中六 數學試卷 (F6 Maths Past Paper)

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5 Mock Examinations - Mathematics
Time Allowed: 2 hours 15 minutes
Diocesan Girls' School
Secondary 6 Mock Examinations (2020-2021)
Instructions:
Mathematics
February 2021
Total marks: 105
1. This paper consists of THREE sections, A(1), A(2) and B.
2. Attempt ALL questions. Write your answers in the spaces provided in this Question-Answer Book.
3. Graph paper and supplementary answer sheets will be supplied on request. Write your name, class
and class number on each sheet, and staple them INSIDE this book.
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
6. The diagrams in this paper are not necessarily drawn to scale.
2020-2021 S.6 Mock Examinations - Mathematics
13. In the figure, a right circular conical vessel of base radius 24 cm is held vertically. The depth of
the water in the vessel is 9 cm and the radius of the water surface is 18 cm.
(i) the volume of water in the vessel in terms of T,
(ii) the wet surface area of the vessel correct to 2 decimal places.
(b) The water in the vessel is now poured into a right cylindrical tank of base radius 23 cm. A
metal sphere of radius 4 cm is then put into the tank. Someone claims that metal sphere will
not be completely immersed in the water. Do you agree with the claim? Explain your answer.
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
14. In the figure, L₁ : y = √3x passes through the origin and L₂ : x + y −20 = 0 cuts the x-axis at B.
L₁ and L₂ intersects at point A. P is a moving point in quadrant I such that the perpendicular
distance from P to L, is equal to the perpendicular distance from P to the x-axis. Denote the
locus of P by I.
(a) Describe the geometric relationship between I and ZAOB.
(b) Find the equation of I.
(c) Circle C lies inside AOAB and it touches OA and OB only. The centre of C is Q(k√3, k).
Prove that Q lies on I.
Find, in terms of k, the equation of C in general form.
Find the greatest integral value of k.
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
SECTION B (35 marks)
15. If 4 boys and 4 girls randomly form a queue,
(a) find the probability that the boys and the girls stand alternatively in the queue;
find the probability that no girls are next to each other in the queue.
2020-2021 S.6 Mock Examinations - Mathematics
16. In the figure, D is a point on AB. AC and DE intersect at F. AB = DE and
ZABC = ZBDC = ZEDC.
(a) Prove that AABC = AEDC.
(i) Name a cyclic quadrilateral in the figure.
(ii) If BD = 1 and BC = 3, find ZCAE.
2020-2021 S.6 Mock Examinations - Mathematics
17. The general term P(n) of sequence P is the sum of two parts, one part varies as n and the other
part varies as n². Suppose P(1)=-10 and P(3)=18.
(b) Find the two smallest consecutive terms in sequence P such that their difference is greater
(c) Sequence T is defined by T(n)=P(n+1) - P(n). Eason claims that sequence T'is an
arithmetic sequence. Do › you agree? Explain your answer.
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
18. Let f(x)=(x²
-(x² + (4k-8)x+1-6k), where k is a constant.
(a) Using the method of completing the square, express the coordinates of the vertex of the graph
of y = f(x) in terms of k.
(b) The graph of y = g(x) is obtained by reflecting the graph of y = f(x) with respect to the
y-axis, and then translating the resulting graph leftward by 1 unit and upwards by
units. Let B be the vertex of the graph of y = g(x). Denote the origin by O.
Show that B lies on the graph of y = 2x.
Denote the point (5, -14) by A. Find k such that B is the orthocenter of AOAB.
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
Section A1 (35 marks)
1. Simplify
2. Factorize
and express your answer with positive indices.
(a) 2a²b-5ab+3b,
(b) 2a²b-5ab+3b+4a²-9.
3. (a) Round off 0.891302 to 5 significant figures.
(b) Round up 34.594 to 2 decimal places.
(c) Round down 56213 to the nearest 1000.
2020-2021 S.6 Mock Examinations - Mathematics
19. Figure I shows a rectangular paper. PQ = EG = 6 cm, PR = DG = 8 cm. The corner APQR is cut
off and the paper is folded along AF, BC and DE to form a model standing on the ground as
shown in Figure II. It is known that ARF'F and BQC'C are perpendicular to the plane ABCDEF
and G is underneath the plane ABCDEF. It is given that AB = 3 cm.
Let ZRAF = 0 and the height of the model be h cm. Find AR in terms of
Find the height of the model.
Find the angle between the planes DEG and ABCDEF in Figure II.
Find the angle between the lines AB and RQ in Figure II.
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
2020-2021 S.6 Mock Examinations - Mathematics
End of Paper
2020-2021 S.6 Mock Examinations - Mathematics
4. The marked price of a backpack is $400. It is sold at a discount of 15%.
(a) Find the selling price of the backpack.
(b) After selling the backpack, the percentage profit is 25%. Find the cost of the backpack.
5. Consider the compound inequality
->-1 or 3(x-1) ≥ 4x + ·
+ x + 1/1/2
(a) Solve (*).
(b) Find the number of positive integers satisfying (*).
2020-2021 S.6 Mock Examinations - Mathematics
6. The number of candies owned by Carson is 35% more than that of Wilson. If Carson gives 8 of
his candies to Wilson, then the ratio of the number of Carson's candies to the number of Wilson's
candies will be 25 : 22. Find the total numbers of candies that they have.
7. The rectangular coordinates of A are (√3,-1). A is rotated anticlockwise about the origin O
through 60° to A'.
(a) Find the polar coordinates of A and A'.
(b) Describe the geometric relationship between the x-axis and AAOA'.
(c) B is a point on the y-axis. If ZBA'A = 120°, find the rectangular coordinates of B.
2020-2021 S.6 Mock Examinations - Mathematics
In the figure, ABCDE is a circle with CE as a diameter. AB BC =1:1 and ZBEC = 32°.
Find ZADC and ZABE.
2020-2021 S.6 Mock Examinations - Mathematics
9. The manager of a company uses the following pie chart to show the distribution of the numbers
of employees in different departments.
(a) What percentage of the number of employees in department A is that in department B?
(b) It is known that the number of employees in department D is 90 less than that in department
C. Find the total number of employees in this company.
Numbers of employees in different
departments
2020-2021 S.6 Mock Examinations - Mathematics
Section A2 (35 marks)
10. The stem-and-leaf diagram shows the weights (in kg) of 15 women in a group, where a and b are
integers. It is given that the range of the weights of the women is 32 kg.
Stem (10 kg)
Leaf (1 kg)
(a) Find the values of a and b.
(b) If two women are randomly chosen from the group, find the probability that the heavier
woman of the two is heavier than the upper-quartile of the whole group.
2020-2021 S.6 Mock Examinations - Mathematics
11. Place X and place Y are 120 km apart. The figure shows the graphs for Amy and Billy travelling
on the same straight road between X and Y. Amy and Billy leave place X at 1:00 and arrive at
place y at 3:00. Amy travels at a constant speed. After travelling for 60 km, Billy travels at a
speed of 30 km/h for 24 km and then at a speed of 90 km/h for the rest of the travelling.
Distance from place X (km)
(a) Find the speed of Amy in km/h.
(b) Find the speed of Billy in the first 60 km of his journey in km/h.
(c) Find the distance from place X where Amy and Billy meet on the road.
2020-2021 S.6 Mock Examinations - Mathematics
12. Let p(x) be a cubic polynomial. When p(x) is divided by x² +3x+2, the remainder is
44x + 44. It is given that x-2 is a factor of p(x). When p(x) is divided by x, the remainder
(a) Factorize p(x).
(b) Consider the equation p(x) - kx + 2k = 0, where k is a positive real constant. Someone claims
that all the roots of the equation are real. Is the claim correct? Explain your answer.