Program Last Touching

7/27/2019 Program Last Touching

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PROGRAM LAST TOUCHINGADDITIONAL MATHEMATICS

FUNCTIONS

SBP-KEMENTERIAN PELAJARAN MALAYSIA NEGERI JOHOR

1. Diagram shows a function that maps set A to set B.

It is given that the function that maps set A to set B is 3: xxf Find

a. The value of m

b. The value of )3(1ff

[2 marks]

Diagram shows the graph of the function |1|)( xxf , for the

domain 52 x .

Statea. the )5(1

f

b. )3(f f

[2 marks]


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2. Given that 0,4

: xx

xg and the composite function

2: xxgf , find

a. )(xf

b. The value of x when 6)( xfg [4 marks]

Given the functions 17: xxg and 13

: x

xh . Find

a. )8(1g

b. )(xgh

[4 marks]

3. Given that pxxf 8: and5

26:1

xxg

Find

a. )(xg

b. The value of p if )()2( xfxg [4 marks]

Given the functions 2,2

3)(

x

xxf and 45)( xxgf . Find

)(xg

[3 marks]

QUADRATIC EQUATION


4. Given that 2x and3

1x are the roots of the equation

03 2 cbxx , find the value of b and the value of c .

[3 marks]

Given that 3 dan k are the roots of the quadratic equation

pxx 2 . Find the value of k and p .

[3 marks]


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QUADRATIC FUNCTION


5. Find the range of values of x for xx 9202 .

[2 marks]

Diagram shows the graph of cbxaxf 2)()( . State the value of

a, b and c.

[3 marks]

6. Given quadratic function qpxxf ]5)6[()( 2 has a

maximum point )15,3( 2nnT . Express q in terms p

[3 marks]

Given the quadratic function 72)( 2 xxf , find the range of x if

11)( xf .

[3 marks]


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INDICES & LOGARITHMS


7. Solve the equationx

x

625

125

2

[3 marks]

Solve the equation 233 23 xx

[3 marks]

8. Solve the equation 1)2(loglog 33 xx

[3 marks]

Given that 4loglog 93 x , find the value of x .

[3 marks]

9. Given h2log5 and k3log5 , express 90log12 in terms of h

and k .

[4 marks]

PROGRESSIONS

SBP-KEMENTERIAN PELAJARAN MALAYSIA NEGERI JOHOR10. It is given an arithmetic progression is 5, 7, 9,.87. Find the

number of terms of this progression.

[3 marks]

Find the sum to infinity of the geometric series ....4

1

2

11

[2 marks]

11. It is the first three terms of a geometric series are

........81

1

27

1

9

1 Find the sum to infinity of the series .

[3 marks]

Find the least number of terms in the arithmetic progression -5, -1,

3,.so that its sum exceeds 200.

[3 marks]

The first three consecutive positive terms of a geometric progression

are 2, y and 18. Find the common ratio of the progression.

[2 marks]


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LINEAR LAW


12.The variables x and y are related by the equation

qxpxy 522 , where p and q are constants. Diagram shows

a straight line graph )2( xy against 2x .

Find the value of p and of q .

[4marks]

The variables x and y are related by the equationn

my , where m

and n are constants. Diagram shows the straight line obtained by

plotting y2log against x .

Find the value of m and of n .

[3 marks]

COORDINATE GEOMETRY



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13.Diagram shows a straight line PQ with the equation 168

xy

.

Find the equation of the straight kine which is perpendicular to PQ

and passes through the point Q.

[3 marks]

Diagram shows a straight line PMQ.

Given that a point M(4, 4) divides the line segment joining the points

)4,( ttsP and )2,7( stQ in the ratio 2 : 1 . Find the value of sand t .

[4 marks]

14.Diagram shows A, B and C are three points on a straight line.

It is given that 5AC=AB . Find the coordinates of B.

Diagram shows a straight line AB which intersects to the straight line

CD at the point )3,2(T .


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[3 marks]

The equation of the straight line AB is 42

1 xy and the equation

of the straight line CD is 12 xy . Calculate the area of the triangleATD.

[3 marks]

VECTORS


15.Given yxPQ 23 and yxhQR 4)1( . The point P, Q and

R are collinear. Find the value of h

[3 marks]

Diagram shows vectors a and b . Draw the resultant vector of 2ab.

[2 marks]

16.Diagram shows OABC is a parallelogram such that jiOA 34 The vectors PQ and RS are parallel . it is given that baPQ 32


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and jiOB 511

Find the unit vector in the direction of OC [3 marks]

and bakRS2

9)1( . Find

a. The value of k

b. The ratio of PQ : RS

[4 marks]

DIFFERENTION


19. Given that23

)35()( xxxf , find )2('f

[3 marks]Given

4

24

xy , find the approximate value of y, when x changes from

2 to 2.02.

[4 marks]

20. Two viariables P and x are related by the equationx

xP 2

3 .

Given x increases at a constant rate of 4 units per second when

x = 2, find the rate of change of P.

[3 marks]

Find the coordinates of the turning points on the curvex

xy 1

4 and

determine the maximum point.

[4 marks]

INTEGRATIONS



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21.Given3)52(

x

hy and )(xg

dx

dy , find the value of h if

3

2

7]1)([ dxxg

[3 marks]

Given2

32

x

xy

and )(3 xg

dx

dy where )(xg is a function of x .

Find the value of

1

1

)( dxxg .

[4 marks]

STATISTICS


22.The mean of a set data 1,8,32 mm is 7.Find

a. The value of m

b. The new mean if each of the data multiflied by 3.

[3 marks]

Table shows the marks of a group of students in an examination.

Marks 1 2 3 4 5

Number of students 4 6 8 x 7

Find

a. The maximum value of x if the mode mark is 8 .

b. The minimum value of x if the mean of the marks is greater than

3.

[3 marks]

PROBABILITY DISTRIBUTIONS


24. The probability that it will rain on a particular day is5

2 . If X is the

number of rainy days in a week, find.

a. The mean of the distribution of X

b. The standard deviation of the distribution of X

[3 marks]

25.Diagram shows a standardized normal distribution graph. Diagram shows a standard normal distribution graph


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The probability represented by the area of the shaded region is

0.7286.

a. Find the value of k .

b. X is continuous random variable which is normally distributed

with a mean of and a standard deviation of 8 . Find the value

of if X = 70 when the z-score is k .

[ 4 marks]

The probability represented by the shaded region is 0.7514.

a. Find the value of k

b. X is a continuous random variables which is normally distributed

with a mean 80 and a variance of 9. Find the value of X when thez-score is k .

[4 marks]


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FUNCTIONS

NEGERI TERENGGANU NEGERI KELANTAN

1. Diagram shows the function f that maps x to y and the function

g that maps y to z .

Determine

a. )6(1

g

b. )4(gf

[2 marks]

Diagram shows the relation between set A and set B

a. State

i. The value of h

ii. The range of the relation

b. Using the function notation, write a relation between set A and

set B.

[3 marks]

2.The functions f and g are defined by xxf 31: and

3: 2 xxg . Find

a. )2(g

b. )2(1gf

[3 marks]

Given the function mxxf 3: and 3

4:1

nxxf , where m and

n are constants. Find the value of m and of n.

[3 marks]

3. The following information refers to the functions f and g


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a. Given that 10)4( f , find the value of k

b. Find )(xgf [4 marks]

QUADRATIC EQUATIONSNEGERI TERENGGANU NEGERI KELANTAN

4. Find the range of values of p such that the equation

22 2 pxx has no real roots.

[2 marks]

Form the quadratic equation which has the roots -3 and4

1. Give the

answer in the form 02 cbxax , where a, b and c are constants.[2 marks]

5.Given that one of the roots of the quadratic equation

082 kxx is twice the other. Find the values of k .[3 marks]

kxxf 3: , where k is a constant

12

42:

xxg


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QUADRATIC FUNCTIONS


6. Find the range of values of t for ttt 18)4( .[3 marks]

Diagram shows the graph of a quadratic functions khxxf 2)()( ,where h and k are constants.

State

a. The value of h

b. The value of k

c. The equation of the axis of symmetry

[3 marks]

7.Diagram shows the graph of a quadratic function2)(29)( hxxf and the maximum point is (h, k), where h and

k are constants.

Find the range of values of x for 36)5( xx .[3 marks]


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State

a. The value of kb. The value of h

c. The equation of the axis of symmetry

[3 marks]


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COORDINATE GEOMETRY


8. The vertices of a triangle are )3,7(P , )5,1(Q and )1,( kR . Given

that the area of the triangle is 19

2

unit , find the possible valuesof k .

[3 marks]

13. The straight line kxhy 3 passes through point (0, 2) and isparallel to the straight line joining the points (3, 1) and (2, 5).

Find

a. The value of h

b. The value of k

[3 marks]

14. Given that the point P(-3r, 2) divides the line segment A(-4 , t) and

B(r, 8) in the ratio AP : PB = 1 : 4 , find the value of r and of t .

[3 marks]

INDICES & LOGARITHMS

NEGERI TERENGGANU NEGERI KELANTAN9. Solve the equation 3)81(27 2 x

[3 marks]Solve the equation

1

3

9

127

x

x.

[3 marks]

12. Given thatph 3 and qk 3 , express

9log

2

3

kh in terms of p

and q .

[3 marks]

Solve the equation xx 2log3)1(log 22 [3 marks]

13. Given that 1loglog 293 PQ express P in terms of Q .

[4 marks]

STATISTICS



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14. Table shows marks obtained by a group of students in a monthly

test.

Mark 1 - 20 21 - 40 41 - 60 61 - 80

Number of students 6 9 13 12

Without drawing an ogive, find the third quartile mark

[3 marks]

22.The mean of 5 numbers is h2 . The sum of the square of the

numbers is 605 and the variance is2

3k . Express h in terms of k .[3 marks]

DIFFERENTIATIONS


15. Given that the equation of a curve which passes through point P is

2)12( xy . The normal gradient to the curve at point P is2

1 .

Find the coordinates of P.

[3 marks]

19. Given that3

)12(

4)(

xxg , find )1(''g .

[4 marks]

20. Two variables r and s are related by the equation2

3

ssr . Given

that r increase at a constant rate of 5 units per second, find the rate of

change of s when s = 2 .

[3 marks]

INTEGRATIONS


16. Given that 4

1

7)( dxxg , find the value of

a. 1

4

)( dxxg

b. p if 4

1

25])([ dxpxg

[4 marks]

20. Given that 3

1

10)( dxxg , find the value of m if

3

1

6])(2[ mdxmxxg .

[3 marks]


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17. A curve which has gradient function 2kx , where k is a constant,passes through points (0, 10) and (2,0). Find

a. The value of k

b. The equation of the curve

[4 marks]

PROGRESSION


10. Given that three consecutive terms of an arithmetic progression

are 12, 3k -2 and 4 + k , where k is a constant. Find the value of k .

[2marks]

9. The fourth term of an arithmetic progression is 7 . The sum of the

first six terms is 51. Find the first term and the common difference

of the progression.

[4 marks]

11.The first three terms of a geometric progression are 36, 12, 4. Find

a. the fourth term

b.the sum of nfinity, of the geometric progression

[4 marks]

10. Thethn term of a geometric progression is given by

1

5

2

4

1

n

nT

. Find the sum to infinity of the progression.

[3 marks]

11. The three consecutive terms of a sequence are m 9, 2m and 4m

+5. Find the value of m such that the sequence is an arithmetic

progression.

[2 marks]


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VECTORS


18.Diagram shows square grids where aOP , bOQ .

a. Express OR in terms of a and b.

b. In the answer space given, plot and label the point S where

abRS 2 [3 marks]

15. The following information refers to vector p and vector q

Given that p and q are parallel, find the value of m .

[3 marks]

19.Diagram shows a parallelogram PQRS and M is the mid point

of QR.

16. Diagram shows a triangle ABC

mjip 3 where m is a constants

jiq 42


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Given that jiPQ 4 and jiPM 45 , where I and j areunit vector parallel to x-axis and y-axis respectively.

Find

a. QM in terms of I and j.

b. i. PS

ii. the unit vector in the direction of PS [4 marks]

The point S lies on straight line AC. It is given that pAC ,

qAB and AS : SC = 2 : 1. Express in terms of p and / or q.

a. AS

b. BS

[3 marks]

LINEAR LAW


20. Diagram shows a straight line graph of y10log against x10log . 12. Diagram shows a straight line graph of xy against2x .


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Given that21000xy , calculate the value of p and of q .

[3 marks]

Given that2x

ba

x

y , where a and b are constants, find the value

of a and of b.

[3 marks]

DIFFERENTIATIONS


15. Given that the equation of a curve which passes through point P is

2)12( xy . The normal gradient to the curve at point P is2

1 .

Find the coordinates of P.

[3 marks]


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PROBABILITY DISTRIBUTIONS


25.Diagram shows a standard normal distribution graph.

The probability represented by the area of the shaded region is0.7648. Find

a. )( kzP

b. The value of k

[3 marks]

24.Diagram shows the graph of binomial distribution for discrete variable

X.

Find

a. The value of m

b. 1xP [4 marks]

25. X is a continous random variable of a normal distribution with a

mean of 420 and a standard deviation of 12 .Find

a. The z-score when X = 405

b. The value of k when 7881.0)( kzP [4 marks]


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FUNCTIONS

NEGERI PAHANG RAMALAN

1. Diagram shows the relation between set P and set Q.

State

a. The type of relation between set P and set Q

b. The value of w if 12

: x

xf

[2 marks]

The diagram shows the relation between set P and set Q.

Statea. the codomain of the relationb. the range of the relation

c. the type of the relation .

2. Given the function )1(3: xxg , find

a. The value of )4(2g .

[2 marks]

b. The function of f if xxgf 6)( [2 marks]

Given the functionf(x) =x

3, 0x and the composite function gf(x) =

4x. Find

a. g(x),b. the value of xwhen gf(x) = 8.

The diagram shows the inverse function1( ) 3f x x p , where p

is a constant.

9

28

49

3

5

11

7

Set P Set Q


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Finda. the value of p .

b. ( )f x

QUADRATIC EQUATIONS


3. The quadratic equation 03)3(2 pxpx , where p isconstant, has two equal roots. Find the possible values of p .

[3 marks]

Determine the type of roots of thequadratic equation 24 9 12x x

QUADRATIC FUNCTIONS


4. Find the range of values of x for which xx 8)2( 2 .

[3 marks]Find the range of values of x for

which xy > 2 and y

2x + 3 = 0

The diagram shows the straight line6 2y x intersecting the curve 23y x p

15

1f3x px


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at two distinct points.

Find the range of values of p .

INDICES AND LOGARITHMS


5. Given that 26log3log 44 xy , express y in terms of x .[3 marks]

Express 2 15 5 5n n n in the simplest form .

6. Given that px4log , qy32log and

nqmp

y

x

2 , find the

value of m and n .

[4 marks]

Given that 5log 3 0.683 and 5log 7 1.209 , calculate

without using calculator the values ofa. 5l og 4 2

b. 7log 315

Solve 3log 1

5125

x .

PROGRESSION


7. A piece of string of length 12 m is cut into 20 pieces in such a waythat the lengths of the pieces are in arithmetic progressions. If the

length of the longest piece is five times of the length of the

shortest piece, find the length of the longest piece.

[3 marks]

1.Given that 1 0.033333...t

is a recurring

decimal, find the value of t .

8. The fourth and seventh terms of a geometric progression are 18 The sum of the first n terms of an arithmetic

23y x p

6 2y x


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and 486 respectively. Find the third term.

[4 marks]progression is given by 26nS n n . Find

a. the third term.b. the sum of all the terms from the sixth

term to the tenth term.

COORDINATE GEOMETRY


9. Point P(h, 7) divides line the segment joining the points E(3, 10)

and F(8, k) internally such that EP : PF = 1 : 4 . Find the values of h

and k .

[3 marks]

In the diagram, P(0, 2) , Q(2, 3) and R arethree points on a straight line such that PQ :QR 1 : 4.

Find the coordinates of point R .

10. In diagram, OABC is a quadrilateral. The equation of the straight line

AB is 1

46

yx

.

In the diagram, P is a point on the x-axis .

0

P(0, 2)

Q(2, 3)

R

x

x

x

x

y


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Find the area of the quadrilateral OABC.

[3 marks]

Find the coordinates of the point P if thedistance of PQ is 10 units.

VECTORS


11. Given jia 64 , jib and bac 2

1, express c in the

form yjxi .[2 marks]

In the diagram below, PQRSTU is a regularhexagon.

P Q

R

ST

U

0

x

P

Q(1, 8)

x

y

x


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Express PQ PT RS as a single vector .

12. The vector OF has a magnitude of 10 unit and has the same

direction as OE . Given that

4

3OE and

y

xOF , find

the value of x and y .[3 marks]

The square grid in the answer space shows

the vectors OA a and OB b . If 3OP a b

and3

2

2

PQ a b , mark and label clearly

point Q on the square-grid.

13. Diagram shows a triangle ABCThree points A, B and C are such that

5

6OA

,6

4OB

and

2

kOC

, where 0 is the origin.

Find the value of k if the points A, B andC are collinear .

B

O

A

ab


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The point E is the midpoint of AC and D lies on the line BC such

that BC=5DC . Given xAB and yBC 5 , express in term of xand y .

a. AD

b. DE [4 marks]

DIFFERENTIATION


14. Find the value of34

32

2

2

lim

xx

xxx

[2 marks]

Find the value of2

22

2lim

2x

x x

x x

.

15. Volume,3

Vcm , of a solid is given by32

3

28 rrV , r is the

radius. Find the approximate change in V if r increases from 3 to

3.005 cm.

[4 marks]

Given that the function

qxpxdxdy

xy )12(and

)12(2 32

33 .

Find the values of p and q.

LINEAR LAW


16. Diagram shows part of a straight line graph drawn to represent

linear form of the equationx

y 625

.

The diagram shows part of the graph of yagainst x . It is know that x and y arerelated by the linear equation lg lg lgy a x b ,

where a and b are constants.


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Find the values of h and k .

[4 marks]

a.Skecth the straight line graph for theequation.

b.Calculate the values of a and b.

INTEGRATION


17. Find the value of dxx

xx

1

1

4

)6)(6(

[3 marks]

Given that2

13

x

xy

and xk

dx

dy4 with k(x) is a

function in terms of x. Find the value of

1

1

)( dxxk .

18. The gradient function of a curve passing through (1, 2) is given by

2)43(1x

. Find the equation of the curve.

[3 marks]

The diagram shows part of the graph of a

curve.

(2, 40)

(1, 5)x

y

0


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Calculate the value of4 6

0 1

ydx xdy

STATISTICS


The set of data 24, 3,3 5, 1, 3, 6r r r r r r is

arranged in ascending order. If the median is2, find the positive value of r.

PROBABILITY DISTRIBUTION


24. In an examination 40% of the students passed. If a sample of 10

students is randomly selected, find the probability that less than 2

students passed.

[3 marks]

Find the value of k, ifa. ( ) 0.1056P Z k

b. ( ) 0.2033P Z k

25. Diagram shows a standardized normal distribution graph. The diagram shows a normal distribution

graph for a variable X with mean is 4 andstandard deviaton is 1.

(4, 6)

1

x

y

0


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Given that the area of the shaded region is 30.5% of the total

area under the curve, find.

a. )( kzP b. The value of k

[3 marks]

Find the area of the shaded region.

( )f x

x0 3 5

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