MATHEMATICS Compulsory Part PAPER 1 Question

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2015-DSE
MATH CP
PAPER 1
LONGMAN MATHEMATICS SERIES
Candidate Number
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2015
Class
MOCK PAPER
Class Number
MATHEMATICS Compulsory Part
PAPER 1
Question-Answer book
Time allowed : 2 hours 15 minutes
This paper must be answered in English
INSTRUCTIONS
1.
After the announcement of the start of the
examination, you should first write your Candidate
Number, Class and Class Number in the spaces
provided.
2.
This paper consists of THREE sections, A(1), A(2)
and B.
3.
Do not write in the margins. Answers written in the
margins will not be marked.
4.
Unless otherwise specified, all working must be
clearly shown.
5.
Unless otherwise specified, numerical answers should
be either exact or correct to 3 significant figures.
6.
The diagrams in this paper are not necessarily drawn
to scale.
2015-DSE-MATH-CP 1-1
© Pearson Education Asia Limited 2014
SECTION A(1) (35 marks)
2.
Simplify
( x −2 y ) 5
and express your answer with positive indices.
x8 y 3
(3 marks)
Factorize
(a)
a2 + a − 6 ,
(b)
a 2 + a − 6 + ab 2 − 2b 2 .
(3 marks)
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2015-DSE-MATH-CP 1-2
© Pearson Education Asia Limited 2014
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1.
3.
The table below shows the distribution of the number of credit cards owned by some
staff of a company.
Number of credit cards
0
1
2
3
4
Number of staff
8
18
20
12
2
Find the median, the mean and the standard deviation of the above distribution.
4.
The marked price of a watch is 30% higher than its cost. If the watch is sold at a
discount of 20% on its marked price, a profit of $130 is made. Find the cost of the
watch.
(3 marks)
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2015-DSE-MATH-CP 1-3
© Pearson Education Asia Limited 2014
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(3 marks)
5.
Consider the formula
(a)
1 3 5
−
= .
h hk k
Make k the subject of the above formula.
(b) If the value of h is decreased by 4, write down the change in the value of k.
6.
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Answers written in the margins will not be marked.
(4 marks)
In Figure 1, △ABC is an equilateral triangle. D and E are
points on BC and AC respectively such that AE = CD. AD
and BE intersect at the point P.
(a)
Prove that △ABE ≅ △CAD.
(b)
Find ∠BPD.
(4 marks)
Figure1
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2015-DSE-MATH-CP 1-4
© Pearson Education Asia Limited 2014
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7.
In a polar coordinate system, O is the pole. The polar coordinates of points A and B
are (6, 35°) and (6, 155°) respectively. Let L be the line which bisects ∠AOB.
(a)
Is L perpendicular to AB? Explain your answer.
(b)
Find the polar coordinates of the point of intersection of L and AB.
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(5 marks)
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2015-DSE-MATH-CP 1-5
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8.
A pack of salt is termed standard if its weight is measured as 60 g correct to the
nearest 10 g, while a pack of sugar is termed standard if its weight is measured as 55 g
correct to the nearest 5 g .
(a)
Find the maximum absolute errors of the weights of a standard pack of salt and
a standard pack of sugar.
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(b) Is it possible that the total weight of 20 standard packs of salt is greater than that
of 25 standard packs of sugar? Explain your answer.
(5 marks)
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2015-DSE-MATH-CP 1-6
© Pearson Education Asia Limited 2014
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9.
Let g ( x) = x 3 − x 2 − 11x + c , where c is a constant. When g ( x) is divided by x + 1,
the remainder is 12.
(a)
Is x + 3 a factor of g ( x) ? Explain your answer.
(b)
David claims that all the roots of the equation g ( x) = 0 are real. Do you
agree? Explain your answer.
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(5 marks)
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2015-DSE-MATH-CP 1-7
© Pearson Education Asia Limited 2014
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SECTION A(2) (35 marks)
10.
Let $C be the cost of painting a circular tray of radius r m. It is given that C is the sum
of two parts, one part varies directly as r and the other part varies directly as r 2 .
When r = 2, C = 16 and when r = 4, C = 40.
(a)
Find the cost of painting a circular tray of radius 3.5 m.
(4 marks)
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If the cost of painting a circular tray is $72, find the radius of the tray.
(2 marks)
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(b)
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2015-DSE-MATH-CP 1-8
© Pearson Education Asia Limited 2014
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There are 30 students in a class. The stem-and-leaf diagram below shows the test
scores of the 30 students.
Stem (tens)
Leaf (units)
1
a
2
3
1 3
4
5 6 9
5
0 1 2 3 7 9 9
6
0 0 3 4 4 6 6 9
7
1 b 3 5 5 5
8
3 6
9
2
(a)
If the range and the inter-quartile range of these scores are 81 and 22
respectively, find the values of a and b.
(3 marks)
(b)
Due to a mistake in recording, the score of a student should be 11 instead of 71.
Mary claims that for the two statistical measures mentioned in (a), correcting
the score from 71 to 11 will ONLY affect the value of the inter-quartile range.
Do you agree? Explain your answer.
(3 marks)
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2015-DSE-MATH-CP 1-9
© Pearson Education Asia Limited 2014
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11.
Figure 2 shows the graphs of Peter and Susan walking on the same straight road
between city A and city B during the period 5:10 to 11:50 on a certain morning. Susan
travels at a constant speed during that period. It is given that city A and city B are
20 km apart.
Figure 2
(a)
Find the duration of Peter’s resting time.
(1 mark)
(b)
Find the distance between city A and the point where Peter and Susan meet.
(3 marks)
(c)
Peter claims that his average speed is greater than that of Susan during the
period 5:10 to 11:50 on that morning. Do you agree? Explain your answer.
(2 marks)
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2015-DSE-MATH-CP 1-10
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12.
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2015-DSE-MATH-CP 1-11
© Pearson Education Asia Limited 2014
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13.
(a)
Figure 3(a) shows a circle centred at O with radius 5 cm. AB is a chord of the
circle with length 8 cm. Find the area of the shaded region.
Figure 3(a)
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(b)
Figure 3(b) shows a rectangular tank with dimensions 15 cm × 30 cm × 15 cm,
filled with x cm3 of gear oil. A solid cylindrical metal rod of base radius 5 cm
and length 30 cm is fixed horizontally at the bottom of the tank, as shown in
Figure 3(c). The depth of the oil level in Figure 3(c) is 8 cm. Find the value
of x.
Figure 3(b)
Figure 3(c)
(4 marks)
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2015-DSE-MATH-CP 1-12
© Pearson Education Asia Limited 2014
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(4 marks)
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2015-DSE-MATH-CP 1-13
© Pearson Education Asia Limited 2014
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14.
In the rectangular coordinate plane, the coordinates of points A, B, C and R are
(−1, 0), (7, 6), (7, 0) and (3, 3) respectively. P is a moving point in the rectangular
coordinate plane such that AP is perpendicular to BP. Denote the locus of P by Γ.
(a)
(i)
Find the equation of Γ.
(ii)
Describe the geometric relationship between Γ and R.
(4 marks)
(i)
Find the distance between Q and R.
(ii)
Winston claims that the ratio of the area of △ARC to the area of
△AQC is 1 : 4. Do you agree? Explain your answer.
(5 marks)
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2015-DSE-MATH-CP 1-14
© Pearson Education Asia Limited 2014
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The equation of the straight line L is 4x + 3y + 104 = 0. It is found that Γ and L
do not intersect. Let Q be a point lying on L such that Q is nearest to R.
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(b)
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2015-DSE-MATH-CP 1-15
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Section B (35 marks)
15.
On the sports day of a school, the timekeeping group for running events consists of
1 chief judge, 1 referee and 8 timekeepers. The chief judge and the referee are chosen
from 4 teachers while the 8 timekeepers are selected from 12 students. The 12 students
include some secondary 4 and secondary 5 students.
(a)
How many different timekeeping groups can be formed?
(2 marks)
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If there are 9 secondary 4 students among the 12 students, find the probability
that all the timekeepers in the timekeeping group are secondary 4 students.
(2 marks)
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(b)
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2015-DSE-MATH-CP 1-16
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16.
(a)
Let k be a non-zero real constant.
1
in the form a + bi , where a and b are real numbers.
Express
1 − ki
(2 marks)
(b)
The roots of the quadratic equation x 2 + px + q = 0 are
5
5
and
.
1 − 3i
1 + 3i
Find p and q.
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(3 marks)
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2015-DSE-MATH-CP 1-17
© Pearson Education Asia Limited 2014
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17.
(a)
In Figure 4, the equations of the four straight lines are x + y = 17 , x + y = 24 ,
x = 18 and y = 15 . The shaded region (including the boundary) represents the
solution of a system of inequalities. Find the system of inequalities.
Figure 4
(b)
A merchant has two warehouses P and Q. Warehouse P stores 24 tonnes of
flour and warehouse Q stores 16 tonnes of flour. Two cake shops A and B place
orders for 18 tonnes and 15 tonnes of flour respectively. The costs of
transporting each tonne of flour from the warehouses to the cake shops are
shown in the following table:
To
From
Warehouse P
Warehouse Q
Cake shop A
Cake shop B
$50
$20
$30
$10
Suppose the merchant transports x tonnes and y tonnes of flour from
warehouse P to cake shops A and B respectively.
The merchant claims that the transportation cost can be less than $ 900. Do
you agree? Explain your answer.
(4 marks)
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2015-DSE-MATH-CP 1-18
© Pearson Education Asia Limited 2014
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(2 marks)
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2015-DSE-MATH-CP 1-19
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18.
Figure 5 shows a vertical tower AB of height 35 m. A police officer at B is observing
the traffic condition along the straight road PQR. A, P, Q and R are points on the
horizontal ground.
(a)
The police officer at B finds that the angle of depression of P from B is 8°
while the angle of depression of Q from B is 6°. It is known that ∠PAQ = 145°
and Q is due east of P. Find the distance between P and Q.
(4 marks)
(b)
A vehicle travels along the road PR at a uniform speed. The police officer
observes that the vehicle only takes 15 seconds to travel from P to Q and
suspects that it has exceeded the speed limit of the road. At the very moment
when the vehicle passes point Q, a patrol car at A starts moving to R along
road AR to stop the vehicle. After two minutes, the patrol car meets the vehicle
at R. Find the average speed of the patrol car.
(5 marks)
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2015-DSE-MATH-CP 1-20
© Pearson Education Asia Limited 2014
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Figure 5
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2015-DSE-MATH-CP 1-21
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19.
Stanley joins a savings scheme offered by a bank. At the beginning of 2015, Stanley
has to put an initial deposit of $P into his account. Every year afterwards, he has to
deposit 10% more than he did in the previous year. The interest rate is 5% per annum,
compounded yearly.
(a)
(i)
Find, in terms of P, the amounts in Stanley’s account at the end of 2015,
2016 and 2017.
(Note: You need not simplify your answers.)
(ii) Show that the amount in Stanley’s account at the end of the nth year is
$21P (1.1n − 1.05 n ) .
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(b)
Stanley plans to buy a parking space which is worth $500 000 at the end of
2014. Suppose the value of the parking space increases by 5% every year.
Stanley joins the savings scheme with $P = $50 000 at the beginning of 2015.
At the end of which year will Stanley receive an amount from the savings
scheme that is just enough to buy the parking space? Explain your answer.
(4 marks)
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2015-DSE-MATH-CP 1-22
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(7 marks)
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2015-DSE-MATH-CP 1-23
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END OF PAPER
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2015-DSE-MATH-CP 1-24
© Pearson Education Asia Limited 2014
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