Training Pack forHKDSE Questions with “Explain your Answer” (Updated Version)
Section A(1)
1.The price of a stock drops by 20% on Thursday and then rises by 22% on Friday. It is given that the price of the stockon Friday is $24.4. On which day, Wednesday, Thursday or Friday, is the price of the stock the highest? Explain your answer. (3 marks)
2.In the figure, ABCD is a rhombus and E is a point on BC. It is given that , and .
(a)Find a.
(b)Is △CDE an isosceles triangle? Explain your answer.
(4 marks)
3.The figure below shows two points C(8, –2) and D(–3, 6) in a rectangular coordinate plane.Now, C is reflected with respect to they-axis to C′, whileD is rotated anti-clockwise about the origin through 90 to D′.
(a)Writedown the coordinates of C′ and D′.
(b)Is△CD′C′an isosceles triangle?Justify your answer.
(4 marks)
4.The figure shows the box-and-whisker diagram for the cholesterol level of some patients in a health centre.
It is given that the range and the inter-quartile range of the distribution are 4.1 and 1.9 respectively.
(a)Find a and b.
(b)One year later, it is found that the highest cholesterol level of the above patients is decreased by 25%. The doctor claims that at least 50% of the patientshave lowered their cholesterol level. Do youagree? Explain your answer.
(5 marks)
5.A pack of rice is termed normal-size if its weight is measured as 5 kg correct to the nearest 0.1 kg.
(a)Find the greatest possible weight of a normal-size pack of rice.
(b)Is it possible that the total weight of 4normal-size packs of rice is measured as 21 000 g correct to the nearest 1000 g? Explain your answer.
(5 marks)
Section A(2)
6.In the figure, ACBO is a piece of plastic sheet in the shape of a sector of radius 25 cm and . The sheet is folded to join OAand OB to form an upright inverted cone of base radius r cm andheight h cm as shown in the figure.
(a)Find r and h. (3 marks)
(b)A frustum is formed by cutting off a right circular cone of height 12 cm from the lower part of the cone in the figure. Alan claims that the volume of the frustum is greater than 1 000 000 mm3.
Do you agree? Explain your answer.
(4 marks)
7.The following stem-and-leaf diagram shows the ages of the members of a tennis club.
Stem (tens) / Leaf (units)1 / 1 7 8
2 / 0 2 2 2 4 5 a
3 / 1 4 6
4 / b 8 8 9 9
It is known that the inter-quartile range and the median of the distribution are 24 and 27 respectively.
(a)Find the unknown digits a and b. (4 marks)
(b)A few months later, two members with ages 22 and 48 leave the club, and two new members join the club. It is found that both the mode and the inter-quartile range of the distribution remain unchanged. The coach makes the following TWO claims:
Claim 1: The age of the younger new member is 22.
Claim 2: The possible age of the older new member is 45.
Justify whether each of his claims is correct or not. (4 marks)
8.As shown in the travel graph below, Jane cycles from town A to town C via town B, and Tom cycles from town C to town Avia town B. They both cycle at a constant speed throughout the journey. Tom leaves town C at 08:00 and Jane leaves town A afterwards. They meet each other at 12:00. Jane arrives at town C at 14:24 and Tom arrives at town A at 15:30.
(a)How far are they away from town B when they meet each other? (2 marks)
(b)At what the time does Jane leave town A?(3 marks)
(c)Jane has the following claim:
‘If I left town A 30 minutes earlier, I could reach town Bbefore I met Tom.’
Is her claim correct? Explain your answer.(3 marks)
9.The circle Cpasses through the point A(2, 2) and the centre of C is K( , ).
(a)Find the equation of C. (2 marks)
(b)Q(9 , 1) is a point on the circle C. R is a moving point in the rectangular coordinate plane such that it keeps at a fixed distance from Q. Denote the locus of R by . It is known that passes through K.
(i)Describe the geometric relationship between and Q.
(ii)Find the equation of .
(iii)It is known that cuts circle C at S and T. Karen claims that QSKT is a rhombus. Do you agree? Explain your answer.
(5 marks)
10.Let, where k is a constant. It is given that when f(x) is divided by x 3, the remainder is 9.
(a)Find the value of k. (2 marks)
(b)It is given that, where p, q and r are constants. Find the values of p, q and r. (4 marks)
(c)Vicky claims that all the roots of the equationare real numbers. Do you agree? Explain your answer. (2 marks)
11.Let $C be the cost of making a bottle with heightx cm. It is given that C is the sum of two parts, one part is constant and the other part varies directly as x2. When x = 3, C = 4.9; when x = 7, C = 8.9.
(a)Find the cost of making a bottle with height10 cm. (4 marks)
(b)There is a larger bottle which is similar to the bottle described in (a). If the volume of the larger bottle is 64 times that of the bottle described in (a), will the cost of making the larger bottle exceed $200? Explain your answer. (2 marks)
Section B
12.In the figure,ABC is a triangle. AD, CE and BF intersect at Q. It is given that BF AC and
∠ABF=∠ACE.
(a)Show thatA,E,QandFare concyclic.
Hence, determine which line segment is a diameter of the circle passingthrough A,E,QandF. (3 marks)
(b)Does the orthocentre of△BCQlie inside△BCQ? Explain your answer. (2 marks)
13.Two chemists adopt two different scales, namely Scale A and Scale B, to represent the concentration of hydrogen ions in a solution respectively.
Scale / FormulaA /
B /
It is given that P and Q are the concentrations of hydrogen ions in a solution on Scale A andScale B respectively, where a is a positive constant, and m is the number of hydrogen ions in the solution.
Suppose the concentrations of hydrogen ions in solution Xon Scale A and Scale B are 2 and 3 respectively.
(a)Find a.(4 marks)
(b)The concentration of hydrogen ions in solution Yis equal to 4 on Scale A. A chemist claims that the concentration of hydrogen ions in this solution on Scale B is also equal to 4.Is his claim correct? Explain your answer. (3 marks)
14.In a production line of LED television, the serial codes of all the products consist of 5 characters, in which the first 2 characters must be upper or lower case letters, followed by 2 characters which are numbers from 0 to 9, andthe last character must either be 0 or 1. It is known that, for each product in the production line, the serial code is unique and all its characters do not repeat. For example, Ab120 is a possible serial code of a product.
(a)Suppose the number of products in the production linereaches 400000 this year. Are there enough serial codes for all the products this year? Explain your answer. (3 marks)
(b)There are three products A, B and C. The serial codes of A and B are pC490 and aM230 respectively. Find the probability that all the charactersin C’s code do notappear in A’s and B’s, correct to 3 significant figures. (3 marks)
15.A chain store company has 12 stores. Two of them operate in the Mainland China, and the others operate in Hong Kong. StoreA is one of the stores in Hong Kong. This year, it is given that the standard score of store A’s annual profit is 2, the standarddeviation of the annual profits of all the stores is $0.1 million, and the annual profits of the two stores which operate in the Mainland China are both equal to the mean of all stores. If the annual profits of these two stores are excluded in the financial report, determine, with justification, whether each of the following statements is true or not.
(a)The standard deviation of the annual profits of the stores as stated in the financial report is smaller than $0.1 million.
(b)The standard score of the annual profit of storeAas stated in the financial report is smaller than 2.
(4 marks)
16.In Figure (a), ABCD is a thin metal sheet in the shape of rhombus. Eis themid-point ofCD.
AC = 10 cm and ∠BAD= 130°.
(a)Find
(i)the length of a side of the rhombus,
(ii)∠BED.
(Give your answers correct to 3 significant figures.)(5 marks)
(b)Thethin metalsheet ABCD is cut along BE. The sheetABEDobtained is then foldedalong BD such that △BDE islying onthe horizontal ground as shown in Figure (b).It is giventhat the angle between △ABD and △BED is 45°.
Suppose P is the projection of A on the horizontal ground.
(i)Find the area of △PBD, correct to 3 significant figures.
(ii)Is it possible that P lies inside △BDE? Explain your answer.
(4 marks)
17.Johnny joins an investment scheme offered by a bank. At the beginning of 2015, Johnny has to put an initial deposit of $Pinto his investment account. Every year afterwards, he has to deposit 4% more than he did in the previous year. The interest rate is 2% per annum, compounded half-yearly.
(a)(i)Find, in terms of P, the amounts in Johnny’s account at the end of 2015, 2016 and 2017.
(Note: You need not simplify your answers.)
(ii)Show that the amount in Johnny’s investment account at the end of the nth year is
.
(7 marks)
(b)Johnny plans to buy a box of wines which is worth $60000 at the end of 2014. Suppose the value of the box of wines increases by 4% every year. Johnny joins the investment scheme with $P = $5000 at the beginning of 2015. At the end of which year will Johnny receive an amount from the investment scheme that is just enough to pay 40% of the price of the box of wines as the down payment? Explain your answer. (4 marks)
18.Figure (a) shows an inverted cup of height 10 cm, which consists of a hollow cylinder and acircular ring-shaped rim. When threeinverted cups are stacked up (see Figure (b)), the overall height
is 16 cm.
Figure (a) Figure (b)
(a)Find the overall height when ten inverted cups are stacked up. (2 marks)
(b)Now, a number ofinverted cups are stacked up and put into columns.A single cup is put in the firstcolumn, then five cups are stacked up and put in the second column, and then nine cups are stacked up and putin the third column,…, and so on.
(i)Find the total number of cups from the 14th column to the 20th column.
(ii)If the total number of cups available is 200 and there are no incomplete columns, does the sum of the heights of all columns exceed 600 cm? Explain your answer.
(iii)In a factory, the cost of producing a cup is $P(n), where, and n is the number of cups produced. Using the method of completing the square, find the minimum cost of producing a cup. Hencefind the maximum number of complete columns of inverted cups that can be stacked upwhen the production cost of a cup is minimum.
(11 marks)
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