Training Pack for HKDSE Questions with Explain Your Answer (Updated Version)

Training Pack for HKDSE Questions with “Explain your Answer” (Updated Version)

1. Let $x be the price of the stock on Wednesday.

∴ The price of the stock is the highest on Wednesday.

2. (a) ∵ (property of rhombus)

∴ (int. ∠s, )

∴

(b) (property of rhombus)

(alt.∠s,

(∠ sum of △)

∵ All interior angles of △CDE are different.

∴ △CDE is not an isosceles triangle. 1A

3. (a) Coordinates of C′

Coordinates of D′

(b)

∵ The lengths of CD′, C′D′ and CC′ are different.

∴ △CD′C′ is not an isosceles triangle.

4. (a) ∵ Range 1M

∴ 1A

∵ Inter-quartile range

∴ 1A

(b) New highest cholesterol level

∵ The new highest cholesterol level of the new distribution is lower than the median of the original distribution, so, at least half of the patients have lowered their cholesterol level.

∴ The doctor’s claim is correct. 1A

5. (a) The maximum absolute error

The greatest possible weight of a normal-size pack of rice

(b) The greatest possible total weight of 4 normal-size packs of rice

∴ It is impossible that the total weight of 4 normal-size packs of rice is measured as
21 000 g correct to the nearest 1000 g.

Alternative Solution

The minimum total weight of 4 normal-size packs of rice which is measured as 21 000 g,
correct to the nearest 1000 g

∴ The least weight of one normal-size pack of rice

∴ It is impossible that the total weight of 4 normal-size packs of rice is measured as
21 000 g correct to the nearest 1000 g.

6. (a) ∵ ACB = base circumference of the cone

∴

(b) Let r′ cm be the radius of the cut off circular cone.

Considering similar triangles, we have

The volume of the frustum formed

∴ Alan’s claim is correct.

7. (a) ∵ 1M

∴ 1A

∵ 1M

∴ 1A

(b) Since the mode remains unchanged, i.e. 22,

one of the new member’s age should be 22. 1M

Since the median remains unchanged,

the other new member’s age should be at least 22.

∴ Claim 1 is correct. 1A

If the older new member is of age 45,

the inter-quartile range will change from 24 to 23. 1M

∴ Claim 2 is incorrect. 1A

8. (a) The speed of Tom

Distance travelled by Tom when he meets Jane

Distance from town B when they meet each other

(b) The speed of Jane

Time required for Jane to complete the journey

∴ Jane leaves town A at 09:54.

∴ Tom reaches town B at 11:45.

If Jane left town A 30 minutes earlier, so she left at 9:24.

The travelling time of Jane from town A to town B

The time that Jane reaches town B is 11:39.

∵ Jane reaches town B earlier than Tom.

∴ Jane’s claim is correct.

9. (a) The radius of C

∴ The equation of C is

(b) (i) The locus of G is a circle passing through K with centre at Q and
radius equal to 5. 1A

(ii) From (b)(i), we have

radius of G = 5

and coordinates of the centre of G = (-9 , 1).

∴ The equation of G is

(iii) ∵ QS = QT = 5 (radii)

and KS = KT = 5 (radii)

∴ QS = QT = KS = KT

∴ QSKT is a rhombus.

∴ Karen’s claim is correct.

10. (a) When f(x) is divided by x - 3, the remainder is -9.

By the remainder theorem, we have. 1M

∴ 1A

(b) 1M

By comparing the coefficients of x3, we have

By comparing the coefficients of x2, we have

By comparing the constant term, we have

Consider the equation .

∴ The equation has two distinct real roots.

i.e. All the roots of the equation are real numbers.

Therefore, Vicky’s claim is correct. 1A

Alternative Solution

∴ All the roots of the equation are real numbers.

Therefore, Vicky’s claim is correct. 1A

11. (a) Let , where k1 and k2 are non-zero constants.

When x = 3 and C = 4.9, we have

...... (1)

When x = 7 and C = 8.9, we have

...... (2)

Solving (1) and (2), we have k1 = 4 and k2 =.

∴ The required cost

(b) ∵ The larger bottle is similar to the bottle described in (a).

∴

∴ The required cost

∴ The cost of making the larger bottle will not exceed $200.

12. (a) In △ACE,

ÐAEC = 180° – ÐACE – ÐEAC (Ð sum of △)

= 180° – ÐABF – ÐEAC (given)

= ÐBFA (Ð sum of △)

= 90°

∵ ÐAEC + ÐBFA = 180°

∴ A, E, Q and F are concyclic. (opp. Ðs supp.)

Alternative Solution

ÐABF = ÐACE (given)

∴ ÐEBF = ÐECF

∴ B, C, F and E are concyclic. (converse of Ðs in the segment)

∵ ÐBEC = ÐBFC (Ðs in the segment)

= 90° 1M

∵ ÐBEC = ÐAFQ = 90°

∴ A, D, P and E are concyclic. (ext. Ð = int. opp. Ð) 1M

∵ ÐAFQ = 90°

∴ AQ is a diameter of the circle passing through A, E, Q and F.

(converse of Ðs in semi-circle)

(b) ∵ ÐBEC = 90°

∴ BE is an altitude of △BCQ, which lies outside △BCQ.

∵ The altitude BE must pass through the orthocentre of △BCQ.

∴ The orthocentre must lie outside △BCQ.

∴ The orthocentre of △BCQ does not lie inside △BCQ. 1A

13. (a) Consider Scale B,

Consider Scale A,

(b)

Concentration of hydrogen ions on Scale B

∴ The chemist’s claim is correct.

14. (a) The possible number of serial codes

∴ The possible number of serial codes is not enough for all the products in

the production line.

(b) C’s code should end with 1.

∵ Number of letters available

and number of digits available

∴ The required probability

1A for 381 888 – 2

15. (a) Since the profits of the two stores which operate in the Mainland China are equal to the mean, excluding them will make the distribution of data less concentrated about the mean.

i.e. The standard deviation will increase.

∴ It is not true that the standard deviation of the annual profits of the stores as stated
in the financial report is smaller than $0.1 million.

(b) Since the profits of the two stores which operate in the Mainland China are equal
to the mean, the mean will remain unchanged.

From (a), we know that the standard deviation will increase.

Since the mean remains unchanged and the standard deviation will increase, by
the formula of standard score, we know that the standard score of the annual profit
of store A will decrease.

∴ It is true that the standard score of the annual profit of store A as stated in the
financial report is smaller than 2.

16. (a) (i)

(property of rhombus)

∴ Length of a side of the rhombus

(ii) In △BCE, by the cosine formula,

BD (property of rhombus)

In △BED, by the cosine formula,

(b) (i) Perpendicular distance from P to BD

Area of △PBD

(ii) Perpendicular distance from E to BD

∵ The perpendicular distance from P to BD is longer than the perpendicular distance from E to BD.

∴ It is impossible that P lies inside △BDE.

17. (a) (i)

(ii) The total amount that in Alfred’s account at the end of the nth year

(b)

∴ Johnny will first save an amount exceeding 40% of the price of the box of wines at the end of 2020 (i.e. at the end of the 6th year).

18. (a) For each cup, let x cm be the height of the hollow cylinder, while y cm be the height of the rim.

By solving (1) and (2), we have and .

Overall height when ten inverted cups are stacked up

(b) The numbers of cups in all columns form an arithmetic sequence with the first term 1 and the common difference 4.

(i) Total number of cups from the 14th column to the 20th column

(ii) Let n be the maximum number of columns that can be put.

∴ There are at most 10 columns.

Number of cups in the 10th column

Height of the 10th column

∴ The heights of all columns form an arithmetic sequence with the first term 10 and the last term 118.

Sum of the heights of all columns

∴ The sum of the heights exceeds 600 cm.

(iii)

∴ The minimum cost of producing a cup is $3 (when the number of cups produced is 550.)

Since the number of cups in the first n columns is , we have

∴ At most 16 complete columns of inverted cups can be stacked up when the production cost of a cup is minimum.