4E Maths P2 Prelim_2012

SINGAPORE SPORTS SCHOOL SECONDARY 4 EXPRESS

PRELIMINARY EXAMINATION 2012

CANDIDATENAME

MATHEMATICS 4016 / 02

Paper 2 August 2012

2 hours 30 minutesAdditional materials: Writing Paper (6 pieces)

Graph paper (1 sheet)

READ THESE INSTRUCTIONS FIRST

Write your name, index number and class on all the work you hand in.Write in dark blue or black pen on both sides of the paper.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue, correction fluid or correction tape.

Answer all the questions.

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 answer in degrees correct to one decimal place.

For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π.

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.

This question paper consists of 9 printed pages [including this cover page]

100

FOR EXAMINER’S USE

CLASS INDEX NUMBER

2

Mathematical Formulae

Compound interest

Total amount = P (1+ r

100 )n

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 4 πr 2

Volume of a cone =

13

πr2 h

Volume of a sphere =

43

πr3

Area of triangle ABC =

12

a b sin C

Arc length = rθ , where is in radians

Sector area =

12

r2θ, where is in radians

Trigonometry

asin A

= bsin B

= csin C

a2=b2+c2−2 bccos A

Statistics

Mean =

∑ fx

∑ f

Standard deviation = √∑ fx2

∑ f−(∑ fx

∑ f )2

SSP MYE 2012 Sec 4 Express Mathematics Paper 2 4016/02

3

Answer all the questions.

1(a) Naomi swims 4000 metres daily. The ratio of the distances she spends warming up, doing

training drills and warming down is 3: 13: 2. Calculate

(i) the distance she spends warming up. [2]

(ii) the percentage of the 4000 metres she spends doing training drills [2]

(b) In 2012, her time in national schools’ for 200metres (Freestyle) is 2 mins 20.84 s.

(i) Calculate her speed in kilometres per hour. [2]

(ii) In 2013, Naomi is expected to be 15% faster than her time in 2012. Calculate,

in minutes and seconds, the expected time. [2]

(iii) Her 2012 time is 8% faster than her time in 2011. Calculate, in minutes

and seconds, her 2011 time. [2]

2(a) Express as a single fraction in its simplest form.

52 x+7

− 3 x

4 x2−49 [3]

(b) Simplify

6−4 x

2 x2+7 x−15 . [2]

(c) It is given that

c2= 12d−5e3e+9 d .

(i) Find c when d = 2 and e = - 4. [2]

(ii) Express d in terms of c and e. [2]

SSP MYE 2012 Sec 4 Express Mathematics Paper 2 4016/02 [Turn Over

41R

12.4m

S

P Q

6.1m

T

4

3

PQRS is a quadrilateral. Given that SR = 12.4m and QR = 6.1m, angles SPR and PQR are right angles, and angle PRS = 41°, calculate

(a) PR, [2]

(b) PQ, [1]

(c) angle QPS, [2]

(d) cos∠PRT . [2]

4 In the diagram, A, B, C, D and E are points on a circle with centre O (indicated by a

dot). Given that CE is the diameter.

(a) Find

(i) ∠ ACO

[1]

(ii) ∠CDA

[1]

(iii) ∠ ABC

[2]

(bi) Show that triangles FEA and FDC are similar. [2](ii) Given the ratio of AF: FC is 6 : 7 and AE is 10 cm. Find CD. [1]


O

E

F

D

A

B

C

●

44o

5


6

5(a) ξ = { x: x is an integer and 100 ≤ x ≤ 150} A = {integers divisible by 6}

B = {integers that are perfect squares}

(i) Draw a Venn Diagram to illustrate this information.

[1]

(ii) List the element(s) in the set A ∩ B. [1]

(iii) Find n (A ∩ B’ ). [1] (b) In the recent Euro 2012, Spain had a goal scoring pattern. For example, Spain scored 3 goals in all their odd numbered matches e.g. Match 1, 3 , 5 etc. It scored 5 goals in all their even numbered matches e.g. Match 2,4, 6 etc. during the preliminary round. Italy had a different goal scoring pattern. Table 1 shows the goal scoring patterns of Spain and Italy during the preliminary round.

Table 1 (Preliminary round)Match (odd number e.g. 1, 3 or 5) Match (even number e.g. 2, 4 or 6)

Spain 3 5

Italy 2 1

Table 1 can be represented by the matrix P = (3 52 1 )

During the final round (consists of quarter, semi & final matches), the pattern was different. Spain scored 1 goal (match odd) and 4 goals (match even). Italy scored 2 (match odd) and 0 (match even). (i) Represent the goal scoring patterns by the 2 countries in the final round by a matrix Q. [1] (ii) Evaluate R= 3P + 2Q. [1] (iii) Given that in this tournament, there were six preliminary round matches and

four final round matches. State what the elements of R represent. [1] Given that their sponsors rewarded Spain and Italy $20,000 and $15,000 respectively for every goal scored.

(iv) Evaluate S = (20 00 15)R [1]

(v) State what the elements of S represent. [1] (vi) Given that each team had 18 players who shared the reward equally.

( xy )=k ST where x is each team player’s reward (Spain) in thousand dollars, y

is each team player’s reward (Italy) in thousand dollars, T is a 2x1 matrix and k is a constant. Find k and T. [2]


7

6 Answer the whole of this question on a sheet of graph paper.

The following is a table of values for the graph y=14 ( 3

x−x2+4).

x -6 -5 -4 -3 -2 -1.5 -1 -0.5y -8.1 -5.4 -3.2 a -0.4 -0.1 0 -0.6

(a) Calculate the value of a, giving your answers to 1 decimal place. [1]

(b) By taking 2 cm to represent 1 unit on both axes, draw the graph of

y=14 ( 3

x−x2+4) for −6≤x≤−0 . 5 . [3]

(c) By drawing a tangent, find the gradient of the curve at the point where

x= - 4 . [2]

(d) Use your graph to find the largest value of y and the value of x for which this

occurs. [2]

(e) Use your graph to find value(s) of x for which 3x−x2+4=−8 [2]

7 The diagram, Fig. A (not draw to scale), consists of a regular hexagon of side 6cm and a triangle, formed by the AB and DC produced. F is mid-point of one of the sides of the hexagon.

(a) Find the shortest distance from F to BC. [3]

(b) Find the length of EF. [2]

(c) Name the type of triangle CBE is. [1]

(d) A string is used to enclose this diagram (its

perimeter). Find its length. [1]

(e) Show that the area enclosed by the hexagon is 93.5cm², correct to 3 sf. [2]

(f) Below diagram is a zoom-in view of the diagram above. Arc AD is part of a circle with

radius AE (and centre, E). Find the area of the region shaded on the diagram below. [3]


A

B

D

C

B

F

C

AD

E

Fig A

8

8 In the diagram, Q is the foot of a light tower. Ships P and R are anchored as shown below. Bearings of P and R from Q are 316º and 260° respectively.

(a) Calculate (i) PR, [4] (ii) angle QRP [2] (iii) the bearing of R from P. [2]

(b) S is a top end of a long flag pole mounted on the light tower at Q. QS is 250m. Calculate (i) angle of depression of R from S. [2] (ii) the greatest possible angle of elevation of S from a point on RP. [2]


North

1050m

Q

P

750m

R

13

x

9

9

The diagram shows the cross-section of a figure consists of a cone with height 13 cm and

a hemisphere with a radius of x cm. A string is used to go 1 round the edge of this

cross-section.

(a) Show that the length of string needed for the slant height of the cone is √13²+x ² [1]

(b) Find in terms of x, the length of string needed for cross-section of the hemisphere. [1] (c) Given that the length of string needed for the whole cross-section is 64cm.

Form an equation in x and show that it reduces to 72 x2−4928 x+41895=0

[2]

(take π = 22/7).

(d) Solve the equation 72 x2−4928 x+41895=0 , giving both answers correct to 2 decimal

places. [3]

(e) Find the diameter of the hemisphere, giving the answer correct to the nearest millimetre.

[2]

(f) Find the surface area of the figure. [3]


10

10(a) During the recent Euro 2012, there are 30 games in total. The goals (including penalty kicks) per game are shown in the table below.

2 5 0 9 11 3 5 4 3 9 8 7 5 4 34 0 6 5 7 8 4 5 3 6 8 8 2 3 4

(i) Construct a frequency table from this information. [1] (ii) Calculate (a) the mean, [1] (b) the standard deviation. [2]

(b) A bag contains 5 footballs, 1 netball and 3 volleyballs. Two balls are drawn at random, one after the other, from the bag without replacement. (i) Draw a tree diagram to show the probabilities of the possible outcomes. [2] (ii) Find, as a fraction in its simplest form, the probability that (a) both balls are footballs, [1] (b) second ball is netball, [2] (c) one is football and the other is volleyball. [2]

*** END OF PAPER ***


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