Tuen Mun Government Secondary School



Tuen Mun Government Secondary School

Summer Supplementary Exercise (S3 to S4)

Instructions:

1. Use single-lined paper to finish all the following questions.

2. Show working steps clearly.

3. Hand in your assignment on 2nd September, 2013.

4. Please copy the following directory. In case you lose your assignment, you can access it through “School Website > Structure > Subjects > Mathematics website > Assignment”

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Chapter 1: Laws of Integral Indices

1. Simplify the following expressions, and express your answers with positive indices.

| |(a) |[pic] |(b) |[pic] |(c) |[pic] |

| | | | | | | |

| |(d) |[pic] |(e) |[pic] |(f) |[pic] |

| | | | | | | |

2. Evaluate the following and express your answers in scientific notation.

| |(a) |[pic] |(b) |[pic] |

3. It is known that there are [pic] bacteria on a lawn with dimensions of [pic].

(a) Find the number of bacteria on the lawn per [pic].

(b) If the weight of each bacterium is about [pic], estimate the weight of the bacteria on the lawn per [pic].

(Express your answers in scientific notation.)

Chapter 2: More about Factorization

1. Factorize the following expressions.

|(a) |[pic] |(b) |[pic] |(c) |[pic] |

|(d) |[pic] |(e) |[pic] |(f) |[pic] |

|(g) |[pic] |(h) |[pic] |(i) |[pic] |

2. (a) Factorize[pic].

(b) Hence factorize [pic].

3. (a) Factorize the following expressions.

(i) [pic] (ii) [pic]

(b) Hence factorize [pic].

Chapter 3: Study of 3-dimensional Figures

1. Draw one of the nets for the right prism.

2. Draw all the planes of reflection of the following solids and hence find the number of planes of reflection of the solid.

3. Draw all the axes of rotation of the following solid, and hence find the number of axes of rotation of the solid. For each axis of rotation, how many folds of rotational symmetry are there?

Chapter 4: Mensuration

[In this exercise, give your answers correct to 3 significant figures, if necessary.]

1. In the figure, VABCD is a right pyramid. The base ABCD is a square with sides of 5 cm each. The slant edge is 8 cm long.

(a) Find the height VO of the pyramid.

(b) Find the volume of the pyramid.

2. In the figure, VABC is a pyramid. ABC is an isosceles right-angled triangular base where [pic]. The height VA of the pyramid is 20 cm.

(a) Find the area of (VBC.

(b) Find the total surface area of the pyramid.

3. (a) A metal right cylinder with both its base radius and height of 10 cm is melted. [pic] of the metal is recast to form a right circular cone with the base same as the original cylinder. Find the height of the cone.

(b) The rest of the metal is recast to form another right circular cone with the base same as the original cylinder. Find the total surface area of this cone.

4. If the volume of a sphere is [pic], find the surface area of the sphere.

5. A metal hemisphere with the radius of 4 cm is melted and recast to form a metal sphere.

(a) Determine whether the total surface area of the solid increases or decreases.

(b) Find the percentage increase / percentage decrease in the total surface area of the solids.

Chapter 5: Theorems Related to Triangles

1. In the figure, D and E are the points on AB and AC respectively. ED is the angle bisector of (AEB. [pic]. Prove that [pic].

2. In the figure, ADB and AEC are straight lines, CD and DE are the angle bisectors of (ACB and (ADC respectively.

(a) Find (ADE.

(b) Prove that DE is an altitude of (ADC.

3. In the figure, AEB, BDC and CFA are straight lines. AD is the angle bisector of (BAC, DE and DF are altitudes of (ABD and (ACD respectively, AD and EF intersect at G.

(a) Prove that [pic].

(b) Prove that AG is the perpendicular bisector of EF.

4. In the figure, ADB, AFE, BEC and CFD are straight lines. [pic] and [pic].

(a) Prove that [pic].

(b) Prove that [pic].

5. In the figure, [pic] and [pic]. Prove that AD is the angle bisector of (BAC.

Chapter 6: Introduction to Probability

1. There are some black balls and white balls in a bag, of which the number of white balls is 5 more than that of black balls. If a ball is drawn at random from the bag, the probability of getting a black ball is [pic], find the number of black balls.

2. For a batch of raffle tickets, the probability of winning a prize is [pic]. If 240 more tickets are added without prizes, the probability of winning a prize becomes [pic]. Find the original number of raffle tickets.

3. The table shows the distribution of the members of a track and field team of a school last year.

A member is selected at random from the team this year. Estimate the probability of each of the following events happening.

(a) The member is a S1 boy.

(b) The member is a girl.

(c) The member is a S4 or S5 student.

4. To investigate the number of fish in a pond, scientists caught 300 fish and put a ring around their tails before letting them go. After a period of time, the scientists caught another 300 fish again from the pond and discovered that 10 of them were with rings.

(a) Find the relative frequency of the fish in the pond with rings.

(b) Estimate the number of fish in the pond.

5. In each of the bags I and II, there are 1 white ball and 1 black ball. A ball is drawn at random from bag I and put into bag II, then a ball is drawn at random from bag II.

(a) Find the total number of possible outcomes.

(b) Find the probabilities of the following events happening.

(i) The same ball is drawn twice.

(ii) The balls drawn are of the same colour.

6. The following is a game at an amusement park.

4 identical 5 cm ( 5 cm squares are drawn on a 20 cm ( 20 cm table. A participant needs to throw a token with the diameter of 2 cm onto the table. If the token lies on a square without touching the boundaries, 10 tokens are given. If the centre of the token thrown does not fall onto the table, the token will drop to the ground and the participant can play once again.

(a) If a token lies on square ABCD without touching the boundaries of square ABCD, what is the area of the region formed by the possible locations of the centre of the token?

(b) Find the expected value of the number of tokens given in each game.

(c) Will you play the game? Explain briefly.

Chapter 7: More about Percentages

1. It is known that the expense of Nicole today is 15% less than that of yesterday, and the expense yesterday was 80% more than that of the day before yesterday. Find the overall percentage change in the expense of Nicole over these three days.

2. To produce a certain product, 1.5 kg of plastic and 2.4 kg of aluminum are required. The prices of plastic and aluminum last month were $6 per kg and $15 per kg respectively. It is given that the price of plastic this month decreases by 5%, and that of aluminum increases by 12%.

(a) Find the material cost of the product last month.

(b) Find the material cost of the product this month.

(c) Find the percentage change in the material cost of the product over these two months.

3. Jack bought a hi-fi set 4 years ago. If its yearly depreciation rate is 20% and it is currently worth $9 600,

(a) find the value of the hi-fi set 4 years ago.

(b) find the value of the hi-fi set after 3 years.

(c) after how many years will the value of the hi-fi set drop below half of the current value?

4. Mr. Lam borrowed a sum of money from a bank on simple interest at the beginning of year 2005. It is known that the amount he had to repay finally is twice the principal.

(a) If the interest rate per annum was 24%, when did Mr. Lam repay all the amount owed?

(b) If Mr. Lam repaid all the amount owed at the beginning of year 2010, find the interest rate per annum.

5. Emily deposits $39 000 into Wing Fu Bank for 2 years at an interest rate of 4% p.a. compounded half-yearly.

(a) Find the amount obtained after 2 years.

(b) If Emily plans to withdraw the amount in (a) and deposits it in Wah Hing Bank for half a year at an interest rate of 3% p.a. compounded monthly,

(i) find the amount obtained from Wah Hing Bank after half a year.

(ii) find the total interest earned over these [pic] years.

(Give your answers correct to the nearest dollar.)

Chapter 8: Properties of Quadrilaterals

1. In the figure, ACDE is a trapezium. B is a point on AC such that BD ( DC, BD (( AE and (BDC ( 90(. Find (AED.

2. In the figure, ABCD is a square. AFE, DCE and BFC are straight lines. Find (AED.

3. In the figure, BGEH is a parallelogram. ABC, DEF, AGE, BGD, BHF and CHE are straight lines. AG ( FH.

(a) Prove that[pic].

(b) Prove that[pic].

4. In the figure, ABCD is a parallelogram. AGF and DGCE are straight lines. GA ( GF and DG ( CE.

(a) Prove that (AGD ( (BEC.

(b) Prove that BE // GF.

(c) Prove that BEFG is a parallelogram.

5. In the figure, ABC is an isosceles triangle where AB ( AC. D, E and F are the mid-points of AB, BC and AC respectively. Prove that ADEF is a rhombus.

6. In the figure, ABC and DEF are straight lines. Find the perimeter of trapezium ABED.

[ Hint( Join CD. ]

Chapter 9: Coordinate Geometry

1. In the figure, the vertices of (PQR are P(4, 2), Q(3, 6) and R((8, (1). Prove that PQ ( PR.

2. If P((2, (10), Q(0, (4) and R(a, 2) are collinear, find the value of a.

3. It is given that A(a, b) is a point on the graph of the equation y ( 3x ( 5.

(a) Express b in terms of a.

(b) Hence express the coordinates of A in terms of a.

(c) If A also lies on the straight line passing through P(13, 12) and Q((7, (4), find the coordinates of A by using the result of (b).

4. Three vertices of parallelogram ABCD are A(3, 5), B((3, (2) and C(4, (1). If point D(a, b) lies in quadrant I, find the coordinates of D.

5. In the figure, straight lines L1 and L2 intersect perpendicularly at A(4, 7), and L1 and L2 intersect the x-axis at B((2, 0) and C respectively.

(a) Find the slope of L1.

(b) Find the slope of L2.

(c) Find the coordinates of C.

(d) Hence find the area of (ABC.

6. If P(1, 1) is the mid-point of A(a ( 7, 2a) and B(4 ( 2a, b), find the values of a and b.

7. In the figure, E divides line segments AB and CD into two parts in the ratio of 2 : 3 respectively.

(a) Find the coordinates of E.

(b) Find the coordinates of D.

(c) Prove that AC and DB are parallel to each other?

Chapter 10: Applications of Trigonometry

1. In the figure, find the area of trapezium PQRS.

2. Find ( in each of the following.

(a) [pic] (b) [pic] (c) [pic]

3. (a) Simplify [pic].

(b) If [pic], find the value of [pic].

4. Simplify the following.

| |(a) |[pic] |(b) |[pic] |

5. Prove that each of the following is an identity.

| |(a) |[pic] |(b) |[pic] |

| | | |

6. In the figure, the scale of the map is 1 : 100 000, where [pic] and [pic] on the map. It is given that the difference in the heights of every two consecutive contour lines are equal and the gradient of AB is 1:10.

(a) Find the difference in the heights of every two consecutive

contour lines.

(b) Find the angle of inclination of BC.

7. In the figure, the angle of depression of point X from point P at the top of building A is 50(. It is known that the bases of the two buildings and point X are on the same horizontal line, and the distance between the two buildings is 50 m. Find the angle of depression of point X from point Q at the top of building B.

Chapter 11: Use and Misuse of Statistics

1. (a) Find the mean, median and mode of 22, 25, 25, 25, 27, 27, 29, 30, 42.

(b) (i) If each datum in (a) is multiplied by 3, find the new mean, median and mode.

(ii) If each datum in (a) is divided by 2, find the new mean, median and mode.

2. The following shows the time spent (in min) by Zoe in jogging over the past 6 days.

40, 35, 30, 25, 30, 50

It is given that the mean, median and mode of the jogging time are 35 min, 32.5 min and 30 min respectively. If Zoe does not go jogging today, find the mean, median and mode of the jogging time over these 7 days. What will be the changes in these averages?

3. The table shows the ratings of pet food A and B in terms of three areas.

|Area |Pet food A |Pet food B |Weight |

|Nutritional value |16 |11 |x |

|Taste |10 |15 |3 |

|Price |13 |17 |4 |

(a) If the weighted mean rating of pet food A is 13.5, find the value of x.

(b) Find the weighted mean rating of pet food B.

(c) Which pet food has a higher overall rating?

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