MOCK 19(I)

MATH Name
COMPULSORY
PART Class ( )
PAPER 1
OXFORD UNIVERSITY PRESS
MOCK 19(I) Marker’s Examiner’s
Use Only Use Only

Marker No. Examiner No.

MATHEMATICS Compulsory Part
Question No. Marks Marks
PAPER 1 12
34
Question-Answer Book
56
7
(21/4 hours)
8
This paper must be answered in English
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10
11
INSTRUCTIONS
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1. Write your Name, Class and Class Number in the 13
spaces provided on Page 1.
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2. This paper consists of THREE sections, A(1), A(2) 15
and B.
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3. Attempt ALL questions in this paper. Write your
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answers in the spaces provided in this
Question-Answer Book. Do not write in the 18
margins. Answers written in the margins will not
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be marked.
Total
4. Graph paper and supplementary answer sheets will
be supplied on request. Write your Name and mark
the question number box on each sheet, and fasten
them with string INSIDE this book.

5. Unless otherwise specified, all working must be

clearly shown.

6. Unless otherwise specified, numerical answers

should be either exact or correct to 3 significant
figures.

7. The diagrams in this paper are not necessarily

drawn to scale.

© Oxford University Press 2019

MOCK 19(I) MATH COMPULSORY PART PAPER 1 1

 SECTION A(1) (35 marks)

1. Make a the subject of the formula 2(3a  11) = 3a  5b. (3 marks)

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m 6 n 3
2. Simplify and express your answer with positive indices. (3 marks)
( m5 n  4 ) 2

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 5 4
3. Simplify  . (3 marks)
3k  2 2k  7
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4. Factorize

(a) 25x2  4,
(b) 5x2y  17xy + 6y,

(c) 5x2y  17xy + 6y  25x2 + 4.

(4 marks)

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 5x  3
5. (a) Solve the inequality 3(x  4)  .
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5x  3
(b) How many integers satisfy both inequalities 3(x  4)  and 6x + 24 > 0?
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(4 marks)
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6. The marked price of a computer is $7 000 which is 40% above its cost.

(a) Find the cost of the computer.

(b) If the computer is sold at a discount of 12% on its marked price, find the percentage profit.
(4 marks)

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 7. Seven years ago, the ages of Peter and Irene were in the ratio 3 : 2. The ratio now becomes 4 : 3.
Find the present age of Irene.
(4 marks)
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 8. The stem-and-leaf diagram below shows the distribution of the weights of the students in a group.

Stem (10 kg) Leaf (1 kg)

4 1 7 9
5 0 0 a 5 5
6 2
7 0 a a
It is given that the mean of the distribution is 57 kg.

(a) Find a.

(b) Find the range, the inter-quartile range and the standard deviation of the distribution.
(5 marks)
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 9. In Figure 1, O is the centre of the circle ABCD. BAE and CODE are straight lines. It is given that
BDC = 48 and AO = AE.

48
C E
O D

Figure 1

(a) Find AOE.


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(b) Someone claims that AB is shorter than AE. Do you agree? Explain your answer.
(5 marks)

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 SECTION A(2) (35 marks)

10. It is given that f (x) is the sum of two parts, one part is a constant and the other part varies directly
as x2. Suppose that f (1) = 206 and f (3) = 254.
(a) Find f (x). (3 marks)
(b) Solve the equation f (x) = 80x. (2 marks)
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 11. The following table shows the distribution of the numbers of films watched by a group of students
last month, where a > 5, b < 11 and c > 0.

Number of films 0 1 2 3 4 5
Number of students c+1 4 a 8 ba c

The median of the distribution is 2.5.

(a) Find a and b. (3 marks)

(b) It is given that the mode of the distribution is greater than 2. Write down

(i) the least possible value of c,

(ii) the greatest possible value of c.
(2 marks)

(c) Suppose c is the value obtained in (b)(i). If a student is randomly selected from the group,
find the probability that the selected student watched more than 3 films last month. (2 marks)
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 12. Let f (x) = 4x3 + (a + 2)x2 + 2x  3b, where a and b are constants. x + 1 is a factor of f (x). When f (x)
is divided by x  2, the remainder is 9.
(a) Find a and b. (3 marks)

(b) Let g(x) be the quotient when f (x) is divided by x2 + 2x + 3. Someone claims that the
equation kx g(x) = f (x) has more than one real root for all real values of k. Do you agree?
Explain your answer. (4 marks)
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 13. A right circular cylindrical container has base radius 16 cm and height 14 cm. The container is held
vertically and some water is added into it. Then a solid metal sphere of surface area 144 cm2 is put
into the container. It is found that the metal sphere is totally immersed in the water and the water
surface just reaches the top of the container.

(a) Find the volume of the solid metal sphere in terms of . (2 marks)
(b) Find the original depth of water in the container. (3 marks)

(c) An inverted right circular conical vessel of curved surface area 720 cm2 is formed by a paper
sector of arc length 48 cm. Then the vessel is held vertically. The water in the circular
cylindrical container in (b) is now poured into the vessel. Will the water overflow? Explain
your answer. (3 marks)
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 14. In Figure 2, ABCD is a square. AB is produced to F. DF cuts AC and BC at E and G respectively.

A B F

D C
Figure 2

(a) Prove that

(i) △BCE  △DCE,
(ii) △BEG ~ △FEB.
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(4 marks)

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(b) Someone claims that when 0 < AFD < 30, DE < FG. Do you agree? Explain your
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answer. (4 marks)

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 SECTION B (35 marks)

15. There are 8 boys and 5 girls in a dance class. 7 students are selected from the class to form a team.
(a) If exactly 5 boys are selected, how many different teams can be formed? (2 marks)
(b) If more girls are selected than boys, how many different teams can be formed? (2 marks)
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 x2 x
16. Let f (x) =  + 11.
16 2
(a) Using the method of completing the square, find the coordinates of the vertex of the graph of
y = f (x). (2 marks)

(b) The graph of y = g(x) is obtained by reflecting the graph of y = f (x) in the x-axis. The graph of
y = h(x) is obtained by translating the graph of y = g(x) vertically. If the graph of y = h(x)
touches the straight line y = 6, find the y-intercept of the graph of y = h(x). (3 marks)
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 17. Let p and q be the sum of roots and the product of roots of the quadratic equation
(x + 2)(x  2) = 8(x  1) respectively.
(a) Write down the values of p and q. (2 marks)
(b) The 1st term and the 2nd term of a geometric sequence are log q and log p respectively. Find
the greatest value of  such that the sum of the ( + 1)th term and the (2 + 1)th term of the
sequence is less than log 22 020. (4 marks)
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 18. Figure 3 shows a geometric model ABCD in the shape of a tetrahedron. It is given that AD = 15 cm,
BC = 17 cm, CD = 27 cm, ABD = 58, ADB = 65 and ABC = 116.

B
Figure 3

(a) Find AB and AC. (4 marks)

(b) Let K be a point on AD such that BK  AD. Someone claims that BKC is the angle between
the face ABD and the face ACD. Do you agree? Explain your answer. (3 marks)
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 19. Let P be a moving point. G is the circumcentre of △PQR. The coordinates of Q, R and G are (6 , 9),
(a , 11) and (h , 3) respectively, where h > 0.
(a) Express the coordinates of G in terms of a. (2 marks)
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(b) It is given that the slope of RG is . Denote the circumscribed circle of △PQR by C.
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A straight line L: y = kx cuts C at two distinct points S and T, where k > 0. M is the mid-point
of ST.

(i) Find a.

14  3k
(ii) Show that the x-coordinate of M is .
1 k 2

(iii) The shortest distance from the origin O to the line passing through G and M is 2 41 .
Denote the location of P by a point A when P is farthest from M, and denote the
location of P by a point B when P is nearest to the y-axis.
If U is a point below the x-axis such that the area of the circle passing through A, B and
U is the least, are A, M, B and U concyclic? Explain your answer.
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(11 marks)

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END OF PAPER

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