Add Math P2 Trial Spm Zon A Kuching 2008

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3472/2 Matematik Tambahan Kertas 2 2 ½ jam Sept 2008

SEKOLAH-SEKOLAH ZON A KUCHING LEMBAGA PEPERIKSAAN SEKOLAH ZON A PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 2008

MATEMATIK TAMBAHAN Kertas 2 Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1.

This question paper consists of three sections : Section A, Section B and Section C.

2. Answer all question in Section A , four questions from Section B and two questions from Section C.

3. Give only one answer / solution to each question.. 4. Show your working. It may help you to get marks.

5. The diagram in the questions provided are not drawn to scale unless stated. 6. The marks allocated for each question and sub-part of a question are shown in brackets.. 7. A list of formulae is provided on pages 2 to 3.

8. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator.

Kertas soalan ini mengandungi 11 halaman bercetak

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2

The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA 1

x=

− b ± b − 4ac 2a 2

2 am × an = a m + n 3

am ÷ an = a m − n

6 7

log a b =

9

Tn = a + (n − 1)d

n [2a + (n − 1)d ] 2 11 Tn = ar n − 1 a (r n − 1) a (1 − r n ) 12 Sn = , (r ≠ 1) = r −1 1− r a 13 S∞ = , r <1 1− r 10

4 (am) n = a nm 5

log c b log c a

8

log a mn = log a m + log a n m log a = log a m − log a n n log a mn = n log a m

Sn =

CALCULUS dy dv du =u +v dx dx dx du dv u v −u 2 y = , dy dx dx , = v 2 dx v 1 y = uv ,

4 Area under a curve b

∫y

=

dx or

a

b

3

dy dy du = × dx du dx

=

∫ x dy a

5 Volume generated b

2 = ∫ πy dx or a

b

=

∫ πx

2

dy

a

GEOM ETRY

1 Distance =

( x2 − x1) 2 + ( y2 − y1) 2

2 Midpoint y1 + y 2   x1 + x 2  (x , y) =  , 2   2 3

r = x2 + y2

4

 x i  yj r 2 2 3472/2 x  y

5 A point dividing a segment of a line  nx1 + mx 2 ny1 + my 2  ,  (x, y) =  m+n   m+n 6. Area of triangle = 1 ( x1 y2 + x2 y3 + x3 y1 ) − ( x2 y1 + x3 y2 + x1 y3 ) 2

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3 STATISTICS

1 2 3

7 8 9

= =  = =

4

= =

5

m =

6

10

P(AB) = P(A) + P(B)  P(AB)

11

P(X = r) = , p + q = 1

12

Mean , = np

13 14

z=

TRIGONOMETRY 1 Arc length, s = r 2 Area of sector , A = 3 sin 2A + cos 2A = 1 4 sec2A = 1 + tan2A 5 cosec2 A = 1 + cot2 A

9 sin (AB) = sinAcosB cosAsinB 10 cos (AB) = cos AcosB sinAsinB 11 tan (AB) =

6 sin 2A = 2 sinAcosA 7 cos 2A = cos2A – sin2 A = 2 cos2A − 1 = 1  2 sin2A 8 tan2A =

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12 

13

a2 = b2 +c2

14

Area of triangle =

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2bc cosA

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4

SECTION A [40 marks] Answer all questions in this section. 1

Solve the simultaneous equations p  m  1 and p 2  2m 2  pm  8 . Give your answers correct to three decimal places. [5 marks]

2

dS . dr Hence, determine the rate of increase of the surface area of the sphere if the radius is increasing at the rate of 0⋅2 cm s−1 when r = 3. [3 marks]

(a) Given that the surface area, S cm2, of a sphere with radius r is 4 π r2. Find

2

d2y  dy  (b) Given that y = x – 3x +2, find the values of x if +   + 14x – 11 = y. 2 dx  dx  [4 marks] 2

3

Table 1 shows the distribution of scores obtained by a group of students in a competition. Score Number of students

1 4

2 6

3 12

4 5

5 3

TABLE 1 (a) Calculate the standard deviation of the distribution.

[3 marks]

(b) If each score of the distribution is multiplied by 2 and then subtracted by c, the mean of the new distribution of scores is 2⋅8, calculate (i)

the value of c,

(ii) the standard deviation of the new distribution of scores. [3 marks]

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5

Diagram 1 shows a sector AOB with centre O and a radius of 12 cm. A C

O

B DIAGRAM 1

Point C lies on OA such that OC : OA = 3 : 4 and ∠OCB = 90°. [Use π = 3.142] Find

5

(a) the value of ∠ COB, in radian,

[2 marks]

(b) the perimeter of the shaded region,

[3 marks]

(c) the area of the shaded region.

[3 marks]

Diagram 2 shows a square with side of length a cm was cut into four equal squares and then every square was cut into another four equal squares for the subsequent stages.

a cm a cm

Stage 1

Stage 2

Stage 3

DIAGRAM 2 Given that the sum of the perimeters of the squares in every stage form a geometric progression. (a) If the sum of the perimeters of the squares cut in stage 10 is 10 240 cm, find the value of a. [2 marks] (b) Calculate the number of squares cut from stage 5 until stage 10.

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[4 marks]

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6

In Diagram 3, ABC is a triangle. The point P lies on AC and the point Q lies on BC. The straight lines BP and AQ intersect at R. C

P R

Q

A B DIAGRAM 3 It is given that AB = 4 x , AC = 6 y , AP = PC and BC = 3BQ . (a) Express in terms of x and y (i)

BP ,

(ii) CQ . [3 marks] 2 3

(b) Given that BR = (− x + (i)

3 y ) and RP = mBR . 4

State BR in terms of m, x and y .

(ii) Hence, find the value of m. [5 marks]

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7

SECTION B [40 marks] Answer four questions from this section. 7

Use graph paper to answer this question. Table 1 shows the values of two variables, x and y, obtained from an experiment. c The variables x and y are related by the equation y = where c and d are x+d constants. x y

1 2⋅88

2 2⋅30

3 1⋅92

4 1⋅64

5 1⋅44

TABLE 2 (a) Plot xy against y , by using a scale of 2cm to 0.4 unit on the x-axis and 2cm to 1 unit on the y-axis. Hence, draw the line of best fit. [ 5 marks ] (b) Use your graph from 7(a) to find the value of (i)

c,

(ii) d, (iii) x when y =

5 . x [ 5 marks ]

8

(a) Prove that cosec 2 x = tan x + cot 2 x .

[4 marks]

(b) (i) Sketch the graph of y = 2 sin x + 1 for 0 ≤ x ≤ 2π . [3 marks] (ii) Hence, sketch a suitable straight line on the same axes, and state the number of 2 solutions to the equation 2sin x  x for 0 ≤ x ≤ 2π .  [3 marks]

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9

(a) The results of a study shows that 30% of the residents of a village are farmers. If 12 residents from the village are chosen at random, find the probability that (i) exactly 5 of them are farmers, (ii) less than 3 of them are farmers. [5 marks] (b) The age of a group of teachers in a town follows a normal distribution with a mean of 40 years and a standard deviation of 5 years. Find (i) the probability that a teacher chosen randomly from the town is more than 42 years old. (ii) the value of m if 15% of the teachers in the town is more than m years old. [5 marks]

10 Solutions by scale drawing will not be accepted. Diagram 4 shows a straight line AD meets a straight line BC at point D. y C 8

A(7, 7)

D B(12, 2) x

O DIAGRAM 4 Given ∠ADB = 90˚ and point C lies on the y-axis. (a) Find the equation of the straight line AD.

[ 3 marks ]

(b) Find the coordinates of point D.

[ 3 marks ]

(c) The straight line AD is extended to a point E such that AD : DE = 1 : 2. Find the coordinates of the point E. [ 2 marks ] (d) A point P moves such that its distance from point B is always 5 units. Find the equation of the locus of P. [ 2 marks ] 3472/2

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9

11 (a) Diagram 5 shows a curve y = x2 − 4x and a straight line y = x. y

y=x

0

x

DIAGRAM 5 Find the volume of the solid generated when the shaded region is rotated through 360 ° about the x-axis. [6 marks] (b) The gradient of the curve y = px2 − qx at the point (1, 2) is 5. Find (i)

the value of p and of q.

(ii)

the equation of the normal to the curve at the point (1, 2). [4 marks]

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SECTION C [20 marks] Answer two questions from this section. 12 A particle starts moving in a straight line from a fixed point O. Its velocity V ms 1 is given by V  4t 2  8t  3 , where t is the time in seconds after leaving O. (Assume motion to the right is positive) Find (a) the initial velocity of the particle.

13

[1 mark]

(b) the values of t when it is momentarily at rest.

[2 marks]

(c) the distance between the two positions where it is momentarily at rest.

[3 marks]

(d) the velocity when its acceleration is 16 m s−2.

[4 marks]

In the diagram, ABC and EDC are straight lines. E 2 cm D 10 cm 6 cm

7 cm

A

B

5 cm

C

Given that AE = 10 cm, BD = 7 cm, BC = 5 cm, CD = 6 cm and DE = 2 cm. Calculate (a) ∠BCD,

[2 marks]

(b) ∠AEC,

[3 marks]

(c) AC,

[2 marks]

(d) the area of triangle BDE.

[3 marks]

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11

14 Use the graph paper provided to answer this question. Mr. Simon has RM 3 600 to buy x scientific calculators and y reference books. The total number of scientific calculators and reference books is not less than 60. The number of reference books is at least half the number of scientific calculators. The price of a scientific calculator is RM 40 and the price of a reference book is RM 30.

(a) Write three inequalities other than x

0 and y

0 that satisfy the conditions above. [3 marks]

(b) By using a scale of 2 cm to 10 units on both axes, construct and shade the region R that satisfies all the conditions above. [3 marks]

(c) If Mr. Simon buys 50 reference books, what is the maximum balance of money after the purchase? [4 marks] 15 Table 3 shows the monthly expenditure and weightage of Mohd Amirul for the year 2005 and 2007. Item Food Rental Transport Others

Expenditure (RM) Price Index Year 2007 Year 2005 650 500 130 p 600 550 q 250 125 r 360 135 TABLE 3

Weightage 6 5 3 4

(a) Find the values of p, q and r .

[3 marks]

(b) Find the composite index for the year 2007 based on the year 2005.

[3 marks]

(c) Given the composite index for the year 2008 based on the year 2007 is 128, calculate the monthly expenditure of Mohd Amirul for the year 2008. [4 marks]

END OF QUESTION PAPER

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