Jurongville Prelim 2009 Em P2

JURONGVILLE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009

Elementary Mathematics (PAPER 2) Secondary 4 Express/ 5 Normal (Academic) 3rd September 2009 (Thursday)

Duration

Marks Name

: ________________________________ [

1 hour 2 (0800 - 1030)

: 2 :

]

100 Class

: 4___ / 5N ___

Parent’s Signature: _______________________

INSTRUCTIONS: Write your name and index number on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer ALL questions. START A NEW QUESTION ON A FRESH PAGE. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. *Observe our school values of Integrity and Excellence by not cheating and doing your best in this paper

DO NOT OPEN THE BOOKLET UNTIL YOU ARE TOLD TO DO SO Setter: Mdm Lorraine Pek/ Mrs Ong SB

1

This document consists of 11 printed pages.

2

Mathematical Formulae

n

Compound interest

r   Total amount = P1    100 

Mensuration

Curved Surface area of a cone = rl Surface area of a sphere = 4r 2 Volume of a cone =

1 2 r h 3

Volume of a sphere =

4 3 r 3 1 ab sin C 2

Area of triangle ABC =

Arc length = r , where  is in radians Sector area =

Trigonometry

1 2 r  , where  is in radians 2

a b c   sin A sin B sin C

a 2  b 2  c 2  2bc cos A

Statistics

Mean =

 fx f

Standard deviation =

3

 fx f

2

  fx      f   

2

Answer all the questions 1(a) Solve the equation

1 2 1   . x 2x 1 3

[3]

y 1 , express y in terms of x. y 1

(b)

Given that x 

[3]

(c)

Given that x 2  3 x  y 2  3 y and that x  y , find the value of x  y .

2

A, B, C and D are points on level ground, with A due North of B, BAD  38 ,

[2]

ADB  56 , BD = 6 km and CD = 10 km. Given that ADC is a straight line, Calculate (a)

BC,

[2]

(b)

DBC ,

[2]

(c)

the bearing of B from C,

[2]

(d)

the shortest distance from D to BC.

[2] North

A 38° D 10 km

56° 6 km

B

C

A vertical tower stands at D. The angle of elevation of the top of the tower, T, from B is

3.5 . Find (e)

the height of the tower in metres,

[2]

(f)

the maximum angle of elevation of the top of the tower, T, when observed along BC. [2]

4

3

Mr Tan bought some strawberries for $25. He paid $x for each kilogram of strawberries.

(a)

Find, in terms of x, an expression for the number of kilograms of strawberries bought by Mr Tan.

(b)

[1]

Due to humidity, he lost 5 kg of strawberries. He sold the remainder of the strawberries for $1.50 per kg more than what he paid for it. Write down an expression, in terms of x, for the sum of money he received (assuming he managed to sell all the strawberries). [2]

(c)

He made a profit of $15. (i)

Write down an expression in x to represent this information, and show that it reduces to 2 x 2  9 x  15  0

(ii)

[3]

Solve the equation 2 x 2  9 x  15  0 , giving both answers correct to 2 decimal places.

(d)

[3]

Find the amount of strawberries bought by Mr Tan, correct your answer to the nearest kilogram.

4

[2]

A car company buys two Japanese cars A and B for $ 65,000 each. Within a year, road tax costs $1400, insurance costs $1350, and maintenance costs $1000 for each car. Car A is rented out at $120 per day for a total of 300 days this year. At the end of the year, the car is sold at $85,000. (a)

Calculate the profit for the whole transaction for car A.

(b)

If the company wants to increase the profit by 3%, what should the new rental for Car A per day? Give your answer correct to two decimal places.

(c)

[2]

[2]

Car B is rented out at a weekly charge of $950 with an additional mileage charge of 30 cents per kilometre for the distance travelled. The car is rented out for a total of 40 weeks and an average of 600 km per week. Car B is sold for $75000 at the end of the year. Calculate (i)

the total amount of rental the company receives for car B,

(ii)

the percentage of the total profit from selling the two cars. Give your answer correct to 1 decimal place

[2]

[3]

5

5

The dates of the month of September 2009 are shown below. A rectangle is placed in various position to enclose nine of the numbers. A possible position of the rectangle is shown.

(a)

Sun

Mon

6 13 20 27

7 14 21 28

Tue 1 8 15 22 29

Wed 2 9 16 23 30

Thurs 3 10 17 24

Fri 4 11 18 25

Sat 5 12 19 26

A rectangle is placed so that the number in the middle is 14. Find the sum of the nine numbers.

(b)

[1]

Taking the number in the middle as x, (i)

write expression, in term of x, for the number above it.

[1]

(ii)

find and simplify an expression, in terms of x, for the sum of the nine numbers. [1]

(c)

(d)

Another rectangle is placed in a position such that the sum of the nine numbers is 162. (i)

Use your answer in b(ii) to write an equation in x.

[1]

(ii)

By solving this equation, find the number in the middle of the rectangle.

[1]

A third rectangle is placed in a position such that the sum of the numbers along the diagonal is 51. Find the number in the middle of this rectangle.

6

[1]

6

A circle with centre O passes through B, C, D and E. The tangents to the circle at B and C meet at A. COFE is a straight line.

C

D O F 65° E

56° B

A

(a)

Give a brief reason why  OCA is a right angle.

(b)

Given that  CED = 65o and  CAB = 56o, calculate

[1]

(i)

 BOC ,

[1]

(ii)

 BDC ,

[1]

(iii)  OFD ,

[1]

 OBD ,

[1]

(iv) (c)

Given that AB = 8 cm, calculate the radius of the circle.

(d)

A point P, not shown, is on the opposite side of BC from O, so that

[2]

 BPC = 100 o. Does the point P lie on the circumference of the circle, inside the circle or outside the circle ? Give a reason for your answer.

[2]

7

7.

The diagram shows a semi-circle inscribed in the rectangle ABCD. ACE is a sector centered at A. O is the mid-point of AD. Give your answers for angles in radians. B

12 cm

C F

6 cm

A

D

O

E

Find (a) angle CAE,

[2]

(b) area of CDE, the portion of the sector ACE that is protruding out of the rectangle ABCD. [4] (c) angle FOD, [1] (d) length of line segment AF,

[2]

(e) the perimeter of the shaded region.

[3]

8

V

8.

24 cm

A

5 cm

5 cm 10 cm

B

A solid cone of base diameter 20 cm has its vertex, V, 24 cm vertically above the center of its horizontal base. A solid hemisphere of diameter 10 cm is removed from the cone. A vertical cross-section through V of the solid formed is as shown above. (a) Calculate the length of VA.

[1]

(b) Calculate the volume of this solid.

[2]

(c) Calculate the total surface area of the solid.

[3]

(d) In addition, if a further solid cone of height 6 cm is removed from the top, find the volume of the remaining solid. [3]

9

9.

Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y 

x 1 1. x2

Some corresponding values of x and y are given in the following table. x

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

y

5.00

1.00

0.11

p

-0.44

-0.56

-0.63

-0.69

-0.72

-0.76

(a) Calculate the value of p.

[1]

(b) Using a scale of 2 cm to represent 1 unit on the x-axis and 4 cm to represent 1 unit on the y-axis, draw the graph of y  (c)

x 1 1 for values of x in range 0.5  x  5.0 . x2

[3]

Use your graph to (i) (ii)

find the value of x when y = 0.5, x 1 solve 2 1  0. x

[1] [1]

(d) On the same graph, draw the line y  2x. Prove that the intersection of y 

(e)

x 1 1 and y  2x will be the solution to the equation x2

2x3  x2  x 1  0 .

[3]

By drawing a suitable straight line, calculate the gradient at x = 2.

[2]

10

10.(a) The mass m g of each 90 Fuji apples was recorded. The cumulative frequency curve below illustrates the distribution of the masses.

90

80

70

60

50

40

30 Cumulative frequency 20

10

0 60

70

80

90

100 Mass, m (g)

11

110

120

130

(i) Mass, m (g) Number of apples

Copy and complete the grouped frequency table of the mass of Fuji apples. 60  x  80

80  x  90

90  x  100 100  x  110 110  x  120 120  x  130

4

4 [2]

(ii)

(iii)

Using your grouped frequency table, calculate an estimate of (a) the mean mass of Fuji apples produced,

[2]

(b) the standard deviation.

[2]

The apples produced by Royal Gala have the same median but a larger standard deviation. Describe how the cumulative frequency curve will differ from the given curve. [1]

(b) Every Sunday evening, Raymond either goes to watch a movie or goes out to have 3 dinner with his friends. The probability that he goes out to watch a movie is . If he 5 1 watches a movie, the probability that he will wake up late on Monday is . If he goes 3 1 out for dinner with his friends, the probability he will wake up late is . 4 (i)

Copy and complete the tree diagram below.

  

  

[2]

late

Movie  3    5

_     _    

  

  

not late

late

Dinner _    

not late

(ii)

Find the probability that Raymond will not wake up late on Monday.

[2]

(iii)

Find the probability that Raymond will wake up late on just one of two consecutive Mondays.

[2]

END OF PAPER

12

JURONGVILLE SECONDARY SCHOOL Mathematics Department Answer Key Level/ Stream Paper Examination

Questio n 1 a b

2

3

c a b c d e f a b ci

4

cii d a b ci ii

: : :

Secondary 4 Express/ 5 Normal (Academic) 2 Preliminary Examination 2009

Answer

Question

x = 1 or -1.5 2

x 1 x 2 1 x y 3 14.3 cm 60  x  80 35.6  3.49 km 367m 6.0 25 x 75 17.5   5x 2x 75 17.5   5 x  25  15 2x 75  10x 2  45x  0 2 x 2  9 x  15  0 x  1.29 or  5.79 19kg $52250 $125.23 $45200 75.4% y

7

8

9 9

Answer

bii biii

62 93

biv c d a b c d e

37 4.25 cm P outside the circle 0.464 rad 5.73 cm2 0.927 rad 10.7 cm 35.6 cm

a

26 cm

b

2250 cm3

c d a ci cii d

1210 cm2 2.5 cm p = -0.25 1.2  0.1 1.60

x 1 1  2x x2 x  1  x2  2x x2 x  1  x2  2x3 2 x 3  x 2  x  1  0 ( proven )

5

a

126

6

bi bii ci cii d a

x-7 9x 9x = 162 18 x = 17 90 ( radius OC is perpendicular to

e 10

ai aiia aiib iii bi bii

tangent AC)

bi

124

biii

13

Tangent: (-0.8,1) and (3.1, -0.75) Gradient = -0.45  0.1 See table below 100 g 11.3 Less steep curve implies wider spread

7 10 21 50

Answer for 10(ai)

Mass, m (g) Number of apples

60  x  80 80  x  90

4

9

90  x  100 100  x  110

28

14

37

110  x  120

120  x  130

8

4