All form4 questions.docx

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CHAPTER 1 : STANDARD FORM

EXERCISE 2

1.Round off each of the following numbers to 3 significant figures.

(a)2963= __________________

(b)51852= __________________

2.Round off 71.65 to

(a) 3 Significant figures= __________________

(b) 1 Significant figures= __________________

3.Round off 0.06053 to

(a) 3 Significant figures= __________________

(b) 2 Significant figures= __________________

(c) 1 Significant figures= __________________

4.Express each of the following numbers in standard form

(a)48000= ___________________

(b)9200.05= ___________________

(c)0.026= ___________________

(d)0.00000038= ___________________

5.Convert the following numbers in standard form to a single number.

(a)2.19 103= ___________________

(b)8.2 104= ___________________

6.Evaluate each of the following and express your answers in standard form.

(a) 16000000 + 2500000= ____________________

(b) 0.69 + 10.451= ____________________

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7. Evaluate each of the following and express your answers in standard form.

(a) 920000 87000= ____________________

(b) 0.043 0.00095= ____________________


(a) 4.8 700= ____________________

(b) 0.0029 0.065= ____________________


(a) 48.6 0.03= ____________________

(b) 0.000096 0.04= ____________________

10. Given that the mass ofa carbon atom and an oxygen atomare2 1023 gand

2.7 1023 g respectively, one molecule of carbon dioxidegasconsist ofone

carbon atom and two oxygen atoms. Calculate the mass in gram, of one molecule of carbon dioxide gas.

Answer : _________________

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CHAPTER 1 : STANDARD FORM

DIAGNOSTIC TEST

1. Round off 0.0487 correct to two significant figures.

A 0.04 B 0.05 C 0.048 D 0.049

2. 2.7 105 + 77000 =

A 1.04 105 B 1.04 109

C 3.47 105

D 3.47 109

3. The area of a rectangular plot of land is 9.2 km2. Its width is 2300 m. Find the length, in m, of the plot of land.

A 4 103 B 4 104

C 6.9 103

D 6.9 104

4.

5.

Express 0.0000405 in standard form.

A 4.05 10-5 B 4.05 105 C 405 10-7

D 405 107

9 28 102 = (4 103 )2

A 2 32 103

B 2 32 104 C 5 8 103 D 5 8 104

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6. 5 34 107 3 7 108 =

A 164 108 B 164 107

C 4 97 108

D 4 97 107

7. Round off 0 07207 correct to three significant figures.

A 0 07

B 0 072

C 0 0720

D 0 0721

8. Round off 80725 correct to three significant figures.

A 80700

B 80720 C 80730 D 80800

9. Express 9 9263106 as a single number.

A 0 009263

B 0 0009263

C 0 00009263

D 0 000009263

10.0 02=

4000000

A 5 103

B 5 104 C 5 108 D 5 109

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CHAPTER 2 : QUADRATIC EXPRESSIONS AND EQUATIONS

EXERCISE 1 :

Expand :

1. (x - 8) (x + 3)

Answer :

2. (x + 3) (x 3)

Answer :

3. m(x + y) (x + y)

Answer :

4. (2x 1) (x + 3)

Answer : 5. 4( x2 2 ) x( 5x + 1 )

Answer :

6.2x( x 3 ) + 4x( x 6 )

Answer :

7. (x 8 )( 2x + 5 )

Answer :

8. ( 3u 2s )( u s )

Answer : 9. x( 5 x ) 4x2

Answer :

10. 9 ( u 3)( u + 2 )

Answer :

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EXERCISE 2

Expand :

1.(2p q)2 p(p q)

Answer :.

2. (p q)2 (p2 q2)

Answer :.

3. (3f + g)(2f g) g2

Answer :.

4. (6h + k)(k 3h) + (h2 k2)

Answer :.

5. 2x(x 1) + (2x + 1)2

Answer :.6. p( p 4q ) ( 2p q )2

Answer :.

7. ( x 5 )2 ( x + 3 )2

Answer :.8. ( 3x 1 )2 5( x + 1 )

Answer :.9. ( a + 4 )2 64a

Answer :.10. ( k 2m )2 ( m2 4k2 )

Answer :.

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DIAGNOSTIC TEST

1. (2p + q) ( q 2p) =

A. 4p2 q2B. q2 4p22 4p2 4pq q2 3 4p2 4pq + q2

2. (3m n) (2m n) =

A. 6m2 mn + n2 B. 6m2 7mn n2

C. 6m2 7mn + n2

D. 6m2 5mn + n2

3. (x + 5y)2 5xy =

A. x2 + 25y2 B. x2 + 5xy + 25y2

C. x2 5xy + 25y2

D. x2 10xy + 25y2

4. (x + y)2 + (x2 y2) = A. 2x2 + 2xy

B. 2x2 2xy

C. 2x2 + 2xy 2y2 D. 2x2 2xy + 2y2

5. ( 3h 5 )( 2h + 4 ) =

A 6h2 + 2h + 20 B 6h2 + 2h 20

C 6h2 +12h 20

D 6h2 10h + 20

6. m( m 2 ) 2m( m + 3 ) =

A m2 8m B m2 + 8m

C -m2 + 4m

D m2 6m

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7. ( 3p m )( p 4m ) =

10. 3p2 + 4m2 11. 3p2 + 11mp 4m2

12. 3p2 13mp 4m2

13. 3p2 + 12mp + 4m2

8. ( p + q )2 ( 2p2 q2 ) =

A p2 + 2q2 B p2 + 2pq

C p2

D p2 + 2pq + 2q2

9. ( f 2 )( f + 1 ) + ( 2f 3 ) =

A f2 2f B f2 + f 1

C f2 + f 5

D f2 5

10. 3x( x 2y ) ( 2x y )2 =

A 3x2 + 10x + y2 B x2 2xy y2

C x2 10xy y2 D x2 + 2xy + y2

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EXERCISE 1

1. Factorise completely p2 - 2p

2. Factorise completely 4 x2 81

3.Factorise completely r2 4r 12.

4. Solve the quadratic equation k( k 12 ) + 20 = 0

5. Express the area of the triangle in terms of x.

(4x+1)cm

6x cm

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EXERCISE 2

1. Solve the quadratic equation 3b ( 2b + 1 ) = 4 2b

2. Factorise completely 6 17 x - 14 x2

3.Solve the equation2w23=1

5w

4.Solve the value of m for(m + 2)=(m + 2)

m 3

5. Johan is 3 years older than his sister Aishah and the product of their age is 18. Find the age of Johan.

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DIAGNOSTIC TEST

1.Solve the equationy 2 y= 0

3

2.Solve the quadratic equationy 2=y+1

263

3.Solve the equation2 y 23=1

3y3

4. Solve the quadratic equation11x x2= x2+5

2

5.The diagram shows a right-angled triangle ABC.A2x cmC

(a) Form an equation in terms of x using

Teorem Phytogoras and show that9cm

x2 6x = 0.(x + 9) cm

(b) Hence, find the length of AC.

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EXERCISE 2CHAPTER 3: SETS

1. Given P = { 1,4, 6, 8, 9},Q = { 3,4, 6, 7, 10} ,R = { 2, 3,4, 8,9 }

List the elements in theintersection of the followingset.

P Q R =

ANSWER: .

2 Given = A B C .Shade the region represented by the set (A B) C

ANSWER:

AB

C

3.Determine the union ofthe followingpairs of sets.

A ={x : x is a factor12}

B ={x : x is a multiple of 4 less than 16}

ANSWER: .

4.Given that the universal set ={ x :14 x 27, x are integers }.

Set A= {x:xare even numbers }

Set B={x :xareperfectsquarenumbers}

Find thecomplementof the following unionof setsA and B.

ANSWER: .

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4 Shade the complement of the set, (A B C ) ANSWER:

A

B

C

E. Given universal set = P U Q U Rsuch as

Set = { x :4 x 14,x are integers }

Set P = { x: x aremultiplesof6 }

Set Q = { x : x are odd numbers }

Set P U R = { x: x are multiples of 2 }

and P R.

Find n ( P R).

ANSWER: .

7.Given = { x: 2 x 10, x are integers } ,

F = { 4,10 } and

G = { Prime numbers } G)

List the elementsfor set( F

ANSWER: .

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8.The shaded region in the Venn diagram below is

B

A

C

ANSWER: .

E. Given universal set : = {x : 5 x 14, x are integers }

P = { x : x are even numbers}

Q = { x : x are prime numbers }

Theelements in set P Qare

ANSWER: .

6. The diagram shows the universal set , set A and B. The shaded region in the diagram below is represented by the set

AB

ANSWER:

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CHAPTER 3: SETS

DIAGNOSTIC TEST

1Diagram 1 is a Venn diagram which shows the elements of the sets P, Q and R.

PRQ

12 3

4 6 5 7

2

DIAGRAM 1

If the universal set , = P Q R , then set P R Q is

E { 5 , 7 }

F { 2 , 6 }

G { 2 , 3 , 6 }

H { 2 , 3 , 6 , 7 }

2Diagram 2 represents a class of 40 pupils. Set M = {Pupils who play tennis} and set R = { Pupils who play badminton} .

MR

DIAGRAM 2

Given that n (M) = 19 , n (M R ) = 9 and the number of pupils who do not play either game is 7, find the number of pupils who play badminton.

E 10

F 14 G 23 H 25

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14. Given that the universal set = E F G , G F and E G , the Venn diagram that represents these relationships is

A EFBEG

GF

C GD F

EE

FG

E Given that the universal set = {x: 1 x 10, x is an integer}, set L ={1,3,5,9}, set M = {x: x is a perfect square} and set N = {x: x is a multiple of 3}, n ( L M N ) is A 0

E 2 F 3 G 8

5Diagram 3 is a Venn diagram with the universal set = X Y Z.

XYZ

IIIIIIIVV

DIAGRAM 3

The set X ( Y Z ) is represented by the regions

E I , V

F I , II , IV G I , III, V H I , II , IV , V

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6.Diagram 4 isa Venn diagram,withtheuniversal set = A B C.

A

IIIIVB

V

III

C

Diagram 4

The set A C ,is representedbyregions

A.II, IV

B.I,II,IV

C.I,II,III

D.II, III,IV

7.Theshadedregionrepresentstheset

A

B

C

A. A B C

BA B C3. ( A B ) C D. A B C

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8.Giventhat the universal set = { x : 1 x< 20 , x is an integer}

Theset

E = {Perfect squarenumbers less than30}

F = {Multiples of 4 less than 20 }

Listthe elementE F

A. { 4,16 }

B{ 1, 9 }C. { 8, 12 } D. { 1, 4, 8, 9, 12, 16 }

9. Diagram below shows the number of clients in three banks.

Given = A B Cand n() = 300.

The number of clients forBank B is more than those in Bank A

by 40people. Findthevalue of Y.

AB

3050X

402030

YC

A. 30

B. 40 C. 50 D. 60

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10.

PQ

4x2x

5

13

6

R

The Venn diagram shows the number of elements in set P ,Q and R. Given universal set , E = P U Q U R and n(E) = 40 . Find the value of x.

A. 4

B. 5 C. 6 D. 7

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CHAPTER 3: SETS

EXERCISE 1 (Paper 2)

1. The Venn diagram shows set A, B and C. The universal set = A B C . Shade

on separate Venn diagrams in Diagram 1 and Diadram 2, the region that represents each set given below.

a. A B A

C

B

DIAGRAM 1

b. B C

A

C

B

DIAGRAM 2

2. The Venn diagram shows sets P, Q and R. Shade the region in the Diagram 3 and Diagram 4 which represents

a. P Q RPQR

DIAGRAM 3

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b. P Q R'P

Q

R

DIAGRAM 4

3. The Venn diagram shows sets E, F and G. The universal set = E F G . Shade

the region that represent each of the following sets in the Diagram 5, Diagram 6 and Diagram 7.

a. F G

E

GF

DIAGRAM 5

b. (F G)' E

E

GF

DIAGRAM 6

c. F (E G' )

E

GF DIAGRAM 7

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4. The Venn diagram shows the relationship among sets A, B, C and . Find the number of elements in the Diagram 8 and Diagram 9.

a. i) n(AB)

ii) n(ABC)

.3

A.1B

.9.2.4.5.10

.8

.7

.6

C

DIAGRAM 8

b.

B.cA.i

.d.f

.a

.gC

.e.b

.h

DIAGRAM 9

i) n(ACB)

ii) n(AB)

22

5.

M

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P

1

N32

5

7

DIAGRAM 10

The Venn diagram in the Diagram 10 shows the sets M, N and P in the universal set . The number of elements in each set is shown by the number in each region. Find

a. n ( N P)

b. n ( M P)

c. n ( N P)'

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CHAPTER 3: SETS

EXERCISE 2

1. Given the universal set,

= {x:20x35, where x are integers}, A= {multiples of 5},

B= {x:x are numbers where one of the digit is 0} and C= {x:x are numbers where the tens digit > the units digit}.

(a) List the elements of set B and C.

(b) Find n(AB).

(c) Find n(AC).

2. Given set R = {2,4}, S = {1,2,3,4,5,6} and T = {0, 2, 3, 4, 5, 6, 7, 8, 12}. The universal set, = RST.

(a) Draw a Venn diagram to show the relationship amongst R, S and T.

(b) List the elements of S T

(c) Find n(RT).

3. Shade the following related region in Diagram 1 and Diagram 2.

(a) PQ(b) P'QR

PQPQ

RR

DIAGRAM 1DIAGRAM 2

4. Given the universal set,

= {x:20x32, x are integers} set P = {x:x are multiples of 3}, set Q= {x:x are factors of 100} and

set R= {x:x are numbers where the sum of its digits is 5}. Find

(a) Set P, (b) Set Q,

(c) n(QR).

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5. The Venn diagram below shows set P, Q and R. On Diagram 3 and Diagram 4 provided, shade

(a) PQR

PQR

DIAGRAM 3

(b) PQR

PQ

R

DIAGRAM 4

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CHAPTER 3: SETS

DIAGNOSTICS TEST

1. The Venn diagram in the answer space shows sets , P, Q and R. = P Q R.

On the Diagram1 and Diagram 2 in the answer space, shade the set [ 3 marks]Answer:

(a) P Q

P

QR

DIAGRAM 1

(b) ( P Q) R

P

Q

R

DIAGRAM 2

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2. The Venn diagram in the answer space shows sets J, K and L. = J K L.

On the Diagram 3 and Diagram 4 in the answer space, shade the set [ 3 marks]

Answer:

(a) ( J L ) K

JKL

DIAGRAM 3

(b) J ( K L) JKL

DIAGRAM 4

3. The Venn diagram in the answer space shows sets P, S and R.

On the Diagram 5 and Diagram 6 in the answer space, shade the set

[ 3 marks]

Answer:

(a) P S

PR

S

DIAGRAM 5

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(b) ( P R ) S

P

R

S

DIAGRAM 6

4. The Venn diagram in the answer space shows sets A, B and C. = A B C

On the Diagram 7, Diagram 8 and Diagram 9 in the answer space, shade the [ 3 marks]Answer:

(a) set A B

A

BC

DIAGRAM 7

(b)SetC B

A

BC

DIAGRAM 8

(c)set( B C ) A

A

BC

DIAGRAM

5. The incomplete Venn diagram in the Diagram 10 shows sets , J and K. Set L satisfies the following conditions

L

L J and L K

L J = and L K =

Draw the set L in the Diagram 10 in the answer space.

[ 3 marks]

Answer:

JK

DIAGRAM 10

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CHAPTER 4: MATHEMATICAL REASONING

EXERCISE 1

1(a) State whether the following statement is true or false. - 3 > - 4 or 23 = 6

5 Write down two implications based on the following sentence. x is a multiple of 3 if and only if it is divisible by 3

6 Complete the premise in the following argument.

Premise 1:All integers less than zero are negative integers.

Premise 2:________________________________________

Conclusion:y is a negative number.[ 5 marks]

2. (a) Is the following sentence a statement or a non-statement? 1 is a prime number.(b) write down the conclusion in the following argument .

Premise 1 : If the side of the cube is 4 cm, then its volume is 64cm3

Premise 2 : The volume of cube p is not 64cm3

Conclusion : __________________________________________________

(c) 10 2 x 10 3 = 10 2+3

104 x 10 5 = 10 4+5

106 x 10 7 = 10 6+7

Based on the above information, make a general conclusion by induction of 10 m x 10 n

General conclusion: ________________________________________

[ 5 marks]

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3 (a) Form a true compound statement by combining the two statements given below.

52 = 10

14 = 0.25

4. Complete the premise in the following argument. Premise 1 : If x is an angle in a semicircle, then x = 90

Premise 2 : _________________________________________________

Conclusion : x = 90

5. Complete the following sentence using a suitable quantifier to make it a true statement.

_______________ prime numbers are odd numbers.

[ 5 marks]

4 (a) Based on the object and its given property, construct a true statement using an appropriate quantifier.

Object : Even numbers.

Property: Divisible by 4.

(b) State whether the following statements is true or false.

(i) 3m + m = 4m and x + y = xy

(ii) 250 = 2.5 x 102 or 0.0340 has 4 significant figures.

F. Write down the conclusion in the following argument. Premise 1 : If m < 0, then 3m< 2mPremise 2 : 3m > 2m Conclusion:_____________________________________

[ 5 marks]

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5. (a) Determine the following mathematical statement is a statement or a non-statement. Give reason for your answer.

2 3 = 3 2

(b) Complete the following statement using and or or to make it a true statement. 2 is multiple of 4...... x + 2x = 3x

(c) Complete the following arguments.

Premise 1:_______________________________________Premise 2: A rhombus is a quadrilateral. Conclusion: A rhombus has 4 sides.[5 marks]

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EXERCISE 2

1 (a) Write two implications from the following compound statement. x g y g if and only if x y.

Implication 1:.Implication 2:.

E Construct a true statement for the following sentences using the appropriate quantifier.

i).......................multiple of 3 are divisible by 4.ii)......................triangles identical in forms and sizes considered congruent.

[ 5 marks]

7. (a) Complete the following arguments.

Premise 1: If k 3, then 2 k 6. Premise 2: 2 k < 6.

Conclusion:

Premise 1: If a number is a factor of 4, then the number is also a factor of

16.Premise 2:.. Conclusion: 2 is a factor of 16.

(b) Complete the following statement by using and or or to form a true statement.All prime numbers has only 2 factors .......................all prime numbers are

odd.[ 5 marks]

3. (a) Make a general conclusion by induction based on the numerical sequence below 0, 3, 8, 15 ...0 12 1

3 22 1

8 32 1

15 42 1

Conclusion :.....................................................................................

(b) Complete the following premise.

Premise 1:. Premise 2: Angle P is less than 90

Conclusion: Angle P is an acute angle[ 5 marks]

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4. (a) Write two implications for the following compound statement.

tan = 1 if and only if = 45Implication 1:...Implication 2:.....................................................................................................

E If a 0, then 1 x a 0. Write the converse of the implication.

F State whether the converse is true or false.

[ 5 marks ]

5. (a) Complete the conclusion based on the two given premises. Premise 1: If n is an even integer, then n 1 is an odd number. Premise 2: n 1 is not an odd number.

Conclusion:

(b) Complete the following premise.

Premise 1:. Premise 2: ABC is an isosceles triangle.

Conclusion: ABC has two sides of equal length.

I Determine whether m2 2m 3 (m 3)(m 1) is a statement or not. Give reason for your answer.

[ 5 marks ]

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DIAGNOSTIC TEST

1.(a) Determine whether each of the following is a statement or a non statement.(i) x + 3y(ii) 42 = 8

2 Fill in the blank with the symbol > or < to form a false statement.

(i)2332

(ii)- 6

- 4

P

(c )Q

Based on the above Venn diagram, complete the following statement using an appropriate quantifier so that the statement is true.

. elements of set Q are elements of set P

I Write down the conclusion in the following argument. Premise 1 : All prime numbers have only two factors. Premise 2 : 5 is a prime number.Conclusion :..[ 5 marks]

2. (a) Determine whether the following statement is true or false.

(i) 32 = 6 or 52 = 0.4

(ii) -3 x - 4 = 12 and -3 + - 4 = 7

(b) Write down two implications based on the following sentence.

mn = 0 if and only if m = 0 or n = 0

(c ) Complete the following argument.

Premise 1: If the radius of a circle is 5 cm, then its circumference is 10 Premise 2 :

Conclusion: the radius of circle P is not 5 cm.

[ 5 marks]

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3. (a) Based on the object and its given property, construct a true statement using an appropriate quantifier.

Object: odd numbers

Property: prime numbers

(b) Combine the two statements below to form a true statement.

Statement 1: 3 + (- 2) = 5

Statement 2: 16 is a perfect square

(c ) Complete the following argument.

Premise 1:

Premise 2: The sum of interior angles of polygon Q is 540 Conclusion: Q is a pentagon.[ 5 marks]

4. (a) Identify the antecedent and consequent in the following implication. If a triangle has two equal sides, then it is an isosceles triangle.(b) State the converse of each of the following implication and determine if the converse is true or false.

15. If x < 4 , then x < 6(ii)If A B = A, then A B

(c ) Make a general conclusion by induction based on the numerical sequence below.

2, 9, 16, 23,

2 = 2 + 7 (0)

9 = 2 + 7 (1)16 = 2 + 7 (2)23 = 2 + 7 (3).The numerical sequence can be represented by ..

(d) Complete the following argument

Premise 1 :

Premise 2 : M N M Conclusion: M is not a subset of N[5 marks]

5. (a) Determine whether each of the following statements is true or false.

(i) 17 is a prime number or an even number.

(ii) 5 and 8 are factor of 15

(b) complete the following statement using an appropriate quantifier so that the statement is false.

empty set do not have any elements

(c) Complete the following argument.

Premise 1 :

Premise 2 : x is a natural number. Conclusion: x is greater than zero.

(d) Construct an implication in the form of if and only if from the following pairs of implications.Implication 1 : If n2 is an odd number, then n is an odd number. Implication 2 : If n is an odd number, then n2 is an odd number.

[5 marks ]

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CHAPTER 5 : THE STRAIGHT LINE

EXERCISE 1

7 In the diagram , PQ is a straight line.

Py

4

2

4 3 2 1 012 x

2

FindQ

F. the y-intercept, G. the gradient , of the straight line.

2.

yQ (5,18)

3

0x

Answer :a)..

b)

Find

a) the gradient of PQb) the equation of straight line PQAnswer :a)b)

G. Determine the gradient of the straight line y = 4x +9

Answer :

4.y

F

E(-2,6)

0 G1

8. TheDeterminediagram showstey-intercepttwo straightofhelines,straightEFandlineFG,2y onx a=Cartesian- plane. The gradient of EF is 1 and the distance of FG is 10 units. The x-intercept of FG is

38 Answer :.

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J The gradient of the straight line 2x + 6y = 13

Answer :...

J The x - intercept of the straight line 4x 2y = 12 is

Answer :

16. In the diagram, AB is a straight line.

A-12x -3B

What is the gradient of AB ?

Answer :..F In the diagram, OR = OS. The equation of RS is

Y

S

Rx

-40

Answer:

10. In the diagram , MLT is a right-angled triangle.

.

yL(6,7)

MT(6,4)

0x

Find

H the coordinate of M, I the equation of the straight line LT. Answer :.a).............................b).

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EXERCISE 2

1.y

PQ (3,5)

ORx

The diagram shows a parallelogram OPQR. Given that the gradient of OP = - 53 . The x-intercept of line QR is ......

Answer :

2.

y

Q(0,5)

R(12,3)

0x

S(4,k)

In the diagram, PQRS is a parallelogram. The equation of the straight line RS is 2y = x 6. FindI the value of k, J the gradient of the straight line QR, Answer: a)

b).

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4. In the diagram, OPQR is a parallelogram and O is the origin. Find the gradient of the straight line OR, the equation of the straight line PQ yQ

R(2,4)

P(6,1)

0x

Answer: a)b).

4. In the diagram PQRS is a parallelogram

P(-2,12) y

Q

S0x

R

D. Given that the gradient of line PQ is -3, find the equation of PQ E. Find the coordinate of point R. Answer a).................................b) ................................

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10. In the diagram, LH is a straight line.

Given that the gradient of the straight line LH is -1.

y

L(-3,9)

H(0,k)

0x

Find

E. the value of k, F. the equation of the stragiht line LH.

Answer: a)

b).

6. In the diagram above, ABCD is a parallelogram and O is the origin. The straight

line BD is parallel with the y-axis and the equation of line AB is y =2x + 4.

3

yB(6,10)

AC

0Dx

FindE. the x-intercept of the straight line AB, F. the gradient of AD, Answer: a)b)

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y

Q(4,8)

S(10,2)

0P(2,0) Rx

b. In the diagram, PQ is parallel to RS and O is the origin. Find the gradient of RS the x- intercept of RS

Answer: a)

b)

8. In the diagram OM = 13 OA and the length of the straight line OA is 9 units.

y

A B

0x

(a) Find the coordinate of point B.

(b) Calculate the gradient of line MB.Answer a)..................................

b)..................................

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9. In the diagram, CD is parallel to EF.

yD

y = 3x+5F

x

C

FindE(-1,-4)

(a) the x-intercept of line CD, (b) the equation of straight line EF. Answer: a)b)

10. In the diagram, EFGH is a parallelogram. Given that the gradient of line EF is 52 . Find the coordinate of point G.

y

EH(8,5)

.

F0 Gx

Answer:

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DIAGNOSTIC TEST

1. In Diagram 1, MN is a straight line. M y

6

08X

Diagram 1

N

Find the gradient of MN.

A -2

B 3

4

C - 34

D 2

2. Given that the equation of a straight line is x 2y + 3 = 0. Find the x-intercept of the straight line. A -1 B -2 C -3 D -4

3. In Diagram 2, the length of OC is 2 units.

y A(10,6)

B

Diagram 2

OCx

The gradient of line BC is

A 3 B 4 C -4

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D-3yP(8,2)

Ox

Diagram 3

R(-2,-8)

4. Based on Diagram 3, find the equation of line PR. A y = -x 6 B y = -x + 6 C y = x + 6 D y = x 6 5. In Diagram 4, the equation of the line WZ is y = 34 x. Q (8, k) is a point on the line WZ.

y

W Q (8, k)

Diagram 4

0xZ

Find the value of k.

A 6 B 7 C 8 D 9

6. The equation of a straight line passing through the point ( -3, 8 ) and parallel to the line y = 23 x + 5 is A y = 23 x + 8

B 3y = 2x + 10 C 3y = 2x + 30

D y = 23 x -3

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y

E

Diagram 5

0Gx

7. In Diagram 5, OE = OG. Find the gradient of line EG.

A 1 B -1 C 2 D -2

yU(-3,6)

0

Diagram 6

x

8. In Diagram 6, find the gradient of the line OU.

A 1 B -1 C 2 D -2

9. The y-intercept of the straight line 4y = 3 2x is

A 3 4

B - 12

C - 34

D 1 2

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y

RJ ( 4, 8 )Diagram 7

0x

P

6. In Diagram 7, the length of OP is 10 units. The equation of line PR is A y = 54 x + 8 B y = - 54 x + 8 C y = 54 x +4 D y = - 54 x + 4

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EXERCISE 1

1.yR(5,7)

Q(1,4)

x

P(-1,0)SDiagram 1

In Diagram 1, straight lines PQ and RS are parallel. Find the

2. gradient of the line PQ ,

3. equation of the line RS , 4. x-intercept of the line RS.

2. y. N

Kx

OM(5,0)

L Diagram 2

In Diagram 2, straight lines KL and MN are parallel. The equation of the line KL is y + 3x + 3 = 0. Find the

(b) gradient of the line KL ,

(c) equation of the line MN , (d) y-intercept of the line MN.

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3.yF(3,k)G(7,10)

E

OHx

Diagram 3

In Diagram 3, straight line FG is parallel to the x-axis while the lines EF and GH are parallel. The gradient of the line EF is 2.

State the value of k ,

Find the equation of the line GH , Find the coordinates of the point E .

4. y

M (5,20)

L

O x

K

Diagram 4

In Diagram 4, OKLM is a parallelogram and line ML is parallel to the y-axis.Point L is on the x-axis.

a) State the coordinates of the point L ,

b) Find the y-intercept of the line KL, c) Find the equation of the line KL.

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5.

y

V(8, k)

U

T(-2,1)

xS O W Diagram 5

In Diagram 5 , the equation of the straight line STUV is 2y = x + 4. The lines OT and UW are parallel. Find the

a) value of k ,

b) y-intercept of the line STUV , c) equation of the line UW .

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EXERCISE 2

1.y

B(-3,10)

C

AO

x

Diagram 1

The equation of the straight line AB in Diagram 1 is y = 2x +16.

a) State the equation of the straight line BC,

b) Find the x-intercept of the straight line AB, c) Find the equation of AC.

Y

U

Z (6,5)

C (-2,3)

V

O

D (8,-2)

2.

Diagram 2

In Diagram 2, the straight line UV is parallel to CD. O is the origin.

a) Calculate the gradient of line CD,

b) Find the equation of line UZV, c) Find x- intercept of line UZV.

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3.y

R

Q (h , 3 )

x

P( -8, 0 )0S ( 6, 0 )

Diagram 3

In Diagram 3, the gradient of the straight line PQR is 3.Find the

a) value of h

b) y-intercept of the line RS c) equation of the line RS

6. y

B (2 , 8 )

C

A (2 , 3 )

x

0

Diagram 4

In Diagram 4, OABC is a parallelogram. Find the

a) coordinates of C

b) equation of the straight line BC

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5.

YQS

12

-43X

P

Diagram 5

In Diagram 5, straight lines PQ and RS are parallel.

a) State the gradient of PQ

b) Find the equation of RS c) Find the intercept of RS

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DIAGNOSTIC TEST

y

1.

A

B (6, 5)

D

OC (6, 0)x

Diagram 1

In Diagram 1, A and D are located on the y-axis and ABCD is a parallelogram. If the gradient of AB = - 12 ,a) State the gradient of CD. b) Determine the coordinates of D. c) Find the equation of the straight line AB.

[ 5 marks ]

2.

P (0, 9)

Q (5, 7)

Ox

R (0, -5)

Diagram 2

Diagram 2 shows a triangle PQR.

a) State the y-intercept of the straight line PQ,

b) Find the equation of the straight line QR, c) Find the equation of a straight line that passes through P and parallel to the straight line QR.

[ 5 marks ]

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y

3.

M

P

ONx

Diagram 3

In Diagram 3, MN = 12 units and parallel to the y-axis. Given NO = 6 units and PO = 23 MN,

a) State the coordinates of P, b) Find the gradient of the straight line PM, c) Find the equation of the straight line PM. [ 5 marks ]

4.y

D

Ay = 2x + k

OEx

CB

Diagram 4

In Diagram 4 , two straight lines AB and CD intersect at point (-1, 3). Given the gradient of AB is -1, find the

a) value of k,

b) equation of the straight line AB, c) coordinates of point E. [ 5 marks]

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5.y

P (-3, 6)Q (h, k)

M (4, 3)

ORx

Diagram 5

In Diagram 5, OQ and PR intersect at M which is midpoint of OQ. Given O is the origin. Find

a) the value of h and k,

b) the equation of the straight line PR .

[ 5 marks]

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CHAPTER 6: STATISTICS

EXERCISE 1 (Paper 1)

8 Given that the class interval of a set of data is 6 9, 10 13, 14 17, .. . Determine the upper boundary of the class interval 10 13.

Answer:..

2. Calculate the size of class interval 36 40.

Answer:..

3. Calculate the mean for the following data

1094303029581118

21432271310218626

Answer:..

Questions 4 and 5 are based on the table 1. Table 1 is the frequency table which shows the marks obtained by 10 students in Mathematics quiz.

Mark1 56 1011- 1516 20

Frequency5401

Table 1

4. Determine the modal class of the data.

Answer:..

5. Calculate the mean of the data.

Answer:..

H. Given that the mean of a set of data 8, 10, 7, x, 5, 5 is 6.5. Calculate the median of the same set data.

Answer:..

H. Find the range of the following set of ungrouped data 2.44, 3.69, 2.74, 1.68, 1.1

Answer:..

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Questions 8-10 are based on the table 2.

8. Find the range of class interval the following set of grouped data in table 1

Breadth(cm)11 - 1617 2223 2829 3435 - 40

Frequency47892

TABLE 2

Answer:..

9. Find the modal class of the data.

Answer:..

10. Find the midpoint of the modal class.

Answer:..

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EXERCISE 2

1.Given a set of numbers 2,4,3,4,5,6,8,x,5. Find the value of x if the (a ) median is 4

(b) mean is 5

Answer: (a).(b).

2. The table 1 below shows the scores obtained by a group of students in a quiz competition.

Score123456

Number of2312832

students

TABLE 1

Find:9. the mode 10. the median 11. the mean

Answer: (a)..(b)...(c )

3.(a) Complete the following frequency in table 2

Distance (m)FrequencyMidpoint

1-32

4-66

7-912

10-125

13-153

TABLE 2

(b) Based on table 2, calculate the estimated mean distance of the data.

Answer: (a).(b).

4. Find the range of the following set of data

(a) 3, 5, 8, 11, 14(b) 12, 13, 10, 8, 19, 25

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

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5.The table 3 below shows the marks obtained by group of students in a Mathematics examinationMarks4060758388

Number of46106x

students

TABLE3

K If the mean mark is 75, find the value of x. L State the minimum value of x if the mode is 88

Answer: (a)(b).

6.16,25, 13, 26, 15, 16, 18, 17, 20, 24

For the above data, find the

K range L mean

Answer: (a)(b)..

7. Diagram 1 is a pie chart which shows the total number of boys and girls in two clubs. Table 4 shows the number of boys and girls of these clubs, but is incomplete.

ClubsBoysGirls

Chess6050Boys

Debate(a)(b)

Total100(c)1500

TABLE 4GirlsDIAGRAM 1

Complete the table.

Answer: (a)..(b)..(c)..

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8. Diagram 2 is a pictograph showing the number of blood donors at a blood donation campaign over a period of 3 days.Given that the number of blood donors on Monday make up 30 % of the total blood donors over the period.

Monday

Tuesday

Wednesday

Represents 60 blood donors

DIAGRAM 2

Calculate17. the number of blood donors on Wednesday 18. the total number of blood donors the 3 days.

Answer: (a).(b)

9. Table 5 shows the mass, in kg, of 40 parcels.

Mass(kg)Frequency

6-104

11-1510

16-207

21-2510

26-305

31-354

TABLE 5

. Calculate the:

G mean mass of the parcels , in kg. H midpoint of the third class

Answer: (a)(b).

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J Table 6shows the record of overtime done by a group of workers, in hours, in a particular month.

Overtime (hours)Frequency

10-195

20-2914

30-3910

40-499

50-5912

TABLE 6

(a)State the modal class

(c) Find the midpoint of the modal class

Answer(a)..(b)

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DIAGNOSTIC TEST

K A class interval has an upper limit of 15 and lower limit of 10. The lower boundary is A 9.5 B 10.5 C 14.5 D 15.5

L The table 1 shows the frequency distribution of the scores of a group of players

Score123456

Frequency439x32

TABLE 1

If 3 is the modal score, the maximum value of x is

10

9 8 7

F. If the median of a set of integers, 3,8,9,x and 7 is x, the probable value of x is

5

6 8

9

G. The diagram 1 is a pictograph showing the number of durians of different grades sold on a particular day. The information in the pictograph is represented by a pie chart.

Grade A durians

Grade B durians

Grade C durians

Represents 50 duriansDIAGRAM 1

Calculate the angle of the sector which represents the number of grade C durians sold.

A 900 B 112.50 C 1350

D 157.50

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Questions 5 and 6 are based on table 2.

Table 2 shows the scores obtained by 12 students.

35708090

91455282

74465388

TABLE 2

5. Find the range of the score

A 55

B 56 C 57 D 5

6. If x mark is added to each student as a bonus and the mean is 70 16 . Find the value of x.

A 2

B 3 C 4 D 5

7. Which of the following class interval has a size of 5?

A 1.1-1.5 B 2.05-2.10 C 5-9 D 15-20

8 Find the mode for the following data

2, 2, 1, 3, 4, 1, 2

c. 1 d. 2 e. 3 f. 4

9

19, 18, 16, 15, 20, 15, 18, 16

The median for the above set of numbers is

A.16B.17b. 18 c. 20

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10

1111 111

The diagram above shows a tally chart. The symbol represents the value

d. 5

e. 7 f. 8 g. 13

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CHAPTER 6 : STATISTICS

EXERCISE 1 ( PAPER 2)

b. The data in Diagram 1 shows the marks obtained by 30 students. By using five class intervals, construct a frequency table for the data.

112431252221

313610442827

353635301335

421418183016

173434322037

DIAGRAM 1

2. Complete Table 1

ClassLower limitUpper limitLowerUpperClass size

boundaryboundary

55 60

61 66

67 72

73 - 78

TABLE 1

D Diagram 2 shows the masses of tomatoes, in kg, yielded by a farm for a period of 30 days.

59596850424660577147

62598062745556764053

36717451836444555151

DIAGRAM 2

6. Construct a frequency table with class intervals 36 43, 44 51 and so on and then find the midpoint of each class. 7. State the modal class. 8. Calculate the mean mass of the tomatoes yielded by the farm per day.

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2 The data in Diagram 3 shows the number of papayas sold by Pak Ali per day for a period of 30 days.

2545423632

2620253238

3731352232

3140302726

2428213339

3028342933

DIAGRAM 3

(a) Based on the data in Diagram 3 and by using a class interval of 5, complete Table 2.

Class intervalFrequencyMidpoint

20 24

25 29

TABLE 2

(b) Based on Table 2, calculate the estimated mean number of papayas sold.

24103619192526331630

17313511313215332738

24184035112023273734

DIAGRAM 4

5. The data in Diagram 4 shows the heights, in cm, of 30 seedlings in a nursery.

(a) State the range of the data. (b) Based on the data in Diagram 4, complete Table 3.

Height (cm)FrequencyMidpointUpper

boundary

10 16

17 23

24 30

31 37

38 44

TABLE 3

(c) Based on Table 3,

i) state the modal class ii) calculate the mean height of the seedlings.

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EXERCISE 2

1.Given a set of numbers 2, 4, 3, 4, 5, 6, 8, x , 5, . Find the value of x if the

( a )median is4

( b )mean is5

2 Table 1 shows the score obtained by a group of students in a quiz competition.

Score123456

Number of students2312832

FindTable 1

( a )the mode

( b )the median

( c )the mean

3. The data below shows the marks obtained by 40 students in a monthly test.

9988759258758070

7032705890685078

4589459361815865

6976885891677152

5540808039466169

(a)Using a class interval of 10 marks , complete the following table.

MarkFrequencyMidpoint

21-30

31-40

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(b) For this part of the question, use the graph paper

By using a scale of 2 cm to 10 marks on x axis and 2 cm to 1 student on y axis , draw a frequency polygon based on the data.

( c ) From your answer in (a),

(i) determine the modal class, (ii) calculate the estimated mean of the group of students.

4. The data below shows the mathematics test marks of 40 students.

8698729694907680

9286938781808367

8593728472868688

7475838588699079

8290917668968978

(a) By using the a class interval of 5 marks , complete the following table.

MarkFrequencyMidpoint

65 - 69

70 - 74

(b) From the table in (a)

(i) state the modal class, (ii) calculate the estimated mean mark of test.

(c) For this part of the question, use the graph paper.

By using a scale 2 cm to 5 marks on x-axis and 2 cm to 1 student on y-axis, draw a

histogram for the data.

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5.

Mass ( gm )Frequency

20-240

25-298

30-3410

35-3936

40-4448

45-4940

50-5427

55-5911

The table above shows the frequency distribution of mass of books . ( a ) State the midpoint of the modal class

( b ) Based on the table above , construct a cumulative frequency table. ( c ) For this part of the question, use the graph paper.

By using a scale of 2 cm to 5 gm on x axis and 2 cm to 20 books

on y axis , draw an ogive for the data . (d ) From your ogive in ( c ), find

i. the interquartile range,

ii. the number of books with length greater than 50 cm.

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DIAGNOSTIC TEST

1. Data below shows the number of papaya trees planted by 50 farmers.

60777081736979697567

65716266787664717379

70666489816173787368

68777463716587676374

74807072758276816874

(a) (i) Using size of class interval 5 , complete the Table 1 below.

ClassUpper boundaryFrequencyCumulative

IntervalFrequency

55 59

Table 1

(ai) Hence , state the modal class

(c) By using a scale of 2 cm to represent 5 trees on the x axis and 2 cm to represent 5 farmers on the y axis , draw an ogive for the data above.

(d) Based on the ogive in (b) , Osman make a conclusion that 25% of the farmers planted less than 56 trees.

Determine whether the conclusion is correct or not and give a reason.

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5. Encik Shamsudin reared a total of 148 turtles. The distribution of the length of the turtles is shown in Table 2.

Length (cm)Frequency

5 99

10 1419

15 1929

20 2443

25 2930

30 3414

35 394

Table 2

(a) By stating the answer correct to two significant figures, calculate the mean length of the turtles reared.

(b) Construct a cumulative frequency table.

By using a scale of 2 cm to represent 5 cm on the x axis and 2 cm

to represent 20 turtles on the y axis , draw an ogive for the data above

(c) From the ogive drawn in (b), find

(i) the median, (ii) the first quartile, (iii) the third quartile,

(iv) the interquartile range.

(b) The closing price, in sen, of the 50 counters traded at Bursa Malaysia on a day is given in Figure below..

200150189175255130214161230217

169196208249124121180155144158

146218154234162241193187254184

250178259198146182201160186183

136258142163186204156245194164

(a) Determine the range of price of the 50 counters.

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(b) Using size of class interval 20 , complete the Table 3 below.

ClassMid pointFrequency

Interval

121-140

Table 3

(c) Based on the frequency table constructed in (b) , draw a histogram for a data.

4. The heights, in cm, of 50 plants are distributed as shown in the following table.

(a)

Height(cm)MidpointFrequency

40 443

45 495

50 5410

55 5916

60 648

65 696

70 792

2 Copy and complete the above table

3 Hence , calculate the mean height of the plants.

(b)

Upper39.544.5

Boundary

(cm)

Cumulative0

frequency

(i) Based on the information from the table in (a) , copy and complete the above table.

(ii) Using a scale of 2 cm to represent 5 cm on the x axis , and 2 cm to represent 5 plants on the y-axis, draw an ogive for the distribution.

From the ogive , find

(a) the median (b) the third quartile

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5. (a)

Number of8910111213

Television

sets sold

Number of2581096

Shops

The above table gives the numbers of televisyen sets sold by 40 electrical shops on a certain day.Find

(i) the mode (ii) the mean of the distribution.

(b)

Time(minutes)Number ofUpper boundaryCumulative

participantsFrequency

6.1 7 .08

7.1 8.014

8.1 9.022

9.1 10.046

10.1 11.038

11.1 12.020

12.1 13.08

13.1 14.04

The above table shows the frequency distribution of the times , in minutes, taken by 160 participants of a jogathon.Copy and complete the table.

(c) Using a scale of 2 cm to represent 1 minute on the x axis and 2 cm to represent 20 participants on the y-axis, draw an ogive for this distribution. From the ogive , find

(i) the median (ii) the interquartile range (iii) the number of prize winners , given that participants who clocked less than 8.0 minutes were given prizes.

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CHAPTER 7 : PROBABILITY 1

EXERCISE 1

9 A dice is rolled. List the elements in the sample space that satisfy the following conditions. A An odd number is obtained B A multiple of 3 is obtained

Answer:(a)..

(b).

2.

NE R 2 9 3 8

DIAGRAM 1

Diagram 1 shows seven cards are labeled. A card is picked at random from the seven cards.

P = event of selecting a card with a letterQ = event of selecting a card with a consonant or multiple of 3 List all the elements of6. set P 7. set Q

Answer: (a).

(b)

3.

PR O B A B I L I T Y

DIAGRAM 2

Diagram 2 shows cards with the letters are put in a box. A card is picked randomly from the box. The total number of elements in the sample space is

Answer:

..

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(d) Two coins are tossed. List all the elements of the sample space which satisfy the conditions below. Two faces are the same Two faces are different

Answer : (a)

(b) ..

G A box contains several marbles. Eight of the marbles are blue. If a marble is randomly selected from the box, the probability that it is a blue marble is 72 . Therefore, the total number of marbles in the box is..

Answer :

3 Based on the weather record of a region, it is found that the probability of a day being rainy is 52 . Find the expected number of rainy days in a year in the region.

Answer : ..

5. During her study in Ireland, Junaidah received 5 Hari Raya cards her family, 10 Hari Raya cards from her relatives and 15 Hari Raya cards from her friends. If a Hari Raya card that Junaidah received is selected at random, state the probability that the Hari Raya card

(b) is sent by her family

(c) is not sent by her relatives

Answer : (a) ..

(b)

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I A box contains 280 marbles of which 120 are red, and the rest are green or white. If a marble is picked at random from the box, the probability of picking a green marble is 14 . How many white marbles are there?

Answer :

K In a group of 115 students, 80 are girls. A further 5 boys then join the group. If a student is chosen at random from the group, state the probability that the student chosen is a boy.

Answer : ..

10.

Favourite ColourRedGreenBlue

Frequency7590x

TABLE 1

Table 1 shows the results of a survey on the favourite colours of students in a school.

M If a student is chosen from the school, the probability that the students favourite colour is red is 14 . Find the value of x.

N If 100 students were chosen from the school, predict the number of students whose favourite colour is green

Answer : (a) ..

(b)..

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EXERCISE 2

1 . A bag contains 3 blue balls, 4 white balls and 2 green balls. A ball is drawn from the bag. The number of elements in the sample space is .

Answer: .

2 . A letter is picked at random from the word P R O B A B I L I T Y . Write the element of event which the vowel letters are picked.

Answer: ..

3 . A coin is tossed 3 times and the outcomes are recorded. List down the event of getting 2 heads and a tail.

Answer: .

4 . A box contains 5 red balls, 7 black balls and 3 yellow balls. If a ball is picked at random from the box , state the probability that it is red in colour.

Answer: .

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5 . In a quiz, the probability of a team entering semi-final is1. If there are 28 teams

4

taking part in the quiz, find the number of teams expected to enter the semi-final.

Answer: ..

6 . The probability of an employee coming late to work on Monday is1.How

5

many employee are expected to come late to work on Monday if there is a total of 550 employees working on that day ?

Answer: ..

7 . A bag contains a number of balls, 24 of which are green and the remainder blue. If a ball is chosen at random, the probability that it is green is 0.8. Find the number of blue balls in the bag.

Answer: .

8 . A bag contains 36 number cards, some of which are even and the rest odd. The probability of a card, drawn at random being even is 16 . Find the number of odd

Answer: ..

80

find the number of blue pens in the box.

is drawn at random from the box. If the probability of drawing a black pen isppr maths nbk

9 . A box contains 30 red pens, 50 black pens and the remainder are blue pens. A pen

165 ,

Answer: ..

10 . If we choose one number from among the first 30 counting numbers, 1, 2, 3,, 30 find the probability that the chosen number is divisible by 4.

Answer

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DIAGNOSTIC TEST

5. In a group of 45 students, 35 are girls. A further 5 boys then join the group. If a student is chosen at random from the group, state the probability that the student is chosen is a boy

6. 3 10

7. 1

7

8. 1

10 1 3

G. A jar contains 270 sweets of orange, lychee and coffee flavour. There are 48 orange flavoured sweets. If a sweet is picked at random from the jar, the probability of picking a lychee flavoured sweet is 13 . How many coffee flavoured sweets are there ?

A 90

B 132 C 106 D 16

3.

161822286084

DIAGRAM 1

The diagram 1 shows some number cards. If a card is picked randomly, find the probability that a multiple of 4 is picked.A 2

3

B 1

2

C 1 3

D 1

6

82

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(c) During Christmas, Angela sent 6 greeting cards to her friends, 5 greeting cards to her relatives and 2 greeting cards to her brothers. If a greeting card sent by Angela is randomly selected , the probability that it was not sent to her brother is

2 2 13

3 5

13

4 1

6 D 11

13

5. A class has 18 boys and k girls. If a student is picked at random from the class,

the probability that the student picked is a girl is 74 . Find the value of k.

18

24 30 42

6. The probability of Hafiz choosing a green coloured shirt is 15%. If Hafiz has 20 shirts, how many shirts are green?

A 3 B 4 C 5 D 6

7. In a group of tourist, 15 of them wear watches. The probability of choosing at random a tourist from the group who wears a watch is 53 . If three tourists who wear watches leave the group, the probability of choosing at random a tourist who wears a watch is

A 1 22 B 1225

C 6 11 D 1525

83

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(d) A box contains 40 red buttons and some white buttons. The probability of choosing at random a white button from the box is 75 . Find the number of white buttons in the box.

A 56 B 100 C 120 D 140

(e) In a football match between team P and team Q, the probability that team P

will win is1and the probability that team Q will win is2. The probability

27

that the match will be a draw is

A 17

B 73

C 3

14

D 4 21

5. Eggs are categorized into grades A, B and C. A basket contains a total of 100 eggs. If an egg is picked at random from the basket, the probability that the

eggs are grade A and B are 14 and 52 respectively. Find the number of eggs of grade C in the basket.

(d) 25

(e) 35 (f) 40 (g) 45

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CHAPTER 8 : CIRCLES III

EXERCISE 1

1. Given PQR is a tangent to the circle QSTU. TFind the value of

ya) x

SU

35

xb) y

PQR

Answer: a)..

b)..

2. In the diagram, ABC is a tangent to the circle BEFG. EFind the value of

A y 40 110Fa) x

Bx

b) y

CAnswer: a)..

b)..

3. PQR is a tangent to the circle QSTU with centre O.

Find the value of

TUa) x

y O

xRb) y

S40

30

Q

PAnswer: a)..

b)..

4. In the diagram, PQR is a tangent to the circle with centre O.

35Find the value of

a) x

O60

Pxyb) yAnswer : a)..

QRb).

85

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5. In the diagram, ABC is a tangent to the circle with centre O. Find the value of

a) p

q 34 Opb) q80C

BAAnswer: a)..b)..6. In the diagram, FGH is a tangent to the circle centre O, at point G. Find the value of y.

E

100y D

F 40O

G

HAnswer: 10 In the diagram, PQR is a tangent to the circle centre O, at point Q. Find the value of y.

35 OPy

42

Q

RAnswer: .I. In the diagram below, PQR is a tangent to the circle centre O, at point Q. PTOS is a straight line. Find the value of y.

TOS

P y42

Q

RAnswer: ..

86

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9. In the diagram below, PQR is a tangent to the circle centre O, at point Q. Find the

value of y.T

SP

26

y Q

R

Answer: a)

10. In the diagram, EFG is a tangent to the circle FLMN. Find the value of y. M

105

L35 G

yF

E

Answer: a)

87

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EXERCISE 2

1. In the diagram, DEFG is a tangent to the circle with centre O. EOJ, FKJ and GKOHare straight lines.Find the value of

Ja) p

H

p

Ob) q

30 K

DqrG c) r

EF

Answer: a)..b).c)..

2. In the diagram, ABC is a tangent to the circle with centre O. EODC is a straight line.FFind the value of

Epa) p

Orb) q

48DC

ABqc) r

Answer: a)..

b).c)..3. In the diagram, PQ is a tangent to the circle with centre O. POR and SOQ are straight lines. S Find the value ofa) xOy Rb) y

xzc) z

24Q

PAnswer: a)..

b).

c)..

88

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4. In the diagram, PTQ is a tangent to the circle to the circle with centre O. PAB is a straight line.Find the value of

Ba) x

x

Ozb) y

80Ac) z

y30

QTP

Answer: a)..b).c)..

I. In the diagram, VST is a tangent to the circle with centre O. PORV, TOQ and QRS are straight lines. Find the value of

Qa) x

P

34 ORb) y

TxyV

S

Answer: a)..b).6. In the diagram below, PQR is a tangent to the circle at point Q. The centre of the circle is O. Find the value of x.

75

X

o

Answer: .

89

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7. In the diagram below, PQR is a tangent to the circle at point Q. TOQ is the diameter of the circle and O is the centre of the circle. Find the value of x.

xo

34Answer:

8. In the diagram below, EFG is a tangent to the circle centre O, at point F. Find the value of x.

62 o

Answer:

9. In the diagram below, PQ and RS are common tangents to the circles centres O and T, at P, R, Q and S respectively. Find the value of x.

150oxo

Answer:. 10 In the diagram , EF is a common tangent to the circles centres O and Q, at points E and F respectively. OHQ is a straight line. Find the value of x.

25

Answer:

90

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DIAGNOSTIC TEST

1. In the diagram, tangent PQ touches the circle at Q. Find the value of x.TA 30 o

B 70 o

30 100 SC 80 o

PD 100 o

x

R

b In the diagram, PQ is a tangent to the circle with centre O at P. Find the value of y.

S y

O 120 R

40 QP

6. 40 o 7. 50 o 8. 60 o D 70 o

3. In the diagram, PQ is a tangent to the circle at Q. Find the value of x.

PQA 30o

50 x

B 40 o

T80 RC 50 o

D 80 o

S

4. In the diagram, PQR is a tangent to the circle with centre O at Q. Find the value of y.

S

T 65O 100

y P Q R

(e) 40 o (f) 50 o

(g) 65 o

(h) 115 o

91

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5. In the diagram, RS is a tangent to the circle at S and PQR is a straight line. Find the value of x. A 20PB 25 oC 30 o

D 40 o

Q 40 x

S R

6. In the diagram below, EFG and HFJ are common tangents to the circles centre O and Q, at J, G, E and H respectively. Find the value of x.

A. 76

B. 52 C. 104

D. 90

xo

7. In the diagram below, LMN is a tangent to the circle at M. POQ is the diameter of the circle. Find the value of y.

A. 42

B. 132C. 138

D. 48

42

92

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8. In the diagram below, LMN is a tangent to the circle centre O, at the point M. LQOP is a straight line. Find the value of x.A. 65

B. 15C. 50D. 40

25

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CHAPTER 9: TRIGONOMETRY II

EXERCISE 1

11 In Diagram 1, ABC is a straight line.

D

6 cm10 cm

AxoC

B

DIAGRAM 1

Find the value of sin x.

J. In Diagram 2, ABC is a straight line.

D

5 cm13 cm

yo

ABC

DIAGRAM 2

Find the value of cos y.

3. Find the value of tan 0o + 2 sin 90o 3 (cos 180o). 4. Diagram 3 shows the curve y = f(x)?

1y=f(x)

090180270360

-1

DIAGRAM 3

State the name of the curve y = f(x).

94

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J. State the angle in the first quadrant that is equivalent to 200.

K. In Diagram 4, BCD is a straight line.

A

7 cm

5 cm3 cm

BCD

DIAGRAM 4

Find tan .

12. In Diagram 5, BCD is a straight line.

A

2 cm

B2 cm C 2 cm D

DIAGRAM 5

Find cos .

M In Diagram 6, ABC is a right angled-tringle.

AB

25cm

C

DIAGRAM 6

Given that sin = 54 , find the length of BC

95

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M In Diagram 7, BDC is a straight line.

A

10 cmxy

BDC

DIAGRAM 7

Given that sin x = 53 and tan y = 54 , find, the length, in cm, of BC.

10. In diagram 8, BCD is a straight line.

A

16 cm xy20 cm

BCD

DIAGRAM 8

Given that cos y = 0.6, find the value of cos x.

96

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EXERCISE 2

1.In Diagram 1, PQR is a straight line and PQ =1QR

3

S

8 cm

xo

P5 cm QR

Find the value of sin xo.

Answer: ..

19. In Diagram 2, PQR is a straight line.

R

Q

6 cm

P8 cmS

DIAGRAM 2

Find the length of QS, in cm.

Answer : .

J Given that tan = 0.7382 and 180o 360o, find the value of .

Answer : .

97

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L In Diagram 3, O is the centre of a unit circle.

y

1

( -0.8,0.6) xo

-1O1x

-1

DIAGRAM 3

Find the value of tan xo.

Answer: ..

O In Diagram 4, PQRS is a rectangle.

PQ

x

10 cmM

S

Find the value of cos xo.

12 cmR

DIAGRAM 4

Answer :

98

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6. Diagram 5 shows the graph of y = cos xo.

y

1

Opx

-0.5

-1

DIAGRAM 5

Determine the value of p .

Answer : .

9. In Diagram 6, PTU and PQRS are straight lines.

U

T

17 cm

x

P Q R S

15 cm

DIAGRAM 6


Answer :

2. In Diagram 7, PQR is a straight line.

S

12 cm

x

PQ R

DIAGRAM 7

It is given that cos xo = 54 . Find the length of SP, in cm.

Answer : .

99

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3. In Diagram 8, PRS is a straight line and RS = 12 cm.

T

25 cm

15 cm

x

PRS

15 cm

Q

DIAGRAM 8Find the value of sin xo.

Answer : ..

6. In Diagram 9, QRST is a straight line.

P

17 cm8 cm

x

QRST

21 cm

DIAGRAM 9

Find the value of sin xo.

Answer : ..

100

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DIAGNOSTIC TEST

1. Given that, sin =3and 90o 180o , find the value of cos .

5

E 54

F 5

3

G 54

H 34

g. Determine the equivalent of sin 46o 24 .

cos 133o 36 Bsin133o36

Csin226o24

A sin 313o 36

4. Given that, cos 45o = 0.7071. Find the value of cos 225o

A -0.7071

B -0.6929 C 0.6929 D 0.7071

5. In Diagram 1, BCD is a straight line. Find the value of cos ro. roB 4 cm A C

3 cm

DIAGRAM 1D

E 54

F 3 5

G 53

H 34

101

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5. By using a calculator, find the value of cos 216o 24

0.8048

0.8049 -0.8049 -0.8048

2. In Diagram 6, PQR is a straight line and PR = 10 cm.

P

x Q

S

12 cm

R


2 135

3 1213

4 5 13

5 12

13

DIAGRAM 6

102

7.In Diagram 7, PRS is a straight line and tan x =3 .

4

S

y

R

T6 cm

5 cm

xQ

P

DIAGRAM 7

Find the value of tan yo.

12

85

1 2

5

8

8. Which of the following graphs represents y = cos x ?

AyBy

11

O90180xO

-1-1

CyDy

11

O90180xO

-1-1

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90180

90180

103

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(h) In Diagram 9, O is the centre of a unit circle.

y

xx

O

(0 .5,-0.866)

DIAGRAM 9

Determine the value of sin xo.

0.5

-0.866 -1.732 -0.5774

10. Given that cos x = cos 110o and 180o x 360o, find the value of x.

A 200o B 250o

C 290o

D 340o

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CHAPTER 10 : ANGLES OF ELEVATION AND DEPRESSION

EXERCISE 1

1. Diagram 1 shows a post QR which is vertical to the ground. An observer P looks up at R

R

PQ

DIAGRAM 1

. Name the angle of elevation concerned.

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

12 In the diagram 2, ADC represents a vertical tower. Given that the angles of elevation of D and C from B are 30o and 50o respectively.

C

D

A B 100m

DIAGRAM 2

Calculate the distance from D to C.

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

105

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K. In the diagram 3, PQ and SR are two vertical poles on a horizontal plane.The angle of elevation of R from P is 52.

R

Ph m

2 m

2.5

QS

DIAGRAM 3

Calculate the value of h.

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

4. In the diagram 4, XY and KLN are vertical poles on a horizontal plane. K

X

L

4.62

2

Y10N

The angle of elevation of K from X isDIAGRAM 4

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

106

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5. In the diagram 5, BCD represents a vertical pole. The angle of elevation of D from A is 50o. D

C

20o

A B 20 m

DIAGRAM 5

Calculate the distance from C to D.

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

6. In the diagram 6 PQST is a straight line.

R

160 o

PQST

DIAGRAM 6

The angle of depression of Q from R is

107

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

7. In the diagram 7, EF and UV are two vertical poles. The angle of elevation of U from F is 32o while the angle of depression of U from E is 28o.

E

U

4 m

FV

DIAGRAM 7

Calculate the height of the pole EF in m, correct to 1 decimal place.

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

8. The height of a tower is 400m. A car is parked 250m away from the foot of the tower. Calculate the angle of elevation of the top of the tower from the car.

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

9. A bird is flying a height of 800m above sea level. At a particular moment, the horizontal distance between the bird and a ship is 650m. Calculate the angle of depression of the ship from the bird.

108

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

10. The diagram 8 shows a vertical wooden structure PQRS erected on a horizontal ground. The angle of the elevation of P from S is 50.

P

Q

20 o

RSDIAGRAM 8

Find the angle of depression of S from Q

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

109

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EXERCISE 2

L. A point P is 30 from the foot of a flagpole. The angle of elevation of the top of the pole from P is 25. Calculate the height of the pole, correct to 2 significant figures.

Answer:.......

13. A tower is 100 m tall. From the top of the tower, the angle of depression of a car is 50. Calculate the distance of the car from the base of the building, correct to 2 significant figures.

Answer:.......

N A man, who is 1.7 m tall, stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 24. Calculate the height of the tree, correct to the nearest metre.

Answer:.......

N A point P on horizontal ground is 40 m from a building. The angle of depression of X from the top, T, of the building is 60. Calculate the height of the building.

Answer:....... 5. In Diagram 1, A and B are two balls 10 m apart on horizontal ground. From the top of a tower, D, the angles of depression of A and B are 35 and 45 respectively .

110

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D

A10 mBC

. Calculate the height of the tower CD, correct to the nearest metre.

Answer:.......

111

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6. Diagram 2 shows two vertical iron bars, KM and ST. The angle of elevation of S from M is 56 and the angle of depression of T from K is 37.

S

K

MT

80 m Diagram 2

Calculate the difference in length between the two iron bars in m..

Answer:.......

7. Diagram 3 shows two identical lamp posts which are 300 m apart. The angle of elevation of E from A is 45 while the angle of depression of A from G is 71.

EG

FHA

Diagram 3Calculate20. the height of each lamp post to the nearest metre. 21. the angle of elevation of E from H.

Answer (a):.......

Answer (b):.......

112

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8. In Diagram 4, RST is a vertical pole.

R

S

40 m18 o

TU

Diagram 422 o

Find

K the angle of elevation R from U.

L the angle of depression of U from S. M the distance between R and S in m. Answer (a):.......Answer (b):.......Answer (c):.......

9. The diagram 5shows two vertical poles GH and IJ. The angle of elevation of I from H is 36.

I

G

5 m

3 m

HJ

Diagram 5

Find

M the horizontal distance between the two poles. N The angle of depression of G from I.

Answer (a):.......

Answer (b):.......

113

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114

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10. The diagram 6 shows 3 vertical sticks, AB, CD and EF. The angles of elevation of C from A and E are 60 and 70 respectively.

C

A

E

3 m

BDF

5 m 4 m Diagram 6

Find

P the height of the two sticks, CD and EF. Q The angle of depression of F from C.

Answer (a):.......

Answer (b):.......

115

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DIAGNOSTIC TEST

1. The angle of depression from a river bank to a sampan is 25o. If the vertical distance from the bank to the sampan is 40m, calculate the horizontal distance between the sampan and the river bank.

7. 18.65m

8. 29.79m 9. 47.67m 10. 85.78m

2.

T

6 m

R8 mS

Diagram 1

In diagram 1, find the angle of elevation from R to T.

10. 36o 52 11. 41o 25

12. 48o 35

13. 53o 08

3.

U

10 m

V7 mW

Diagram 2

In diagram 2, find the angle depression from U to W.

3. 30o

4. 44o 36

5. 45o 34

6. 135o 34

116

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117

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4. A bird is at a distance of 50m horizontally from a helicopter. The bird is flying at a height of 60m vertically below the helicopter. Find the angle of depression of the helicopter to the bird.

4. 33o 33 5. 39o 48

6. 50o 12

7. 56o 27

5. A student is observing a ceiling fan. If the angle of elevation of the student to the fan is 20o and the horizontal distance between the two is 6m, calculate the height of the fan the floor ( ignore the height of the student).

7. 2.05m

8. 2.18m 9. 3.82m 10. 5.63m

6. In the diagram 3, PQ and RS are two vertical pillars standing on horizontal ground. The angle of elevation of Q from S is 22.

Q

S

h m2 m

P8 mR

Diagram 3

Calculate the value of h..

8. 3.2

9. 19.8 10. 5.2 11. 21.8

118

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7. In the diagram 4, PTS and QR are two vertical poles on horizontal ground.

S

T R

PQ

Diagram 4

The angle of elevation of point S from point R is.

A. RST

B. SRP C. SQP D. SRT

8. In the diagram 5, PQ and RS are two vertical poles. The angle of depression of P from S is 12.

S

P20 m

Q25 mR

Diagram 5

Calculate the height, in m, of the pole PQ.

A. 5.31

B. 14.69 C. 14.80 D. 97.62

119

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9. In the diagram 6, UV and PQ are two vertical poles on a horizontal plane. The angle of depression peak U from peak P is 26.

P

U

12 m

V18 mQ

Diagram 6

Calculate the height of pole UV.

A. 2.49

B. 3.22 C. 4.11 D. 4.18

10. The diagram 7 shows two vertical flag posts, KL and MN, on a horizontal plane. The angle of elevation of vertex M from vertex K is 48.

M

K

8 m

L N 6 m

Diagram 7

Calculate the angle of depression of the foot N from the vertex K.

A. 12 33

B. 23 25 C. 36 10 D. 48 23

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CHAPTER 11 : LINES AND PLANES IN 3-DIMENSION

EXERCISE 1 (PAPER 2)

13 The diagram shows a pyramid with a triangular base LMN

K

3 cm

L5 cm

N

3 cm

M

The apex K is vertically above point L. Calculate

L. the angle between line KN and the base LMN M. the angle between the planes KLM and KLN

Answer:(a)..........

(b)......

M. The digram shows a pyramid with vertex V which is 4 cm vertically above N.

V

A

NB

5 cm

D12 cmC

The digram shows a pyramid with vertex V which is 4 cm vertically above N. Calculate the angle between the edge VC and the plane ABCD.

Answer:.

121

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14. The diagram shows a cuboid.

DC

AB 3 cm

EH

8 cm

F 6 cmG

Calculate

O the length of EG

P the angle between line DG and the base EFGH

Answer:(a)...

(b)..

O The diagram shows a right pyramid with rectangular base PQRS and VT = 5cm. V

S

R

T6 cm

P8 cmQ

Calculate

22. the angle between VP and plane PQRS 23. the angle between plane VQR and plane PQRS

Answer: (a)

(b)..

122

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N The diagram shows a cuboid. S and T are the mid points of BF and AE respectively.

H8 cmG

EF

TS

30 cm

DC6 cmAB

Calculate

O the length of BT

P the angle between line CT and the base ABCD

Q the angle between the planes BCT and BCGF

Answer:

.

6.The diagram shows pyramid with a rectangular base PQRS.

(a).

(b).

(c)

M

SR

8 cm.P6 cm.Q

Given that MS = 8 cm. , calculate the angle between the line MQ and the base PQRS .

123

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Answer :

7.The diagram shows a cuboid.

D

AC

S2

BR

P

3 cm.5 cm.

Q

Calculate the angle between the line DQ and the plane BCRQ.

Answer :.

R The diagram shows a cuboid.

MPQ

DC

6NSR

A12 cm.B10 cm.

and N are the midpoints of DP and AS respectively.

H. Name the angle between planes ABM and ABCD.

I. Calculate the angle between line BM and plane ABRS.

Answer :(a)............

(b)

124

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E The diagram shows a cuboid with a horizontal rectangular base EFGH

AB

D4 cm.

C

E

F

3 cm.

H8 cm.G

Calculate the angle between;

2 plane HGB and base EFGH.

3 line BH and plane ABCD.

Answer :(a) .............

(b)..

E The diagram shows a pyramid and right angled triangle RST is horizontal P24 cm.R

7 cm.

QS

30 cm.

T

E Name the angle between planes QST and PRT

F Calculate the angle between line PT and plane PQSR.

125

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Answer :(a)

(b)..

CHAPTER 11 : LINES AND PLANES IN 3-DIMENSION

DIAGNOSTIC TEST

(d) The diagram shows a cube with a horizontal base ABCD

EH

FG

DC

AB

Name the angle between line AF and the plane ABE5 EAB

6 EAF 7 EBA 8 EBF

2.BC

AD

Q

R

PSM

The diagram shows a cuboid. Name the angle between line BM and the plane PRQS

2 BRQ

3 BMQ 4 BMR 5 BMS WV

3.

TU

SR

PQ

126

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The diagram shows a cuboid. The angle between line SU and plane PSWT isA. USP

B. USQ C. UST D. USW

4. The diagram shows a right pyramid with a quardrilateral base PQRS.

V

QPR

S

What is the angle between the line VQ and the base PQRS?

A VQR

B VQP C VQS

D QVR

7. The diagram shows a cuboid with a horizontal base GHIJ

PQSRHIGJName the angle between line QI and the plane JPI

E QJP

F QPI G QIH H QIP

6.W

T

S

P

V

U

R

127

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The diagram shows a cuboid. Name the angle between the two planes PSTW and VSP

3 VPW

4 VSW 5 VSP 6 VPT

7.

EH

K

FG

PS

QR

The diagram shows a cuboid . The angle between the two planes PQKH and GHSR is

(iii) PHK

(iv) PHS (v) PHE (vi) QKS

TR

8

P

Q

S

The diagram shows a pyramid with a horizontal triangular base PQR. RSTP is a vertical plane. The angle between the two planes TPQ and SRQ is

(d) PRQ

(e) SQR (f) PQS (g) TQS

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9. P R M

Q

ACN

B

The diagram shows a right prism with triangle ABC as its uniform cross section. The angle between the two planes AMN and ABQP is

A MAN

B MAQ C MAB D BAN

10.K

G

H

EF

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The diagram shows a pyramid with a rectangular base EFGH . HK is normal to the base. The angle between the two plane FGK and EHK is

A. EKF11. EKG

12. HKG 13. HGK

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