Algebraic Component Module

8/14/2019 Algebraic Component Module

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TOPICAL MODULE

ALGEBRAIC COMPONENTTOPICS 2003 2004 2005 2006 2007 2008

P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2

Functions 6 0 9 0 8 0 4 6 7 0 10 0Quadratic Equations 3 0 2 0 3 0 3 0 4 0 2 0

Quadratic Functions 3 8 6 0 6 0 5 0 6 0 6 7

Simultaneous Equation 0 5 0 5 0 5 0 5 0 5 0 5

Indices & Logarithms 8 0 9 0 10 0 9 0 7 0 6 0

Progressions 7 0 9 8 8 6 7 7 8 7 7 7

Linear Law 4 10 3 10 4 10 4 10 3 10 4 10

Total Marks By Paper 31 23 38 23 39 21 32 28 35 22 35 29

TOTAL MARKS 54 61 60 60 57 64

FUNCTIONS

Paper 1

1. Given the function : 5 3 g x x . Find the value of p ,if g(3)=4p+ 2 .

2. Given the function2 3

: , 11

xh x x

x

. Find the function of 1( )h x .

3. Given the function ( ) 2 3 f x x and 4( ) , 33 g x xx . Find the

composite function of gf .

4. Given the function6

( ) , 11

f x xx

. Find an expression for the function

of 2 ( )f x .

5. Given the function1 5( ) 6 , 0 g x x x

x

. Find the value of 1(10)g .

6. Given the function ( ) 3 f x x and ( ) 2 5 gf x x . Find the function ofg .

7. Given the function ( ) 5 f x x and ( ) 5 3 g x x . Find the value of( 1)gf .

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8. Given the function ( ) 3 f x x and 2( ) g x ax b . If 2( ) 2 12 13 gf x x x , findthe value of a and b .

9. Given the function ( ) g x px q with p and q being constants. Find thevalue of p and q if ( 2) 9g and 1(7) 6g .

10. Given the function ( ) 3 2 f x x and 2( ) 3g x x , Find the compositefunction of fg .

11. Given the function ( ) 4 5 f x x . Find the value of p if (2 ) f p p .

12. given the function ( ) g x px q with p and q being constants. If g(4)=5 and g(-6)= 0, find the value of p and q .

13. Given the function: 2 1 g x x

and other functionf

with conditionthat 2( ) 2 5 gf x x . Find the function f .

14. Find the range of the function ( ) | 2 1| f x x for the domain -1 x 1.

15. Given the function : 3 4 f x x and : 2g x x p . Find the value of pif (3) 8gf .

16. Given the function ( ) 4 3 f x x . Find the value of x , if2

( ) ( ) f x f x .

17. Given the function ( ) 3 2 f x x and 12 7

( )3

xf g x

. Find the function of

g .

18. Find the range of x for |1 2x|< 11 .

19. Given the function ( ) | 4 3 | 6 f x x . Find the possible values of x .

20. Given the function ( ) 2 3 f x x and ( ) 5 2 g x x . Find the value of x, if2 2f g .

21. Given the function15

( ) , 33

x f x x

x

. Find the value of x in which it

maps to itself.

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Paper 2

1. The function gis defined by 3: xxg and the function f is suchthat .76: 2 xxxfgFinda. the function f ,

b. the values of k for which ( ) 3082 += kkf .

2. A quadratic functions f is defined by ( ) 542 = xxxf . Sketch thegraph of function ( )xf .

3. a. Given the function f and g are defined by 56: xxf and4:

2 + xxg , find the value ofx for which ( ) ( )xgxf = .

b.

The diagram 1 above shows the relation between Set C and Set D.State the

i. domain of this relation,ii. range of this relation.

4

court

hos ital

restaurant

school

student

teacher

doctor

SET A

SETB

Diagram 1


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4. a. Given that ( ) 49 += xxf and ( ) 83 += xxg . Findi. ( )xfg 1 ,

ii. the value of x such that ( ) 23= xgf .

b.

The above arrow in Diagram 4 represents a functionkx

hxf

: .

Find

i. the value of h and of k,

ii. the value of m such that ( ) mmf3

5= .

5. a. Given the function qxpxf : . Findi. ( )xf 1 in terms ofp and q ,

ii. the value of p and ofq if ( ) 181 =f and ( ) 24 =f .

b. Functions fand g are define by 93: xxf and 32: + xxg .Find 11 gf .

5

xkx

h

f

7

- 5

5

- 1

Diagram 2


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6. a. In Diagram 1, set Ashows the images of certain elements of set B

Diagram 3

i. State the type of relation between set Aand set B

ii. The range of the relation

Diagram 4

b. Diagram 2 shows the function 0,32

:

xx

xxf , find the value of a

6

a

5

xx

x 32

3

2

1

-1

f

g

SetA SetB

e

h


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7. a. Given thatf-1: x 1

x+ 2, x 0, andf : x

a

x b, x -b, where aand b

are constants, find the values ofaand b.

b. Diagram 2 shows the function g : x

2

xm ,

x2

xm

g 34

Diagram 5

Find value of m

8. a. Given the functions g : x 2x - 1 and gf : x 6x + 7 , find the function

f .

b. Given the functions h : x 3x + 1 and hf : x 7 9x2 , find the

function f .

c. Given that xmxg 3: + and342:1 kxxg , find the values of m and k..

9. a. Given the inverse function2

32)(1

=

xxf , find

i. the value of f(4),ii. the value of k if f1 (2k) = k 3 .

b. Given that g(x) = mx + n and g2 (x) = 16x 25 , find the values of m andn.

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QUADRATIC EQUATIONS

Paper 1

1. Given that 4 is a root of the quadratic equation 22 5 0 x x p , findthe value of p .

2. Solve the equation ,2 6

5

xx

.

3. Given the roots of the quadratic equation 2 9 4 0 px x is 4 and1

2.

Find the value of p .

4. Given the quadratic equation 2 9 0 x px has two equal roots. Findthe value of p .

5. Given the difference between the two roots of the quadratic equation22 14 0 x x p is 3 . Find the value of p .

6. A root of the quadratic equation2

0 x x p is 3. Find the value ofthe other root .

7. Without solving the quadratic equation, determine the nature of the

roots for 24 6 3 0x x .

8. Express2

2( 1) 5 3x x in the general form of a quadratic equation .

9. Find the roots of the equation , 22 5 12x x .

10. Given that a root is twice the other root of the quadratic equation24 18 0 x x p , find the value of p .

11. Find the roots of the quadratic equation 22 5 8x x . Give your answercorrect to 3 decimal places.

12. Given the equation 2 3 0 x x k has the roots of and such that

2

. Find the value of k .

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13. Given y = mx is the equation of tangent to the curve2 5 9y x . Find

the possible values of m .

14. Given p + 2 is a root of the equation 2 5 6x x . Find the value of p .

15. The quadratic equation 2 2 1 ( 2) x x k x has two real and equal roots.Find the possible values of k .

16. The quadratic equation 2 3 6 3 4mx mx x has a root of1

m. Find the

value of m .

17. If the quadratic equation2( 1) ( 1) 2 0k x k x has two real and equal

roots, prove that k = 7 .

18. Solve the equation (5x 3)(x + 1) = x (2x 5) . Give your answer correct tofour significant figures .

19. The quadratic equation2

( 5) 8 1 p x x has two distinct roots . find therange of p .

20. Form the quadratic equation which has the roots of 2 and1

2. Give

your answer in the form of 2 0ax bx c where a, b and c areconstants .

21. Given that m and n are the roots of ( x 2) (x + 1) + k = 0 andmn = - 6 . find the value of k, m and n .

22. Given the quadratic equation 24 3 5 0 x kx k . Find

a. the value of k if the quadratic equation has two real and equalroots .

b. the range of k if the quadratic equation has two real anddistinct roots .

23. The quadratic equation 2 2 1 0 x kx k has the roots of m and n .Given that 2 2 4m n , find the value of k .

24. The quadratic equation 2 5 9 0 px qx p has two real and equal roots.Given that p and q are positive, find the ratio of p to q. hence, solvethe equation .

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25. The quadratic equation,23 2 9 0 x px q has the roots -5 and 2 . Find

a. the value of p and q

b. the range of k such that23 2 9 x px q k has no real roots.

Paper 2

1. a. Determine if 1,0 and 3 are the roots of the quadratic equations( ) 3218 2 =x .

b. Given that 2 is one of the roots of quadratic equations 032 =+ pxx ,

where pis a constant . Find the values of the root.

2. a. Given the quadratic equations 0322 =+ nxmx has two equal roots,express n in terms of m .

b.

The Diagram 2 above shows a window with two parts. The top part is arectangle and the bottom part is a square. Given that the area of the

window is 22000cm .

i. Show that 02000252 =+ xx ,ii. Find the value of x ,iii. Find the area of the rectangle.

3. a. Given that p and q are roots of the equation 0124 2 =+ xx . Form aquadratic equation which has the roots 12 p and 12 q .

b. The quadratic equation ( ) ( )432 += mmxx has equal roots. Find valueof m .

11

25 cm

cm

Dia ram 2


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4. a. Given that the roots of the quadratic equation 2x2 + (3 k)x + 8p = 0are p and 2p , p 0. Find k and p.

b. Find the range of values of k if the quadratic equation x2 + 2kx + k + 6

= 0 has equal roots.

5. a. The equation x2 6x + 7 = h(2x 3) has roots which are equal. Findthe values of h.

b. Given that and are roots of the equation x2 2x + k = 0 , while 2and 2 are the roots of the equation x2 + mx + 9 = 0. Determine thepossible values of k and m.

6. a. Given2

pand

2

qare roots of the equation kx(x 1) = m 4x. If p + q

= 4 and pq = - 5, find the values of k and m .

b. The quadratic equation 2x2 + px + q = 0 has roots -2 and 3. Find the

value of p anf q so that 2x2 + px + q = k has real roots.

7. a. Given and -5 are roots of a quadratic Equation.Write down thequadratic equation in the form ax2 + bx + c = 0.

b. Find the range of values of x for which the equation x2 + kx + 2k 3 = 0has no real roots.

8. a. The quadratic equation 2 4 3 x mx x+ + = has two equal roots. Find thepossible values ofm.

b. Form the quadratic equation which has the roots 3 and2

1. Give your

answer in the form 02 =++ cbxax , where a, band care constants.

c. Solve the quadratic equation ).2)(1()4(2 += xxxx Give your answer

correct to four significant figures.

9. a. A quadratic equation 432 +=+ kxxx has distinct roots. Find the

possible values ofk

b. The roots of the quadratic equation 0452 2 =++ kxx are h and 3 find

the value of k and h

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10. a. Find the range of the values of p if the quadratic equation2 6 2 0 px x+ = has no roots .

b. Solve the quadratic equation 2 ( 4) (1 )( 2) x x x x = + . Give your answer tofour significant figures.

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QUADRATIC FUNCTIONS

Paper 1

1. Find the range of x that satisfies the quadratic inequality of 23 4x x >

.

2. Given 4x 2y + 3 = 0 . Find the range of x if y . 2 .

3. Find the range of x if 8x 23x .

4. Given the graph 22( ) y p q x= has a minimum point of (3, -4) . findthe value of p and q .

5. Skecth the graph of 2( ) ( 3) 6 f x x= for the domain 0 x 6 .

6. Find the range of x if (2x + 1)(x 3) < 0 .

7. Find the range of k such that 22 5 x x k + is always positive .

8. Find the range of x for 3y + 1 = 4x and 2y > 1 + x .

9. Find the value of m such that the equation2(1 ) 3 3 0m x mx+ + = has

equal roots .

10. Find the equation of the axis of symmetry for the graph of22 5 3 y x x= + + .

11. Find the range of x given 3 2x < 13 and 3x 2< 4 .

12. Find the range of m if y = mx +1 does not intersect with the curve of2 3 5 y x x= + .

13. Find the range of x if2

3x 75 .

14. State the coordinates of the minimum point of2

2(5 3) 4y x .

15. Given y = 2x + 3 intersects with the curve of 23 6 y x mx at twodifferent points. Find the range of m .

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16. Find the range of x, if x (6x - 7 ) > 10 .

17. Find the value of p , if the graph of2 5 y x px is a tangent to line

y = 4 .

18. Determine the equation of the curve 25 ( 3)y x after it is reflected atthe x-axis.

19. Find the range of p if y = 2x + p does not intersect at the curve of2 2 5x y .

20. Find the range of x, if2

02 1

x

x

.

21. Find the range of x, if ( x 3)(x + 5) > (x 3)(3x 2) .

22. Given8

3

yx

. Find the range of x if y > 10 .

23. Given 2 2( ) 2 5 ( )f x x x a x b c .a. find the value of a, b and c .

b. sketch the graph of 2( ) 2 5 f x x x .

24. Given the coordinates of the minimum point of2( ) 2 f x x px q is

(2, 5) . Find the values of p and q .

25. Given the graph 2(3 ) y p x q has a maximum point at2

,63

.

a. find the value of p and q .b. sketch the graph .

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Paper 2

1. a. Diagram below shows the graph of a quadratic function 2)(2 kxy ,

where k is a constant.

Find

i. the value of k,

ii. the equation of the axis of symmetry,

iii. the coordinates of the maximum point.

b. Solve the inequality22 3 2 0x x

2. a. Given ( ) ( ) mxmxxf += 12 . The curve ( )xkfy = cut the y -axis at thepoint ( )15,0 . If 3=m find value of k.

16

( )3,2( 3

O x

y

f(x)

x

)11,6(11

0

Diagram 6


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b. The Diagram 5 above shows the graph of a quadratic function

( ) ( ) 22 += hxxf ,where h is a constant.Findi. the value of h ,

ii. the equation of the axis of symmetry,iii. the coordinates of the minimum point.

3. a. Diagram 1 shows the graph of a quadratic function 2( ) 4 , f x px x q= + where p and q are constants.

Diagram 7

The curve ( ) y f x= has a maximum point ( -1, 5 ). State

i. the value ofpii. the value ofq

b. Given that the quadratic functions ( ) 24383 xxxf =+ .Express ( )xf in form 2)( rxqp ++ where qp, and r areconstants.

17

y(-1, 5 )

0x


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4. a.

Diagram 3

In the above Diagram 3, the point

9,

21 is the maximum point of the

curve ( ) 21= qxpy where p and qare constants.Findi. the value of p and q ,ii. the coordinates of the points A and B.

b. The graph of the function f(x) = 2x2 + (3 k)x + 8 does not touchthe x-axis. Determine the range of k.

5. a. Find the range of x if 3x (2x + 3) 4x + 1b. Solve 5 + m2 > 9 3m.

c. A quadratic functions f is defined by ( ) 542 = xxxf .Sketch the graph of function ( )xf .

6. a. Given ( ) ( ) mxmxxf += 12 . The curve ( )xkfy = cut the y -axis at the point ( )15,0 . If 3=m find value of k.

18

9,2

1

Bx

A

y

0

2)1( = qxpy

f(x)

x

)11,6(11

0

Diagram 5


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( 4, )

b. The Diagram 5 above shows the graph of a quadratic function

( ) ( ) 22 += hxxf ,where h is a constant.Find

i. the value of h ,ii. the equation of the axis of symmetry,

iii. the coordinates of the minimum point.

7. a. Given f(x) = 2x2 8. Find the range of x so that f(x) is positive.b. Find the range of x which satisfy the inequality (x 1)2 > x 1

8. a. The graph of the function 125222 ++= ppxxy cut the x-axis

twice.Find the range of the values ofp.

b. Find the range of values of x for 1)-x)(3+x2(>21)-(x

9. a. Given y = h + 4kx 2x2 = q 2(x + p)2 .Find p and q in termsof h and / or k.

b. If h = -10 and k = 3,i. State the equation of the axis of symmetry,ii. Sketch the graph of y = f(x)

10. Diagram 2 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 (4, q),where qis a constant.

State

a. the value of p,b. the value of q,c. the equation of the axis of symmetry.

Diagram 8

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SIMULTANEOUS EQUATIONS

Paper 2

1. Solve the simultenous equations, y = 4 2x and2

4 0y x .

2. Solve the simultenous equations, y = 2 x and 2 22 8 0x y .

3. Solve the simultaneous equations, y = 5 x and 2 2 1 2 x y xy

4. Solve the simultaneous equations, x + 3y = 1 and2 3 5 20 x x y

5. Solve the simultaneous equations, x + y = 1 and2 2

6 2x y

6. Solve the simultaneous equations, x = 3 + y and2 22 14 x xy y

7. Solve the simultaneous equations, x = 2 2y and 22 6 0 y xy

8. Solve the simultaneous equations, 2x + y = 8 and2 2

4 3 52x y

9. Solve the simultaneous equations, x + 2y = 0 and 2 2 8 10 8 0 x y x y

10. Solve the simultaneous equations, y 4x = 3 and 2 2 8 x y xy

11. Solve the simultaneous equations, x - 7y = 2 and 2 234 7 16 x y xy

12. Solve the simultaneous equations, x + 2y = 5 and 2 25 4 12 29 x y x

13. Solve the simultaneous equations, 5x - 2y = 6 and22 6 5 xy x .

14. Solve the simultaneous equations, 3x 2y = 5 and ( ) 5 ( )y x y x x y .

15. Solve the simultaneous equations,2

4 8 2 x y x x y

16. Solve the simultaneous equations, 2 22( ) 1 2 11x y x y x y

17. Solve the simultaneous equations, 2y = x 2 and6

5x y

y x

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18. Solve the simultaneous equations, 1 and ( 1) 5 22 4

x y x y x

19. The straight line x y = 5 intersects the curve 2 22 9 x xy y atpoint P and point Q . Find the coordinates of P and Q .

20. Given point ( -1 , 2p ) is the solution of the simultaneous equation2 29 4 x ky kx xy , where k and p are constants. Find the value of

k and p .

21. Given the perimeter and the area of a rectangular field are 80 m

and 396 2m respectively . Find the length and the breadth of the

field.

22. Given the straight line 2x - 3y =4 intersects with the curve2 2 x xy y k at point (-4, h ). Calculate

a. the value of h and k .b. the other intersection point .

23. Solve the simultaneous equations,

x - 4y = 8 and2 8 4 y y x .

24. Solve the simultaneous equations 3p + q = 5 and 2 2 5p q . Give youranswer correct to three decimal places .

25. Solve the simultaneous equations,

3x + y = 3 and2 1

23x y

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INDICES AND LOGARITHMS

Paper 1

1. Solve2 6 23 9 0x x .

2. Express2 1 2 2 1

5 5 15(5 )n n n to its simplest form .

3. Solve 5log4 64x .

4. Solve 23 5 0x x .

5. Solve 4 13(9 ) 27x x .

6. Evaluate the expression3 1

2

3 3

4(3 )

n n

n

.

7. Given 3 33 log 2 logx y . Express x in terms of y .

8. Given log 2m x and log 5m y . Express log 12.5m m in terms of x andy .

9. Solve 2 41

log log 2 3

2

x x .

10. Find the value of y given log 5 log 135 3y y .

11. Solve 9 3 16log log (2 1) log 4.x

12. Find the value of m , if log 27 9m .

13. Evaluate 3log 3 3 .

14. Solve1

3

25125

5

x

x

.

15. Solve 3 3 3log log ( 4) 1 log ( 1) p p p .

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16. Given 2 45

log log2

k w . Express k in terms of w .

17. Given 3 3log 5 and log 7m n . Find the value of p if 33

log2

m np

.

18. Given 2 4log and log x m y n . Express each of the following in termsof m and n .

a.2x y

b. 2x

y

19. Given ylog 9 and log 27x m n . Express 27log xy in terms of m and n .

20. Solve1 1 2

3 9 27 y y y

.

21. Solve 10 10 10log (2 1) log log (2 3). x x x

22. Solve 3log4 8x .

23. Given 2 23 1.582 5 2.322log and log . Without calculator , find the value of

.a. 2log 60

b. 15log 8

24. Given2

2 (8 ) 2 and 9 (3 ) 27 x y x y . Find the value of x and y .

25. Given the curve 6n y ax passes through (5, 1.5) and (25, 31.5). Find the value of a and n .

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Paper 2

1. a.` Solve the equation 02781 321 xx .b. Given log7 2 = hand log7 5 = k. Express log7 2.8 in terms of hand k.

2. a. Given that 2log7 = m and 5log7 = 5 n . Express 8.2log7 in

terms of m and n..

b. Solve the equation 4log3log2)48(log 555 x .c. Solve the equation 12log)93(log 33 tt .

3. a. Solve the equation 2 42 x =2

16x .

b. Solve the equation4 28 4x x

c. Solve the equation 2 2log 6 log ( 3) 2x x

4. a. Solve the equation 066 123 = + xx

b. Given that ma =2log and na =7log , express aa 5.3log in terms of m

and n .

c. Solve the equation xx 33 log2)45(log =

5. a. Solve the equation1

125 525

x

x=

b. Given 32 1log ,5

n = find the value ofn

c. Solve the equation2

2 2log log ( 3) 4x x =

6. a. Solve the equation 3 3log log ( 3) 1x x =

b. Solve the equation2

32

4

18

+

=x

x

c. Solve the equation )1(log2log 33 += xx

7. a. Given that 1loglog 416 = xy ,express y in term of x.b. Given that 3010.02log10 = and 8451.07log10 = . Find the value of

28log10 .

c. Solve the equation 2loglog 42 = xx .

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8. a. Without using calculator ,evaluate 32log2log310log2 555 + ,

b. If a=7log2 express 49log2

17log2 168 in terms of a .

c. Solve ( ) ( ) ( )34log5log12log2 222 ++=+ xxx .

9. a. Solve the equation42

2

25

1125

+

+

=x

x

b. Given that xyxy 444 loglog21log3 ++= , express y in terms ofx .

c. Solve the equation xx 22 log3)14(log +=+ .

10. a. Solve the equation 51205 12 =x .

b. Find the value of xof 2loglog2 93 = xx .

c. Solve the equations 1)9(27 =x .

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PROGRESSIONS

Paper 1

1. Find the 16th term of the arithmetic progression

1 2,1 ,3,.......

3 3

2. Find the number of the multiples of 3 between 20 and 110 .

3. For the arithmetic progression 5, 8, 11,. Which terms is equal to320 .

4. Given that 4k, 12,2

2 4k k+ are three consecutive terms of anarithmetic progression, find the possible values of k .

5. The 12th term and the 17th term of an arithmetic progression are38 and 53 respectively. Find the common difference.

6. Find the sum of the first n terms of the arithmetic progression

11, 7, 3, .

7. How many terms of the arithmetic progression 12, 16, 20..must be

taken for the sum to be equal to 132 .

8. The second term of an arithmetic progression is 8 while the sum ofthe first six terms is 30. Find the common difference .

9. The sum of the first 6 terms of an arithmetic progression is 72 while the sum of the first 9 terms is 81. Find the commondifference .

10. If the sum of the first n terms of an arithmetic progression is given

by (2 3)nS n n= + , find the common difference .

11. Calculate how many months it will take to repay a debt of RM5800by monthly payments of RM100 initially with an increase of RM20for each month after that.

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12. Find the sum of first 6 terms of the geometric progression

3, 2,4

3,.

13. Find the 8th term of the geometric progression 3, -6, 12, -24,..

14. For the geometric progression 3, 6, 12,.calculate the least numberof terms that must be taken for the sum to exceed 1000.

15. If the second term of a geometric progression is 3 and the seventh

term is32

81, find the first term.

16. The third term and the sixth term of a geometric progression are27 and 8 respectively . Find the second term .

17. The second term and the third term od a geometric progression are

1

2and 1 respectively . Find the sum of the first three terms .

18. Given that x + 2, 2x , 2x + 3 are the first three terms of anarithmetic progression, finda. the value of x .

b. the sum of the next 20 terms.

19. In a certain arithmetic progression, the sum of the first n terms is

given by2

6 3nS n n= + . Finda. the n th term of n .b. the common difference.

20. The first term of an arithmetic progression is 24 . It consists of 23terms. Given that the sum of the last 3 terms is 5 times the sumof the first 3 terms, finda. the common differenceb. the sum of the first 11 terms .

21. If the sum to infinity of a geometric progression is 64 and the first

term is 16, find the common ratio.

22. Write the recurring decimal 0.459459459..as a single fraction inits lowest terms.

23. A geometric progression is such that the second term is 6 and thesum to infinity is 32 . Find the possible values of the common ratio.

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24. Given that the first term of a geometric progression is 18 and thesum of the first three terms is 38, find the possible values of thecommon ratio.

25. The sum of the first 6 terms of an arithmetic progression is 39

and the sum of the next 6 terms is 69. Finda. the first term and the common difference.b. the sum of all the terms from the 15th term to the 25th term .

Paper 2

1. The mass of chemicals produced by a factory increases by 8 % every year. 6 000 tonnes of chemical were produced in 1993 .a. Calculate the mass of chemicals produced by the factory in

1999.b. Determine in which yea did the total mass of chemicals

produced exceeds 85000 tonnes for the first time.

2. Diagram 2 shows a part of a series of squares with ABCD as thefirst square with sides 24 cm, AEFG the second square, AHIJ thethird square and so on . Each square is drawn in such a way thatthe length of each side is half that of the previous square.

Diagram 2

a. Find the area of the fifth square in the series .b. Estimate the total area of all the squares that can be drawn

in this manner.c. Find which square tat has its length of sides less than 0.05cm for the forst time.

24

cm

24 cm C

B

D

G

J

A H E

I

F

2852

65P

41.6

O


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3.

Diagram 3

a. The above diagram shows a pendulum released from the position OP. It

swings freely through the angle of 65, 52, 41.6 and so on. Calculatethe total angle it covers in 8 swings.

b. The sequence -11, -5, 1,is an arithmetic progression. State thethree consecutive terms of this arithmetic progression where the sumof these three terms is 93.

4. Diagram 4 shows an arrangement of these right-angled triangles for the

infinite series of similar triangles.

Diagram 4

The base and the height of the first right-angled triangle is xcm and ycm

respectively. The base and the height of the subsequent triangle are a

quarter of the base and half of the height of the previous triangle.

29

cm

x cm


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a. Show that the areas of the triangles form a geometric progression. State

the common ratio of the progression.

b. Given x= 160 cm and y= 320.

i. Determine which triangle has an area of1

64

cm.

ii. Find the sum to infinity for the area, in cm, of all the triangles.

5. Diagram 1 shows the first four semi-circles of the 16 semi-circlesformed from a piece of wire. The radius of the first semi-circle is pcm and the radius of the subsequent semi-circles decrease uniformlyby 3 cm. Given the radius of the smallest semi-circle is 5 cm.

Diagram 1

Calculatea. the value of p.

b. the length of the wire (in terms of ) needed to form the above16 semi-circles.

6. a. An arithmetic progression consists of 10 terms. The sum of thelast 5 terms is 5 and the fourth term is 9 . Find the sum ofthis progression.

b. The price of a house in a certain residential area is RM200 000.Its price increases by 6 % each year. Find the minimum number

of years needed for the price to exceed RM500 000 for thefirst time.

7. Ramesh has 189 new stamps. He puts 3 stamps in the firstalbum, 6 stamps in the second album, 12 stamps in the thirdalbum and so on .

pp - 3 p - 6 p - 9

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a. Show that the number of new stamps in each albumforms a geometric progression.

b. How many albums are used.

8. The sum of the fourth and the fifth terms of a geometric

progression is 12 . If the fourth term of the progression is 8, finda. the common ratio and the first term of the progression.b. the sum to infinity of the progression.

9. If 10log p , 10log pq and2

10log pq are the first three terms of a

progression.

a. Show that it forms an arithmetic progression.b. Find the sum of the first 5 terms of the progression.

10. In the beginning of year 2005, Azmi inversted a sum of RM 1500in a cooperative fund that gives an annual interest of 6.0 %. Atthe end of each year Azmi reinvests both the principal sum andits interest in the fund.

a. Calculate the sum invested at the end of the third year.b. At the end of which year will the total investment first

exceed RM3000.

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LINEAR LAW

Paper 1

1. Reduce the non-linear equation2p y qx to the linear form, where p

and q are constants.

2. If the equation (10 )qxy p , where p and q are constants, is reducedto the linear form, find

a. the gradient .b. the Y-intercept, in terms of p and or q .

3. If a straight-line graph of y againsty

xis drawn to represent the

equation hx + ky = xy, where h and k are constants, finda. the gradient.b. the Y-intercept, of the straight line in terms of h and/ or k .

4. Explain how a straight line graph can be drawn to represent the

equation (3)bxy a

, where a and b are constants.

5. If a straight line graph of1

yagainst x is drawn to represent the

equationc

yd x

, where c and d are constants, find

a. the gradientb. the Y-intercept, of the straight line in terms of c and/ or d .

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6. The below figure shows part of a straight-line graph drawn to

represent the equation 3 y hx kx , where h and k are constants. Find

the value of h and of k.

7. The below figure shows part of a straight line graph drawn to

represent the equation n y mx , where m and n are constants. Findthe value of m and of n .

8. When the graph of 10log y is plotted against 10log x to represent the

equationx

yk

, where k is a constant, a straight line that passes

through the point (4, 1) is obtained . Find the value of k .

9. When the graph of 2log y is plotted against 2log ( 2)x to representthe equation ( 2)ny k x , where k and n are constants, a traight linethat passes through the point (-1, 2) and (2, -1) is obtained. Find thevalue of k and of n .

y

x

2x0

( 1, 6)

( 4, 18)

2log y

2log x0

( 2, 3)

( 6, 1)

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10. When the graph of 3log y is plotted against 3log x to represent the

equation3mx

yk

, where m and k are constants, a straight line with

a gradient of 3 and passes through the point (2, 3) is obtained .Find the value of m and of k .

11. The variable x and y are related in such a way that wheny

xis

plotted against1

x, a straight line that passes through the points (1,

-3) and (4, 9) is obtained, as shown in the below figure . Express yin terms of x .

12. The variables x and y are related in such a way that when1

yis

plotted against1

x, a straight line that passes through the points (0,

4) and (8, 2) is obtained, as shown in the figure . Express y interms of x .

y

x

1

x0

( 1, -3)

( 4, 9)

1

y

1

x0

( 8, 2)

( 0, 4)

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13. The below figure shows part of a straight line graph drawn to

represent the equation6 1

2y

x . Find the value of h and of k .

14. The below figure shows the graph of 5log y against 5log x . The values

of x and y are related by the equation n y ax , where a and n areconstants . Find the value of a and of n .

15. The variables x and y are related in such a way that when y is

plotted against 10log x , a straight line that passes through the points

(3, 0) and (6, 2) is obtained, as shown in the above figure . Calculatethe value of y when x = 1000.

xy

x0

( 8, k)

( 0, h)

5log y

5log x0

( 3, 7)

( 1, 1)

y

5log x0

( 6, 2)

( 3, 0)

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16. The following diagram shows the straight line graph of2

y

xagainst x

.

Given that 2 37 y x x , calculate the value of h and of k .

17. x and y are related by the equation

2

y px qx , where p and q areconstants. A straight line is obtained by plotting

y

xagainst x , as

shown in the diagram below .

Calculate the value of p and of q .

2

y

x

x0

( k, 3)

( 2, h)

y

x

x0

( 10, 2)

( 2, 6)

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18. Diagram 2 shows the graph of1

yagainst

1

x.

Diagram 2

Given that

3

1 6

xy

x , find the values of p and q .

19. The experimental values of x and y are related by the equation2 2

12 3

y x

a b . When 2y is plotted against 2x , a straight line is

obtained with a gradient of - 2.5 and a 2y -intercept of 5 .

Calculate the values of anda b .

20.

The diagram above shows part of a straight line obtained by

plotting 10log y against x . If the straight line passes through

3,7 and 1,1 , express y in terms of x

1

y

1

x

13

( , )p

(3, )q

0

3,7

1,1

lg y

x0

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Paper 2

1. Table shows the values of two variables, x and y, obtained from an

experiment. Variable x and y are related by the equationxpky = 1 where

p and k are constants.

x 0.30 0.40 0.50 0.60 0.70 0.80 0.90

y 2.24 2.64 3.04 3.50 4.03 4.64 5.34

Table 1

a. Plot ylog against )1( x , using a scale of 2 cm to 0.1 unit on the

)1( x axis and 2 cm to 0.05 unit on the ylog axis. Hence, draw

the line of best fit.

b. Use your graph from (a) to find the value of

i. p

ii. k

2. Table 1 below shows the value of two variables, xand y, obtained from an

experiment. Variables xand y are related by the equation 1xy mn , where

m and n are constants.

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

Table 2

a. Complete the table below and plot log10 yagainst (x+ 1). Hence, drawthe line of best fit.

x + 1 2 4 5

y 0.60 0.76

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Table 3

b. Use your graph, to find the value of

i. m

ii. n

3. The variables a and y are known to be related by the equation2 2 y ax bx

, where a and b are constants. Table 1 shows the corresponding values ofx and y obtained from an experiment.

Table 4

a. By using a scale of 2cm to represent 1 unit on both axes, plot2y

against xx

. Hence , draw the line of best fit.

b. Use the graph in (a) to find,i. the values of a and b

ii. the values ofx satisfying 2( ) 0a b x bx

4. Use graph paper to answer this question.

x 20 10 6.25 4.55 3.23

y 1.19 1.45 2.22 2.94 16.7

Table 5

Table 2 above shows the data of an experiment for two variables x and y,

which are related by a law in an equationx

y= p qx , where p and q are

constants. One of the data has been recorded wrongly.

a. From the graph, findi. the value of y which is recorded wrongly and the value it

supposed to be,ii. the values of p and q,iii. the value of x when y = 12.

x 2 4 6 8 10

y 2.82 4.91 7.01 9.02 11.2

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function for x and/or y respectively. State X and Y in terms of x.

5. Use a graph paper to answer this question.

Table 1 shows the values of two variables, x and y, obtained from an

experiment. The variables xand yare related by the equation , where

hand kare constants.

x 1 2 3 4 5 6

y 6.31 10.01 15.49 25.12 39.81 63.10

Table 7

a. Plot log10yagainst x, by using a scale of 2 cm to 1 unit on x-axis and

2 cm to 0.2 unit on y-axis.Hence, draw the line of best fit.

b. Use the graph from (a) to find the value of

i. h

ii. k

6. Use the graph paper provided to answers this questionsThe following corresponding variables ofxand ywere measured

x 2 3 4 5 6 7y 2 4.5 5.6 6.3 6.8 7.1

Table 8

The variables xand yare related by the equation ,baxxy where aand bareconstants

a. Plot xy against x, by using a scale of 2cm to 1 units on the x-axis and2cm to 5 units on the y-axis. Hence, draw the line of best fit

b. Use the graph from (a) to find the value of

i. aii. b

7. Use the graph paper to answer this question.

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Table below shows the values of two variables, x and y , obtained from an

experiment. Variables x and y , are related by the equation xpky , where

p and k

x2 4 6 8 10 12y 3.16 5.50 9.12 16.22 28.84 46.77

x 1.0 1.5 2.0 2.5 3.0 3.5

y 0.81 2.72 5.76 9.92 15.21 21.62

x 1 2 3 4 5

y 1.32 1.76 2.83 5.51 13.00

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b. Use the graph in (a) to find the value of .i. aii. b

10. The table below shows the values of two viarables, x and y, obtainedfrom an experiment . The variables x and y are related by the

equation x y pk , where p and k are constants.

x 2 4 6 8 10 12

y 5.18 11.64 26.20 58.95 132.63 298.42

a. Plot 10log y against x by using a scale of 2 cm to 2 units on

the x-axis and 2 cm to 0.2 units on the y axis . Hence, draw

the line of the best fit.b. Use your graph from (a) to find the value ofi. pii. k

Algebraic Component Module

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