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Secondary 3

INSTRUCTIONS

Belilios Public School

Half-yearly Examination, 2015-2016

Mathematics

2. This paper consists of THREE sections, A(1), A(2) and B.

1. After the announcement of the start of the examination, you should first write your Name, Class and

Class Number in the spaces provided on Page 1.

Time allowed: 1-hours

Maximum marks: 105

3. Attempt ALL questions in this paper. Write your answers in the spaces provided in this

Question-Answer Book. Do not write in the margins. Answers written in the margins will not be

5. Unless otherwise specified, all working must be clearly shown.

4. Graph paper and supplementary answer sheets will be supplied on request. Write your Name, Class,

Class Number and question number on each sheet, and fasten them with a paper clip INSIDE this

7. The diagrams in this paper are not necessarily drawn to scale.

6. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant

Answers written in the margins will not be marked.

12. Rebecca wants to deposit $80 000 in a bank for 4 years. Bank A offers a simple interest at

5% p.a. while bank B offers an interest rate of 4.5% p.a. compounded monthly.

(a) Find he interest eceived ŉ bank A.

Find the interest received in bank B.(Give your answer correct to the nearest $1.)(3 marks)

(c) Which bank should Rebecca deposit the money in order to get larger interest?

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

SECTION B (35 marks)

Mrs Lee wants to book coaches for students joining a school picnic. The following shows

two types of coach available.

Maximum number of passengers

It is given that at least 250 students join the picnic and the budget for booking the coaches is

$8100. Mrs Lee decides to book 3 coaches of type A and x coaches of type B.

(a) By considering the number of students joining the picnic and the budget for booking the

coaches, set up two inequalities in x.

(b) Mrs Lee claims that she can book 6 coaches of type B in the school picnic.

Is she correct? Explain your answer.

(c) (i) At least how many coaches of type B should be booked?

(ii) If the least number of coaches of type B is booked, find the total booking fee.

Answers written in the margins will not be marked.

Booking Fee

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the figure, line segment AB cuts the x-axis and the y-axis at two points P and Q

respectively. The coordinates of A and B are (4, 4) and (-1, -6) respectively.

(a) Find AP : PB.

Find the coordinates of P.

(c) Find PQ: QB.

Hence find AP : PQ: QB.

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Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the year 2014/15, Joseph earned $16 000 per month and was eligible for a salaries

tax allowance of $110 000. In the same year, Matthew's salary was higher than that of

Joseph by 5%, but Matthew's eligible allowance was lower than that of Joseph by 7%.

(a) Find the salaries tax payable by Joseph.

(b) Find the salaries tax payable by Matthew.

(c) By what percentage did Matthew's salaries tax exceed that of Joseph?

(Give your answer correct to the nearest integer.)

The table below shows the salaries tax rate:

Net chargeable income

On the first $40 000

On the next $40 000

On the next $40 000

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the figure, A, B and C are three islands. Ship P and ship Q depart from island A

at the same time. Ship P travels in a direction of N50°E with a speed of 60 km/h,

while ship Q travels in a direction of S70°E with a speed of 50 km/h. They arrive

at island B and island C respectively after 3 hours.

(a) (i) Find the distance between A and B.

(ii) Find the distance between A and C.

(b) Find the distance between B and C.

(c) Find the compass bearing of C from B. (2 marks)

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Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

End of Paper

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

INSTRUCTIONS

Belilios Public School

Half-yearly Examination, 2015-2016

Mathematics

(Suggested Solution)

2. This paper consists of THREE sections, A(1), A(2) and B.

1. After the announcement of the start of the examination, you should first write your Name, Class and

Class Number in the spaces provided on Page 1.

3. Attempt ALL questions in this paper. Write your answers in the spaces provided in this

Question-Answer Book. Do not write in the margins. Answers written in the margins will not be

5. Unless otherwise specified, all working must be clearly shown.

Time allowed: 1-hours

Maximum marks: 105

4. Graph paper and supplementary answer sheets will be supplied on request. Write your Name, Class,

Class Number and question number on each sheet, and fasten them with a paper clip INSIDE this

7. The diagrams in this paper are not necessarily drawn to scale.

6. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant

Answers written in the margins will not be marked.

SECTION A(1) (35 marks)

(a) Solve the inequality 2x - 5 < 7 − 3(1 + 2x).

(b) Find the greatest possible integer x that satisfies the inequality in (a).

2. The sum of three consecutive numbers is smaller than 114. Find the greatest value of

the smallest consecutive integer.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

SECTION A(1) (35 marks)

Solve the inequality 2x − 5 < 7 - 3(1 + 2x).

Find the greatest possible integer x that satisfies the inequality in (a).

2x-5<7- 3 - 6x

2x + 6x <7-3+5

(b) .. The greatest possible integer is 1.

The sum of three consecutive numbers is smaller than 114. Find the greatest value of

the smallest onsecutive integer.

Let the smallest consecutive integer be x,

then the remaining integers are x + 1 and x + 2.

x + (x + 1) + (x+2) < 114

3x + 3 < 114

The greatest value of the smallest consecutive integer is 36.

(a) Let (a,0) and (0, b) be the coordinates of A and B respectively

By the mid-point formula, we have

a = -12 and b=10

Coordinates of A = (-12, 0)

In the figure, A and B are points on the x-axis and the y-axis respectively. M(-6, 5) is the

mid-point of AB.

(a) Find the coordinates of A and B.

(b) Is OM perpendicular to AB? Explain your answer.

Coordinates of B = (0, 10)

1M (remove bracket)

1M (put x on one side)

1M sum of hree numbers) + 1A K 114)

1M (any one)

1A (any one)

(b) Slope of OM

Slope of AB =

4. It is given that cos

By definition,

. Slope of OM × slope of AB #−1

... OM is not perpendicular to AB.

without evaluating 0. (Leave your answer in surd förm.)

By Pythagoras' theorem,

BC=√√AC² - AB²

= √√4² − (√3)²

Construct ABC as shown with cos 0 =

5. Prove the following identities.

(b) sin cos (1 + tan² 0) = tan

(a)_2sin (90° 0 ) − sin²0 +2 = (1 + cos 0)²

where is an acute angle. Find the value of

= 2 cos 0 - 1 + cos²0 +2

= 1 + 2 cos 0 + cos²0

= (1 + cos 0)²

L.H.S. = 2 sin (90° −0) - sin² 0 +2

= 2 cos 0 - (1 - cos²0)+2

1M (correct formula)

+1A (any correct answer)

2 sin (90° -0)- sin² 0 +2 = (1 + cos ()²

1M (sin(90°-0)=cos0)

1M (sin²0 +cos³0 = 1) (either in part (a) or (b))

1M (factorize or expand)

L.H.S. = sin cos 0 (1 + tan²0)

= sin cos 1+

= sin cos x

cos² 0 + sin² 0

sin cos (1+tan²0) = tan

(a) Find the gradient of road AB in fraction.

(b) Find the inclination of road AB.

In the figure, the scale of the contour map is 1:35 000. AB is a straight road. AB is measured

to be 2 cm on the map.

(a) Vertical distance of AB = (350-250) m

Since the scale of the map is 1 : 35 000,

horizontal distance of AB = 2 × 35 000 cm

= 70 000 cm

Gradient of road AB

(b) Let 0 be the inclination of road AB.

Gradient of AB = tan 0

08.13° (cor. to 3 sig. fig.)

The inclination of road AB is 8.13⁰.

1M(tan²0 = sin²0 / cos²0)

The value of a new flat is $6 000 000 now. Its value is increased by 10% per year for the

first 3 years and it becomes 15% per year afterwards.

(a) What is the value of the flat after 5 years?

(b) Find the overall percentage change in the value of the flat in these 5 years.

(a) The value of the flat after 5 years

(b) Overall percentage change

$(10561485-6000000)

= +76.0% (cor. to 3 sig. fig.)

SECTION A(2) (35 marks)

= $6000 000×(1 +10%) ³ × ( 1+15

= $10561485

$10600000 (cor. to 3 sig. fig.)

3(3x −7) > 4(2x − 5) + 12(1)

9x21 8x - 20+12

9x8x-20 + 12 +21

+1 and represent the solution graphically.

If x is an integer that satisfies both inequalities in (a) and 7x +5 ≤ 6x + 21,

find all the possible values of x.

Graphical representation:

1M (1+10%) or (1+15%) + 1A

1A (either exact value or answer correct to 3 sig.

1M (put x on one side)

Referring to the figure, ABCD is a parallelogram. A and B are points on the y-axis and

the x-axis respectively. K(15, 6) is a point on BC such that DK 1 BC.

(a) Find the value of d.

(b) Find the coordinates of B.

(c) Find the area of parallelogram ABCD. (3 marks)

7x+5 ≤ 6x + 21

All he possible vdues of x æe 14, 15 ml 16.

DKL BC, i.e. DK 1 KC

Slope of DK × slope of KC = -1

= √√25 units

(b) Let (0, a) and (b, 0) be the coordinates of A and B respectively.

. B, K and C are collinear.

Slope of BC = slope of KC

... Coordinates of B = (7,0)

DK = √√(15−12)² + (6−10)² units

3² + (-4)² units

1M (slope formula)

1M (for equal slope)

1M (for distance formula)

1A (either DK or BC)

BC = √(19 −7)² + (9 − 0)² units

√12² +9² units

Area of ABCD = BC x DK

In the figure, AC = 2, ZCAB = 45° and CB 1 AB. D and E are points on AC and AB

respectively such that ZCEB= 60° and ED LAC.

(a) (i) Find BC and EB.

(ii) Find AE.

(b) Find DE.

(Leave your answers in surd form.)

(a)(i) In ABC,

BC 2 sin 45°

(ii) In ABC,

AB = 2 cos 45°

= 15 × 5 sq. units

= 75 sq. units

√/23 (or √6-√2)

DE = √√2

In the figure, AD and BE are two towers. The angles of elevation of A and B from C are

60° and 48° respectively. If the horizontal distance between the two towers is 360 m and

the height of tower BE is 100 m, find

(a) the height of tower AD,

(b) the angle of depression of B from A.

With the notations in the figure,

consider ABCE.

DC = DE - CE

Consider AADC.

AD 269.9596 tan 60° m

= 468 m (cor. to 3 sig. fig.)

.. The height of tower AD is 468 m.

≈ (269.9596 tan 60° – 100) m

≈367.5837 m

Consider AGB.

(a) Interest received from bank A

$[80000×5% × 4]

ZABG ≈ 45.5972°

12. Rebecca wants to deposit $80 000 in a bank for 4 years. Bank A offers a simple interest at

5% p.a. while bank B offers an interest rate of 4.5% p.a. compounded monthly.

(a) Find the interest eceived in bank A.

(b) Find the interest received in bank B.(Give your answer correct to the nearest $1.)(3 marks)

(c) Which bank should Rebecca deposit the money in order to get larger interest? (1 mark)

ZFAB = ZABG (alt. Zs, AF // GB)

= 45.6° (cor. to 3 sig. fig.)

The angle of depression of B from A is 45.6°.

(b) Interest received from bank B

- 58000×(1+4.55%)

5745 (cor. to the nearest $1)

(c) Rebecca should deposit the money in bank A.

1M for (1 +r%/n)nt + 1M (for finding interest)

Answers written in the margins will not be marked.

SECTION B (35 marks)

13. Mrs Lee wants to book coaches for students joining a school picnic. The following shows

two types of coach available.

Maximum number of passengers

It is given that at least 250 students join the picnic and the budget for booking the coaches is

$8100. Mrs Lee decides to book 3 coaches of type A and x coaches of type B.

(a) By considering the number of students joining the picnic and the budget for booking the

coaches, set up two inequalities in x.

(b) Mrs Lee claims that she can book 6 coaches of type B in the school picnic.

Is she correct? Explain your answer.

(c) (i) At least how many coaches of type B should be booked?

(ii) If the least number of coaches of type B is booked, find the total booking fee.

(a) By considering the number of students joining the picnic, we have

3(45) + x(30) ≥ 250

By considering the budget for booking the coaches, we have

3(1140) + x(900) ≤ 8100

(b) From (2), 3(1140) + x(900) ≤ 8100

1140 + 300x ≤ 2700

300x ≤ 1560

(c) (i) From (1), 3(45) + x(30) ≥ 250

135+30x ≥ 250

.. Owing to the budget, at most 5 coaches of type B can be booked.

So, she is not correct.

30x ≥ 115

Booking Fee

1M (put x on one side) (in part (b) or (c)(i))

Answers written in the margins will not be marked.

(a) Since point P is on the x-axis, the y-coordinate of P is 0.

Let AP PB = 1 : r.

.. At least 4 coaches of type B should be booked.

If we use section formula to consider the y-coordinate of P, then

r(4) +1(-6)

When 4 coaches of type B is booked,

total booking fee

= $[3(1140) + 4(900)]

In the figure, line segment AB cuts the x-axis and the y-axis at two points P and Q

respectively. The coordinates of A and B are (4, 4) and (-1, −6) respectively.

(a) Find AP : PB.

Find the coordinates of P.

(c) Find PQ: QB.

(d) Hence find AP : PQ: QB.

AP: PB = 2:3.

(b) x-coordinate of P

The coordinates of P are (2, 0).

(c) Since point Q is on the y-axis, the x-coordinate of Q is 0.

Let PQ: QB = 1 : s.

If we use section formula to consider the x-coordinate of Q, then

3(4) + 2(-1)

1M (consider y-coordinate of P is 0 and let the ratio) +1A

1M (consider x-coordinate of Q is 0 and let the ratio)

Answers written in the margins will not be marked.

In the figure, A and B are points on the x-axis and the y-axis respectively. M(-6, 5) is the

mid-point of AB.

(a) Find the coordinates of A and B.

(b) Is OM perpendicular to AB? Explain your answer.

Answers written in the margins will not be marked.

4. It is given that cos 0 = where is an acute angle. Find the value of

without evaluating 0. (Leave your answer ŉ surd förm.)

Answers written in the margins will not be marked.

·· PQ: QB = 2:1.

(d) AP: PB = 2 : 3 and PQ : QB = 2:1

• AP: PQ: QB = 2 :2:1

The table below shows the salaries tax rate:

In the year 2014/15, Joseph earned $16 000 per month and was eligible for a salaries

tax allowance of $110 000. In the same year, Matthew's salary was higher than that of

Joseph by 5%, but Matthew's eligible allowance was lower than that of Joseph by 7%.

(a) Find he salaries ax payable by bseph.

(b) Find the salaries tax payable by Matthew.

(c) By what percentage did Matthew's salaries tax exceed that of Joseph?

(Give your answer correct to the nearest integer.)

Net chargeable income

On the first $40 000

On the next $40 000

On the next $40 000

(a) Joseph's net chargeable income for the year 2014/15

= $(16 000×12-110 000)

= $(40 000 + 40 000 + 2000)

.. Joseph's salaries tax payable

$(40 000×2% +40 000×7% +2000 × 12%)

Matthew's net chargeable income for the year 2014/15

= $[16 000×(1+5%)×12−110 000×(1-7%)]

= $(40 000 + 40 000 +19300)

... Matthew's salaries tax payable

$(40 000 × 2% +40 000 × 7% +19 300×12%)

1M (1 + 5%) or (1-7%) +1A

= 54% (cor. to the nearest integer)

The required percentage

(b) Find the distance between B and C.

16. In the figure, A, B and C are three islands. Ship P and ship Q depart from island A

at the same time. Ship P travels in a direction of N50°E with a speed of 60 km/h,

while ship Q travels in a direction of S70°E with a speed of 50 km/h. They arrive

at island B and island C respectively after 3 hours.

(a) (i) Find the distance between A and B.

(ii) Find the distance between A and C.

(c) Find the compass bearing of C from B. (2 marks)

(a) (i) With the notations in the figure,

AB = (60×3) km

The distance between A and B is 180 km.

(ii) AC = (50×3) km

... The distance between A and C is 150 km.

(b) cos 70°:

AD = 150 cos 70⁰ km

DC = 150 sin 70° km

AE = 180 cos 50⁰ km

EB 180 sin 50⁰ km

any 1 correct method for finding AD, DC, AE or EB(1M)

any 2 correct expressions for AD, DC, AE or EB (1A)

all correct(2A)

BC2 CF² + BF²

= (150 cos 70° +180 cos 50°) km

BF =DC - EB

= (150 sin 70° – 180 sin 50°) km

(Pyth. Theorem)

BC =√(150 cos 70° +180 cos 50°)² + (150 sin 70° - 180 sin 50°) ² km

(c) tan ZCBF =

= 167 km (cor. b 3 ig. fg)

1M (Find CF or BF)

End of Paper

ZCBF = 88.948°

The compass bearing of C from B is S(90°- 88.948)E = S1.05°E.

Answers written in the margins will not be marked.

5. Prove the following identities.

(a) 2sin (90° -0 ) − sin²0 +2= (1 + cos 0)²

(b) sin cos 0(1+ tan² )= tan

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the figure, the scale of the contour map is 1 : 35 000. AB is a straight road. AB is measured

to be 2 cm on the map.

(a) Find the gradient of road AB in fraction.

Find the inclination of road AB.

Answers written in the margins will not be marked.

The value of a new flat is $6 000 000 now. Its value is increased by 10% per year for the

first 3 years and it becomes 15% per year afterwards.

(a) What is the value of the flat after 5 years?

(b) Find the overall percentage change in the value of the flat in these 5 years.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

SECTION A(2) (35 marks)

+1 and represent the solution graphically.

(b) If x is an integer that satisfies both inequalities in (a) and 7x+5 ≤

find all the possible values of x.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Referring to the figure, ABCD is a

the x-axis respectively. K(15, 6) is a point on BC such that DKI BC.

(a) Find the value of d.

(b) Find the coordinates of B.

Find the area of parallelogram ABCD. (3 marks)

parallelogram. A and B are points on the y-axis and

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the figure, AC = 2, ZCAB = 45° and CBL AB. D and E are points on AC and AB

respectively such that ZCEB= 60° and ED LAC.

(a) (i) Find BC and EB.

(ii) Find AE.

(b) Find DE.

(Leave your answers in surd form.)

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In the figure, AD and BE are two towers. The angles of elevation of A and B from C are

60° and 48° respectively. If the horizontal distance between the two towers is 360 m and

the height of tower BE is 100 m, find

(a) the height of tower AD,

the angle of depression of B from A.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.