Name:
Index Number:
Class:
DUNMAN HIGH SCHOOL Promotional Examination Year 5
MATHEMATICS (Higher 2) Paper 1 Additional Materials:
9740/01 01 October 2015 3 hours
Answer Paper Graph paper List of Formulae (MF15)
READ THESE INSTRUCTIONS FIRST Write your Name, Index Number and Class 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 an approved 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.
For teachers’ use: Qn Q1 Q2 Q3 Q4 Score Max Score
5
6
7
7
Q5
Q6
Q7
Q8
7
7
7
7
Q9 Q10 Q11 Q12 Q13 Total 8
8
9
10
© DHS 2015 This question paper consists of 7 printed pages (including this cover page).
12
100
2 1
2
n 2 ( n + 1) Given that ∑ r = , without expanding ( r − 1)3 , find 4 r =1 n
3
3
3
n
3
∑ ( r − 1) .
[2]
r =1
n
2
∑r
By considering ( r −1) = r − 3r + 3r −1 and the earlier result, find a formula for
. You
r =1
should factorise your answer fully.
2
2
[3]
⎛ 2 ⎞ ⎛ −2 ⎞ The position vectors of the points A and B relative to the origin O are ⎜ −1 ⎟ and ⎜ 5 ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎝ ⎠ ⎝ ⎠ respectively. The point P lies between A and B such that AP = λ AB where 0 < λ < 1 . (i) Find the position vector of P in terms of λ .
[1]
(ii) If OP is perpendicular to AB , find the value of λ .
[2]
Given that λ = 13 ,
3
(iii) Find the area of triangle OPA.
[2]
(iv) Write down the ratio of the area of triangle OPB to the area of triangle OPA.
[1]
x2 y 2 The curve C has equation − = 1. Sketch the curve, stating the coordinates of any points of 4 16 intersection with the axes and the equations of any asymptotes.
[2]
The region R is bounded by C, the asymptote of C with positive gradient, the line y = 2 and the x-axis. Find the exact value of the volume of revolution formed when R is rotated completely about the y-axis. [5]
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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3 4
The diagram below shows the graph of y = f ( x ) with asymptotes y = 2 and x = 0. The curve has turning points at ( −2, 2) and (4, −1). y
y = f ( x) y=2
×
( −2, 2) O
2
×
6
x
( 4, −1)
On separate diagrams, sketch the graph of 1 (i) y = and f ( x)
[3]
(ii) y = f ( x )
[2]
By adding a suitable graph to the graph in part (ii), find the number of distinct roots of the 2
equation x2 + ⎡⎣f ( x )⎤⎦ = 4. 5
[2]
A sequence of numbers {xn} is defined by the relation
n2 xn for n = 1, 2, 3, … x1 = 1 and xn+1 = n +1
(i) Find the values of xn for n = 2, 3, 4 and 5.
[2]
(ii) By considering nxn , make a conjecture for xn in terms of n.
[1]
(iii) Prove your conjecture in part (ii) by induction for n = 1, 2, 3, …
[4]
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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4 6 y
y = 2 x + 2x
1 O
... ... 1 n
2 n
n − 2 n −1 n n
3 n
1
x
The diagram shows the curve with equation y = 2 x + 2x for 0 ≤ x ≤ 1. The region bounded by the curve, the axes and the line x = 1 is denoted by S. The area of region S can be approximated by 1 the total area of n rectangles, A, as shown in the diagram. Each rectangle is of width . n (i) Show that A can be given by 1 2 n −1 ⎞ ⎫ 1 ⎧ n − 1 ⎞ ⎛ ⎪ ⎛ 1 2 ⎪ n n n ⎨2 ⎜ + + ... + ⎟ + ⎜1 + 2 + 2 + ... + 2 ⎟ ⎬ . n ⎪ n ⎠ ⎝ ⎠ ⎪ ⎩ ⎝ n n ⎭
(ii) Hence, find A in the form p +
q + n
r n ( 2 − 1) 1 n
[2]
, where p, q and r are constants to be
determined.
[3]
(iii) By considering region S, find the limit of A as n → ∞ , giving your answer to 2 decimal places. [2]
7
(a) Given that
dy = cos 2 3 y and y = 14 π when x = −1 , find an expression for y in terms of x. dx [4] ax
(b) The general solution of a differential equation is given as y = x + e , where a is a constant. Sketch, on a single diagram, two non-linear members of the family of solution curves. [3]
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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5 8
A certain type of cell is grown for research. The number of cells, in millions, at the end of n days is denoted by un and the proportion of cells that died over a day is denoted by m, where 0 < m ≤ 1. The sequence u0 , u1, u2 , ... is given by
un+1 = (1 − m) un + 5 . It is known that u0 = 10 and u2 = 15.4. (i) Find the value of m. (ii) If un <
[3]
5 5 , show algebraically that un < un +1 < for n ≥ 0. m m
[2]
Hence or otherwise, explain algebraically why the number of cells will not grow beyond 25 million. [2] 9
[It is given that a cone of radius r and height h has volume 13 πr 2 h.] A
h cm
D
a cm
B
O
C
R cm
A sphere of constant radius a cm is inscribed in a right circular cone of radius R cm and height h cm, such that the sphere is in contact with the centre of the circular base, O , and the slant surface of the cone. (i) By
R=
considering ha 2 √ ( h − 2ha )
two
different
expressions
for
AC or
.
otherwise,
show
that [3]
(ii) As h varies, find the minimum volume of the cone in terms of a.
[5]
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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6
10
The function f is given by f : x constant.
2x2 + c for x ∈ R, x ≠ −1, x ≠ 1 where c is a positive x2 − 1
(i) Sketch the graph of y = f ( x ) . If the domain of f is restricted to x ≥ k such that f −1 exists, state the minimum value of k . [3] In the rest of the question, the function f is now defined by
f :x
2 x2 + 1 , x > 1. x2 −1
Another function g is given by e− x , x > 0.
g: x
(ii) Show that the composite function gf exists. Find the range of gf, giving your answer in exact form. Show your working clearly. [3] (iii) Without using f 2 ( x) , find a solution to the equation f 2 ( x) = x.
11
(i) Express f ( x) =
[2]
5x A Bx + C as + . 2 (1 + 2 x ) (1 + x ) 1 + 2 x 1 + x 2
[2]
(ii) Hence obtain the series expansion of f ( x ) up to and including the term in x 3 .
[3]
1
⌠ x (iii) Using your result in part (ii), find an approximate value for ⎮ dx. 2 ⌡0 (1 + 2 x ) (1 + x ) Give your answer in its lowest fraction. [2] 1
⌠ x (iv) Use your calculator to evaluate ⎮ dx. Justify whether your answer in 2 ⌡0 (1 + 2 x ) (1 + x ) part (iii) is an appropriate approximation. [2]
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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7 2
12
1 ⎛ 4 x ⎞ ⌠ (a) By means of the substitution x = tan θ , evaluate ⎮ ⎜ dx exactly. ⌡0 ⎝ 1 + x 2 ⎟⎠
(b) (i) By
considering
x2 3 2 2
(1 − x )
x ⎛ = x ⎜ 3 ⎜ (1 − x 2 ) 2 ⎝
⎞ ⎟ , ⎟ ⎠
use
integration
x2 ⌠ find dx. 3 ⎮ 2 2 ⌡ (1 − x )
by
parts
to
[3]
(ii) It is given that a varies with t such that
da = dt
1 4
. Find the rate of change of
a x2 ⌠ 1 3 dx with respect to t at a = 2 . ⎮ 2 2 ⌡0 (1 − x )
13
[4]
[3]
A curve C has parametric equations
x = eθ cos θ , y = eθ sin θ where 0 ≤ θ ≤ 12 π . (i) Find the point on C whose tangent is parallel to the y -axis.
[4]
(ii) Sketch C , stating the coordinates of the point found in part (i) and the points where the curve meets the axes. [2] (iii) Find the equation of the tangent to curve C at the point where θ = 12 π .
[2]
(iv) Hence find the area bounded by the tangent in part (iii), the curve C and the [4] x -axis.
DHS 2015 Year 5 H2 Mathematics Promotional Examination
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