27, 29}, … Find the last term in the 100th set.
[3] [Turn over
Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 3 of 7
5
1 + 2k − kx k is true for all real values < ( x − 1) ( x − 1)2
Find the range of values of k for which of x, except x = 1.
[5]
6
y
x=2
y = f (1 − x) A(6, 4)
B(0,3) 2
0
y =2 2
C( 3 ,0)
x
The diagram shows the graph of y = f (1 − x) . The curve has a maximum point at A(6,4) and axial intercepts at B(0,3) and C(3,0) . The lines x = 2 and y = 2 are the asymptotes. Sketch the graph of (a)
y = f(x + 1),
[2]
(b)
y=
1 , f (1 − x)
[2]
(c)
y = −f (1 − x) ,
[2]
stating the equations of any asymptotes, coordinates of any stationary points and any points of intersection with the axes. 7
The diagram below shows how the concentration of a drug in the bloodstream varies with time, 2 minutes after a dose of the drug is taken orally. The drug is absorbed rapidly into the blood stream reaching a maximum concentration of 30 micrograms per litre in the first 10 minutes. After 10 minutes, the concentration of the drug begins to decrease as shown in the diagram below. x (micrograms per litre)
o
2
t(minutes) Sketch a graph showing the rate of change of concentration of the drug in the body against time for t > 2. [2] Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 4 of 7
8
The curve C has equation f ( x ) =
λ x 2 + 2λ x + 4λ x+2
where λ is a constant and λ > 0.
(i) Without using a calculator, find in terms of λ the coordinates of the stationary points of C.
[3]
(ii) Sketch the graph of y = f ( x) stating the coordinates of the stationary points and the equations of any asymptotes.
[2]
1
(iii) Find
∫ f (− x )dx exactly in terms of λ .
[3]
−1
9
A curve C has parametric equations x = at3 ,
y=
a (1 + t 2 )
where a is a positive constant and t is a real parameter . (i)
Sketch C, showing clearly the asymptote(s) and coordinate(s) of any intersection with the axes.
(ii)
[2]
Find the equation of the tangent L to the curve C at A( a, 12 a ), leaving your answer in the form y = mx + c where m and c are constants.
(iii)
[4]
For t > 0 , find the exact value of the area of the region bounded by C, L and the y − axis .
10 (a) Show that
[6]
r +1 A B , where A and B are constants to be found. = 2+ 2 r (r + 2) r (r + 2)2 2
2 3 4 n +1 in terms of n. + 2 2+ 2 2+ + 2 2 (1) (3) (2) (4) (3) (5) (n) (n + 2)2 (There is no need to express your answer as a single algebraic fraction.) Hence find
[1]
2
2 4 6 + 2 2+ 2 2+ 2 (1) (3) (3) (5) (5) (7) for all positive integers n.
(b) Prove, by induction,
2
+
[3]
2n n(n + 1) = 2 2 (2n − 1) (2n + 1) (2n + 1) 2 [5]
(c) From the result found in (a) and the result proven in (b), find the sum of the series 3 5 7 21 , showing your workings clearly. [3] + 2 2+ 2 2+ + 2 2 (2) (4) (4) (6) (6) (8) (20)2 (22)2 [Turn over
Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 5 of 7
⎞ d ⎛ 1 ⎜ ⎟ = − dx ⎜⎝ x 2 − 4 ⎟⎠
11 (a) (i) Show that
(ii) By using the substitution u =
x
(x
2
−4
1 , find y
Using the result in (i) and (ii), find
∫
∫
3
.
[1]
)
1 u u2 − 4 x ln 2 x
(x
2
−4
3
du .
[4]
dx .
[2]
)
y
(b)
P
R 3
O A curve C has equation y 2 =
C
4
x
2 4 ln x and the parabola P has equation ( y − 2 ) = x − 2 . x
The diagram above shows the region R, bounded by the parts of C and P, and the lines x = 3 and x = 4 .
Find the volume of revolution when R is rotated 2 π radians about the x-axis.
12
[3]
A curve is defined by equation Ax 2 + By 2 + Cx + Dy + 16 = 0 , where A, B, C , D are real constants. A stationary point on the curve lies on the y-axis. The line
3 y + 2 x + 8 3 + 6 = 0 intersects the curve at y = −2 and is parallel to the tangent to the curve at the point ( −4,6 ) . Form a system of linear equations and find the equation of the curve.
[5]
Sketch the graph of the curve, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.
Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 6 of 7
[3]
13
The function f is defined by
f:x
x − k , − 2k < x < 2k where k > 2 .
The diagram shows the graph of the function, y = g ( x ) , where −2 ≤ x ≤ 2 . The graph crosses the x – axis at x = −2 , x = −1 , x = 1 and x = 2 , and has turning points at y ( −1.5, −1) , (0, 4) and (1.5, −1) . 4
| −2 | −1.5
|
−1
O
|
1|
1.5
|
2
x
−1
(i)
Explain why the composite function fg exist.
(ii)
Find in terms of k,
(iii)
(a)
the value of fg( −1 ),
[1]
(b)
the range of fg.
[2]
Given that k = 3 , sketch the graph of y = fg ( x) , stating the coordinates of the turning points, if any.
14
[2]
[2]
A movie screen on a vertical wall is 6m high and 4m above the horizontal floor. A boy who is standing at x m away from the wall has eye level at 1m above the floor as shown in the diagram below. The viewing angle of the boy at that position is θ and the angle of elevation of the bottom of the screen is y. (i) State tan y in terms of x.
[1]
(ii) By expressing θ in terms of x or otherwise, find the stationary value of θ giving your answer in exact form and determine if the value is a maximum or minimum value showing your working clearly.
[6]
6m
θ
y
4m
1m xm ~ End of Paper ~ Anglo-Chinese Junior College H2 Mathematics 9740: 2015 JC 1 Promotional Examination Page 7 of 7
