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KOLEKSI SOALAN SPM

KERTAS 1

NAMA

...........................................................................................

Koleksi Soalan Peperiksaan Sebenar SPM (Matematik Tambahan Kertas 1)

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TOPIC: FUNCTION

1. SPM 2007P1Q1

2.SPM 2007 P1 Q2

Given that f : x 3x , find the values of a such that f(x) =5. [2marks]

3. SPM 2007 P1 Q3

The following information is about the function h and the composite function h2.

4. SPM 2003 P1 Q1

5.SPM 2003 P1 Q2

Given that g : x 5x+1 and h : x x2 -2x+3. find

(a) g-1 (3) (b) hg (x) [4 marks]

Diagram shows the linear function h

(a) state the value of m (b) (b) by using the function notation, express h in

terms of x [ 2marks]

h : x ax+b, where a and b are

constants, and a >0

h2 : x 36x -35

Find the value of a and b [3marks]

P= { 1, 2, 3 }

Q = { 2, 4,6, 8,10 }

Based on the above information, the relation between P

and Q is defined by the set of ordered pairs

{(1,2),(1,4),(2,6),(2,8) State

(a) the image of 1 (b) the object of 2 [2marks]


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6. SPM 2004 P1Q1

7.SPM 2004 P1 Q2

Given that the functions h : x 4x+m and h -1 : x 2kx +5

8, where m and k are constants, find the

value of m and of k. [3marks]

8.SPM 2004 P1 Q3

Given that function h: x 6

, 0xx

and the composite function hg(x) =3x, find

(a) g(x) (b) the value of x when gh(x) =5 [4marks]

9.SPM 2005 P1 Q1

Diagram below shows the relation between set P and set Q

Diagram shows the relation between set P and set Q.

State

(a) the range of the relation (b) (b) the type of the relation [2 marks]

State

(a) the range of the relation (b) the type of the relation [2 marks]


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10. SPM 2004 P1 Q2

Given that the function h:x 4x+m and h-1 : x 2kx +8

5, where m and k are constants, find the value

of m and of k. [3 marks]

11.SPM 2004 P1 Q 3

Given that function h(x) = 0,6

xx

and the composite function hg9x) =3x, find

(a) g(x) (b) the value of x when gh(x) =5 [4 marks]

12.SPM 2008 P1 Q 1

Diagram below shows the graph of function f(x) = 2 1x , for the domain 0 5x

State

(a) the value of t (b) the range of f(x) corresponding to the given domain. [3 marks] [ ½, 0 ( ) 9f x ]

13. SPM 2008 P1 Q2

Given that functions g:x 5x+2 and h:x x2 - 4x + 3, find

(a) g-1(6) (b) hg(x) [ 4/5, 25x2 -1]

14. SPM 2008 P1 Q3

Given the functions f(x) = x-1 and g(x) = kx+2, find

(a) f(5) (b) the value of k such that gf(5) =14 [3 marks] [ 4, 3 ]

0 t 5

1

y

x


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TOPIC : QUADRATIC EQUATIONS

1.SPM 2003 P1 Q3

Solve the quadratic equation 2x(x-4) =(1-x)(x+2). Give your answer correct to four significant figures.

(ans: 2.591, -0.2573) [3 marks]

2.SPM 2003 P1Q4

The quadratic equation x(x+1) = px-4 has two distinct roots. Find the range of values of p. (ans: 5, -3)

[3 marks]

3.SPM 2004 P1Q4

From the quadratic which has the roots -3 and ½. Give your answer in the form ax2 +bx+c=0, where a, b

and c are constants. (ans:2x2+5x-3=0) [2 marks]

4.SPM 2005 P1Q4

The straight line y = 5x-1 does not intersect the curve y=2x2 +x+p. Find the range of values of p.

[3 marks] (ans: p<1)

5.SPM 2005 P1 Q5

Solve the quadratic equation x(2x-5) =2x-1. Give your answer correct to three decimal places.

[3 marks](ans: 8.153, 0.149)

6.SPM 2006 P1Q3

A quadratic equation x2 +px+9 =2x has two equal roots. Find the possible values of p. [3 marks]

(ans: 8, -4)

7. SPM2006 P1Q5

Find the range of the values of x for (2x-1)(x+4)>4+x. (ans:x<-4,x>1) [2 marks]

8.SPM 2004 P1Q5

Find the range of values of x for which x(x-4) 12 . [3 marks]

(ans: -2 6)x


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9.SPM 2007P1Q4

(a) Solve the following quadratic equation : 3x2 +5x -2 =0

(b) The quadratic equation hx2 +kx+3 =0, where h and k are constants, has two equal roots, express h in

terms of k. [4marks]

10. SPM 2007 P1 Q5

Find the range of values of x for which 2x2 1+x. [3 marks]

11. SPM 2008 P1Q4

It is given that -1 is one of the roots of the quadratic equation x2-4x-p=0. Find the value of p.

[ ans 5] [2marks]

7. SPM 2008 P1Q6

Find the range of values of x for (x-3)2 < 5-x [ 3marks]

[ ans 1<x<4]


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TOPIC : QUADRATIC FUNCTIONS

1.SPM 2003 P2 Q2

The function f(x) = x2 -4kx+5k2 +1 has a minimum value of r2 +2k, where k are constants.

(a) By using the method of completing the square, show that r =k -1 [4marks] (b) Hence, or otherwise, find the values of k and r if the graph of the function is symmetrical about

x= r2 -1 [4marks] [k =3,r = -1]

2.SPM 2004 P1Q6

3. SPM 2005 P1 Q6

Diagram below shows the graph of a quadratic function f(x) = 3 (x+p)2 +2, where p is a constant.

The curve y=f(x) has the minimum point (1,q), where q is a constant. State

(a) the value of p.

(b) the value of q

(c) the equation of the axis of symmetry. [3marks]

[ -1, 2, x = 1 ]

Diagram shows the graph of the function

y = - ( x- k )2 – 2, where k is a constant.

Find

(a) the value of k (b) the equation of the axis of symmetry (c) the coordinates of the maximum point. [3marks]

[ k = 1, x =1, (1, -2) ]


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4. SPM 2006 P1 Q4

Diagram below shows the graph of a quadratic function y = f(x) . The straight line y = -4 is a tangent to

the curve y = f(x)

(a) Write the equation of the axis of symmetry of the curve.

(b) Express f(x) in the form of (x+b)2 +c , where b and c are constants. [3marks]

[ x=3, f(x) = (x-3)2 -4 ]

5. .SPM 2007 P1 Q6

The quadratic function f(x) = x2 +2x-4 can be express in the form f(x) = (x+m)2- n, where m and n are

constants. Find the value of m and n. [3marks]

[ ans : 1, 5]

6. SPM 2008 P1 Q5

The quadratic function f(x) = p(x+q)2 +r , where p,q and r are constants, has a minimum value of -4. The

equation of the axis of symmetry is x=3.

State

(a) the range of value of p (b) the value of q (c) the value of r [ 3marks] [ ans p>0, -3, -4]


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TOPIC : CIRCULAR MEASURES

1.SPM 2003P1Q19

2.SPM 2004 P1 Q19

3. SPM 2005 P1Q18

The length of the arc RS is 7.24 cm and the

perimeter of the sector ROS is 25 cm. Find the value

of , in rad. [3 marks]

(ans:0.8153]

Given that length of the major arc AB is 45.51 cm,

find the length , in cm, of the radius.

(use 3.142 ) [3 marks]

(Ans:7.675]

The length of the minor arc AB is 16 cm and the angle of the major

sector AOB is 290 . Using =3.142, find

(a) the value of , in radians,

(b) the length , in cm, of the radius of the circle. [3 marks]

(ans: 1.222, 13.09]


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4.SPM 2006 P1Q16

5. SPM 2007 P1 Q18

6. SPM 2008 P1 Q18

Diagram shows sector OAB with centre O and sector AXY with

centre A.

Given that OB=10 cm,AY=4 cm, XAY=1.1 radians and the

length of arc AB= 7 cm, calculate

(a) the value of in radian. (b) The area, in cm2, of the shaded region. [4 marks]

(ans: 0.7 rad, 26.2]

Diagram shows a sector BOC of a circle with centre O.

1.85BOC rad

It is given that AD =8 cm and BA=AO=OD=DC=5 cm.

Find

(a) the length, in cm, of the arc BC (b) the area, in cm2, of the shaded region [4marks]

Diagram beside shows a circle with centre O and radius 10

cm.

Given that P, Q and R are points such that OP=PQ and OPR

=900, find

(a) QOR, in radians

(b) the area, in cm2, of the coloured region. [4marks]

Ues =3.142

( /3, 30.7)


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TOPIC: INDICES AND LOGARITHMS

1. SPM 2003P1Q5

Given that 2 4log log 3T V , express T in terms of V. [4 marks]

( ans: T= 8 V )

2.SPM 2003P1Q6

Solve the equation 2 14 7x x [4 marks]

(ans:1.677)

3.SPM 2004 P1Q7

Solve the equation 4 8 632 4x x [3 marks]

(ans: 3)

4.SPM 2004 P1Q8

Given that 5 5log 2 , log 7 ,m and p express log 5 4.9 in terms of ma and p.

(ans:2p-m-1) [4 marks]

5.SPM 2005 P1Q7

Solve the equation 4 32 2 1x x [3 marks]

(ans:x = -3)

6.SPM 2005 P1Q8

Solve the equation 3 3log 4 log (2 1) 1x x [3 marks]

(ans: 3/2)

7.SPM 2005 P1Q9

Given that 27

log 2 , log 3 , exp log ( )4

m m m

mp and r ress in terms of p and r.

(ans:3r-2p+1) [4 marks]


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8.SPM 2006 P1Q6

Solve the equation 2 3

2

18

4

x

x

(ans:1) [3 marks]

9.SPM 2006 P1Q7

Given that 2 2 2log 2 3log log ,xy x y express y in terms of x. [3 marks]

(ans:y = 4x)

10.SPM 2006 P1Q8

Solve the equation 2+ log 3 (x-1) =log 3 x (ans:9/8) [3 marks]

11. SPM 2007P1 Q7

Given that log2 b = x and log2 c= y, express log4 8b

c

in terms of x and y. [4marks]

[ 3/2 + x/2 –y/2 ]

12.SPM 2007 P1Q8

Given that 9(3n-1) = 27n, find the value of n. [3marks]

[1/2]

13. SPM 2008 P1 Q7

Solve the equation: 162x-3 = 84x [ 3marks]

(ans: -3 )

14. SPM 2008 P1 Q8

Given that log4 x = log2 3, find the value of x [3marks]

( ans: 9 )


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TOPIC: STATISTICS

1.SPM2007 P1 Q22

A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares of the

numbers is 800.

Find, for the five number

(a) the mean (b) the standard deviation [3 marks]

(ans : 12, 4)

2. SPM 2008 P1Q22

A set of seven numbers has a mean of 9.

(a) Find x

(b) when a number k is added to this set, the new mean is 8.5. find the value of k. [3marks]

{ ans : 63, 5 }


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TOPIC: DIFFERENTIATION

1.SPM 2003 P1Q16

Given that y = x2 +5x, use differentiation to find the small change in y when x increases from 3 to 3.01.

[Ans:0.11] [3marks]

2.SPM 2004 P1Q20

Differentiate 3x2(2x-5)4 with respect to x [3marks]

[ 6x(6x-5)(2x-5)3 ]

3.SPM 2004 P1Q21

Two variables, x and y are related by the equation y = 3x+2

x. Given that y increases at a constant rate of

4 units per second, find the rate of change of x when

x=2 [3marks]

[Ans: 8/5]

4. SPM 2005 P1Q19

Given that h(x) =2

1

(3 5)x , evaluate h’’ (1) [4marks]

[ 27/8]

5.SPM 2005 P1Q20

The volume of water, V cm3, in a container is given by 318 ,

3V h h where h cm is the height of the

water in the container. Water is poured into the container at the rate of 10 cm3s-1. Find the rate of

change of the height of water, in cm s-1-, at the instant when its height is 2 cm. [3 marks]

[ 0.8333]

6.SPM 2006 P1Q17

The point P lies on the curve y=(x-5)2. It is given that gradient of the normal at P is -1/4. Find the

coordinates of P. [3 marks]

[ (7,4)]


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7.SPM 2006 P1 Q18

It is given that y= 72

3u , where u=3x-5. Find

dy

dxin terms of x. [4marks]

[ 14(3x-5)6 ]

8. SPM 2007 P1 Q19

The curve y = f(x) is such that 3 5dy

kxdx

, where k is a constant. The gradient of the curve at x =2

is 9. Find the value of k [2 marks]

[ 2/3]

9.SPM 2007 P1 Q 20

The curve y = x2 -32x+64 has a minimum point at x = p, where p is a constant. Find the value of p.

[3marks]

[ Ans: 16]

10.SPM 2008 P1 Q19

Two variables, x and y , are related by the equation 2

16y

x . Express, in terms of h, the approximate

change in y, when x changes from 4 to 4+h, where h is a small value. [3marks]

( ans : 1

2h )

11. SPM 2008 P1 Q20

The normal to the curve y=x2 -5x at point P is parallel to the straight line y = -x+12. find the equation of

the normal to the curve at point P. [4marks]

{ ans : y = -x-3}


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TOPIC: PROGRESSIONS

1.SPM 2003 P1Q7

The first three terms of an arithmetic progression are k-3, k+3, 2k+2. Find

a) the value of k b) the sum of the first 9 terms of the progression. [3 marks]

[ans: 7, 252]

2.SPM 2003 P1Q8

In a geometric progression, the first term is 64 and the fourth term is 27. calculate

(a) the common ration, (b) the sum to infinity of the geometric progression. [4 marks]

[ans: ¾, 256]

3.SPM 2004 P1Q9

Given a geometric progression y,2,4

y,p……, express p in terms of y. [2 marks]

[ans: p= 8/y2 ]

4.SPM 2004 P1 Q 10

Given an arithmetic progression -7, -3, 1, ……, state three consecutive terms in this progression which

sum up to 75. [3 marks]

[ans: 29, 25, 21 ]

5. SPM 2004 P1 Q 11

The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres of water is added to

the tank everyday. Calculate the volume, in litres. Of water in the tank at the end of the 7th day.

[2 marks]

[ans : 510]

6.SPM 2004 P1Q12

Express the recurring decimal 0.969696… as a fraction in its simplest form. [4 marks]

[ans : 32/33]


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7.SPM 2005 P1 Q 10

The first three terms of a sequence are 2, x, 8. Find the positive value of x so that the sequence is

(a) an arithmetic progression (b) a geometric progression. [2 marks]

[ans:5,4]

8. SPM 2005 P1 Q 11

The first three terms of an arithmetic progression are 5, 9, 13. Find

(a) the common difference of the progression, (b) the sum of the first 20 terms after the 3rd term. [4 marks]

[ans:4, 1100]

9.SPM 2005 P1 Q 12

The sum of the first n terms of the geometric progression 8, 24, 72, … is 8744. Find

(a) the common ration of the progression (b) the value of n [4 marks]

[ans: 3, 7 ]

10.SPM 2006 P1 Q9

The 9th term of an arithmetic progression is 4+5p and the sum of the four terms of the progression is 7p-

10, where p is a constant.

Given that common difference of the progression is 5, find the value of p. [3 marks]

[ans: 8]

11.SPM 2006 P1 Q 10

The third term of a geometric progression is 16. The sum of the third term and the fourth term is 8. Find

(a) the first term and the common ratio of the progression. (b) The sum to infinity of the progression. [4 marks]

[ans: 64, 42 2/3 ]

12.SPM 2007 P1 Q9

(a) determine whether the following sequence is an arithmetic progression or a geometric progression.

(b) Give a reason for the answer in part (a) [2 marks]

Answer :GP, the ratio of two consecutive terms of the sequence is a constant


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13.SPM 2007 P1 Q10

Three consecutive terms of an arithmetic progression are 5-x, 8, 2x.

Find the common difference of the progression. [3 marks]

Answer :14

14. SPM 2007 P1 Q 11

The first three terms of a geometric progression are 27, 18,12.

Find the sum to infinity of the geometric progression. [3 marks]

Answer : 81

15.SPM 2008 P1 Q9

It is given that the first four terms of a geometric progression are 3, -6, 12 and x. Find the value of x.

[2marks]

[ ans -24 ]

16. SPM 2008 P1 Q10

The first three terms of an arithmetic progression are 46, 43 and 40. the nth term of this progression is

negative. Find the least value of n. [3marks]

[ans 17]

17. SPM 2008 P1 Q 11

In a geometric progression, the first term is 4 and the common ratio is r. Given that the sum to infinity of

this progression is 16, find the value of r. [2marks]

[ ans : ¾]


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TOPIC: LINEAR LAW

1. SPM 2003 Paper 1 Q10

x and y are related by the equation y = px2 + qx, where p and q are constants. A straight line is obtained

by plotting y/x against x as shown in the diagram below.

Calculate the values of p and q [4marks]

Solution:

Step 1: Change non linear equation 2y px qx to linear equation

( )y

p x qx : ,

Step 2: Find the gradient , p =9 1

22 6

Step 3: Find the y intercept, substitute any one point which the line passes through,

(6,1) 1=-2(6) +q, q=13

Exercises:

1.SPM 2004 P1Q13

Diagram below shows a straight line graph against x.

Given that y=6x-x2, calculate the value of k and

of h . [3marks]

{ans:h=3, k=4}


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2. SPM 2005 P1 Q 13

3. SPM 2006P1 Q11

The first diagram shows the curve y = 3x2+5. The second graph shows the straight line graph obtained

when y = -3 x2 +5 is expresses in the linear form Y= 5X+c. Express X and Y in terms of a and/or y

(ans:X=9/25 x2, Y=-3/5 y) [3 marks]

4. SPM 2007 P1 Q12

The variables x and y are related by the equation

y=kx4, where k is a constant.

(a) convert the equation y=kx4 to linear form. (b) Diagram shows the straight line obtained

by plotting log10y against log10x. Find the value of

(i) log10k (ii) h {ans:1000,11}

The variables x and y are related by the equation

y2=2x(10-x).

A straight line graph is obtained by plotting 2y

x

against x

Find the value of p and of q [3 marks]


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5.SPM 2008 P1 Q11

The variables x and y are related by the equation 5x

ky , where k is a constant. Diagram below shows

the straight line graph obtained by plotting log 10 y against x.

(a) Express the equation 5x

ky in its linear form used to obtained the straight line graph shown in

the diagram above. (b) Find the value of k [ 4marks] [ ans log y = -x log 5 + log k, 0.01 ]

(0, -2)

log10 y

x 0


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TOPIC: INTERGRATION

1.SPM 2003 P1Q18

Diagram shows the curve y=3x2 and the straight line x=k.

2.SPM 2004 P1Q22

Given that

1

(2 3) 6

k

x dx

, where k>-1, find the value of k. [4 marks]

[ANS: 5]

3.SPM 2005 P1Q 21

Given that

6 6

2 2

( ) 7 (2 ( ) ) 10f x dx and f x kx dx , find the value of k.

[ 4 marks]

4.SPM 2005 P1Q 21

Given that

6 6

2 2

( ) 7 (2 ( ) ) 10f x dx and f x kx dx , find the value of k.

[ ANS: ¼] [ 4 marks]

5.SPM 2008 P1 Q 21

Given that 2 3(6 1)x dx px x c , where p and c are constants, find

(a) the value of p,

(b) the value of c if 2(6 1) 13x dx where x =1. [3marks]

{ans : 2, 10}

If the area of the shaded region is 64 unit2, find

the value of k. [3 marks]

[ans: 4]


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6. SPM 2006 P1Q 20

7.SPM 2006 P1 Q21

Given that

5

1

( ) 8,g x dx find

(a) the value of

1

5

( )g x dx

(b) the value of k if 5

1

[ ( )] 10kx g x dx [4 marks]

[ -8, 3/2]

8.SPM 2007 P1Q21

Given that

7

2

( ) 3h x dx , find

(a)

2

7

( )h x dx

(b)

7

2

[5 ( )]h x dx [4 marks]

[ -3, 22]

Given that the area of the shaded region is 5 unit2 ,

find the value of 2 2 ( )

b

a

f x dx

[2 marks]

[ -10]


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TOPIC: VECTORS

1. SPM 2003 P1 Q12

2.SPM2003 P1Q13

(ans : -13)

3. SPM 2003 P1 Q 14

(ans: -6p+4q, 2p+ 8q/3}

4 SPM 2004 P1 Q16

Given that O(0,0) A(-3,4) and B(2,16), find in terms of unit vectors, i and j ,

(a) AB

(b) the unit vector in the direction of AB [4marks]

(ans : 5 51

,12 1213

)

Diagram shows two vectors, OP andQO .

Express

(a) OP in the form x

y

(b) OQ in the form xi yj [2marks]

( 5

3

, -8i+4j)

p = 2a +3 b

q = 4 a – b

r = ha + (h-k) b, where h and k are

constants

Use the information given to find the

values of h and k when r = 3p -2q

[3marks]

Diagram shows a parallelogram ABCD with BED as

a straight line.

Given that 6 , 4AB p AD q and DE =2EB,

express in terms of p and q

(a) ( )BD b EC [4marks]


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5 SPM 2004 P1 Q17

Given that A(-2,6), B(4,2) and C(m,p), find the value of m and of p such that 2 10 12AB BC i j

[4marks]

(ans : m=6, p=-2 )

6 SPM 2005 P1Q15

7 SPM 2005 P1 Q 16

8. SPM 2006 P1 Q 13

Diagram shows vector OA drawn on a Cartesian plane.

(a) Express OA in the form x

y

(b) Find the unit vector in the direction of OA [2marks]

(ans : 12 121

,5 513

)

Diagram shows a parallelogram, OPQR, drawn on a

Cartesian plane.

It is given that 6 4OP i j and 4 5PQ i j

Find PR [3marks]

(ans : -10i+j)

Diagram shows two vectors, OA and AB

Express

(a) x

OA in the formy

(b) AB in the form xi yj [2marks]

(ans : 4

,3

-4i-8j )


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9. SPM 2006 P1 Q 14

The point P,Q, and R are collinear. It is given that 4 2 3 (1 )PQ a b and QR a k b , where k is a

constant. Find

(a) the value of k, (b) the ration of PQ : QR [4marks]

(ans : -5/2, 4:3 ]

10.SPM 2007 P1 Q16

The following information refers to the vectors a and b

2 1,

8 4a b

, find

(a) the vector 2a b ,

(b) the unit vector in the direction of 2a b [4 marks]

(a)

12

5(b)

12

5

13

1

11. SPM 2007 P1 Q15

yx4

154

27

12.SPM 2008 P1 Q15

The vectors a and b are non zero and non parallel.

It is given that (h+3) a = (k-5) b , where h and k are constants.

Find the value of

(a) h

(b) k [2marks]

[ans ; -3, 5

Diagram shows a rectangle OABC and the point

D lies on the straight line OB. It is given that

OD=3DB.

Express OD , in terms of x and y [3marks]


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13. SPM 2008 P1 Q16

Diagram below shows a triangle PQR

The point T lies on QR such that QT: TR =3:1

Express in terms of a and b :

(a) QR (b) PT [4marks]

[ans: 4a-6b, 3a+3b/2 ]

P

Q

R

T

4 a

6 b


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TOPIC: TRIGONOMETRIC FUNCTIONS

1.SPM 2007 P1 Q 17

.Solve the equation cot x + 2 cos x =0 for 0 360x [4 marks]

[ 90, 210, 270,330]

2.SPM 2003 P1 Q20

Given that tan = t, 90o , express , in terms of t :

(a) cot

(b) sin ( 90 - ) [ 3marks]

[ans : 1/t, 1/2 1t ]

3. SPM 2003 P1Q21

Solve the equation 6 sec2 A -13 tan A =0, 0 360 [4 marks]

[ans : 33.69, 213.69, 56.31, 236.31 ]

4. SPM 2003P2Q8(a)

Prove that tan + cot = 2 cosec 2 [4 marks]

5. SPM 2004 P1 Q18

Solve the equation cos2 x –sin 2 x = sin x for 0 360 [4 marks]

[ans : 30,150,270]

6.SPM 2005 P1 Q17

Solve the equation 3 cos 2x = 8 sin x -5 for 0 360x [4 marks]

[ ans : 41.81, 138.19 ]


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7. SPM 2006 P1Q 15

Solve the equation 15 sin 2 x = sin x +4sin 30 for 0 360x [4 marks]

[ ans : 23 35, 156 25 , 199 28 , 340 32 ]

8. SPM 2008 P1 Q17

Given that sin p , where p is a constant and 90 180 . Find in terms of p

(a) cosec (b) sin 2 [ 3 marks]

(ans; 1/p, -2p 21 p )

9. SPM 2008 P1 Q4

Prove that 2

2 tantan 2

2 sec

xx

x

[2 marks]


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TOPIC: PERMUTATION AND COMBINATION

1. SPM 2007 P1 Q23

A coach wants to choose 5 players consisting of 2 boys and 3 girls to form a badminton team. These 5

players are chosen from a group of 4 boys and 5 girls.

Find

(a) the number of ways the team can be formed, (b) the number of ways the team members can be arranged in a row for a group photograph. If the

three girls sit next to each other. [4 marks] [ 60, 36]

2.SPM 2003 P1 Q22

A B C D E 6 7 8

Diagram above 5 letters and 3 digits. A code is to be formed using those letters and digits. The code

must consists of 3 letters followed by 2 digits. How many codes can be formed if no letter or digit is

repeated in each code. [360]

3. SPM 2003 P1 Q23

A badminton team consists of 7 students. The team will be chosen from a group of 8 boys and 5 girls.

Find the number of teams that can be formed such that each team consists of (a) 4 boys (b) not more

than 2 girls. [4marks]

(answers : 700, 708)

4.SPM 2004 P1 Q 23

Diagram below shows 5 cards of different letters.

H E B A T

(a) Find the number of possible arrangements, in a row, of all the cards. [120]

(b) Find the number of these arrangement in which the letter E and A are side by Side. [48]

5. SPM 2004 P1 Q 24

A box contains 6 white marbles and k black marbles. If a marble is picked randomly from the box, the

probability of getting a black marble is 3/5. Find the value of k

[9]. [3marks]


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6. SPM 2005 P1 Q 22

A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant

monitors and 6 prefects. Calculate the number of different ways the team can be formed if

(a) there is no restriction

(b) the team contains only 1 monitor and exactly 3 prefects. [4marks]

[ 792, 160]

7.SPM 2005 P1 Q 24

Colour Number of cards

Black 5

Blue 4

Yellow 3

[ 19/66]

8. SPM 2006 P1 Q 22

Diagram shows seven letter cards.A four letter code is to be formed using four of these cards. Find

(a) the number of different four-letter codes that can be formed.

(b) the number of different four-letter codes which end with a consonant.

[4marks]

[ 840, 480 ]

9.SPM 2006 P1 Q 23

The probability that Hamid qualifies for the final of a track event is 2/5 while the probability that Mohan

qualifies is 1/3. Find the probability that

(a) both of them qualify for the final (b) only one of them qualifies for the final [3marks]

[ 2/15, 7/15 ]

Table shows the number of coloured cards in a box. Two cards are drawn at random from the box. Find

the probability that both cards are of the same colour

[3marks]

U N I F O R M


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10. SPM 2008 P1 Q23

Diagram below shows six numbered cards.

A four-digit number is to be formed by using four of these cards. How any

(a) different numbers can be formed?

(b) different odd numbers can be formed ? [4 marks]

{ ans : 360, 240}

11 SPM 2008 P1 Q24

The probability of sarah being chosen as a school prefect is 3

5while the probability of Aini being chosen

is 7

12. Find the probability that

(a) neither of them is chosen as a school prefect

(b) only one of them is chosen as a school prefect. [4 marks]

{ 1/6, 29/60}

3 5 6 7 8 9


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TOPIC: PROBABILITY DISTRIBUTION

1.SPM 2003 P1 Q 24

Diagram below shows a standard normal distribution graph.

If P(0< z < k)=0.3128, find P(z > k) [2 marks]

[0.1872]

2.SPM 2003 P1 Q 25

In an examination, 70% of the students passed. If a sample of 8 students is randomly selected, find the

probability that 6 students from the sample passed the examination.

[0.2965] [3 marks]

3.SPM 2004 P1 Q25

X is a random variable of a normal distribution with a mean of 5.2 and a variance of 1.44. Find

(a) the Z score if X =6.7,

(b) P(5.2 6.7)X [4 marks]

[ 1.25, 0.3944]

4. SPM 2005 P1 Q 25

the mass of students in a school has a normal distribution with a mean of 54kg and a standard deviation

of 12 kg. find

(a) the mass of the students which gives a standard score of 0.5 (b) the percentage of student a with mass greater than 48 kg [4marks]

[ 60, 0.69146]

F(z)

k 0 z


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5.SPM 2006P1 Q25

[ 1.03, 82.09]

6. SPM 2007 P1 Q24

The probability that each shot fired by Ramli hits a target is 1/3

(a) If Ramli fires 10 shots, find the probability that exactly 2 shots hit the target. (b) If Ramli fires n shots, the probability that all the n shots hit the target is 1/243. Find the value of

n. [4 marks] [ 0.1951, 5]

7. SPM 2007 P1Q25

X is a continuous random variable of a normal distribution with a mean of 52 and a standard deviation of

10.

Find

(a) the z-score when X= 67.2 (b) the value of k when P(z<k) = 0.8849 [4 marks]

[ 1.52, 1.2]

8. SPM 2008 P1 Q25

The masses of a group of students in a school have a normal distribution with a mean of 40 kg and a

standard deviation of 5 kg. Calculate the probability that a student chosen at random from this group

has a mass of

(a) more than 45 kg (b) between 35 kg and 47.8 kg [ 4 marks] {ans : 0.1587, 0.7819}

Diagram shows a standard normal distribution graph.

The probability represented by the area of the shaded region is

0.3485.

(a) Find the value of k (b) X is a continuous random variable which is normally

distributed with a mean of 79 and a standard deviation of 3. Find the value of X when the z-score is k

[4marks]

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