spm add math 2009 paper 1extra222

SPM ADD MATH 2010 Paper 1 1

SIJIL PELAJARAN MALAYSIA 2009

ADDITIONAL MATHEMATICSPaper 13472/12 hours


Diagram 1 shows the relation between set X and set Y in the graph form.

1

State(a) the objects of q,(b) the codomain of the relation.

( 2 , q )

( 2 , s )

( 4 , p )

( 4 , r )

( 6 , q )

( 2 , q )

( 6 , q )

(a)the objects of q 2 , 6

(b) the codomain of the relation. { p , q , r , s }


Given the functions g:x 2x - 3and h:x 4x, find(a) hg(x),(b) the value of x if hg(x) = ½ g(x).

2

h g (x )

= h ( 2x - 3 )

= 4 ( 2x - 3 )

= 8x - 12

(a) (b) hg(x) = ½ g(x)

8x - 12 = ½ ( 2x - 3 )

16x - 24 = 2x -3

14x = 21

x = 3/2


3 Given the function g : x 3x -1, find(a) g(2),(b) the value of p when g-1(p) = 11.

g : x 3x -1

(a) g(2)

= 3( 2 ) -1

(b) g-1(p) = 11

= 5

3(11) -1 = p

p = 33-1

= 32


4 The quadratic equation x2 + x = 2px - p2, where p is a constant, has two different roots.Find the range of values of p.

x2 + x = 2px - p2

x2 + x -2px + p2 = 0

x2 + x( 1 -2p) + p2 = 0

b2 - 4ac > 0

( 1-2p )2 - 4(1)(p2) > 0

1 – 4p + 4p2 - 4p2 > 0

1 – 4p > 0

4p < 1

p < ¼


5 Diagram shows the graph of a quadratic function f(x) = - (x + p)2 + q, where p and q are constants.

State(a)the value of p, (b) the equation of the axis of symmetry.

f(x) = - (x + p)2 + q

x = -3x + 3 = 0

x + 3 = x + p

p = 3

(a)

(b) x = -3


6 The quadratic function f(x) = -x2 + 4x + a2, where a is a constant, has maximum value 8.Find the values of a.

f(x) = -x2 + 4x + a2

= -( x2 - 4x ) + a2

= -[ x2 - 4x + ( -2 )2 - ( -2 )2 ] + a2

= -( x - 2 )2 + ( -2 )2 + a2

( -2 )2 + a2 = 8 a2 = 4

a = 2


7 Given 3n - 3 x 27 n = 243, find the value of n.

3n - 3 x 27 n = 243

3n - 3 x ( 33 ) n = 35

3n - 3 x 33n = 35

3n – 3 + 3n = 35

n – 3 + 3n = 5

4n = 8 n = 2


8 Given that log8 p - log2 q = 0, express p in terms of q.

log8 p - log2 q = 0

log8 p = log2 q

qp

22

2 log8log

log

log2 p = 3log2 q

log2 p = log2 q3

p = q3


9 Given the geometric progression -5, 10/3 , - 20/9 ,..., find the sum to infinity of the progression.

10 205 , , ,...

3 9

a = -5

1035

r 2

3r

5

21

3

S

55

3

3


10 Diagram 10 shows three square cards.

The perimeters of the cards form an arithmetic progression. The terms of the progression are in ascending order.

(a)Write down the first three terms of the progression.

(b)Find the common difference of the progression.

(a) 4(3) , 4(5), 4(7)

= 12 , 20, 28

(b) d = 20 - 12

= 8


11 The first three terms of a geometric progression are x, 6, 12. Find(a)the value of x, (b) the sum from the fourth term to the ninth term.

(a) x , 6 , 12

6

126

x6

2x

6

2x

= 3

(b)

a = 3 r = 2

12

123 9

9

S = 3( 29 – 1 )

= 1533 3

3

3 2 1

2 1S

= 3( 23 – 1 )

= 21

4 9 1533 21S = 1512


12 Diagram 12 shows a sector BOC of a circle with centre 0.

It is given that BOC= 1.42 radians, AD = 8 cm and 0A = AB = OD = DC = 5cm.Find(a) the length, in cm, of arc BC, (b) the area, in cm2, of the coloured region.

(a) arc BC

= 10 ( 1.42 )

= 14.2

(b) the area

= ½ (10)2 ( 1.42 )

- ½ (8) (3)

= 71 - 12

= 59


Given that a = 13 i + j and b = 7 i – k j, find(a) a - b in the form x i + y j,(b) the values of k if | a - b | = 10.

13

a = 13 i + j , b = 7 i – k j,

(a) a - b

= 13 i + j – ( 7 i – k j )

= 6 i +(1+k) j

(b) | a - b | = 10

1016 22 k

36 + 1 + 2k + k2 = 100 k2 + 2k – 63 = 0

( k +9 ) ( k -7 ) = 0

k = -9 , 7


14 Diagram 14 shows a triangle PQR.

Given and point S lies on QR such that QS : SR = 2 : 1, express in terms of a, and b

= -3 a + 6 b

SR RP ��

13 6 6

3a b b

2 6a b b

4a b


15 Diagram shows a straight line AC.The point B lies on AC such that AB : BC = 3 : 1.Find the coordinates of B.

3 4 1 2 3 0 1 3,

3 1 3 1B

12 2 0 3,

4 4

10 3,

4 4

5 3,2 4


16 Solve the equation 3 sin x cos x - cos x = 0 for 00 x 3600.

3 sin x cos x - cos x = 0

cos x ( 3 sin x – 1 ) = 0

cos x = 0 3 sin x – 1 = 0

x = 90o, 270o sin x = 1/3

x = 19.47o, 160.53o


17 It is given that sin A = 5/13 and cos B = 4/5 where A is an obtuse angle and B is an acute angle.Find(a) tan A,(b) cos(A - B).

A513 B

5

412

(a) tan A

5

12

3

(b) cos( A – B )

= cosA cosB – sinA sinB

12 4 5 3

13 5 13 5 48 15

65 65 33

65


18Given that and

m

dxxf5

6)( 5

( ) 2 14m

f x dx find the value of m.

m

dxxf5

6)( 5

( ) 2 14m

f x dx

5 5

( ) 2 14m m

f x dx dx

5

6 2 14m

dx 52 8

mx

2 ( m – 5 ) = 8

m – 5 = 4

m = 9


19 The gradient function of a curve is = kx - 6, where k is a constant. It is given that the curve has a turning point at (2, 1).Find(a)the value of k, (b) the equation of the curve.

dy

dx

6dy

kxdx

(a)

0 2 6k

2 6k 3k

3 6dy

xdx

(b)

236

2

xy x c

2

3 21 6 2

2c

7c 236 7

2

xy x


20 A block of ice in the form of a cube with sides x cm, melts at a rate of 972 cm3 per minute. Find the rate of change of x at the instant when x = 12cm.

3V x

23dV

xdx

23 12

432

dV dV dx

dt dx dt

972 432dx

dt

972

432

dx

dt

2.25 /cm s


21 Diagram shows part of the curve y = f(x) which passes through the points (h, 0) and (4, 7).

Given that the area of the coloured region is 22 unit2, find the value of h4 f(x)dx.

4

)(h

dxxf

= 4(7) - 22

= 6


22 There are 4 different Science books and 3 different Mathematics books on a shelf.Calculate the number of different ways to arrange all the books in a row if(a) no condition is imposed,(b) all the Mathematics books are next to each other.

S1 S2 S3 S4 M1 M2 M3

(a)7

7p

7!5040

(b) S1 S2 S3 S4M1 M2 M35 3

5 3p p

5! 3! 120 6 720


23 The probability that a student is a librarian is 0.2. Three students are chosen at random.Find the probability that(a) all three are librarians,(b) only one of them is a librarian.

P( x = 3 )(a)

= 1 ( 0.2)3 1

= 0.008

X ~ Bin ( 3 , 0.2 )

= 3C3 (0.2)3 (0.8)0

P( x = 1 )(a)

= 3 ( 0.2) 0.64

= 0.384

= 3C1 (0.2)1 (0.8)2


24 A set of 12 numbers x1, x2, ... , x12 , has a variance of 40 and it is given that x2 = 1 080.Find(a)the mean, (b) the value of x.

(a) 22

2 xN

x

2108040

12x

240 90 x 250x

7.071x

(b)12

xx

12(7.071)x 84.853


25 The masses of apples in a stall have a normal distribution with a mean of 200 g and a standard deviation of 30 g.(a) Find the mass, in g, of an apple whose z-score is 0.5.(b) If an apple is chosen at random, find the probability that the apple has a mass of at least 194g.

X ~ N ( 200 , 302 )

(a) XZ

2000.5

30

X

15 200X

215X

(b) ( 194)P X 194 200

( )30

P Z

( 0.2)P Z 1 ( 0.2)P Z 1 0.4207 0.5793

spm add math 2009 paper 1extra222

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