Secondary School Mathematics & Science Competition 2016
Mathematics
Date
: 24 Apr, 2016
Total no. of pages
: 17
(excluding the current page)
Time allowed : 9:30 - 10:45 a.m. (1 hour 15 minutes)
1.
Total marks
: 100
Write the Candidate Number, Exam Centre Number, Seat Number, Name in English, Name
of School, Form, Language and Subject in the spaces provided on the Part A MC Answer
Sheet and the Part B Answer Sheet.
2.
When told to open this question paper, the candidate should check that all the questions are
there. Look for the words ‘END OF PAPER’ after the last question.
3.
Answer ALL questions in Part A.
(a) The candidate is advised to use an HB pencil to mark all answers on the MC
Answer Sheet.
(b) There are twelve questions carrying 2 marks each and twenty questions carrying
3 marks each.
(c) The candidate should mark only ONE answer for each question.
If the candidate
marks more than one answer, his/her will receive NO MARK for that question.
4.
Answer ANY TWO questions out of six questions in Part B.
(a) Each question carries 8 marks.
(b) Unless otherwise specified, answers may be exact values or mathematical
expressions.
(c) Answers should be written in the space provided on the Part B Answer Sheet.
5.
No mark will be deducted for wrong answers.
6.
The diagrams in this paper are not necessarily drawn to scale.
© The Hong Kong Polytechnic University 2016 All Rights Reserved.
Part A:
1.
Multiple Choice Questions (84 marks)
 1 


 2016 
A.
B.
C.
D.
2016
 2016 
224 9

1.
1.
2016 .
2016 .
(2 marks)
2.
5a  5b  a 2  b2 
A.
 a  b5  a  b .
B.
 a  b5  a  b .
C.
 a  b5  a  b .
D.
 a  b5  a  b .
(2 marks)
3.
If
A.
B.
C.
D.
1 1 2
  , then g 
f g h
h
f .
2
h f
.
2
fh
.
2f h
fh
.
2f h
(2 marks)
Page 1
4.
In the figure, the diameter of a cylindrical pole is 6 cm and its length is 16 cm . A and B
are points on the edge of the two bases, where AB  16 cm , and C is the mid-point of AB.
The pole is decorated by a string around the pole from A to B via C.
length of the string.
B
C
Find the shortest
A
6 cm
16 cm
A.
B.
C.
16 cm
D.
32 cm
20 cm
22 cm
(3 marks)
5.
Which of the following compound inequalities have no solutions?
A.
7 x  1  7

9 x  5  3
B.
7 x  1  7

9 x  5  3
C.
7 x  1  7

9 x  5  3
D.
7 x  1  7

9 x  5  3
(3 marks)
6.
Let k be a constant.
A.
B.
C.
D.
If x 2  kx  k  1  0 for all real values of x, then
k  2 .
k  2 or k  2 .
2  k  2 .
there are no real solutions for k.
(3 marks)
Page 2
7.
 a 2  2a  k  0
If  2
, a  b and a 2  2b  5 , then k =
 b  2b  k  0
A.
B.
C.
D.
9 .
1.
0.
1.
(2 marks)
8.
Let  and 
be the roots of the equation 2 x2  3x  4  0 .
1
1
equations has roots
and
?


A.
B.
C.
4 x 2  3x  2  0
D.
4 x 2  3x  2  0
Which of the following
4 x 2  3x  2  0
2 x 2  3x  4  0
(3 marks)
9.
Which of the following functions can be represented by the graph in the figure?
y
3
2
y  f  x
(30, 2)
1
0
A.
f  x   sin  x  30 1
B.
f  x   sin  x  30  1
C.
f  x   cos  x  30  1
D.
f  x   cos  x  30  1
x
90
180
270
360

(2 marks)
Page 3
10. The figure shows the graph of y  ax 2  bx  c .
Which of the following cannot be
possible values of a, b and c?
y  ax 2  bx  c
A.
B.
C.
D.
a  1, b  1, c  1 .
a  1, b  2, c  3 .
a  4, b  1, c  4 .
a  1, b  4, c  4 .
(2 marks)
11.
Which of the following best represents the graph of y  2 log 2 x ?
A.
C.
B.
D.
(2 marks)
Page 4
12. Let 2h  5k  100 p , where h, k and p are non-zero real numbers.
A.
B.
C.
D.
Find the value of
p p
 .
h k
1
2
1
log 7
log 7
2
(3 marks)
1
1
13. If log 64 x  log 64 y  , then
2
3
1
1
x y .
A.
2
3
B.
1
x y .
3
C.
x
D.
x4 y.
1
y.
3
(2 marks)
14. Let x, y  0 .
A.
B.
C.
D.
If log x  log 5 y  A , then
x

y
A2 .
A5 .
2A .
5A .
(3 marks)
3
2
15. It is given that f  x   x  3x  3 . Find the remainder when f  x  1 is divided by x 1 .
A.
B.
C.
D.
3
7
17
24
(3 marks)
Page 5
16. If a polynomial f  x  is divisible by x  5 , which of the following polynomials must be
a factor of f  2 x  3 ?
A.
B.
C.
D.
x4
x 8
2x  3
2 x  13
(3 marks)
17. If the H.C.F. of 4x 2 y and P is 2x 2 y , while the L.C.M. of 8xy and P is 8x 3 y 2 , then P is
A.
2x 2 y 2 .
B.
2x 3 y 2 .
C.
2x 2 y .
D.
2x 3 y .
(3 marks)
18. Which of the following equations cannot be reduced to a quadratic equation?
A.
4x  31 2 x   2016  0
B.
1
1
 2
1
x  2016 x  20162
C.
2  log 2016 x   log 2016 x3  2  0
D.
2016 x6  20 x 4  1  0
2
(2 marks)
Page 6
19. For the equation
3
2
 x
 3 , which of the following statements must be true?
2 1 2 1
x
I. The equation has two distinct real solutions.
II. The equation has at least one irrational solution.
III. The real solution(s) of the equation is/are positive.
A.
B.
C.
D.
I only
III only
I and II only
II and III only
(3 marks)
20. If x varies directly as y and inversely as the square of z, which of the following must NOT be
constant?
A.
y
xz 2
B.
x2 z 4
y2
C.
xy
z2
D.
x3 z 6
y3
(2 marks)
21. W is partly varies as t and partly varies as t 2. When t  4 , W  32 . When t  2 ,
W  0 . Find the greatest value of W.
A.
0
B.
C.
D.
4
8
32
(3 marks)
Page 7
22. In the figure, O is the centre of the circle.
parallelogram.
AB is a diameter of the circle, and OBCD is a
Find ABC.
A
O
B
D
C
A.
B.
C.
D.
45o
50o
60o
65o
(3 marks)
23. In the figure, O is the centre of the circle ABCD.
PQ intersects the circle at A and B.
QR
intersects the circle at C and D. If OPQ = 13, AB = CD and OP = 2OR, find ORQ.
(Give the answer correct to 3 significant figures.)
A.
B.
C.
D.
26.4
26.5
26.6
26.7
(3 marks)
Page 8
24. In the figure, PQ is parallel to the base BC of
ABC . APB and AQC are straight lines.
The circle passing through P and tangent to AC at Q cuts AB again at R. Which of the
following points are concyclic?
A
P
Q
R
B
A.
B.
B, C, Q, P
B, C, Q, R
C.
D.
C, Q, P, R
None of the above
C
(3 marks)
25. The figure shows the graph of the straight line ax + 1 = by. Which of the following(s) is /
are true?
I. b > 0
II. a > 0
III. a  1
A.
II only
B.
C.
D.
III only
I and II only
I and III only
(2 marks)
Page 9
26. Let x  2 y  12  0 be the equation of straight line L1 .
y-axis at A and B respectively.
L1 intersects the x-axis and
If C  2, 0  is a point on the x-axis, find the coordinates
of the orthocentre of ABC.
A.
 0, 4
B.
 0, 3
C.
1, 3
D.
 4, 0
(3 marks)
27. The equations of the straight lines L1 and L2 are 7 x  3 y  1 and 4 x  4 y  5
respectively.
Let P be a moving point in the rectangular coordinate plane.
If the
perpendicular distance from P to L1 is equal to the perpendicular distance from P to L2 ,
then the locus of P is
A.
B.
a straight line.
a circle.
C.
D.
a parabola.
a pair of straight lines.
(3 marks)
28. The standard deviation of a set of 2016 data
a1, a2 , a3 , ... , a2016
is  . Which of the
following set of data has standard deviation 2016 ?
A.
a1   , a2   , a3   , ... , a2016   
B.
a1  2016 , a2  2016 , a3  2016 , ... , a2016  2016 
C.
2016a1, 2016a2 , 2016a3 , ... , 2016a2016
D.

2016
a1 ,  2016 a2 ,  2016a3 , ... ,  2016a2016 
(3 marks)
Page 10
29. In a meeting, each of the participants shakes his/her hand exactly once with every other.
there are totally 66 handshakes amongst men and 28 handshakes amongst women, how
many handshakes took place between a man and a woman?
A.
B.
C.
D.
If
94
96
108
190
(3 marks)
30. Jacky walked from his home to a cinema to buy tickets. After a period of time, he walked
back to his home at a lower speed. Which of the following graphs shows the distance of
Jacky from his home vs. time?
A.
Distance
O
B.
Distance
O
C.
Time
Distance
O
D.
Time
Time
Distance
O
Time
(2 marks)
Page 11
31. In the figure, a square paper ABCD is folded along EF such that the corner C meets the
mid-point of AD. C ' and B ' are the new positions of C and B respectively. C ' B ' cuts AB
at G. Find AG : DE .
G
A
B'
F
B
C'
D
A.
B.
C.
D.
E
2:1
3:2
12 : 7
16 : 9
C
(3 marks)
32. In the figure, the four small circles are identical and tangent to each other. They are inscribed
inside a larger circle. The diameters of the small circles are 2 cm. Find the area of the larger
circle.
A.
 2  3 2 
cm2
B.
3  2 2 
cm2
C.
4 2  3 2  cm2
D.
4 3  2 2  cm2




(3 marks)
Page 12
Part B (8 marks per question for total of 16 marks) (Answer ANY TWO questions)
33. Figure 1(a) shows a sculpture of a four-sided pyramid by Sol LeWitt (1928 – 2007).
Source: https://tagreen.files.wordpress.com/2010/06/pyramid.jpg
Figure 1(a)
Figure 1(b)
The pyramid has 12 vertical layers from front to back made of cubic stones. Figure 1(b)
shows the front 3 layers of visible stones of the pyramid. The layer most in front is the 1st
layer.
(a) Write down the number of visible stones in the nth layer in terms of n.
(2 marks)
(b)
Find the total number of visible stones in these 12 layers.
(2 marks)
(c)
A number starting from 1 is assigned to each visible stone as shown in Figure 2.
16
10 14 15
9 13 4 11 7
5 6 1 2 3 12 8
Figure 2
The number on the top visible stone of the 1st layer is 4. The number on the top
visible stone of the 2nd layer is 16 and so on. Write down the number on the top
visible stone of the nth layer in terms of n.
(2 marks)
(d)
On which layer is the visible stone with number 387?
Page 13
(2 marks)
34.
Let C : x 2  y 2  2 x  4 y  4  0 be the equation of a circle. Suppose that the straight line
L : y  x  b intersects C at two distinct points A  x1 , y1  and B  x2 , y2  .
(a)
Express x1  x2 , y1  y2 , x1 x2 and y1 y2 in term of b.
(b)
Suppose that the circle having the segment AB as its diameter passes through the origin.
Find the value(s) of b.
(4 marks)
(4 marks)
 2x  4 
35. Let y  
 , where 2  x  4 .
 x 1 
3
(a)
dy a(2 x  4) n

It is known that
, where a, n and m are constants. Write down the
dx
( x  1)m
values of a, n and m.
(3 marks)
(b)
Find the least value of y for 2  x  4 .
(c)
(i) Find
d2y
.
dx 2
 2x  4 
(ii) Is the graph of y  

 x 1 
(2 marks)
(2 marks)
3
concave upward or downward for 2  x  4 ?
(1 mark)
Page 14
36. (a)
(b)
Let y 
e 3t
.
1  et
dy
.
dt
Find
(2 marks)
Let N be the number of certain species of rabbits. The rate of change of the number
of rabbits can be modelled by
dN 3e3t  2e4t  1  2et  e 2t
, where t is the time

2
dt
1  et 
measured in weeks. There are 100 rabbits when t = 4. Express N in terms of t.
(c)
(4 marks)
Estimate the number of rabbits when t  2 . Give your answer correct to the nearest
integer.
(2 marks)
37. Let m and n be positive integers, and x  0 .
1  x   1  x 
n
n 1
 1  x 
n2

It is given that
 1  x 
nm
1  x 

n  m 1
 1  x 
n
x
is an identity.
(a)
If Cnn  Cnn1  Cnn2 
 Cnnm  Crk , express k and r in terms of m and n. (6 marks)
(b)
Hence, or otherwise, evaluate
30
 r  r  1 r  2  r  3 .
r 4
Page 15
(2 marks)
38. In Figure 3, City H plans to build a highway and a bridge from A on the shore to B on the
island. The length of the straight shoreline AC is 9 km , AC  BC and BC  6 km .
It costs $M per km to build the highway on the shore and $N per km to build a bridge across
the sea to B, where N  M . The total cost of building the highway together with the bridge
is denoted by $T. Suppose that the bridge comes out of the shore x km from C.
Shore
x km
A
C
Highway
Bridge
B
Island
Figure 3
(a)
Express T in terms of x, M and N.
(b)
Solve
(c)
Find
(d)
Find the maximum value of
(1 mark)
dT
 0 and express the answer in terms of M and N.
dx
d 2T
in terms of N.
dx 2
(2 marks)
(2 marks)
M
when the total cost T is minimum.
N
(Give your answer in surd form.)
(3 marks)
END OF PAPER
Page 16
FORMULAS FOR REFERENCE
sin( A  B )  sin A cos B  cos A sin B
cos( A  B )  cos A cos B sin A sin B
tan( A  B ) 
tan A  tan B
1 tan A tan B
sin A  sin B  2sin
A B
A B
cos
2
2
sin A  sin B  2 cos
A B
A B
sin
2
2
cos A  cos B  2 cos
A B
A B
cos
2
2
cos A  cos B  2sin
A B
A B
sin
2
2
2sin A cos B  sin( A  B )  sin( A  B)
2 cos A cos B  cos( A  B )  cos( A  B )
2sin A sin B  cos( A  B)  cos( A  B )
Page 17