DOWNS EDUCATION
唐氏綜合教育
MATHEMATICS Compulsory Part
Mock 1
Paper 1
Time Limit: 2.5 hours
This paper has to be answered in English
Name:
A1
A2
B
Total
35
35
35
105
SECTION A(1) (35 marks)
1. Simplify(𝒂−𝟐 𝒃−𝟓 )(𝒂𝟒 𝒃𝟐 )𝟑 and express your answer in positive indices. (3 marks)
𝟓
𝟑
𝒏
𝒎
2. Make 𝒎 the subject of the formula +
= 𝟐 (3 marks)
3. Factorize
(a) 𝟒𝒙𝟐 − 𝟓𝒙𝒚 − 𝟔𝒚𝟐
(b) 𝟒𝒙𝟐 − 𝟓𝒙𝒚 − 𝟔𝒚𝟐 − 𝟑𝒙 + 𝟔𝒚
(3 marks)
4. The marked price of a pen is 20% higher than its cost.
Harry sold a pen with a 30% off discount from its marked price.
Harry lost $64. Find the cost of the pen.
(4 marks)
5.
𝟏
(a) Find the range satisfying 𝟑(𝒙 − 𝟏) ≤ 𝟒 (𝟐 − 𝒙) 𝒂𝒏𝒅 𝟏𝟎 − 𝟐𝒙 ≤ 𝟏𝟑.
𝟐
(b)Write down the integer(s) satisfying (a). (4 marks)
6. A boy and a girl have 90 candies in total. If the boy give 7 candies to the girl,
the number of candies owned by the girl will be 4 times the number of candies
owned by the boy. Find the number of candies owned by the boy. (4marks)
7.
O is the center of the circle. EF is a tangent of the circle touching B.
∠𝑶𝑩𝑪 = 𝟒𝟎𝒐 . 𝑶𝑨 // 𝑪𝑩. Find ∠𝑨𝑩𝑬 and ∠𝑪𝑩𝑭 (6 marks)
8. Coordinate of A and B are ( 6,4) and (8,9) respectively.
A’ is the reflected image of A with respect to the X axis.
B is translated leftwards by 2 units into B’.
(a) Write down the Coordinates of A’ and B’.
(b)Write down the geometric relationship between A, A’ and B’.
(3 marks)
9. The weight of a coconut is measured as 100g, corrected to the nearest 10 g.
(a)Find the minimum possible weight of a coconut.
(b)Enoch claims that 1195 coconut can be measured as 113 kg, corrected to the
nearest kg. Do you agree? Explain your answer. (5 marks)
SECTION A(2) (35 Marks)
10. An Object is formed by a cylinder and a hemisphere with identical radii.
The hemisphere is put on top of the cylinder so the flat circular surfaces of the
cylinder and the hemisphere fit completely. Given that the height and radius of
the cylinder are 18 cm and 6 cm respectively.
(a) Find the total surface area of the object. (3 marks)
Express your answer in terms of 𝜋.
(b)The hemisphere is later recast into a new cylinder with half the original radius.
Are the 2 cylinders similar? Explain your answer. (5 marks)
11. The Cost $C of a paper is varied partly constant and partly to square root of
x. When $C=40, x=36. When $C=25, x= 9.
(a) Express $C in terms of x. ( 4 marks)
(b)Find the change of $C when x increase from 49 to 121 (2 marks)
12. Given that coordinates of A, B and C are (2,8), (8,2) and (10,10).
(a) Find the equation of circle with AB as the diameter. (3 marks)
(b) Proof that C lies outside the circle. (2 marks)
Given that OC touches the circle at 2 distinct point E and F. Denote origin as O.
(c)Find the area of ∆𝐸𝐹𝐴 (4)
13. Given 𝑓(𝑥 ) = 6 𝑥 3 − 𝑝 𝑥 2 + 𝑞𝑥 + 13. Where 𝑝 and 𝑞 are constants.
𝑓(𝑥) ≡ (2𝑥 2 + 𝑐𝑥 + 𝑑)(𝑒𝑥 + 13). Given that x-1 is a factor of 𝑓(𝑥).
(a)Find 𝑝 and 𝑞. (3 marks)
(b)Is 2x-1 a factor of 𝑓(𝑥 )? (2 marks)
(c)John claims that 𝑓(𝑥 )
has only 2 real root. Do you agree? Explain Your answer (3 marks)
14. Given the Standard score of Ernest in the math exam is -3.
If the mean and standard deviation of the math exam are 60 and 5 respectively.
(a) What is the score of Ernest in the math exam? (2 marks)
(b)Assuming the scores of students in the math exam is in normal distribution.
If a student is randomly selected, what is the probability of the student having
a higher score than Ernest? (2 marks)
SECTION B ( 35 marks)
15. Jason and Oscar roll a fair dice. The First person who rolls 2 or 3 wins. If
neither 2 or 3 is rolled, another person will roll the dice. This process repeats
until 2 or 3 is rolled. Jason rolls the dice first.
(a) Find the probability that Oscar wins the game. (2 marks)
(b)Find the probability that Jason wins the game. (2 marks)
16. The third and fifth term of an arithmetic sequence are A(3)= 9 and A(5)= 15
respectively.
(a) Find the first term of the sequence. (2 marks)
(b)Given that the first term of a geometric sequence is G(1)=27 and the common
ratio is 27.
Find the minimum possible value of n in the following formula:
log 3 ( 𝐺 (1)𝐺 (2)𝐺 (3) … . 𝐺(𝑛)) ≥ 7240
(4 marks)
17. Given 𝐺 (𝑥 ) = 𝑥 2 + 2𝑘𝑥 + 𝑥 − 3𝑘 + 4 . Where 𝑘 is a real constant.
Denote y=20 by L.
(a) Prove that L and 𝐺 (𝑥 ) intersect at 2 distinct points. (2 marks)
(b) Denote points of intersection of L and 𝐺 (𝑥 ) be A and B.
(i)Let a and b be x coordinates of A and B respectively, show that
(𝑎 − 𝑏)2 =4𝑘 2 + 16𝑘 + 65 (3 marks)
(ii) Is it possible that length of AB equals to 5? Explain your answer. (2 marks)
18. ABCD is a trapezium shaped paper where AB//CD. given that AB =12cm and
∠𝐴𝐶𝐷 = 40𝑜 . ∠𝐴𝐵𝐶 = 80𝑜 .
(a) Refer to fig. 1, find AC. (2 marks)
(b)Refer to fig. 2. The paper is now folded along AC. Given AD= 10 cm.
∠𝐷𝐴𝐵 = 36𝑜 .
(i)Find DB
(ii)Find ∠𝐷𝐶𝐵
(6 marks)
19. Given that centre of circle C is O(24,32), and point G(54,2) lies outside the
circle. GA and GB are tangents to the circle at point A and B respectively. It is
given that C passes through E(6,56).
(a) Find the equation of C. (2 marks)
(b)Find the point of intersection of OG and AB (3 marks)
(c) Find the equation of the circumcircle of ∆𝐺𝐴𝐵. (3 marks)
(d)Find the ratio of area between circle C and circumcircle of ∆𝐺𝐴𝐵. (2 marks)
End of paper