Add Math P1 Trial Spm Zon A Kuching 2008

SULIT 3472/1 Matematik Tambahan Kertas 1 Sept 2008 2 Jam

1 Name : ………………..…………… Form : ………………………..……

SEKOLAH MENENGAH ZON A KUCHING LEMBAGA PEPERIKSAAN PEPERIKSAAN PERCUBAAN SPM 2008 MATEMATIK TAMBAHAN

Kertas 1 Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1

This question paper consists of 25 questions.

2. Answer all questions. 3. Give only one answer for each question. 4. Write your answers clearly in the spaces provided in the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question and sub-part of a question are shown in brackets. 9. A list of formulae is provided on pages 2 to 3. 10. A booklet of four-figure mathematical tables is provided. . 11 You may use a non-programmable scientific calculator. 12 This question paper must be handed in at the end of the examination .

For examiner’s use only Marks Question Total Marks Obtained 1 2 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 4 11 3 12 4 13 3 14 3 15 3 16 4 17 3 18 3 19 4 20 4 21 4 22 3 23 3 24 3 25 3 TOTAL

80

Kertas soalan ini mengandungi 15 halaman bercetak 3472/1

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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 am ÷ an = a m − n

3

5

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

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

log c b log c a

8

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

dx or

a

b

=

dy dy du = × dx du dx

3

∫y

=

∫ 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

r



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

x i  yj

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3 STATISTICS

1 2 3

7 8 9

= = σ = =

4

σ= =

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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Answer all questions. 1

Diagram 1 shows the linear function f. x

f

f(x)

0

5

1

4

5



n



4

9

DIAGRAM 1 (a) State the value of n. (b) Using the function notation, express f in terms of x. [ 2 marks ]

Answer : (a) ……………………..

1

(b) ……………………... 2.

Two functions are defined by

2

f : x  x  1 and g : x  x 2  3 x  1 . Given that

gf : x  x 2  ax  b , find the value of a and of b. [ 3 marks ]

2 3

Answer : …………………….....

For examiner’s use only

3

The function of p is defined as p(x) 

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x3 ,x  h. 1  2x

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Find (a) the value of h, (b) p −1 ( x) . [ 3 marks ]

Answer : (a) ……………………..

3

(b) ……………………...

3

4

Find the range of values of t if the following quadratic equation has no roots (t + 2) x2 + 6x + 3 = 0. [ 3 marks ]

4 3

Answer : .........………………… For examiner’s use only

5

Given that α and β are the roots of the quadratic equation 2 x 2  3 x  7 . Form the quadratic equation whose roots are 2α and 2 β .

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[ 3 marks ]

5 3

Answer : ................................. ___________________________________________________________________________ 6

Diagram 2 shows the graph of a curve y = a(x + p)² + q that passes through the point (0, 5) and has the minimum point (2, 3). Find the values of a, p and q. y [ 3 marks ] (0, 5) (2, 3) O

x

DIAGRAM 2

Answer : p = ……........................ q = ……........................ For examiner’s use only

7

a = .................................. Find the range of values of x for which x(x − 2) ≤ 15. [3 marks]

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6 3

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7

7 3

Answer : ..................................

8

Solve

3 x −1 = 27 9x [ 3 marks ]

8 3

Answer : ...................................

9

Given that lg 2 = 0 ⋅ 3 and lg 17 = 1 ⋅ 23 , find, without using scientific calculator or mathematical tables, find the value of log 2 34 . [ 3 marks ]

9 3

Answer : ...................................... For examiner’s use only

10 The n th term of an arithmetic progression is given by Tn = 5n − 1. Find (a) the first term and the common difference, (b) the sum of the first 15 terms 3472/1

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of the progression. [4 marks]

10 Answer : (a) …………………………. 4

(b) ....……………...………..

2 2 2 , , , . . . . 19683 6561 2187 Find the three consecutive terms whose product is 157464. [ 3 marks ]

11 The first three terms of a geometric progression are

8

11 3

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Answer : ............................................ 12 Diagram 3 shows the straight line obtained by plotting log10 y against log 10 x.

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3

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(4, ) (0, 6)

log10 x DIAGRAM 3 The variables x and y are related by the equation y = kx 4 , where k is a constant. Find the value of (a) k, (b) h. [ 4 marks ]

12

Answer : (a)…...….………..…....... 4

(b) .................................... ___________________________________________________________________________ 13 The coordinates of the vertices of a triangle PQR are P(2, −h), Q(−1, 0) and R(5, h). If the area of the ∆PQR is 9 units 2 , find the values of h. [ 3 marks ]

13 3

Answer : h = …………………….

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14

x y − =1 5 p 10 x + 12 y − 3 = 0, find the value of p.

If

the

straight

line

is

perpendicular

to

the

straight

line

[ 3 marks ]

14 Answer : .…………………

3

15 Given the vectors a = 3i − mj , b  8i  j and c  5i  2 j . If vector a  b is parallel to % % %% %% % % % % % c vector ~ , find the value of the constant m. [ 3 marks ]

15

Answer : .…………………. For examiner’s use only

16 The diagram 4 shows a parallelogram ABCD drawn on a Cartesian plane. y B 3472/1 2008 Hak Cipta Zon A Kuching A O D C x

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3

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4

DIAGRAM 4





It is given that AB  3i  2 j and BC  4 i  3 j . % % % % Find



(a) BD , (b)



AC . [ 4 marks ]

16

Answer : (a) ……….…….…………... 4

(b) ………………………….. ___________________________________________________________________________ 17 Solve the equation sin 2   5cos   3  cos 2  for 0 0    360 0 .

[ 3 marks ]

17 3

Answer : …...…………..…....... 18 Given that sin x =  3472/1

3 and 90 < x < 270, find the value of sec 2x. 5 2008 Hak Cipta Zon A Kuching

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[ 3 marks ]

18 Answer : …...…………..…....... ___________________________________________________________________________ 19 The diagram 5 shows a semicircle of centre O and radius r cm. C

DIAGRAM 5 A

O

B

The length of the arc AC is 72 cm and the angle of COB is 2692 radians. Calculate (a) the value of r, (b) the area of the shaded region. [Use π = 3.142] [ 4 marks ]

Answer : (a) …………………….. For examiner’s use only

(b) …………………….. 20 Find the coordinates of the turning points of the curve y = x + 3x2 – 2 . [4 marks] 3

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20 3

Answer : …...…………..…....... ___________________________________________________________________________ 21

Given that y = 3m2 and m = 2x + 3. Find dy (a) in terms of x, dx (b) the small change in y when x increases from 3 to 3⋅01. [ 4 marks ]

21

Answer : (a) …………………….. (b) ……………………..

4

22 Find

∫

3 1 − 3x

dx [ 3 marks ]

22 3

Answer : …………………….. 23 Ben and Shafiq are taking driving test. The probability that Ben and Shafiq pass the test 1 2 are and respectively. 5 3 Calculate the probability that at least one person passes the test. 3472/1

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For examiner’s use only

[ 3 marks ]

23 Answer : ………………………..

3

___________________________________________________________________________ 24 A committee of 5 members is to be selected from 6 boys and 4 girls. Find the number of ways in which this can be done if (a) the committee has no girls, (b) the committee has exactly 3 boys. [ 3 marks ]

24 Answer : (a) …………………….. For examiner’s use only

25

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(b) …………………….. A random variable X has a normal distribution with mean 50 and variance σ 2 . Given that P[X > 51] = 0⋅288, find the value of σ . [ 3 marks ]

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15

25 3

Answer : …...…………..….......

END OF QUESTION PAPER

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