chapter 1 number systems 1a p.2 chapter 2 equations of

1

Chapter 1 Number Systems

1A p.2

1B p.10

Chapter 2 Equations of Straight Lines

2A p.26

2B p.40

2C p.56

Chapter 3 Quadratic Equations in One Unknown

3A p.64

3B p.73

3C p.80

3D p.87

3E p.98

3F p.107

3G p.112

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2

F4A: Chapter 1A

Date Task Progress

Lesson Worksheet

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(Full Solution)

Book Example 1

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(Video Teaching)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 1A Level 1


Teacher’s Signature

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Maths Corner Exercise 1A Multiple Choice

○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature

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E-Class Multiple Choice Self-Test

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3

4A Lesson Worksheet 1.1 & 1.2 (Refer to Book 4A P.1.3)

Objective: To understand the real number system and the complex number system.

Real Numbers, Rational Numbers and Irrational Numbers

(a) Real numbers consist of rational numbers and irrational numbers.

(b) A number that can be expressed as a ratio of two integers, i.e.q

p, where p and q are integers with

q ≠ 0, is called a rational number; otherwise, it is called an irrational number.

1. Identify the type of each of the following numbers. Put a tick ‘�’ in the correct space in the table.

3 4 2π 5− 1.6 1.2 ɺ

(a) Rational

number

(b) Irrational

number

(c) Integer

(d) Real number

Instant Example 1 Instant Practice 1

Convert 7.0 ɺ into a fraction.

Let x = 7.0 ɺ .

x = 0.777 7… …………… (1)

10x = 7.777 7… …………… (2)

(2) – (1): 9x = 7

x = 9

7 ∴ 7.0 ɺ =

9

7

Convert 3.0 ɺ into a fraction.

Let x = 3.0 ɺ .

x = _____________ ……… (1)

( )x = _____________ ……… (2)

(2) – (1): ( )x = _________

x =

= � Simplify the

answer.

∴ 3.0 ɺ =

Convert each of the following into a fraction. [Nos. 2–3]

2. 2.1 ɺ �Ex 1A: 5–8 3. 81.2 ɺɺ �Ex 1A: 18–21

Let x = 2.1 ɺ . Let x = 81.2 ɺɺ .

x = _________________ …… (1) x = _________________ …… (1)

( )x = _________________ …… (2) ( )x = _________________ …… (2)

( ) – ( ): ( ) – ( ):

( )x = ( )x =

All integers, fractions, terminating decimals and recurring decimals are rational numbers.

Consider 100x.

4

real part imaginary part

∴ 2.1 ɺ = ∴ 81.2 ɺɺ =

Imaginary Numbers

If N is a positive real number, then iNN =− , where i = 1− .

Complex Numbers

A complex number can be written in the form a + bi, where a and b are real numbers with i = 1− .

a + b i

Express each of the following in the form bi, where b is a real number. [Nos. 4–6]

4. 7− 5. 9− 6. 49−−

= ( )i � i = 1− = ( )i =

=


Identify the real part and the imaginary part of

1 + 4− .

1 + 4−

= 1 + 4 i

= 1 + 2i ∴ The real part is 1 and the imaginary part is

2.

Identify the real part and the imaginary part of

9− + 5.

9− + 5

= ( ) + ( )i

= ∴ The real part is ( ) and the imaginary

part is ( ).

Identify the real part and the imaginary part of each of the following complex numbers. [Nos. 7–8]

7. – 1− + 3 8. – 1 – 2 4− �Ex 1A: 9–12

= ( ) – ( )i =

=

∴ The real part is ( ) and

the imaginary part is ( ).

��Level Up Question��

9. (a) Determine whether π− is a purely imaginary number.

(b) Suggest another real number a such that a is a purely imaginary number.

� Rearrange the order of the real part and the imaginary part.

5

1 Number Systems

Consolidation Exercise 1A

Level 1

1. Determine whether each of the following statements is true (T) or false (F).

(a) Zero is a non-negative real number.

(b) All fractions are rational numbers.

(c) All non-terminating decimals are irrational numbers.

(d) All recurring decimals can be converted into fractions.

(e) All integers are natural numbers.

(f) All terminating decimals are rational numbers.

(g) All purely imaginary numbers are not complex numbers.

(h) All real numbers are complex numbers.

2. Consider the numbers 4,3

1− , 8 , 1.37, −π, 93 , −5, 31.2 ɺ ,

7

4, 12, −1 + 3 , 0 and

3

6.4 ɺ.

(a) Write down all the natural numbers.

(b) Write down all the non-positive integers.

(c) Write down all the rational numbers.

(d) Write down all the irrational numbers.

3. Determine whether each of the following fractions can be expressed as a terminating decimal or a

recurring decimal.

(a) 8

7 (b)

6

1−

(c) 7

12 (d)

25

16−

4. Determine whether each of the following numbers can be expressed as a recurring decimal or a non-

terminating and non-recurring decimal.

(a) 13

4 (b) 7

(c) 2

π (d)

9

10−

6

5. Classify the following numbers. Use the letters shown on the right to

show the answers.

(a) 20 (b) 5

3π

(c) 54.1 ɺɺ (d) 36−

(e) 12

11 (f) 32 −+

Convert each of the following recurring decimals into a fraction. [Nos. 6–11]

6. 2.0 ɺ 7. 4.1 ɺ

8. 6.2 ɺ 9. 45.0 ɺ

10. 87.1 ɺ 11. 31.3 ɺ

Identify the real part and the imaginary part of each of the following complex numbers. [Nos. 12–19]

12. 5 − 2i 13. 6i + 7

14. −3 − 4i 15. −8 + 9i

16. 32

1− 17. 10i

18. 4 +3

5 19. i2−

Level 2

20. Determine whether each of the following statements is true (T) or false (F). If it is false, give an

example to justify your answer.

(a) The product of two irrational numbers cannot be a rational number.

(b) The sum of two natural numbers must be a natural number.

(c) The difference between two irrational numbers must be an irrational number.

21. Determine whether each of the following statements is true (T) or false (F).

(a) 33 −+ is a real number.

(b) 4 + 0i is not a real number.

(c) − i5 is a purely imaginary number.

(d) i76 − is an irrational number.

N: Natural numbers

Z: Integers

Q: Rational numbers

R: Real numbers

C: Complex numbers

7

22. Determine whether the result of each of the following is a rational number or an irrational number.

(a) 13 + 3 (b) 2

9

2

17−

(c) 1 − 25 (d) π2 − 2π

(e) (2 − 2 )2 (f)

+

2

31 (2 − 3 )

23. Do the following equations have solutions in rational numbers, real numbers or complex numbers?

Put a tick ‘✓’ in the correct space in the table.

Equation

Rational

numbers

Real

numbers

Complex

numbers

(a) 2x − 5 =

0

(b) x2 = −4

(c) x2 = 8

(d) x2 =16

Convert each of the following recurring decimals into a fraction. [Nos. 24–29]

24. 23.0 ɺɺ 25. 96.1 ɺɺ

26. 450.0 ɺɺ 27. 844.2 ɺɺ

28. 101.1 ɺɺ 29. 654.3 ɺɺ

30. (a) Convert each of the following recurring decimals into a fraction.

(i) 6.0 ɺ

(ii) 35.0 ɺ

(b) Hence, solve the equation 6.0 ɺ x = 35.0 ɺ .

31. (a) Convert each of the following recurring decimals into a fraction.

(i) 308.0 ɺ

(ii) 3.1 ɺ

(b) Hence, solve the equation x2 = 3.1308.0 ɺɺ× .

32. If a = 4.0 ɺ and b = 38.0 ɺ , find the value of a + 2b and express the answer in a fraction.

33. If m = 61.2 ɺ and n = 5.0 ɺ , find the value of m − 3n and express the answer in a fraction.

34. The real part of the complex number (2x + 5) + (4 – 3x)i is −3, where x is a real number.

(a) Find the value of x.

(b) Find the imaginary part of the complex number.

8

35. The imaginary part of the complex number (6 − 3y) − (4y – 5)i is 1, where y is a real number.

(a) Find the value of y.

(b) Find the real part of the complex number.

36. Let z = (2 − x) − (3x + 6)i be a complex number, where x is a real number. Find the value of x such

that

(a) z is a real number,

(b) z is a purely imaginary number.

37. Let z = (4 + m) + (3 − 2m)i be a complex number, where m is a real number. Find z for each of the

following conditions.

(a) z is a real number.

(b) z is a purely imaginary number.

38. Let z = (4 − 2x) + (x − 2)i be a complex number, where x is a real number.

(a) If z is a real number, find the value of x.

(b) Is there a real number x such that z is a purely imaginary number? Explain your answer.

39. Let z = (6y + 3) − (2 + y)i be a complex number, where y is a real number.

(a) If z is a purely imaginary number, find the value of y.

(b) Is there a real number y such that z is a real number? Explain your answer.

40. Let z = (x + 4) − (2x + 5)i be a complex number, where x is a real number. If the real part and the

imaginary part of z are equal, find the value of x.

41. Let z = (3 − 2k) + (4k − 9)i be a complex number, where k is a real number. If the sum of the real part

and the imaginary part of z is 0, find the value of k.

9

Answers


1. (a) T (b) T (c) F

(d) T (e) F (f) T

(g) F (h) T

2. (a) 4, 93 , 12

(b) −5, 0

(c) 4,3

1− , 1.37 , 93 , −5, 31.2 ɺ ,

7

4,

12, 0,

3

6.4 ɺ

(d) 8 , −π, −1 + 3

3. (a) terminating decimal

(b) recurring decimal

(c) recurring decimal

(d) terminating decimal

4. (a) recurring decimal

(b) non-terminating and non-recurring

decimal

(c) non-terminating and non-recurring

decimal

(d) recurring decimal

5. (a) N, Z, Q, R, C (b) R, C

(c) Q, R, C (d) C

(e) Q, R, C (f) C

6. 9

2 7.

9

13

9

41or

8. 3

8

3

22or 9.

90

49

10. 90

161

90

711or 11.

15

47

15

23or

12. real part = 5, imaginary part = −2

13. real part = 7, imaginary part = 6

14. real part = −3, imaginary part = −4

15. real part = −8, imaginary part = 9

16. real part = 32

1− , imaginary part = 0

17. real part = 0, imaginary part = 10

18. real part =3

54 + , imaginary part = 0

19. real part = 0, imaginary part = 2−

20. (a) F,π

1π× (or other reasonable answers)

(b) T

(c) F, 2 − ( 2 − 2) (or other reasonable

answers)

21. (a) F (b) F

(c) T (d) F

22. (a) irrational number

(b) rational number

(c) rational number

(d) irrational number

(e) irrational number

(f) rational number

23. Rational numbers

Real numbers

Complex numbers

(a) � � �

(b) �

(c) � �

(d) � � �

24. 99

32 25.

33

56

33

231or

26. 55

3 27.

165

404

165

742or

28. 999

100 1

999

1011or 29.

333

1511

333

1523or

30. (a) (i) 3

2 (ii)

15

8

(b) 5

4

31. (a) (i) 12

1 (ii)

3

4

3

11or

(b) 3

1±

32. 9

19

9

12or 33.

2

1

34. (a) −4 (b) 16

35. (a) 1 (b) 3

36. (a) −2 (b) 2

37. (a) 2

11 (b) 11i

38. (a) 2 (b) no

39. (a) 2

1− (b) yes

40. −3

10

F4A: Chapter 1B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

11

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)

Maths Corner Exercise 1B Level 1



___________ ( )




___________ ( )

Maths Corner Exercise 1B Multiple Choice


___________ ( )



_________

12

4A Lesson Worksheet 1.3A (Refer to Book 4A P.1.14)

Objective: To perform operations of imaginary numbers.

Operations of Imaginary Numbers

Let a and b be real numbers, and i = 1− .

(a) a × bi = bi × a = abi (b) ai ± bi = (a ± b)i

(c) ai × bi = (a × b)i2 = ab(–1) = –ab (d) b

a

bi

ai= (where b ≠ 0)

Simplify the following expressions. [Nos. 1–6]

1. 5(3i) 2. 6i + 2i 3. 7i – 5i

= ( × )i = (____ + ____)i =

= =

4. i × 4i 5. 2i × 8i 6. i

i

3

9

= ( × )i2 = =

= ( )( )

=


Simplify 2(5i – i).

2(5i – i) = 2(5 – 1)i

= 2(4)i

= 8i

Simplify (2i + i) × i.

(2i + i) × i = ( + )i × i

= ( )i( )

= ( )( )

=


7. 2i2 × 2i 8. i

i37

�Ex 1B: 1, 2

= ( × )(i2)( ) =

=

Powers of i

(a) i2 = –1 (b) i3 = i2 × i = (–1) × i = –i

(c) i4 = i2 × i2 = (–1) × (–1) = 1 (d) i5 = i4 × i = 1 × i = i

�

i

i3

= i3 – 1

� i2 = –1

The multiplication (or division) of two imaginary numbers gives

a real number. �

13


9. i3 × i2 10. 4

7

i

i 11. i10 �Ex 1B: 10

= ( )( ) = i( ) – ( ) = i4 × ( ) + ( )

= = i ( ) = (i4) ( )i( )

= = ( )( ) × ( )

=


Simplify 4− + 1− .

4− + 1− = 4 i + i = 2i + i = (2 + 1)i = 3i

Simplify 9− – 25− .

9− – 25− = ( )i – ( )i

= ( )i – ( )i = ( )i =


12. 9− 1− 13. 16− ÷ 4− �Ex 1B: 3, 4

= =


14. ( 2 i)( 2 i3) 15. i6 – 4i4 + i �Ex 1B: 14

= =


16. (a) Find the remainder of 2 014 ÷ 4.

(b) Hence, simplify i2 014.

For a positive real number N,

iNN =− .

14

4A Lesson Worksheet 1.3B(I) (Refer to Book 4A P.1.16)

Objective: To perform the addition and subtraction of complex numbers.

Addition of Complex Numbers

If a, b, c and d are real numbers, then

(a + bi) + (c + di) = (a + c) + (b + d)i. � Add the real part and the imaginary part separately.

Simplify the following expressions and express the answers in the form a + bi. [Nos. 1–2]

1. (1 + 3i) + 2 2. (4 – i) + 6i

= ( ) + i = + ( )i

= =


Simplify (2 + i) + (1 + 5i).

(2 + i) + (1 + 5i)

= (2 + 1) + (1 + 5)i

= 3 + 6i

Simplify (5 + 3i) + (2 – i).

(5 + 3i) + (2 – i)

= ( ) + ( )i

=


3. (8 + 2i) + (7 – 3i) 4. (–6 – 4i) + (3 – 2i) �Ex 1B: 5

= =

Subtraction of Complex Numbers

If a, b, c and d are real numbers, then

(a + bi) – (c + di) = (a – c) + (b – d)i. � Subtract the real part and the imaginary part separately.


Simplify (1 + 2i) – (5 + 3i).

(1 + 2i) – (5 + 3i)

= (1 – 5) + (2 – 3)i

= –4 – i

Simplify (7 + i) – (3 – 2i).

(7 + i) – (3 – 2i)

= ( ) + ( )i

=

�

15


5. 7 – (2 + 5i) 6. (12 – i) – 6i

= ( ) + ( )i = ( ) + [(–1) – ]i

= =

7. (–3 + i) – (6 + 2i) 8. (2 – 9i) – (4 – i) �Ex 1B: 6

= =

9. (3 – 4i) + (5 – 6i) + 2 10. –8i + (7 – 6i) – (1 + 4i) �Ex 1B: 16

= ( ) + ( )i = ( ) + ( )i

11. (1 – 3i) + (2 + i) + (4 + 2i) 12. (5 + 9i) – (1 + 3i) – (4 + 5i)

= =


13. (a) Simplify (4 – 2i) – (–3 + 7i) + (5 – 3i) and express the answer in the form a + bi.

(b) Hence, write down the real part and the imaginary part of (4 – 2i) – (–3 + 7i) + (5 – 3i).

Distinguish between real numbers and imaginary numbers carefully.

16

4A Lesson Worksheet 1.3B(II) (Refer to Book 4A P.1.17)

Objective: To perform the multiplication of complex numbers and understand the concept of conjugate

complex numbers.

Review: Multiplication of Polynomials


1. 4(3 + x) 2. (2x – 1) × 5x

= ( )( ) + ( )( ) = ( )( ) – ( )( )

= =

3. (2 + x)(1 + 3x) 4. (1 + x)(1 – x)

= ( )(1) + ( )(3x) =

= ( ) + ( ) + ( ) + ( )

=

Multiplication of Complex Numbers


5. 3(1 + 2i) 6. i(2 + i) �Ex 1B: 7(a)

= ( )( ) + ( )( ) = ( ) + i( )

= = ( ) + ( )

=


Simplify (1 + i)(1 + 4i) and express the answer

in the form a + bi.

(1 + i)(1 + 4i) = (1 + i)(1) + (1 + i)(4i)

= 1 + i + 4i + 4i2

= 1 + 5i + 4(–1)

= –3 + 5i

Simplify (2 – i)(3 + i) and express the answer in

the form a + bi.

(2 – i)(3 + i) = ( )(3) + ( )i

=


7. (4 + i)(1 + 5i) 8. (1 – 2i)(7 – i) �Ex 1B: 7(b), (c)

= =

�

(a + b)(a – b) = a2 – b2

� i2 = –1

4(3 + x)

(2 + x)(1 + 3x)

3(1 + 2i)

(2x – 1) × 5x

17

9. (1 + i)2 10. (1 – 3i)2 �Ex1B: 8(a), (b)

= ( )2 + 2( )( ) + ( )2 =

=

Conjugate Complex Numbers

Let a and b be real numbers. a + bi and a – bi are called conjugate complex numbers.


11. (6 + i)(6 – i) 12. (1 – 4i)(1 + 4i) �Ex 1B: 8(c)

= ( )2 – ( )2 =

=


13. (1 + 2i)2 – i 14. (1 + 5i)(1 – 5i)i �Ex 1B: 17, 20

= =


15. If a is a real number, express the real part of (2a – i)(a + i) in terms of a.

(x + y)2 = x2 + 2xy + y2 (x – y)2 = x2 – 2xy + y2

(x + y)(x – y) = x2 – y2

18

4A Lesson Worksheet 1.3B(III) (Refer to Book 4A P.1.18)

Objective: To perform the division of complex numbers.

Review: Dividing a Polynomial by a Real Number


1. 2

22 x+ 2.

3

36 x−

= ) (

) (+ x

) (

) ( =

=

Dividing a Complex Number by a Real Number

If a, b and c are real numbers, then ic

b

c

a

c

bia±=

±. � Divide the real and imaginary parts separately.


3. 2

24 i− =

) (

) (– i

) (

) ( 4.

3

73 i+ =

) (

) (+ i

) (

) (

=

=

Division of Complex Numbers

Let a, b, c and d be real numbers.

(a) i

i

ci

bia

ci

bia×

+=

+

(b) dic

dic

dic

bia

dic

bia

−

−×

+

+=

+

+ � c – di is the conjugate of c + di.


5. i

1 =

i

1×

i

i 6.

i3

5 =

i3

5×

) (

) ( 7.

i

i

2

2 − =

= ) (

) ( =

=

�

xc

b

c

a

c

bxa+=

+

� Express the answer in

the form a + bi.

i2 = –1

19


Simplifyi+1

1and express the answer in the

form a + bi.

i+1

1 =

i

i

i −

−×

+ 1

1

1

1

= 221

1

i

i

−

−

= )1(1

1

−−

− i

= i2

1

2

1−

Simplifyi−2

1and express the answer in the

form a + bi.

i−2

1 =

) (

) (

2

1×

− i

= 22 ) () (

) (

−

=


8. i+3

4=

) (

) (

3

4×

+ i 9.

i41

2

− = �Ex 1B: 9

=

10. i

i

−1 = 11.

i

i

32

13

+ = �Ex 1B: 18


12. (a) Simplifyi

i

−

−

4

53and express the answer in the form a + bi.

(b) Hence, simplify )72(4

53i

i

i−−

−

−.

(x + y)(x – y) = x2 – y2

� The conjugate of 2 – i

is ( ).

The conjugate of 3 + i is

( ).

The conjugate of 1 – 4i is

( ).

The conjugate of 1 – i is ( ).

The conjugate of 2 + 3i is ( ).

� The conjugate of 1 + i is 1 – i.

20

4A Lesson Worksheet 1.3C (Refer to Book 4A P.1.21)

Objective: To find unknowns from the equality of complex numbers.

Equality of Complex Numbers

Let a, b, c and d be real numbers. If a + bi = c + di, then a = c and b = d, and vice versa.

In each of the following, find the values of the real numbers x and y. [Nos. 1–2]

1. x + 4i = 5 – yi 2. (x + 2i) + yi = 4 + 3i �Ex 1B: 11

x = and ( ) + ( )i = 4 + 3i

x = and


If (x + y) + (x – y)i = –1 – 3i, find the values of

the real numbers x and y.

−=−

−=+

(2) .............

(1) .............

3

1

yx

yx

(1) + (2): 2x = –4

x = –2

Substitute x = –2 into (1).

–2 + y = –1

y = 1

If (y – x) + (2x + y)i = 4 + i, find the values of the

real numbers x and y.

=

=

(2) .............

(1) .............

) () (

) () (

(2) – (1): ( )x = ( )

x =

Substitute x = ( ) into (1).

y – ( ) = ( )

y =

In each of the following, find the values of the real numbers x and y. [Nos. 3–4]

3. 2y + (3x + y)i = 8 – 5i 4. (x + 3y) – (x + 2y)i = 2i �Ex 1B: 12, 13

=

=

(2) .............

(1) .............

) () (

) () (

=

=

(2) .............

(1) .............

) () (

) () (

From (1), y = ( ) + ( ):

Substitute y = ( ) into (2).


5. (a) Express (1 + i)(x + yi) in the form a + bi, where x and y are real numbers.

(b) If (1 + i)(x + yi) = 2 + 8i, find the values of the real numbers x and y. �Ex 1B: 24, 26

�

Rearrange the real and imaginary

parts first.

21

New Century Mathematics (Second Edition) 4A

1 Number Systems

� Consolidation Exercise 1B

Level 1


1. (a) 3

4× i (b) i × (−3)

(c) 5i + 6i (d) 13i − 4i

(e) 3i − 7i (f) −10i − i

2. (a) 4i × i (b) 3i × 4i

(c) −6i × 9i (d) (−5i)(−2i)

(e) i

i

7

14 (f)

i

i

9

18−

(g) i

i

12

8

− (h)

i

i

15

6

−

−

3. (a) 6(2i + 5i) (b) 3(6i – i)

(c) −2(7i − 11i) (d) 5i + 7i + 9i

(e) 6i + 3i − 10i (f) 12i − 8i − i

4. (a) 3i(4i + 5i) (b) (3i + 4i)(4i − 3i)

(c) (8i − 2i)(2i − 3i) (d) i5

(e) i2 × 2i (f) 9i × 3i3

5. (a) 2

4

5

10

i

i (b)

2

5

12

8

i

i−

(c) 3

55

14

34

i

ii + (d)

5

43

15

52

i

ii ×

6. (a) 41 −+− (b) 925 −−− (c) 14481 −−−

7. (a) 19 −− (b) 4916 −− (c) 64121 −÷−


8. (a) 2 + (1 + 6i) (b) 5i + (3 − 4i)

(c) (5 + 3i) + (4 + 7i) (d) (–7 + 3i) + (2 – i)

(e) (8 + 2i) + (–11 + 5i) (f) (9 – i) + (–3 – 6i)

22

9. (a) 5i – (2 + 3i) (b) 13 – (9 − 6i)

(c) (3 + 6i) – (4 + 2i) (d) (–1 + 8i) – (−3 + 7i)

(e) (6 – 4i) – (5 – 9i) (f) (12 − 7i) – (−5 − 10i)

10. (a) 9(2 + 5i) (b) i(6 – 3i)

(c) −2i(4 − 8i) (d) (1 + i)(2 + 4i)

(e) (–3 + 9i)(8 – 3i) (f) (7 + 4i)(–2 – 6i)

11. (a) (5 + 2i)2 (b) (3 − 4i)2

(c) (−6 − 5i)2 (d) (6 + i)(6 − i)

(e) (4 − 3i)(4 + 3i) (f) (8 − 7i)(7 – 8i)

12. (a) i

3 (b)

i

5−

(c) i

i21+ (d)

i

i54 −

(e) i

i

2

6 +− (f)

i

i

3

73 −−

13. (a) i31

1

+ (b)

i41

17

−

(c) i52

29

− (d)

i34

25

+−

(e) i76

34

+ (f)

i65

122

−−

14. Simplify the following expressions, where n is a positive integer.

(a) i6 (b) i9

(c) i4n + 2 (d) i4n − 1

15. If x + 5i = 2 + yi, find the values of the real numbers x and y.

16. If –7 – pi = –q + 2i, find the values of the real numbers p and q.

17. If (a + 5i) + (4 − bi) = 6 + 2i, find the values of the real numbers a and b.

18. If (x − 4i) + (y − xi) = −1 − 5i, find the values of the real numbers x and y.

19. If (2a + bi) − (b − 3ai) = 8 + 7i, find the values of the real numbers a and b.

20. If (c − 2di) + (5 + 3c)i = 3d + (c + 2)i, find the values of the real numbers c and d.

23

Level 2


21. (a) ( i63 )( i32 ) (b) 5

3

22

4

i

i (c) i(4i3 + 5i5 − 2i)

22. (a) 272 −+− (b) 7548 −−−

(c) 4045 −− (d) 32

50

−

−

(e) 16

832

−

−−− (f)

128

2724

−

−−


23. (a) (6 + 3i) + (5 − 5i) + (−3 + 2i) (b) (11 + 2i) − (13 − 12i) + (1 + 6i)

(c) (7 − 8i) + (−9 − 4i) − (−15 + 10i) (d) (−8 + 13i) − (5 + 8i) − (−14 − 7i)

24. (a) (5 + 2i)(1 + 4i) (b) (4 − 3i)(5 + 8i)

(c) (6 − 7i)(−5 − i) (d) (−3 + 2i)(2 − 7i)

25. (a) (2 + i)(4 − 3i)(−1 + i) (b) (5 + 2i)(−2 − 5i)(3 − i)

26. (a) i

i

23 + (b)

i

i

45

41

−

(c) i

i

21

1

+

+ (d)

i

i

37

25

−

+

(e) i

i

52

25

+

− (f)

i

i

38

84

−

−

27. (a) ai +i

ia 23 +, where a is a real number.

(b) i

ki

−1

4− (3 − 2ki), where k is a real number.

28. (a) (11 + 6i) + 4(−3 − 2i) (b) (16 − 9i) − 3i(4 + i)

29. (a) (5 + 3i)(3 + 5i) + (2 − 7i) (b) (2 − 3i)2 − (7 + 6i)

30. (a) i

i

2

42 −+ (5 + 6i) (b) (7 − 8i) −

i

i95 +

31. (a) i

i

32

74

+

+−− (1 − 8i) (b) (4 − 5i) +

i

i

−

+

4

13

24

32. (a) i

i

31

23

+

−−

i

i

31

23

−

+ (b)

i

i

24

45

−

−+

i

i

21

6

+

−

33. (a) Simplifyi

i

21

81

+

+−and express the answer in the form a + bi.

(b) Using the result of (a), simplify

2

21

81

+

+−

i

iand express the answer in the form a + bi.

34. Simplifyi

i−7+

2

23

512

−

+

i

iand express the answer in the form a + bi.

35. If (4 − 3i)(a + 2bi) = −25i, find the values of the real numbers a and b.

36. If (m + 5i)(5 + 3i) = ni, find the values of the real numbers m and n.

37. If the result of (k − 4i)(3 − 2i) is a real number, find the value of the real number k.

38. If the result of

ia

i

−

+ 63is a purely imaginary number, find the value of the real number a.

39. If (3x − yi) −i

i+2= y − (x + 1)i, find the values of the real numbers x and y.

40. If (2a + 3i) − 2

+

+

i

bi

1

2= (a + 1) + 4i, find the values of the real numbers a and b.

41. Let a and b be non-zero real numbers.

(a) Simplifyaib

bia

+

−.

(b) Simplifyaib

bia

aib

bia

+

−−

−

+.

42. Let a and b be non-zero real numbers.

(a) Is (a + bi)(b + ai) a purely imaginary number? Explain your answer.

(b) Let m = (a + bi)(b + ai) and n = (a − bi)(b − ai).

(i) Show that m + n = 0.

(ii) Hence, simplify

−

−+

+

+ iiii

3

1

5

2

5

2

3

1

3

1

5

2

5

2

3

1.

25

Answers

Consolidation Exercise 1B

1. (a) i3

4 (b) −3i

(c) 11i (d) 9i

(e) −4i (f) −11i

2. (a) −4 (b) −12

(c) 54 (d) −10

(e) 2 (f) −2

(g) 3

2− (h)

5

2

3. (a) 42i (b) 15i (c)

8i (d) 21i (e) −i (f)

3i

4. (a) −27 (b) −7 (c) 6

(d) i (e) −2i (f)

27

5. (a) −2 (b) i3

2

(c) 2

1− (d)

3

2−

6. (a) 3i (b) 2i (c)

−3i

7. (a) −3 (b) −28 (c)

8

11

8. (a) 3 + 6i (b) 3 + i

(c) 9 + 10i (d) −5 + 2i

(e) −3 + 7i (f) 6 − 7i

9. (a) −2 + 2i (b) 4 + 6i

(c) −1 + 4i (d) 2 + i

(e) 1 + 5i (f) 17 + 3i

10. (a) 18 + 45i (b) 3 + 6i

(c) −16 − 8i (d) −2 + 6i

(e) 3 + 81i (f) 10 − 50i

11. (a) 21 + 20i (b) −7 − 24i

(c) 11 + 60i (d) 37

(e) 25 (f) −113i

12. (a) −3i (b) 5i

(c) 2 − i (d) −5 − 4i

(e) 2

1+ 3i (f)

3

7− + i

13. (a) 10

1− i

10

3 (b) 1 + 4i

(c) 2 + 5i (d) −4 − 3i

(e) 5

12− i

5

14 (f) −10 + 12i

14. (a) −1 (b) i

(c) −1 (d) −i

15. x = 2, y = 5

16. p = −2, q = 7

17. a = 2, b = 3

18. x = 1, y = −2

19. a = 3, b = −2

20. c = 4, d = 3

21. (a) 218− (b) 2− (c) 1

22. (a) i27 (b) i3−

(c) 230− (d) 4

5

(e) 2

2 (f) i

4

9

23. (a) 8 (b) −1 + 20i

(c) 13 − 22i (d) 1 + 12i

24. (a) −3 + 22i (b) 44 + 17i

(c) −37 + 29i (d) 8 + 25i

25. (a) −9 + 13i (b) −29 − 87i

26. (a) 13

2+ i

13

3 (b) −4 + 5i

(c) 5

3− i

5

1 (d)

2

1+ i

2

1

(e) −i (f) 73

56− i

73

52

27. (a) 2 − 2ai (b) −(2k + 3) + 4ki

28. (a) −1 − 2i (b) 19 − 21i

29. (a) 2 + 27i (b) −12 − 18i

30. (a) 3 + 5i (b) −2 − 3i

31. (a) 10i (b) 7 − 4i

32. (a) i5

11− (b)

5

11− i

10

29

33. (a) 3 + 2i (b) 5 + 12i

34. −6 + 5i

35. a = 3, b = −2

36. m = 3, n = 34

37. −6

38. 2

39. x = 2, y = 5

40. a = 4, b = 1

41. (a) −i (b) 2i

42. (a) yes (b) (ii) 0

26

F4A: Chapter 2A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)



(Full Solution)




___________ ( )


○ Complete and Checked ○ Problems encountered


___________

27

○ Skipped ( )



___________ ( )



_________

28

4A Lesson Worksheet 2.0 (Refer to Book 4A P.2.3)

Objective: To review the distance formula, the slope of a straight line, parallel lines, perpendicular lines

and point of division.

Distance Formula

(a) AB = 212

212 )()( yyxx −+−

Slope of a Straight Line

(b) Slope m of L =12

12

xx

yy

−

− (where x1 ≠ x2)

(c) m = tan θ (θ is called the inclination of L.)

In each of the following, find the distance between the points A and B. [Nos. 1–4] � Review Ex: 1

(Leave the radical sign ‘√’ in the answers if necessary.)

1. A(0 , 2), B(4 , 5) 2. A(3 , 2), B(8 , 14)

AB = 22 )]()[()]()[( −+− AB = 22 )()( +

= =

3. A(2 , –1), B(3 , 3) 4. A(1 , –5), B(–4 , –7)

In each of the following, find the slope m and the inclination θ of the straight line passing through the

points P and Q. [Nos. 5–6]

(Give the answers correct to 3 significant figures if necessary.)

5. P(2 , 1), Q(4 , 3) 6. P(1 , 0), Q(4 , 1) � Review Ex: 2–4

m =) () (

) () (

−

−=

m =

tan θ = ( ) tan θ = ( )

θ = θ = , cor. to 3 sig. fig.

Parallel Lines and Perpendicular Lines

(a) (i) If 1ℓ // 2ℓ , then m1 = m2. (b) (i) If 1ℓ ⊥ 2ℓ , then m1 × m2 = –1.

(ii) If m1 = m2, then 1ℓ // 2ℓ . (ii) If m1 × m2 = –1, then 1ℓ ⊥ 2ℓ .

7. Four points A(0 , 2), B(2 , 0), C(9 , –3) and D(3 , 3) are given. Determine whether

29

(a) AB is parallel to CD, (b) AD is perpendicular to CD. � Review Ex: 5

Slope of AB =) () (

) () (

−

−= ( ) Slope of AD =

Slope of CD =) () (

) () (

−

−= ( ) Slope of CD =

∵ Slope of AB ( = / ≠ ) slope of CD

∴ AB ( is / is not ) parallel to CD.

Point of Division

(a) Mid-point Formula (b) Section Formula

x =2

21 xx + x =

sr

rxsx

+

+ 21

y =2

21 yy + y =

sr

rysy

+

+ 21

In each of the following, find the coordinates of the mid-point P of the line segment AB. [Nos. 8–9]

8. A(13 , 6), B(3 , 4) 9. A(7 , –2), B(–3 , 10) � Review Ex: 6

P =

++

2

) () (,

2

) () (

= ( , )

In each of the following, find the coordinates of the point P which divides the line segment AB in the given

ratio. [Nos. 10–11] � Review Ex: 7

10. A(1 , –4), B(5 , 8), AP : PB = 3 : 1 11. A(4 , 12), B(–3 , –2), AP : PB = 2 : 5

P =

+

+

+

+

13

) (3) (1,

13

) (3) (1

= ( , )


12. (a) Find the slope of the straight line L passing through the points P(2 , 3) and Q(3 , 6).

(b) The inclination of a straight line L1 is 60°. Is L1 parallel to L? Explain your answer.

�

� �

30

4A Lesson Worksheet 2.1A & B (Refer to Book 4A P.2.6)

Objective: To find the equation of a straight line passing through two points.

Review: Slope of a Straight Line

In each of the following, find the slope of the straight line passing through P and Q. [Nos. 1–2]

1. P(0 , 0), Q(3 , 2) 2. P(1 , 6), Q(2 , 4)

Slope of PQ =0) (

0) (

−

−=

Slope of PQ =

) () (

) () (

−

−=

General Form of the Equation of a Straight Line

Ax + By + C = 0, where A, B and C are constants with A and B not both zero.

Express each of the following equations of straight lines in the general form. [Nos. 3–6]

3. y + 1 = −5x 4. y = 2(x + 3)

( )x + y + ( ) = ( ) y = ( )x + ( )

( )x − y + ( ) = ( )

5. 2(y − 1) = 5(1 − x) 6. 2

3

+

−

x

y = 2

( )y − ( ) = ( ) − ( )x ( ) = ( )x + ( )

= =

Two-point Form

The equation of the straight line passing through A(x1 , y1)

and

B(x2 , y2) is given by:

1

1

xx

yy

−

−=

12

12

xx

yy

−

− (where x1 ≠ x2)

In each of the following, write down the equation of the straight line L in the two-point form. [Nos. 7–8]

7. 8.

) (

) (

−

−

x

y=

) () (

) () (

−

−

) (

) (

−

−

x

y=

) () (

) () (

−

−

� Order of terms: x, y, constant

B(–1 , 1)

x O

y

A(1 , 4)

L

A(–3 , 0) x

O

y

B(3 , –6)

L

31


Find the equation of the straight line passing

through the points (0 , 2) and (4 , 1).

The required equation is

0

2

−

−

x

y =

04

21

−

−

x

y 2− = –

4

1

4y – 8 = –x

x + 4y – 8 = 0


through the points (3 , 5) and (1 , –1).


) (

) (

−

−

x

y=

) (

) (

) (

) (= ( )

( ) = ( )

=

In each of the following, find the equation of the straight line passing through A and B. [Nos. 9–10]

9. A(2 , 3), B(0 , 4) 10. A(1 , –2), B(5 , 6) �Ex 2A: 3, 4

11. (a) Find the equation of the straight line L passing through (–1 , 4) and (3 , 8).

(b) Does P(2 , 7) lie on L? Explain your answer. �Ex 2A: 13


12. (a) Find the equation of the straight line L passing through (–2 , 1) and (2 , 5).

(b) If (1 , k) lies on L, find the value of k.

(0 , 2)

x O

y

(4 , 1)

(3 , 5)

x O

y

(1 , –1)

� Express the answer in the general form.

32

4A Lesson Worksheet 2....1C (Refer to Book 4A P.2.9)

Objective: To find the equation of a straight line from a given slope and the coordinates of a point on it.

Point-slope Form

The equation of the straight line with slope m and passing

through A(x1 , y1) is given by:

y – y1 = m(x – x1)

In each of the following, write down the equation of the straight line ℓ in the point-slope form. [Nos. 1–3]

1. 2. 3.

The equation of ℓ is Slope of ℓ = tan ( )

y – 1 = 2( – ) The equation of ℓ is =

The equation of ℓ is

______________________



through (1 , 1) and with slope 3.


y – 1 = 3(x – 1)

y – 1 = 3x – 3

3x – y – 2 = 0 � Express the answer in the general

form.


through (5 , 2) and with slope –4.


y – ___ = _________

=

In each of the following, find the equation of the straight line passing through the point P and with slope m.

[Nos. 4–5]

4. P(3 , 2), m = –2 5. P(–3 , 1), m =3

1 �Ex 2A: 7, 8


y – = ( – )

=

ℓ

(1 , 3)

y

x 45°

O

(0 , 1)

y

x

slope = 2

ℓ

O

0 3

y

x

slope =2

1 ℓ

ℓ passes through (3 , ).

y – ( ) =

33

In each of the following, find the equation of the straight line L1. [Nos. 6–9]

6. 7. �Ex 2A: 10, 11

∵ L1 ( // / ⊥ ) L2

∴ Slope of L1 ( = / × ) slope of L2 =

The equation of L1 is

y – = ( – )

=

8. 9.


10. (a) Find the equation of the straight line L passing through P(2 , –3) and with slope –2.

(b) ℓ is another straight line passing through P and perpendicular to L. Find the equation of ℓ .

x

y

L1

O L2

(–4 , 3)

45°

(1 , 2)

y

x

L1

O

L2

(3 , 8) (2 , 7)

(–5 , 1)

y

x

L1

O

L2: slope = –1

(2 , 3)

y

x

L2: slope = –2

O

L1

First find the slope of L2.

34


2 Equations of Straight Lines


Level 1

In each of the following, find the equation of the straight line which passes through the points A and B.

[Nos. 1–5]

1. (a) (b)

2. (a) A(–1 , 1), B(2 , 3) (b) A(3 , –2), B(–2 , –1)

3. (a) A(5 , 7), B(–4 , –5) (b) A(–1 , 5), B(3 , –7)

4. (a) A(–3 , 1), B(1 , 3) (b) A(2 , –7), B(–2 , 1)

5. (a) A(10 , 7), B(–8 , –2) (b) A(–8 , –1), B(5 , –4)

In each of the following, find the equation of the straight line passing through the point P and with slope m.

[Nos. 6–10]

6. (a) (b)

7. (a) P(3 , 2), m =3

1 (b) P(2 , 0), m =

2

1−

8. (a) P(–2 , –3), m =3

2 (b) P(4 , –6), m =

5

2−

9. (a) P(–1 , –3), m =2

1 (b) P(–4 , 7), m =

3

4−

10. (a) P

2

1,1 , m =

4

1− (b) P

−− 2,

3

1, m =

5

3

y

x

P(4 , 3)

m = 2

O x

P(−1 , 2)

m = −3

O

y

x

B(4 , −3)

O A(−2 , 1)

y

x

A(1 , 2)

B(5 , 4)

O

y

35

In each of the following, find the equation of the straight line ℓ . [Nos. 11–12]

(Leave the radical sign ‘√’ in the answers if necessary.)

11. (a) (b)

12. (a) (b)

In each of the following, the straight lines ℓ and L are parallel to each other. Find the equation of ℓ .

[Nos. 13–15]

13. (a) ℓ passes through (3 , 1) and the slope of L is 1.

(b) ℓ passes through (–1 , 5) and the slope of L is –2.

14. (a) ℓ passes through (–6 , –3) and the slope of L is 2.

(b) ℓ passes through (0 , 0) and the slope of L is –4.

15. (a) ℓ passes through (4 , –1) and the slope of L is

3

1− .

(b) ℓ passes through (7 , 5) and the slope of L is

3

2.

In each of the following, the straight lines ℓ and L are perpendicular to each other. Find the equation of ℓ .

[Nos. 16–18]

16. (a) ℓ passes through (–1 , 2) and the slope of L is 2.


17. (a) ℓ passes through (2 , –4) and the slope of L is

3

1.


18. (a) ℓ passes through (–3 , –4) and the slope of L is

3

1− .

(b) ℓ passes through (1 , 8) and the slope of L is

5

2.

y

x 0

45°

3

ℓ

y

x 0

135°

−3

ℓ

y

x 0

120°

−−5

ℓ

y

0

30°

−2

ℓ

x

36

Level 2

19. In the figure, the straight line L passes through (4 , 1) and (1 , –4).

(a) Find the equation of L.

(b) Determine whether P(–2 , –8) lies on L. Explain your answer.

20. The straight line L passes through (5 , 3) and (–4 , −2).


(b) If L cuts the x-axis at (p , 0), find the value of p.

21. The straight line L passes through (2 , −2) and (−2 , 4).


(b) If P(a , 7) lies on L, find the value of a.

22. The straight line L with slope 3 passes through (1 , 2).


(b) Does L pass through Q(−1 , –4)? Explain your answer.

23. The straight line L with slope

3

2− passes through (−3 , −2).


(b) If L cuts the y-axis at (0 , q), find the value of q.

24. The straight line L with slope

2

5 passes through (2 , −1).


(b) If Q(6 , b) lies on L, find the value of b.

25. R(a , −4) lies on the straight line L1: 5x + 8y − 3 = 0.

(a) Find the value of a.

(b) Find the equation of the straight line L2 passing through R and S(1 , −1).

26. S(−5 , b) lies on the straight line L1: x − 4y + 9 = 0.

(a) Find the value of b.

(b) Find the equation of the straight line L2 passing through S and with slope 4.

O x

y

(4 , 1)

(1 , –4)

L

37

27. In the figure, the coordinates of the point A are (−3 , 2). A is translated

upward by 3 units to the point B. Then B is translated to the left by

4 units to the point C. The straight line L passes through A and C.

(a) Write down the coordinates of B and C.

(b) Find the equation of L.

28. A(5 , 7) is reflected in the x-axis to the point B. C(−3 , 1) is reflected in the y-axis to the point D.

The straight line L passes through B and D.

(a) Write down the coordinates of B and D.

(b) Find the equation of L.

29. A(−5 , −4) is reflected in the y-axis to the point B. Find the equation of the straight line L with slope

5

3− and passing through B.

30. In the figure, the coordinates of the point P are (−5 , −6). P is rotated

anticlockwise about the origin O through 90° to the point Q. Find the

equation of the straight line L with slope

6

1 and passing through Q.

31. A(4 , −5) is rotated anticlockwise about the origin O through 180° to the point B. The straight line L

with slope −5 passes through B.


(b) If L cuts the x-axis at (p , 0), find the value of p.

32. A(k , −6) lies on the straight line L1: x – 2y – 14 = 0.

(a) Find the value of k.

(b) A is rotated anticlockwise about the origin O through 270° to the point B. Find the equation of the

straight line L passing through B and C(6 , −8).

33. In the figure, the straight line L1 cuts the x-axis and the y-axis at (−4 ,

0) and (0 , −5) respectively. The straight line L2 passes through (1 ,

−2).

(a) Find the slope of L1.

(b) If L2 // L1, find the equation of L2.

x

y

O

A(–3 , 2)

x

y

P(–5 , –6)

O

x O

L1

L2

(1 , –2)

y

(–4 , 0)

(0 , –5)

38

34. In the figure, the straight line L1 passes through (−4 , 2) and cuts the

y-axis at (0 , 8). The straight line L2 cuts the x-axis at (4 , 0).

(a) Find the slope of L1.

(b) Find the equation of L1.

(c) If L2 ⊥ L1, find the equation of L2.

35. The straight line L1 with slope m passes through A(3 , −2m) and B(1 , −6).

(a) Find the value of m.

(b) L2 is a straight line passing through C(−2 , 7). If L2 // L1, find the equation of L2.

36. The straight line L1 with slope –1 passes through A(−7 , −1) and B(k , k).


(b) If the straight line L2 passes through B and L2 ⊥ L1, find the equation of L2.

37. In the figure, M is the mid-point of the line segment joining A(5 , 6)

and B(−7 , 0).

(a) Find the coordinates of M.

(b) Find the slope of AB.

(c) The straight line L is the perpendicular bisector of AB. Find the

equation of L.

38. In the figure, A(3 , 5), B(−8 , 4) and C(5 , −3) are the vertices of △ABC. BM is a median of △ABC.

(a) Find the coordinates of M.

(b) Find the equation of BM.

� (c) Find the coordinates of the centroid G of △ABC.

[Hint: If BM is a median of △ABC, then BG : GM = 2 : 1.]

� 39. A(5 , 4), B(–9 , 2) and C(3 , –7) are the vertices of △ABC. D is a point on BC such that BD : DC = 2 : 1.

(a) Find the coordinates of D.

(b) Is AD an altitude of △ABC? Explain your answer.

(c) Find the equation of AD.

� 40. In the figure, A(10 , 2) and B(–6 , 4) are two vertices of △ABC. The

coordinates of the orthocentre H of △ABC are (–5 , 2).

(a) Find the equation of AC.

(b) Find the coordinates of C.

(c) Find the area of △ABC.

M

x

y

O B(–7 , 0)

A(5 , 6)

A(3 , 5)

x O

B(–8 , 4)

C(5 , –3)

y

M

y

x O

B(–6 , 4)

C

H A(10 , 2)

x

y

O

(–4 , 2)

(0 , 8)

(4 , 0)

L1

L2

39

Answers


1. (a) y = 3x − 3 (b) y = −2x + 5

(c) y =2

1− x – 1

2. (a) 2x − 3y − 6 = 0 (b) x + 4y − 12 = 0

3. (a) 10x − 4y + 3 = 0 (b) 4x + 6y + 3 = 0

4. (a) 5x + 3y − 15 = 0 (b) 9x − 6y − 2 = 0

5. (a) 7x + 8y + 2 = 0 (b) 9x − 2y + 14 = 0

6. (a) 143

=+yx

(b) 122

=−

+yx

7. (a) 5x − 3y + 15 = 0 (b) x + 2y + 8 = 0

8. (a) 2x + 3y − 1 = 0 (b) 9x − 10y − 6 = 0

9. (a) x + 2 = 0 (b) y – 6 = 0

(c) 9x + 5y = 0

10. slope:4

3− , x-intercept:

3

2− , y-intercept:

2

1−

11. slope:3

5, x-intercept:

5

1− , y-intercept:

3

1

12. slope: −3, x-intercept:6

7, y-intercept:

2

7

13. slope:5

4, x-intercept:

2

1− , y-intercept:

5

2

14. slope:5

8− , x-intercept: 5, y-intercept: 8

15. slope:4

7, x-intercept: 4, y-intercept: −7

16. slope:4

1− , x-intercept: 2, y-intercept:

2

1

17. slope: 2, x-intercept:2

1− , y-intercept: 1

18. (a) slope of L1: 4, slope of L2: 4 (b) yes

19. no

20. (a) −4 (b) 4x + y – 8 = 0

21. (a) 3 (b) x + 3y + 9 = 0

22. (a) 2

3 (b) 7x + 2y – 3 = 0

23. 7x – 15y – 21 = 0 24. 3x – y + 2 = 0

25. (a) x – 3y – 9 = 0 (b) 9x + 7y – 21 = 0

26. (a) 3x – y + 12 = 0 (b) x + 3y + 1 = 0

27. (a) x – 2y + 2 = 0 (b) –2

28. (a) 2x + 3y + 9 = 0 (b) 1

29. (a) 4

3− (b) 3x + 4y + 12 = 0

30. (a) 3

4 (b) 3x + 4y – 12 = 0

31. (a) 12

(b) x-intercept:4

9, y-intercept:

4

3

32. (a) 10 (b) slope: 2, y-intercept: 10

33. (a) a = 6, b = 5 (b) 5

6−

34. (a) p = –4, q = 40 (b) 10

35. –3 36. 2

37. (a) 3 (b) 7x + 3y + 11 = 0

38. (a) slope:3

4, y-intercept: –4

(b) 3x + 4y + 16 = 0

39. (a) 2 (b) 5x – 2y + 4 = 0

40. (a) 8x – 5y – 40 = 0 (b) 5x + 8y + 64 = 0

41. (a) 2x + y + 1 = 0 (b) yes

42. (a) 5x + 3y – 21 = 0 (b) no

43. (a) y + 8 = 0 (b) −2

44. (a) 2x + 3y + 12 = 0 (b) 3x − 2y − 17 = 0

45. (a) (0 , 7), (0 , −7)

(b) x + y − 7 = 0, x + y + 7 = 0

46. (a) P

2

3,0 , Q

0,

4

9

(b) 12x − 14y + 21 = 0

47. (a) A(−8 , 0), B(0 , 15) (b) 40

48. (a) x + 2y = 0 (b) (8 , −4)

(c) 2x − y − 20 = 0

49. (a) 3x + y − 3 = 0 (b)

2

3,

2

1

50. (a) 5x – y + 1 = 0 (b) (1 , 6)

(c) yes

51. (a) –1 (b) 3x + y – 9 = 0

(c) (4 , –3)

52. (a) k = −4, a = −13 (b) 2

39

53. (a) a = 6, b = −6 (b) 3x + 2y + 18 = 0, –6

(c) 1 : 1

54. (a) 3x + 4y + 20 = 0 (b) (i) 15 (ii) no

40

F4A: Chapter 2B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)




___________ ( )




___________

41

○ Skipped ( )



___________ ( )



_________

42

4A Lesson Worksheet 2.1D (Refer to Book 4A P.2.17)

Objective: To find the equation of a straight line from a given slope and the y-intercept of the straight line,

and to find the slope, the x-intercept and the y-intercept of a straight line.

Review: y-intercept

In each of the following, find the y-intercept of the straight line L. [Nos. 1–2]

1. 2.

Slope-intercept Form

The equation of the straight line with slope m and y-intercept b

is given by:

y = mx + b

In each of the following, write down the equation of the straight line L in the slope-intercept form. [Nos. 3–4]

3. 4. �Ex 2B: 1


Find the equation of the straight line with slope

3 and y-intercept 5.


y = 3x + 5

3x – y + 5 = 0 � Express the answer in the general

form.

Find the equation of the straight line with

slope –4 and y-intercept 8.


y = ( )x + ( )

( ) = ( )

In each of the following, find the equation of the straight line according to the given information. [Nos. 5–8]

5. Slope = 6, y-intercept = 4 6. Slope = 2, y-intercept = –3 �Ex 2B: 2


y = ( )x + ( )

( ) = ( )

7

x

y

L

0

slope =4

1−

6

x

y L

0

slope = 1

2

4

x

y L

O 3

4 x

y

L 0 Be careful of the +/– sign.

y-intercept = y-intercept =

y = x + ( ) y = ( ) + ( )

43

7. Slope = –5, y-intercept =5

1 8. Slope =

3

4− , y-intercept = –1


y = ( ) + ( )

( )y = ( ) + ( )

( ) = ( )

Finding the Slope, the x-intercept and the y-intercept of a Straight Line

For a straight line Ax + By + C = 0, slope =B

A− , x-intercept =

A

C− and y-intercept =

B

C− .


Find the slope, the x-intercept and the y-

intercept of the straight line 3x + 2y + 1 = 0.

Slope =2

3−

x-intercept =3

1−

y-intercept =2

1−

Find the slope, the x-intercept and the y-

intercept of the straight line x – 2y + 4 = 0.

Slope =) (

1− =

x-intercept =) (

) (− =

y-intercept =) (

) (− =

Find the slope, the x-intercept and the y-intercept of each of the following straight lines. [Nos. 9–10]

9. x – 5y = 0 10. 2x + y – 6 = 0 �Ex 2B: 7, 8

Slope =) (

) (− =

x-intercept =) (

) (− =

y-intercept =) (

) (− =


11. (a) Find the equation of the straight line L with slope –2 and y-intercept 1.

(b) Find the x-intercept of the straight line L.

A =

B =

C =

A =

B =

C =

A = 3

B = 2 C = 1

� Multiply both sides

by 5.

A =

B =

C =

44

4A Lesson Worksheet 2.1E (Refer to Book 4A P.2.22)

Objective: To find the equation of a straight line from the given intercepts.

Review: The x-intercept and the y-intercept of a Straight Line

Find the x-intercept and the y-intercept of each of the following straight lines. [Nos. 1–2]

1. 3x + y + 6 = 0 2. 2x + 5y – 10 = 0

x-intercept = –) (

) (= x-intercept = –

) (

) (=

y-intercept = –) (

) (= y-intercept = –

) (

) (=

Intercept Form

The equation of the straight line L with x-intercept a and y-intercept b

is given by:

a

x+

b

y= 1 (where a ≠ 0 and b ≠ 0)

In each of the following, write down the x-intercept and the y-intercept of the straight line L. [Nos. 3–5]

3. L:3

x+

4

y= 1 4. L:

5

x–

2

y= 1 5. L: –x –

6

y= 1

x-intercept of L = x-intercept of L = x-intercept of L =

y-intercept of L = y-intercept of L = y-intercept of L =

In each of the following, write down the equation of the straight line L in the intercept form. [Nos. 6–7]

6. 7. �Ex 2B: 4

) (

x+

) (

y= 1

) (

x+

) (

y= 1


Find the equation of the straight line with x-

intercept 2 and y-intercept 3.


2

x+

3

y = 1

2

6x+

3

6y= 6

3x + 2y = 6

3x + 2y – 6 = 0

Find the equation of the straight line with

x-intercept 4 and y-intercept –7.


) (

x+

) (

y= 1

) (

) ( x+

) (

) ( y= ( )

( )x – ( )y = ( )

= � Express the answer in the general form.

3

x

y

0 5

L

2

x

y

0 –4

L

� Multiply both sides by 6.

Ax + By + C = 0

x-intercept = –A

C

y-intercept = –B

C

45

In each of the following, find the equation of the straight line according to the given information.

[Nos. 8–11]

8. x-intercept = –6, y-intercept = 5 9. x-intercept = 3, y-intercept = –12 �Ex 2B: 5, 6

The required equation is The required equation is

) (

x+

) (

y= 1

) (

x+

) (

y= 1

) (

) ( x+

) (

) ( y= ( )

) (

) ( x+

) (

) ( y= ( )

( )x + ( )y = ( ) =

=

10. Passing through A(6 , 0) and B(0 , –2) 11. Passing through A(–3 , 0) and B(0 , –9)

x-intercept = x-intercept =

y-intercept = y-intercept =

12. (a) Find the x-intercept and the y-intercept of the straight line L:4

x–

3

y= 1.

(b) Find the area of the shaded region in the figure.


13. The x-intercept and the y-intercept of a straight line L are 6 and –8 respectively.


(b) Does the point (2 , –5) lie on L? Explain your answer.

x O

y

L:4

x–

3

y= 1

46

4A Lesson Worksheet 2.1F (Refer to Book 4A P.2.24)

Objective: To find the equation of straight lines parallel to the y-axis or the x-axis, or passing through the

origin.

Review: Coordinates of Points on a Horizontal or Vertical Line

1. In the figure, L1 and L3 are horizontal lines, and L2 is a vertical line.

Write down the values of a, b, c and d.

Straight Lines Parallel to the y-axis (Vertical Lines)

The equation of a vertical line is given by:

x = h

In each of the following, find the equation of the vertical line L. [Nos. 2–3]

2. The equation of L is 3. The equation of L is

x = ( )

x – ( ) = 0

Straight Lines Parallel to the x-axis (Horizontal Lines)

The equation of a horizontal line is given by:

y = k

In each of the following, find the equation of the horizontal line L. [Nos. 4–5]

4. The equation of L is 5. The equation of L is

y = ( )

y – ( ) = 0

Straight Lines Passing through the Origin

The equation of the straight line passing through the origin and the point (a , b)

is given by:

y =a

bx (where a ≠ 0)

x h

y

O

x = h

x

(a , b)

y

O

y = xa

b

x

k

y

O

y = k

–2 x

0

y

S(1 , d)

3L1

L3

L2

P(5 , a)

R(c , –1)

Q(b , 2)

x

y L

0 1

P(3 , 2)

x

y L

O

P(5 , –1)

x

y

O

L x

y

L

0

2

47

In each of the following, find the equation of the straight line passing through the origin and the point P.

[Nos. 6–7]

6. P(1 , 2) 7. P(–4 , 3)


y =) (

) (x

y = ( )x

=

In each of the following, find the equation of the straight line L. [Nos. 8–10] �Ex 2B: 11

8. 9. 10.

The equation of L is

11. Find the equation of the straight lines L1, L2 and L3 in the figure. �Ex 2B: 12

(Express the answers in terms of a if necessary.)


12. (a) Find the equation of the straight line L passing through the origin and the point (1 , 3).

(b) If L passes through the point (–3 , 3a), find the value of a.

x

y

O

L x

y

(2 , –3)

O

L

x

y

O

L

(–5 , 1)

(–5 , 6)

(–2 , 4) (3 , 4)

x

y

O

(–2a , –a)

L1

L3

L2

x

y

P(1 , 2)

O

48




Level 1

1. In each of the following, find the equation of the straight line ℓ in the slope-intercept form.

(a) (b) (c)


2. (a) Slope =3

2, y-intercept = –2 (b) Slope =

4

1− , y-intercept = 3

3. (a) Slope =2

5, y-intercept =

4

3 (b) Slope =

3

2− , y-intercept =

2

1−

4. (a) Passing through (0 , 5) with slope

3

5− (b) Passing through

−

3

1,0 with slope

2

3

5. (a) Passing through

−

4

1,0 with slope

8

7− (b) Passing through (0 , 7) with slope

2

9

6. In each of the following, find the equation of the straight line ℓ in the intercept form.

(a) (b)


7. (a) x-intercept = –3, y-intercept = 5 (b) x-intercept = –8, y-intercept = –4

8. (a) Passing through

0,

2

1 and

3

1,0 (b) Passing through

0,

3

2 and

−

5

3,0

x

y

0

–1

ℓ

m =2

1−

y

x 0

−2

2

ℓ

y

x 0

4

3 ℓ

y

x 0

m = –2

5

ℓ

y

x 0

m = 3

−3

ℓ

49

9. In each of the following, find the equation of the straight line ℓ .

(a) (b) (c)

Find the slope, the x-intercept and the y-intercept of each of the following straight lines. [Nos. 10–17]

10. 3x + 4y + 2 = 0 11. 5x – 3y + 1 = 0

12. 6x + 2y – 7 = 0 13. –4x + 5y – 2 = 0

14. 8x + 5y – 40 = 0 15. 7x – 4y – 28 = 0

16. x + 4y – 2 = 0 17. 2x – y + 1 = 0

18. The straight lines L1: 8x – 2y – 3 = 0 and L2: 4x – y + 7 = 0 are given.

(a) Find the slopes of L1 and L2.

(b) Is L1 parallel to L2? Explain your answer.

19. Are the straight lines L1: y =2

9− x + 6 and L2: 2x + 9y – 1 = 0 perpendicular to each other? Explain

your answer.

20. In the figure, the equation of the straight line L is 4x + y + 5 = 0.

The y-intercept of the straight line L1 is 8.

(a) Find the slope of L.

(b) If L1 // L, find the equation of L1.

21. The equation of the straight line L is 6x – 2y + 1 = 0.

(a) Find the slope of L.

(b) The y-intercept of the straight line L1 is –3. If L1 ⊥ L, find the equation of L1.

22. The equation of the straight line L is 8x + 6y – 9 = 0.

(a) Find the y-intercept of L.

(b) The slope of the straight line L1 is

2

7− . If the y-intercepts of L1 and L are the same, find the

equation of L1.

y

x O

(5 , –9) ℓ

y

x O

(3 , 6) (–6 , 6) ℓ

y

x O

(–2 , –4)

(–2 , 5)

ℓ

8

x

y

0 L1

L

50

23. The equation of the straight line L is 8x – 5y – 7 = 0. The x-intercept of the straight line L1 is 3. If the

y-intercepts of L1 and L are the same, find the equation of L1.

24. The equation of the straight line L is 9x – 4y + 6 = 0. The y-intercept of the straight line L1 is 2. If the

x-intercepts of L1 and L are the same, find the equation of L1.

25. The equation of the straight line L is x + 3y – 9 = 0.

(a) L1 is a straight line passing through (–6 , –5). If the x-intercepts of L1 and L are the same, find the

equation of L1.

(b) L2 is a straight line passing through (7 , –6). If the y-intercepts of L2 and L are the same, find the

equation of L2.

26. The equation of the straight line L is 3x – y + 4 = 0.

(a) Find the equation of the straight line L1 passing through (–5 , –3) and parallel to L.

(b) Find the equation of the straight line L2 passing through (5 , –2) and perpendicular to L.

Level 2

27. The slope and the y-intercept of the straight line L are

2

1 and 1 respectively.


(b) Find the x-intercept of L.

28. The slope and the y-intercept of the straight line L are

3

2− and –3 respectively.


(b) If L passes through (–6 , k), find the value of k.

29. The slope of the straight line L1 is m. L1 cuts the x-axis at (8 , 0) and its y-intercept is 6.


(b) If the straight line L2 is parallel to L1 and passes through (0 , –3), find the equation of L2.

30. The slope of the straight line L1 is m. L1 cuts the y-axis at (0 , –8) and its x-intercept is 6.


(b) If the straight line L2 is perpendicular to L1 and passes through (0 , 3), find the equation of L2.

31. The slope of the straight line L: 4x + ky – 9 = 0 is

3

1− , where k is a constant.


(b) Find the x-intercept and the y-intercept of L.

51

32. The x-intercept of the straight line L: 2x – y + k = 0 is –5, where k is a constant.


(b) Find the slope and the y-intercept of L.

33. The x-intercept and the y-intercept of the straight line L: ax + by + 30 = 0 are –5 and –6 respectively,

where a and b are constants.

(a) Find the values of a and b.

(b) Find the slope of L.

34. The slope and the x-intercept of the straight line L: 5x + py + q = 0 are

4

5 and –8 respectively, where p

and q are constants.

(a) Find the values of p and q.

(b) Find the y-intercept of L.

35. Consider the straight lines L1: 2x + ky – 17 = 0 and L2: 4x – 6y + 9 = 0, where k is a constant. If L1 // L2,

find the value of k.

36. Consider the straight lines L1: x + 2y – 11 = 0 and L2: kx – y – 10 = 0, where k is a constant. If L1 ⊥ L2,

find the value of k.

37. The y-intercept of the straight line L1: 7x + by – 21 = 0 is 7, where b is a constant.

(a) Find the value of b.

(b) If the straight line L2 is parallel to L1 and passes through (–5 , 8), find the equation of L2.

38. In the figure, the y-intercepts of the straight lines L1: 4x – 3y – 12 = 0

and L2 are the same. L2 is perpendicular to L1.

(a) Find the slope and the y-intercept of L1.


39. The x-intercept of the straight line L1: ax + 5y – 10 = 0 is 5, where a is a constant.

(a) Find the value of a.

(b) If the straight line L2 is perpendicular to L1 and the y-intercepts of the two lines are the same, find

the equation of L2.

40. The slope and the x-intercept of the straight line L1 are

5

8 and 5 respectively.

(a) Find the equation of L1.

(b) The straight line L2 is perpendicular to L1. If they intersect at a point on the y-axis, find the

equation of L2.

x

y

L1: 4x – 3y – 12 = 0 L2

O

52

41. The y-intercept of the straight line L is –1. L is parallel to the straight line L1 which passes through

(8 , –5) and (9 , –7).


(b) Does (2 , –5) lie on L? Explain your answer.

42. The y-intercept of the straight line L is 7. L is perpendicular to the straight line L1 which passes

through (7 , 2) and (–8 , –7).


(b) Does L pass through (2 , 4)? Explain your answer.

43. (a) Find the equation of the straight line which passes through (–6 , –8) and is parallel to the x-axis.

(b) If (1 , 4b) lies on the straight line obtained in (a), find the value of b.

44. The x-intercept and the y-intercept of the straight line L1 are –6 and –4 respectively. The straight line

L2 passes through (9 , 5) and is perpendicular to L1.



45. The slope of the straight line L is –1. L cuts the y-axis at the point P. It is given that OP = 7, where O

is the origin.

(a) Write down the two possible coordinates of P.

(b) Hence, find the possible equations of L.

46. In the figure, two straight lines L1: 4x + 6y – 9 = 0 and L2 intersect at a

point P on the y-axis. L1 and L2 cut the x-axis at the points Q and R

respectively.

(a) Find the coordinates of P and Q.

(b) If QR = 4, find the equation of L2.

47. In the figure, the straight line L: 15x – 8y + 120 = 0 cuts the x-axis and

the y-axis at the points A and B respectively.

(a) Find the coordinates of A and B.

(b) Find the perimeter of △OAB.

48. The straight line L1 with slope

2

1− passes through the origin. L1 intersects the straight line L2: x = 8 at

the point P.


(b) Find the coordinates of P.

(c) Find the equation of the straight line which passes through P and is perpendicular to L1.

y

x O

Q

P

R

L2

L1: 4x + 6y – 9 = 0

x O

A

B L: 15x – 8y + 120 = 0

y

53

49. In the figure, the straight lines L1: x – 2y – 1 = 0 and L2 intersect at a

point on the x-axis. L2 passes through Q(3 , –6). P is a point lying on

L2 such that PQ = PR, where the coordinates of R are (–7 , –1).



50. The equation of the straight line L1 is 5x – y – 2 = 0. The straight line L2 cuts the y-axis at M(0 , 1) and

L2 // L1. P is a point lying on L2 such that MP = NP, where the coordinates of N are (6 , 5).



(c) Is △MPN a right-angled triangle with ∠MPN = 90°? Explain your answer.

51. The slope of the straight line L1: 4x + ky + 9 = 0 is 4, where k is a constant. The straight line L2

intersects L1 at a point on the y-axis and the x-intercept of L2 is 3. Two points P(4 , 7) and Q(–4 , 3) are

given.



(c) If R is a point lying on L2 such that PR = QR, find the coordinates of R.

52. The straight lines L1: 3x – 2y = 0 and L2 intersect at P(k , –6). L2 is perpendicular to L1 and cuts the x-

axis at Q(a , 0).

(a) Find the values of k and a.

(b) Let O be the origin. If M is the mid-point of PQ, find the area of △OPM.

53. Let a be a constant. The straight line L1: 6x – y + a = 0 cuts the y-axis at A(0 , 6) and intersects the

straight line L2 at B(–2 , b). L2 cuts the y-axis at C(0 , –9).


(b) Find the equation of L2. Hence, find the x-intercept of L2.

(c) If CM is a median of △ABC, find the ratio of the area of △BCM to the area of △ACM.

54. A(8 , 1) is reflected in the y-axis to the point B. The straight line L passes through B and C(4 , –8).


(b) It is known that L cuts the y-axis at a point P.

(i) Find the length of AP + CP.

� (ii) Is it possible to find a point Q on the y-axis such that AQ + CQ < AP + CP? Explain your

answer.

x

y

O

R(–7 , –1)

L1

L2

Q(3 , –6)

54

Answers


1. (a) y = 3x − 3 (b) y = −2x + 5

(c) y =2

1− x – 1

2. (a) 2x − 3y − 6 = 0 (b) x + 4y − 12 = 0

3. (a) 10x − 4y + 3 = 0 (b) 4x + 6y + 3 = 0

4. (a) 5x + 3y − 15 = 0 (b) 9x − 6y − 2 = 0

5. (a) 7x + 8y + 2 = 0 (b) 9x − 2y + 14 = 0

6. (a) 143

=+yx

(b) 122

=−

+yx

7. (a) 5x − 3y + 15 = 0 (b) x + 2y + 8 = 0

8. (a) 2x + 3y − 1 = 0 (b) 9x − 10y − 6 = 0

9. (a) x + 2 = 0 (b) y – 6 = 0

(c) 9x + 5y = 0

10. slope:4

3− , x-intercept:

3

2− , y-intercept:

2

1−

11. slope:3

5, x-intercept:

5

1− , y-intercept:

3

1

12. slope: −3, x-intercept:6

7, y-intercept:

2

7

13. slope:5

4, x-intercept:

2

1− , y-intercept:

5

2

14. slope:5

8− , x-intercept: 5, y-intercept: 8

15. slope:4

7, x-intercept: 4, y-intercept: −7

16. slope:4

1− , x-intercept: 2, y-intercept:

2

1

17. slope: 2, x-intercept:2

1− , y-intercept: 1

18. (a) slope of L1: 4, slope of L2: 4

(b) yes

19. no

20. (a) −4 (b) 4x + y – 8 = 0

21. (a) 3 (b) x + 3y + 9 = 0

22. (a) 2

3 (b) 7x + 2y – 3 = 0

23. 7x – 15y – 21 = 0

24. 3x – y + 2 = 0

25. (a) x – 3y – 9 = 0 (b) 9x + 7y – 21 = 0

26. (a) 3x – y + 12 = 0 (b) x + 3y + 1 = 0

27. (a) x – 2y + 2 = 0 (b) –2

28. (a) 2x + 3y + 9 = 0 (b) 1

29. (a) 4

3− (b) 3x + 4y + 12 = 0

30. (a) 3

4 (b) 3x + 4y – 12 = 0

31. (a) 12

(b) x-intercept:4

9, y-intercept:

4

3

32. (a) 10

(b) slope: 2, y-intercept: 10

33. (a) a = 6, b = 5 (b) 5

6−

34. (a) p = –4, q = 40 (b) 10

35. –3

36. 2

37. (a) 3 (b) 7x + 3y + 11 = 0

38. (a) slope:3

4, y-intercept: –4

(b) 3x + 4y + 16 = 0

39. (a) 2 (b) 5x – 2y + 4 = 0

40. (a) 8x – 5y – 40 = 0 (b) 5x + 8y + 64 = 0

41. (a) 2x + y + 1 = 0 (b) yes

42. (a) 5x + 3y – 21 = 0 (b) no

43. (a) y + 8 = 0 (b) −2

44. (a) 2x + 3y + 12 = 0 (b) 3x − 2y − 17 = 0

45. (a) (0 , 7), (0 , −7)

(b) x + y − 7 = 0, x + y + 7 = 0

46. (a) P

2

3,0 , Q

0,

4

9

(b) 12x − 14y + 21 = 0

47. (a) A(−8 , 0), B(0 , 15)

(b) 40

48. (a) x + 2y = 0 (b) (8 , −4)

(c) 2x − y − 20 = 0

49. (a) 3x + y − 3 = 0 (b)

2

3,

2

1

50. (a) 5x – y + 1 = 0 (b) (1 , 6)

(c) yes

51. (a) –1 (b) 3x + y – 9 = 0

(c) (4 , –3)

52. (a) k = −4, a = −13 (b) 2

39

55

53. (a) a = 6, b = −6

(b) 3x + 2y + 18 = 0, –6

(c) 1 : 1

54. (a) 3x + 4y + 20 = 0

(b) (i) 15 (ii) no

56

F4A: Chapter 2C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)



(Full Solution)

Maths Corner Exercise 2C Level 1



___________ ( )

Maths Corner Exercise 2C Level 2



___________ ( )

Maths Corner Exercise 2C Multiple Choice


___________ ( )



_________

57


Objective: To find the number of points of intersection of two straight lines.

Review: The Slope and the y-intercept of a Straight Line

In each of the following, find the slope and the y-intercept of the straight line. [Nos. 1–2]

1. y = 3x + 1 2. 2x + y – 7 = 0

Slope =

Slope = –) (

) (=

y-intercept =

y-intercept = –) (

) (=

Number of Points of Intersection of Two Straight Lines

Let m1 and b1 be the slope and the y-intercept of 1ℓ respectively.

Let m2 and b2 be the slope and the y-intercept of 2ℓ respectively.

(a)

(b) (c)

Relationship m1 ≠ m2 m1 = m2, b1 ≠ b2 m1 = m2, b1 = b2

Number of points of intersection

1 0 infinitely many

In each of the following, find the number of points of intersection of the straight lines L1: y = m1x + b1 and

L2: y = m2x + b2. [Nos. 3–4]

3. m1 = 1, b1 = 2; m2 = 2, b2 = 4 4. m1 = –3, b1 = 7; m2 = –3, b2 = 5

)many infinitely / one / no ( have and 21 LL )many infinitely / one / no ( have and 21 LL

on.intersecti of point(s) on.intersecti of point(s)


Find the number of points of intersection of the

straight lines L1: y = 2x and L2: 4x + y + 1= 0.

Slope of L1 = 2

Slope of L2 = –1

4= –4

Slope of L1 ≠ slope of L2 ∴ on.intersecti ofpoint one have and 21 LL

Find the number of points of intersection of the

straight lines L1: y = x + 9 and L2: 5x + y = 0.

Slope of L1 = ( )

Slope of L2 = –) (

) (= ( )

Slope of L1 ( = / ≠ ) slope of L2 ∴ on.intersecti ofpoint ) ( have and 21 LL

Ax + By + C = 0

slope = –B

A

y-intercept = –B

C

Steps Check if (i) m1 = m2,

(ii) b1 = b2.

y = mx + b

slope y-intercept

58

In each of the following, find the number of points of intersection of the straight lines L1 and L2. [Nos. 5–8]

5. L1: 5x – y – 9 = 0, L2: x + 5y + 3 = 0 6. L1: 2x + 8y – 1 = 0, L2: –x – 4y + 6 = 0


) (= ( ) Slope of L1 =


) (= ( ) Slope of L2 =

Slope of L1 ( = / ≠ ) slope of L2

y-intercept of L1 =

y-intercept of L2 =

7. L1: 3x – 7y – 7 = 0, L2: 6x – 14y + 1 = 0 8. L1: 5x + y – 1 = 0, L2: 10x + 2y – 2 = 0

In each of the following, if two straight lines L1 and L2 have no points of intersection, find the value of a.

[Nos. 9–10]

9. L1: y = ax + 16, L2: 5x – y + 11 = 0 10. L1: 2ax – y + 9 = 0, L2: 18x + 3y + 7 = 0

a = –) (

) (

=


11. The straight lines L1: y = 4x + m and L2: nx + 2y – 6 = 0 have infinitely many points of intersection.

Find the values of m and n.

�Ex 2C: 1–8

� Slope of L1 = slope of L2

59



Consolidation Exercise 2C

Level 1

In each of the following, find the number of points of intersection of the straight lines L1 and L2.

[Nos. 1–12]

1. L1: x + 3y – 5 = 0, L2: 2x + 5y + 1 = 0 2. L1: 2x + 2y + 7 = 0, L2: 3x + 3y – 8 = 0

3. L1: 3x + 2y + 2 = 0, L2: 6x + 4y + 4 = 0 4. L1: 5x – 7y + 5 = 0, L2: –3x + 7y – 3 = 0

5. L1: 4x – 2y – 3 = 0, L2: –12x + 6y + 9 = 0 6. L1: 6x + 9y – 2 = 0, L2: 12x + 18y + 5 = 0

7. L1: 3x – 2y + 3 = 0, L2: 9x – 6y + 4 = 0 8. L1: 5x – y + 3 = 0, L2: 20x – 4y + 12 = 0

9. L1: 2x + 5y – 1 = 0, L2: 5x – 3y + 8 = 0 10. L1: –4x + 3y + 4 = 0, L2: 8x – 6y – 8 = 0

11. L1: 8x – 8y + 9 = 0, L2: 6x – 6y + 7 = 0 12. L1: 3x – 9y – 1 = 0, L2: 2x + 4y – 7 = 0

In each of the following, find the coordinates of the point of intersection of the straight lines L1 and L2.

[Nos. 13–18]

13. L1: 5x – 2y = 3, L2: x + 3y = 4 14. L1: y = 3x + 2, L2: 2x + 5y + 7 = 0

15. L1: 3x + y = 10, L2: 2x + 7y + 6 = 0 16. L1: 4x + 3y – 6 = 0, L2: 9x + 5y – 3 = 0

17. L1: 3x – y + 16 = 0, L2: 2x – 7y – 21 = 0 18. L1: x – y – 5 = 0, L2: 9x – y + 27 = 0

In each of the following, find the number of points of intersection of the straight lines L1 and L2

(where k is a non-zero constant) and describe the case of intersection. [Nos. 19–22]

19. L1: 5kx – ky + 4 = 0, L2: y = 5x – 3k 20. L1: 2kx + ky + 9 = 0, L2: y = –2x + 8k

21. L1: 3kx + 2y – k = 0, L2: yk

2= –3x + 1 22. L1: x

k

3– 7y + 7 = 0, L2: 3x = 7ky – 7k

23. Two straight lines L1: 7x + 6y + 3 = 0 and L2: 4x – 5y – 32 = 0 intersect at the point A.

(a) Find the coordinates of A.

(b) Find the equation of the straight line passing through A and with y-intercept –6.

60

24. Two straight lines L1: 5x – 3y – 1 = 0 and L2: 5x – 7y – 9 = 0 intersect at the point P.

(a) Find the coordinates of P.

(b) Find the equation of the straight line passing through P and Q(−3 , 3).

25. Two straight lines L1: 3x – y + 7 = 0 and L2: 3x + 2y – 5 = 0 intersect at the point B.

(a) Find the coordinates of B.

(b) Find the equation of the straight line which passes through B and is parallel to the straight line

x + 2y – 1 = 0.

(c) Find the equation of the straight line which passes through B and is perpendicular to the straight

line x – 3y – 6 = 0.

26. The equation of the straight line L1 is x + 2y + 10 = 0. The straight line L2 passes through (5 , 5) and (–

7 , –4).


(b) Find the coordinates of the point of intersection of L1 and L2.

27. The equation of the straight line L1 is x + y – 6 = 0. The straight line L2 passes through (–1 , –5) and its

slope is 1.



28. The equation of the straight line L1 is 2x + 3y + 19 = 0. The straight line L2 with y-intercept –2 is

perpendicular to L1.



Level 2

29. The straight lines L1: ax – by + 8 = 0 and L2: 5y = –9x are given, where a and b are non-zero constants.

If L1 and L2 have no points of intersection, find the value of

a

b.

30. The straight lines L1: 4x + (k – 1)y + 7 = 0 and L2: 3x + 2ky + 6 = 0 are given, where k is a constant.

(a) Is it possible that L1 and L2 have infinitely many points of intersection? Explain your answer.

(b) If L1 and L2 have no points of intersection, find the value of k.

31. Two straight lines ax + 2y – 6 = 0 and 3x – by – 9 = 0 have infinitely many points of intersection.

Find the values of the constants a and b.

32. Two straight lines –2x + my + 1 = 0 and 4x – 3y + n = 0 have infinitely many points of intersection.

Find the values of the constants m and n.

61

33. The equations of four straight lines are given below:

L1: 125

=+yx

L2: Ax + y + C = 0

L3: y = –mx L4: y = px + q

(a) If L1 and L2 have infinitely many points of intersection, find the slope of L2 and the value of C.

(b) If L1 and L3 are perpendicular to each other, find the value of m.

(c) If L1 and L4 are parallel to each other, and the x-intercept of L4 is three times that of L1, find the

values of p and q.

34. Two straight lines L1: x + ky – 5 = 0 and L2: –2x + 8y + 3 = 0 have no points of intersection, where k is

a constant.


(b) If the equation of the straight line L3 is x + 5y – 14 = 0, find the coordinates of the point of

intersection of L1 and L3.

35. Two straight lines L1: 3x – 4y + k = 0 and L2: 2x + y + 6 = 0 intersect at the point P, where k is a

constant. The y-intercept of L1 is 5.



(c) Find the equation of the straight line with slope 2 passing through P.

36. Two straight lines L1: x – 6y – 9 = 0 and L2: kx + 3y – 9 = 0 intersect at the point A, where k is a

constant. L2 is perpendicular to the straight line 3x + 2y – 5 = 0.


(b) Find the coordinates of A.

(c) A is translated horizontally to the point B such that the slope of OB is –3, where O is the origin.

Find the coordinates of B.

37. Two straight lines L1: x – 3y + 1 = 0 and L2: 3x + 2y + 14 = 0 intersect at the point P.

(a) Find the coordinates of P.

(b) P is reflected in the x-axis to the point R. If the coordinates of the point Q are (−2 , 1), is △PQR

an isosceles triangle? Explain your answer.

38. In the figure, the straight lines L1: x + my – 14 = 0 and L2 cut the y-

axis at the points A and B respectively, where m is a constant. L1

passes through C(2 , 6) and intersects L2 at the point D. The slope and

the

x-intercept of L2 are m and

2

3 respectively.



(c) Find the area of △ABD.

x 0

2

3

C(2 , 6)

D

A

B

L2

L1

y

62

39. The figure shows two given points P(–8 , 11) and Q(16 , 1). The

straight line L passes through R(–1 , –6). L is perpendicular to PQ and

intersects PQ at the point T.

(a) Find the coordinates of T.

(b) Is △PQR an isosceles triangle? Explain your answer.

(c) Find the area of △PQR.

40. In the figure, the straight lines L1 and L2 are perpendicular to each other,

and intersect at A. L1 cuts the y-axis and the x-axis at B(0 , 10) and

C(5 , 0) respectively. C is reflected in the y-axis to the point D. BD

produced meets L2 at the point E. It is given that D is the mid-point of

BE.

(a) Find the coordinates of D and E.


41. In the figure, OABC is a rhombus, where O is the origin and the

coordinates of B are (2 , –6). The diagonals OB and AC intersect at

the point D. The slope of OC is 1.



(c) Find the area of the rhombus OABC.

� 42. In the figure, A(2 , 5), B(6 , 2) and C(–4 , –3) are the vertices of △ABC. D is a point on BC such that AD is an altitude of △ABC.


(b) Find the area of △ABC.

(c) If E is a point on BC such that the area of △ABE is 5, find the

coordinates of E.

� 43. The straight lines L1: 3x – y – 10 = 0 and L2: x + y + 6 = 0 intersect at the point C. A(7 , 11) lies on L1

and B(−5 , −1) lies on L2. D is a point on AC such that AD : DC = AB : BC.

(a) Find the coordinates of C.

(b) Find the coordinates of D.

(c) Find the slopes of AB and BC.

(d) Is BD the angle bisector of ∠ABC? Explain your answer.

x

L

O

T

Q(16 , 1)

P(–8 , 11) y

R(–1 , –6)

x

y

O

C(–4 , –3)

A(2 , 5)

D

B(6 , 2)

y

x O

A

B(0 , 10)

D C(5 , 0)

L1

L2

E

x O

A

B(2 , –6)

C

D

y

63

Answers


1. 1 2. 0

3. infinitely many 4. 1

5. infinitely many 6. 0

7. 0 8. infinitely many

9. 1 10. infinitely many

11. 0 12. 1

13. (1 , 1) 14. (−1 , −1)

15. (4 , −2) 16. (−3 , 6)

17. (−7 , −5) 18. (−4 , −9)

19. 0, the two straight lines are parallel to each

other and have no points of intersection

20. 0, the two straight lines are parallel to each

other and have no points of intersection

21. infinitely many, the two straight lines

coincide with each other and intersect at

infinitely many points

22. infinitely many, the two straight lines

coincide with each other and intersect at

infinitely many points

23. (a) (3 , −4) (b) 2x − 3y − 18 =

0

24. (a) (−1 , −2) (b) 5x + 2y + 9 = 0

25. (a) (−1 , 4) (b) x + 2y − 7 = 0

(c) 3x + y – 1 = 0

26. (a) 3x – 4y + 5 = 0 (b)

−−

2

5,5

27. (a) x – y – 4 = 0 (b) (5 , 1)

28. (a) 3x – 2y – 4 = 0 (b) (−2 , −5)

29. 9

5−

30. (a) no (b) 5

3−

31. a = 2, b = –3

32. m =2

3, n = –2

33. (a) slope of L2:5

2− , C = –2

(b) 2

5− (c) p =

5

2− , q = 6

34. (a) –4 (b) (9 , 1)

35. (a) 20 (b) (–4 , 2)

(c) 2x – y + 10 = 0

36. (a) −2 (b) (−9 , −3)

(c) (1 , −3)

37. (a) (−4 , −1) (b) yes

38. (a) 2 (b) 2x − y − 3 = 0

(c) 20

39. (a) (4 , 6) (b) yes

(c) 169

40. (a) D(–5 , 0), E(–10 , –10)

(b) (6 , –2)

41. (a) (1 , –3) (b) (7 , –1)

(c) 40

42. (a) (4 , 1) (b) 25

(c) (4 , 1)

43. (a) (1 , –7) (b) (3 , −1)

(c) slope of AB: 1, slope of BC: −1

(d) yes

64

F4A: Chapter 3A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)



(Full Solution)




___________ ( )




___________

65

○ Skipped ( )



___________ ( )



_________

66


Objective: To review factorization of polynomials.

Taking out Common Factors

Factorize the following polynomials. [Nos. 1–4]

1. x2 + 2x 2. x2 – 5x

= x(x + ) = x( )

3. 3x2 – 9x 4. 4x2 + 6x

= =

Using the Difference of Two Squares Identity

x2 – y2 ≡ (x + y)(x – y)


5. x2 – 16 6. x2 – 49

= x2 – ( )2 = x2 – ( )2

= (x + )(x – ) = (x + )(x – )

7. 9x2 – 1 8. 81x2 – 4

= ( )2 – ( )2 =

= ( )( )

Using the Perfect Square Identities

(a) x2 + 2xy + y2 ≡ (x + y)2

(b) x2 – 2xy + y2 ≡ (x – y)2


9. x2 + 12x + 36 10. x2 – 16x + 64

= x2 + 2(x)( ) + ( )2 = x2 – 2(x)( ) + ( )2

= (x + )2 =

11. 25x2 – 10x + 1 12. 4x2 + 12x + 9

= =

� Take out the common factor x.

67

Cross-method


13. x2 + 8x + 7 14. 8x2 – 6x + 1

= (x + )(x + ) =

15. 3x2 + 14x – 5 16. 5x2 – 17x – 12

= =


17. 7x2 + 13x 18. 15x2 – 10x �Review Ex: 1–12

= =

19. 9x2 – 100 20. 81x2 – 25

= =

21. 4x2 + 28x + 49 22. 64x2 – 48x + 9

= =

23. x2 – 9x + 20 24. 6x2 + 11x + 4

= =


25. (a) Factorize 3u2 + 2u – 5.

(b) Hence, factorize 3(x + 2)2 + 2(x + 2) – 5.

( ) ( ) ( ) ( )

( ) ( ) = ( )

( ) ( )

( ) ( )

( ) ( ) = –6x

( ) ( ) ( ) ( )

( ) ( ) = ( )

x ( )

x ( )

( ) ( ) = +8x

68


Objective: To solve quadratic equations by the factor method.

Review: Factorization of Polynomials


1. x2 + 2x 2. x2 – 9

= x( ) = ( )( )

3. 4x2 – 4x + 1 4. x2 + 2x – 3

= =

For any real numbers m and n, if mn = 0, then m = 0 or n = 0.

Factor Method

(px + q)(rx + s) = 0

px + q = 0 or rx + s = 0

x = p

q− or x =

r

s−

Solve the following quadratic equations. [Nos. 5–8]

5. (x – 2)(x + 3) = 0 6. x(x – 4) = 0 �Ex 3A: 1–4

( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0

x = or x = x = or x =

7. (x + 5)(x – 7) = 0 8. (2x – 1)(2x + 1) = 0


Solve the quadratic equation x2 + 3x + 2 = 0.

x2 + 3x + 2 = 0

(x + 1)(x + 2) = 0

x + 1 = 0 or x + 2 = 0

x = –1 or x = –2

Solve the quadratic equation x2 – 6x + 8 = 0.

x2 – 6x + 8 = 0

( )( ) = 0

( ) = 0 or ( ) = 0

x = or x =

Take out the common factor x.

a2 – b2 ≡ (a + b)(a – b)

a2 – 2ab + b2 ≡ (a – b)2

x +1 x +2

+x +2x = +3x

x ( ) x ( )

( ) ( ) = ( )

x ( ) x ( )

( ) ( ) = ( )

69

Solve the following quadratic equations. [Nos. 9–16]

9. x2 – 8x = 0 �Ex 3A: 5–7 10. x2 + 9x + 18 = 0 �Ex 3A: 8–14

11. x2 – 4x – 21 = 0 12. x2 + 14x + 49 = 0 �Ex 3A: 15–17

x2 + 2( )( ) + ( )2 = 0

(x + )2 = 0

x + _____ = 0 or x + _____ = 0

x = _____ or x = _____

x = (repeated)

13. x2 – 12x + 36 = 0 14. x2 – 16 = 0 �Ex 3A: 18–20

15. x2 – 9x + 20 = 0 16. 2x2 + 3x – 2 = 0 �Ex 3A: 21, 22


17. (a) Expand (x – 5)(x + 2).

(b) Hence, solve the quadratic equation (x – 5)(x + 2) = 18.

Take out the common factor.

a2 + 2ab + b2 ≡ (a + b)2

a2 – 2ab + b2 ≡ (a – b)2 a2 – b2 ≡ (a + b)(a – b)

( ) ( ) ( ) ( )

( ) ( ) = ( )

x ( ) x ( )

( ) ( ) = ( )

70


3 Quadratic Equations in One Unknown


Level 1

Solve the following equations. [Nos. 1–6]

1. 2x(x – 3) = 0 2. x(5 – x) = 0

3. (x + 6)(x – 6) = 0 4. (2x – 1)2 = 0

5. (4x + 1)(x + 2) = 0 6. (5x + 1)(7x – 3) = 0

Solve the following equations using the factor method. [Nos. 7–28]

7. x2 + 10x = 0 8. 3x2 – 9x = 0

9. 7x2 = 28x 10. –15x = 10x2

11. x2 – 6x + 5 = 0 12. x2 – x – 20 = 0

13. x2 + 7x + 10 = 0 14. x2 + 2x – 15 = 0

15. x2 – 4x – 21 = 0 16. x2 + 13x + 36 = 0

17. 2x2 – x – 6 = 0 18. 3x2 + 4x + 1 = 0

19. 7x2 = 8x – 1 20. 8x + 3x2 + 1 = 4

21. x2 – 12x + 36 = 0 22. 25x2 + 10x + 1 = 0

23. 9x2 + 1 = –6x 24. –4x2 = 9 + 12x

25. 36x2 – 1 = 0 26. 49 – 16x2 = 0

27. 2x2 – 8 = 0 28. 12 – 27x2 = 0

71

Level 2


29. 6x2 + x – 2 = 0 30. 8x2 – 14x + 3 = 0

31. 10x2 – 3x – 4 = 0 32. 12x2 + 17x + 6 = 0

33. (x + 6)(x – 2) = 9 34. 2x – 2 = (x + 8)(x + 1)

35. (3 + 4x)(2x – 3) = 2x2 + x + 1 36. (2x – 1)(5 – x) = 6(x – 2)

37. (5x – 3)(x – 4) = 2 – 4(x + 2) 38. (x + 1)(4x – 3) = (–1 + x)(2x + 3)

39. 49 – (9 – 2x)2 = 0 40. 3(x + 5)2 – 48 = 0

41. (3x + 1)2 – (2 – x)2 = 0 42. 4(x – 3)2 – 9(2 + x)2 = 0

43. (3x – 1)(x – 6) + 2(3x – 1) = 0 44. (3x + 2)(6x – 1) = (6 – x)(2 + 3x)

45. (5x + 3)(2x + 1) = 2(2x + 1)2 46. (9 + 2x)(4x – 3) = 3 – 4x

47. (4x – 1)(8x – 1) = 8x – 2 48. (6x – 8)(3x – 1) = (8x – 6)(3x – 4)

49. 2x(3x + 8) = x2 – x 50. 2(x + 9)(x – 3) = 9 – x2

51. (a) Factorize 4x2 + 11x + 6.

(b) Hence, solve the equation (5x – 9)(4x + 3) = 4x2 + 11x + 6.

52. (a) Solve the equation 2y2 + 7y + 6 = 0.

(b) Using the result of (a), solve the equation 2(x – 3)2 + 7(x – 3) + 6 = 0.

53. 3 is a root of the quadratic equation –x2 + px + 6 = 0.

(a) Find the value of p. (b) Find the other root of the equation.

54. Let b be a constant. Solve the equation (x + 2b)(x + b – 2) = x + 2b and express the answers in terms

of b.

55. Let p be a constant. Solve the equation (x – 2p)2 – 9p2 = 0 and express the answers in terms of p.

72

Answers


1. 0, 3 2. 0, 5

3. –6, 6 4. 2

1(repeated)

5. 4

1− , –2 6.

5

1− ,

7

3

7. 0, –10 8. 0, 3

9. 0, 4 10. 0,

2

3−

11. 1, 5 12. –4, 5

13. −2, –5 14. –5, 3

15. −3, 7 16. –4, –9

17. 2

3− , 2 18. –1,

3

1−

19. 7

1, 1 20. –3,

3

1

21. 6 (repeated) 22. 5

1− (repeated)

23. 3

1− (repeated) 24.

2

3− (repeated)

25. 6

1− ,

6

1 26.

4

7− ,

4

7

27. –2, 2 28. 3

2− ,

3

2

29. 3

2− ,

2

1 30.

4

1,

2

3

31. 2

1− ,

5

4 32.

3

2− ,

4

3−

33. –7, 3 34. –2, –5

35. 6

5− , 2 36. –1,

2

7

37. 5

9, 2 38. 0 (repeated)

39. 1, 8 40. –1, –9

41. 2

3− ,

4

1 42. 0, –12

43. 3

1, 4 44.

3

2− , 1

45. –1,

2

1− 46. –5,

4

3

47. 4

1,

8

3 48.

3

4, 2

49. 0,

5

17− 50. 3, –7

51. (a) (4x + 3)(x + 2) (b) 4

3− ,

4

11

52. (a) 2

3− , –2 (b)

2

3, 1

53. (a) 1 (b) –2

54. –2b, 3 – b 55. –p, 5p

73

F4A: Chapter 3B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )



___________ ( )



_________

74


Objective: To solve quadratic equations by the graphical method.

Solving Quadratic Equations by the Graphical Method

The roots of the quadratic equation ax2 + bx + c = 0 are the x-intercepts of the graph of y = ax2 + bx + c,

where a ≠ 0. The roots obtained are approximate values.

Solve the following equations by the graphical method. [Nos. 1–6]

1. x2 – 5x + 4 = 0 2. –2x2 + x + 1 = 0 �Ex 3B: 1–4

From the graph, the x-intercepts are From the graph, the x-intercepts are

( ) and ( ). ( ) and ( ).

∴ The required roots are ( ) ∴

and ( ), cor. to 1 d.p.

3. –x2 + 4x – 4 = 0 4. x2 + 6x + 9 = 0

From the graph, the x-intercept is ( ). From the graph, the x-intercept is ( ).

∴ The required root is ( ), cor. to 1 d.p. ∴

(repeated).

5. 2x2 + 4x + 3 = 0 6. –3x2 – 9x – 8 = 0

The graph of y = 2x2 + 4x + 3 has ( ) The graph of y = –3x2 – 9x – 8 has ( )

x-intercepts. x-intercepts.

∴ The equation has ( ) real roots. ∴


75

(a) Draw the graph of y = x2 – 4x + 2 from

x = 0 to x = 4.

(b) Hence, solve the equation x2 – 4x + 2 = 0.

(a) y = x2 – 4x + 2

x 0 1 2 3 4

y 2 –1 –2 –1 2

(b) From the graph in (a), the x-intercepts are

0.6 and 3.4.

∴ The required roots are 0.6 and 3.4,

cor. to 1 d.p.

(a) Draw the graph of y = 4x2 – 20x + 25 from

x = 1 to x = 4.

(b) Hence, solve the equation 4x2 – 20x + 25 =

0.

(a) y = 4x2 – 20x + 25

x 1 2 3 4

y 9 1

(b) From the graph in (a), the x-intercept

is ( ).

∴ The required root is ( ), cor. to 1

d.p.

( ).

7. (a) Draw the graph of y = –x2 + 6x – 12 from x = 1 to x = 5. �Ex 3B: 5–10

(b) Use the graph in (a) to solve the equation –x2 + 6x – 12 = 0.

(a) y = –x2 + 6x – 12

x 1 2 3 4 5

y –7 –4

(b)


8. (a) Draw the graph of y = x2 – 5x + 3 from x = 0 to x = 5.

(b) Use the graph in (a) to solve the equation x2 – 5x + 3 = 0.

76




[When drawing graphs, the suggested unit length for the x-axis or the y-axis is 10 divisions (1 cm), unless

stated otherwise.]

Level 1

Solve the following equations by the graphical method. [Nos. 1–6]

1. x2 + 2x – 3 = 0 2. 2x2 – x – 2 = 0

3. 4x2 + 12x + 9 = 0 4. 2x2 – 2x + 2 = 0

15

10

5

x

–3

y

0 – –1

y = 4x2 + 12x + 9

6

4

2

x

–1

y

0 1 2

y = 2x2 – 2x + 2

2

1

x

–1

–2

–1

y

0 1 2

y = 2x2 – x – 2

2

1

x

–1

–2

–3

–4

–3 –2 –1

y

0 1 2

y = x2 + 2x – 3

77

5. y = –x2 – x + 6 6. y = –2x2 + 6x + 9

7. (a) Let y = –x2 – 4x – 3. Complete the following table.

x –4 –3 –2 –1 0

y –3 0

(b) Draw the graph of y = –x2 – 4x – 3 from x = –4 to x = 0.

(c) Hence, solve the equation –x2 – 4x – 3 = 0.

8. (a) Let y = 2x2 + 4x – 1. Complete the following table.

x –3 –2 –1 0 1

y 5 –1

(b) Draw the graph of y = 2x2 + 4x – 1 from x = –3 to x = 1.

(c) Hence, solve the equation 2x2 + 4x – 1 = 0.

y = –x2 – 4x – 3

–2

–4

–4 –3 –2 –1

y

0 x

2

15

10

5

x

–5

–1 1 2

y

0 3 4

y = –2x2 + 6x + 9 6

4

2

x

–2

–3 –2 –1

y

0 1 2

y = –x2 – x + 6

y = 2x2 + 4x – 1

6

4

2

x

–2

–3 –2 –1 0 1

y

78

9. (a) Let y = –4x2 + 8x + 9. Complete the following table.

x –1 0 1 2 3

y –3 9

(b) Draw the graph of y = –4x2 + 8x + 9 from x = –1 to x = 3.

(c) Hence, solve the equation –4x2 + 8x + 9 = 0.

Level 2

10. (a) Draw the graph of y = –1 – 2x – x2 from x = –3 to x = 1.

(b) Use the graph in (a) to solve the equation –1 – 2x – x2 = 0.

11. (a) Draw the graph of y = x2 + x + 1 from x = –2 to x = 1.

(b) Use the graph in (a) to solve the equation x2 + x + 1 = 0.

12. (a) Draw the graph of y = 2x2 – 5x + 3 from x = –1 to x = 3.

(b) Use the graph in (a) to solve the equation 2x2 – 5x + 3 = 0.

[Unit length for x-axis: 10 divisions (1 cm)

Unit length for y-axis: 5 divisions (0.5 cm)]

13. (a) Draw the graph of y = –3x2 – 7x – 2 from x = –3 to x = 1.

(b) Use the graph in (a) to solve the equation –3x2 – 7x – 2 = 0.



14. (a) Draw the graph of y = x2 + 4x + 1 from x = –4 to x = 0.

(b) Use the graph in (a) to solve the equation x2 + 4x + 1 = 0.

15. (a) Draw the graph of y = 2x2 – 3x – 1 from x = –1 to x = 3.

(b) Use the graph in (a) to solve the equation 2x2 – 3x = 1.



16. (a) Draw the graph of y = –x2 + 3x – 3 from x = –1 to x = 4.

(b) Use the graph in (a) to solve the equation 3x = x2 + 3.



y = –4x2 + 8x + 9 15

10

5

x

–5

–1 2 1 0 3

y

79

Answers


1. –3.0, 1.0 2. –0.8, 1.3

3. –1.5 (repeated) 4. no real roots

5. –3.0, 2.0 6. –1.1, 4.1

7. (a) x –4 –3 –2 –1 0

y –3 0 1 0 –3

(c) –3.0, –1.0

8. (a) x –3 –2 –1 0 1

y 5 –1 –3 –1 5

(c) –2.2, 0.2

9. (a) x –1 0 1 2 3

y –3 9 13 9 –3

(c) –0.8, 2.8

10. (b) –1.0 (repeated)

11. (b) no real roots 12. (b) 1.0, 1.5

13. (b) −2.0, –0.3 14. (b) −3.7, –0.3

15. (b) –0.3, 1.8 16. (b) no real roots

80

F4A: Chapter 3C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )



___________ ( )



_________

81


Objective: To solve quadratic equations by the quadratic formula.

Review: General Form of a Quadratic Equation

Rewrite each of the following quadratic equations into the general form ax2 + bx + c = 0, where a > 0.

Then write down the values of a, b and c. [Nos. 1–4]

Quadratic

equation

General form ax2 + bx + c = 0

a b c

1. x2 = –x – 5 x2 + = 0

2. 2x – 6 = –x2

3. 3x2 + 4 = 2x

4. 1 + 7x – 9x2 = 0

Quadratic Formula

The roots of the quadratic equation ax2 + bx + c = 0 (where a ≠ 0) are given bya

acbbx

2

42 −±−= .


Use the quadratic formula to solve x2 + 4x – 5 = 0.

x = )1(2

)5)(1(444 2 −−±−

= 2

364 ±−

= 2

64 ±−

= 2

64 +− or

2

64 −−

= 1 or –5

Use the quadratic formula to solve x2 – 9x

x = ) (2

)( (4) () ( 2 −±−

= ) (

) () ( ±

= ) (

) (

= ) (

) ( or

) (

) (

= or

Solve the following quadratic equations by the quadratic formula. [Nos. 5–8]

5. 2x2 + 7x – 4 = 0 6. 3x2 – 5x – 8 = 0 �Ex 3C: 1–8

x = ) (2

) )( (4) () ( 2 −±− x =

) (2

) )( (4) () ( 2 −±−

= =

7. 16x2 + 8x + 1 = 0 8. 4x2 – 12x + 9 = 0

a = b =

c =

a = b =

c =

a = 1 b = 4 c = –5

a = b = c =

82

x =

= ) (

) () ( ±

=

(repeated)

Solve the following quadratic equations by the quadratic formula. [Nos. 9–12]

(Leave the radical sign ‘ ’ in the answers if necessary. Identify those equations that have no real roots.)

9. x2 + 9x + 2 = 0 10. 4x2 – 3x – 5 = 0 �Ex 3C: 9–13

11. 3x2 – 4x + 6 = 0 12. 8x2 + x + 3 = 0 �Ex 3C: 15, 16

Since ) ( is not a real number,

the equation has no real roots.


13. (a) Rewrite 2x(x + 4) = 3 – x into the general form.

(b) Hence, solve the quadratic equation in (a) by the quadratic formula.

a = b =

c =

a = b =

c =

a = b = c =

a = b = c =

83




Level 1

Solve the following equations by the quadratic formula. [Nos. 1–10]

1. x2 + 8x + 7 = 0 2. x2 + 4x – 12 = 0

3. x2 – 5x – 14 = 0 4. x2 – 8x + 16 = 0

5. 2x2 + 5x + 2 = 0 6. 3x2 – 7x + 4 = 0

7. 4x2 – 9x – 9 = 0 8. 7x2 + 15x – 18 = 0

9. 4x2 – 25 = 0 10. 8x2 + 3x = 0


(Leave the radical sign ‘√’ in the answers if necessary. Identify those equations that have no real roots.)

11. x2 – 3x + 5 = 0 12. x2 + 6x + 4 = 0

13. x2 + 5x – 7 = 0 14. x2 – 4x – 11 = 0

15. 2x2 + 9x – 3 = 0 16. 3x2 + 8x + 1 = 0

17. 4x2 – 7x + 2 = 0 18. 9x2 – 8x – 6 = 0

19. 7x2 – 11 = 0 20. 9x2 + 5 = 0


(Give the answers correct to 2 decimal places if necessary. Identify those equations that have no real roots.)

21. x2 – 7x – 5 = 0 22. x2 + 8x + 3 = 0

23. 4x2 + 3x – 6 = 0 24. 5x2 – x + 11 = 0

84

Level 2



25. 4

1x2 +

6

1x –

8

3= 0 26.

3

8x2 – 4x +

2

3= 0

27. 2x2 + 5 x + 5 = 0 28. 3x2 + 17 x +

3

2= 0

29. 2(x2 + 5) = 3x + 4 30. (x – 1)(3x + 8) = 4

31. (2x + 5)(2x – 5) + 13 = 0 32. 3

2x(x – 2) = x + 3

33. (3x – 1)2 = 2(4x + 3) 34. (x + 4)(3x – 5) = 4(2x – 3)

35. (3 – x)2 – (2x – 1)2 = 3 36. (3x + 2)2 + (4x + 1)2 = 1


(Give the answers correct to 2 decimal places if necessary. Identify those equations that have no real roots.)

37. 4x2 + 21 x – 11 = 0 38. 3x2 +

4

10x + 1 = 0

39. x2 + 8 = 1 – 28 x 40. (x + 4)(x + 5) = 2(x + 2)(x + 3)

41. 2(3x + 1)(x – 3) = (5x – 1)(x + 5) 42. (x + 2)2 – (x + 2)(x – 2) – (x – 2)2 = 0

Solve the following equations by any algebraic method. [Nos. 43–48]


43. 2x(x + 4) = 7 44. 2(x + 1)2 = 7 + x2

45. x2 + x(x + 1) + (x + 1)2 = 0 46. (2x + 3)2 + (x – 4)2 = 26

47. (x + 4)2 – 3(4 – x)2 = 0 48. 3(2x – 1)(3 – x) = 4(1 – x)(3x + 2)

85

Solve the following equations by the quadratic formula. Give the answers in the form a + bi. [Nos. 49–52]

� 49. x2 + 2x + 3 = 0 � 50. x2 – 6x + 13 = 0

� 51. 2x2 + 7x + 9 = 0 � 52. 4x2 – 8x + 13 = 0

53. –2 is a root of the quadratic equation in x: 2x2 – 7cx – 4c2 = 0.

(a) Find the values of c.

(b) If c takes the larger value obtained in (a), solve the quadratic equation (x – c)2 = x(5 – x).

(Leave the radical sign ‘√’ in the answers.)

54. It is given that p and q are the roots of the quadratic equation x2 + 5x + 3 = 0, where p > q.



(b) Harry claims that the roots of the quadratic equation –3 = x(pq + x + 2) are p and q. Do you agree?

Explain your answer.

� 55. It is given that

i

i72 −= a + bi.


(b) Using the results of (a), solve the quadratic equation bx2 – ax + 5 = 0.


� 56. It is given that

i

i

31

5

+= a + bi.


(b) Using the results of (a), solve the quadratic equation x2 – b2 = 1 – a2x.


� 57. (a) Solve the equation x2 – 8x + 3 = 0.

(b) In the figure, the graph of y = x2 – 8x + 3 intersects the x-axis at two points A(a , 0) and B(b , 0),

where a < b. Using the results of (a), find the length of AB.


x

y

y = x2 – 8x + 3

O

A B

86

Answers


1. –1, –7 2. 2, –6

3. 7, –2 4. 4 (repeated)

5. 2

1− , –2 6.

3

4, 1

7. 3,

4

3− 8.

7

6, –3

9. 2

5,

2

5− 10. 0,

8

3−

11. no real roots

12. 2

206 ±− (or –3 ± 5 )

13. 2

535 ±−

14. 2

604 ± (or 2 ± 15 )

15. 4

1059 ±−

16. 6

528 ±−

±−

3

134or

17. 8

177 ±

18. 18

2808 ±

±

9

704or

19. 14

308±

±

7

77or

20. no real roots 21. 7.65, –0.65

22. –0.39, –7.61 23. 0.91, –1.66

24. no real roots

25. 12

2324 ±−

±−

6

582or

26. 4

3 (repeated) 27. no real roots

28. 6

317 ±− 29. no real roots

30. 3

4, –3

31. 2

12± (or 3± )

32. 2

9, –1

33. 18

37614 ±

±

9

947or

34. 6

971± 35. 1,

3

5−

36. 5

2− (repeated) 37. –2.33, 1.18

38. no real roots

39. –2.65 (repeated)

40. 2.37, –3.37

41. 40.02, –0.02

42. –0.47, 8.47

43. 4

1208 ±−

±−

2

304or

44. –5, 1

45. no real roots

46. –1,

5

1

47. 2

19216 ± (or 8 34± )

48. 12

69717 ±−

49. –1 ± i2

50. 3 ± 2i

51. i4

23

4

7±−

52. 1 ±

2

3i

53. (a) 2

1− , 4 (b)

4

4113 ±

54. (a) p =2

135 +−, q =

2

135 −−

(b) yes

55. (a) a = –7, b = –2 (b) 4

897 ±

56. (a) a =2

3, b =

2

1 (b)

8

1619 ±−

57. (a) 2

528 ± (or 4 ± 13 )

(b) 52 (or 132 )

87

F4A: Chapter 3D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)



(Full Solution)

Maths Corner Exercise 3D Level 1



___________ ( )

Maths Corner Exercise 3D Level 2



___________

88

○ Skipped ( )

Maths Corner Exercise 3D Multiple Choice


___________ ( )



_________

89

4A Lesson Worksheet 3.4(I) (Refer to Book 4A P.3.22)

Objective: To understand the relations between the discriminant (∆) and the nature of roots of a quadratic

equation.

Review: Solving Quadratic Equations by the Quadratic Formula


1. 2x2 – 7x + 2 = 0 2. 4x2 – 4x + 1 = 0 3. 2x2 + 5x + 4 = 0

x =)2(2

)2)(2(4)7()7( 2 −−±−− x = x =

=4

) (7 ±

Relations between the Discriminant and the Nature of Roots

For a quadratic equation ax2 + bx + c = 0 (a ≠ 0), discriminant (∆) = b2 – 4ac.

Discriminant

∆∆∆∆ = b2 – 4ac

∆ > 0

( acb 42 − is a

positive real number.)

∆ = 0

∆ < 0

( acb 42 − is not a

real number.)

Number of real roots

(Nature of roots)

Two distinct real roots

One double real root (or two equal real

roots) No real roots


Find the value of the discriminant for x2 – 3x –

4 = 0 and state the nature of the roots of the

equation.

∆ = (–3)2 – 4(1)(–4)

= 25 ∵ ∆ > 0 ∴ The equation has two distinct real roots.

Find the value of the discriminant for x2 – 5x + 7

= 0 and state the nature of the roots of the

equation.

∆ = (–5)2 – 4( )( )

= ∵ ∆ ( > / = / < ) 0 ∴

Find the value of the discriminant for each of the following quadratic equations and state the nature of the roots of the equation. [Nos. 4–5]

4. x2 + 3x – 1 = 0 5. x2 + 6x + 9 = 0 �Ex 3D: 1–6

∆ = ( )2 – 4( )( )

=

a = b = c =

a = b = c =

a = 1 b = –3

c = –4

a = b = c =

a = b =

c =

90


The quadratic equation kx2 – 4x – 2 = 0 has

two equal real roots. Find the value of k. ∵ The equation has two equal real roots. ∴ ∆ = 0

(–4)2 – 4(k)(–2)= 0

16 + 8k = 0

8k = –16

k = –2

The quadratic equation 9x2 – 6x + k = 0 has two

equal real roots. Find the value of k. ∵ The equation has two equal real roots. ∴ ∆ = ( )

( )2 – 4( )( ) = ( )

=

Find the values or the range of values of k for each of the following quadratic equations. [Nos. 6–9] �Ex 3D: 11–13

6. x2 + kx + 1 = 0 has two equal real roots. 7. x2 – 8x – 4k = 0 has no real roots.

∵ The equation has two equal real roots.

∴ ∆ = ( )

( )2 – 4( )( ) = ( )

=

8. kx2 + 10x – 5 = 0 has no real roots. 9. 2x2 + 4x + k – 3 = 0 has two distinct real roots.


10. The quadratic equation x2 + 4x – k = 0 has two equal real roots.


(b) Solve the equation.

a = k b = –4 c = –2

a = b =

c =

a = b =

c =

a = b =

c =

a = b =

c =

a = b = c =

91

4A Lesson Worksheet 3.4(II) (Refer to Book 4A P.3.25)

Objective: To find the number of intersections of the graph of y = ax2 + bx + c (a ≠ 0) and the x-axis by

considering the discriminant of ax2 + bx + c = 0.

Relations between the Discriminant of ax2 + bx + c = 0 (a ≠≠≠≠ 0) and the Number of

Intersections of the Graph of y = ax2 + bx + c and the x-axis

∆∆∆∆ ∆ > 0 ∆ = 0 ∆ < 0

Number of intersections of the graph

of y = ax2 + bx + c and the x-axis

2 1 0

Find the number of x-intercepts of the graph of each of the following equations. [Nos. 1–4] �Ex 3D: 7–10

1. y = x2 + 2x + 2 2. y = 2x2 – 5x + 1

Consider x2 + 2x + 2 = 0. Consider 2x2 – 5x + 1 = 0.

∆ = ( )2 – 4( )( ) ∆ =

= ( > / = / < ) 0 = ( > / = / < ) 0

∴ The number of x-intercepts is ( ). ∴ .

3. y = x2 + 6x + 9 4. y = 4x2 – 7x + 3


The graph of y = x2 – 3x + m does not intersect

the x-axis. Find the range of values of m. ∵ The graph of y = x2 – 3x + m does not

intersect the x-axis. ∴ ∆ < 0

(–3)2 – 4(1)(m) < 0

9 – 4m < 0

–4m < –9

4

9>m

The graph of y = x2 + 4x – m intersects the x-

axis at two points. Find the range of values of

m. ∵ The graph of y = x2 + 4x – m intersects the

x-axis at two points. ∴ ∆ ( > / = / < ) 0

a = b =

c =

92

5. The graph of y = 4x2 – 6x + m intersects the 6. The graph of y = 7x2 + 2x – m intersects the

x-axis at only one point. Find the value of m. x-axis at two points. Find the range of values of m.

∵ The graph of y = 4x2 – 6x + m intersects �Ex 3D: 16–18

the x-axis at only one point.

∴ ∆ ( > / = / < ) 0

7. The graph of y = 4mx2 + 8x + 1 intersects the 8. The graph of y = 3mx2 – 6x – 1 does not

x-axis at only one point, where m ≠ 0. Find the intersect the x-axis, where m ≠ 0. Find the range of

value of m. values of m.

9. The graph of y = x2 – 5x + m – 2 intersects the 10. The graph of y = 2x2 + 4x – (6 – m) intersects

x-axis at only one point. Find the value of m. the x-axis at two points. Find the range of values of m.


11. The graph of y = 2x2 + 3x – k does not intersect the x-axis. If k in an integer, find the maximum value

of k.

93



Consolidation Exercise 3D

Level 1

Find the value of the discriminant for each of the following quadratic equations and state the nature of the

roots of the equation. [Nos. 1–8]

1. x2 + 6x + 2 = 0 2. 2x2 + 8x + 8 = 0

3. 2x2 – 4x + 5 = 0 4. 3x2 – 2x – 7 = 0

5. 4x2 + 7x – 1 = 0 6. –x2 + 2x – 4 = 0

7. 12x – 4x2 – 9 = 0 8. 6x2 + 2 – 7x = 0

Find the number of x-intercepts of the graph of each of the following equations. [Nos. 9–14]

9. y = x2 – 3x + 1 10. y = x2 – 6x – 4

11. y = 4x2 + 2x + 1 12. y = –3x2 + 6x – 3

13. y = 3x2 + 5x 14. y = 4x – x2 – 5

15. Consider the quadratic equation mx2 + 2x + 3 = 0.

(a) Express the discriminant of the equation in terms of m.

(b) If the equation has one double real root, find the value of m.

16. Consider the quadratic equation x2 – kx + 16 = 0.

(a) Express the discriminant of the equation in terms of k.

(b) If the equation has two equal real roots, find the values of k.

17. Consider the quadratic equation 2x2 + 7x – c = 0.

(a) Express the discriminant of the equation in terms of c.

(b) If the equation has two distinct real roots, find the range of values of c.

94

18. Consider the quadratic equation 3x2 + 8x + n = 0.

(a) Express the discriminant of the equation in terms of n.

(b) If the equation has no real roots, find the range of values of n.

19. Each of the following quadratic equations has two equal real roots. Find the value(s) of a.

(a) ax2 + 4x – 2 = 0 (b) 3x2 – 6x + 2a = 0 (c) 4x2 – ax + 9 = 0

20. Each of the following quadratic equations has two distinct real roots. Find the range of values of c.

(a) 5x2 – 3x + c = 0 (b) 3x2 – 4x – 2c = 0 (c) cx2 + 2x + 6 = 0 (where c ≠ 0)

21. Each of the following quadratic equations has no real roots. Find the range of values of k.

(a) –x2 + 4x + k = 0 (b) kx2 – 8x + 2 = 0 (c) 2x2 – 6x + 1 – k = 0

(where k ≠ 0)

22. Each of the following quadratic equations has real root(s). Find the range of values of d.

(a) x2 – 10x – d = 0 (b) x2 + 2x + 2d – 3 = 0 (c) dx2 – 3x + 4 = 0 (where d ≠ 0)

23. The quadratic equation 4x2 – 12x + m = 0 has two equal real roots.

(a) Find the value of m. (b) Solve the equation.

24. The graph of each of the following equations intersects the x-axis at only one point. Find the value(s) of

k.

(a) y = 2x2 – kx + 8 (b) y = 6x2 + 7x + k – 3

25. The graph of each of the following equations intersects the x-axis at two points. Find the range of

values of u.

(a) y = 4x2 – 12x + u (b) y = –ux2 + 8x – 5 (where u ≠ 0)

26. The graph of each of the following equations does not intersect the x-axis. Find the range of values of

t.

(a) y = –3x2 + 3x – 2t (b) y = tx2 – 5x + 6 (where t ≠ 0)

27. The graph of each of the following equations intersects the x-axis. Find the range of values of w.

(a) y = 3x2 + 6x + w – 1 (b) y = 2wx2 – 10x – 1 (where w ≠ 0)

28. The graph of y = –3x2 + 12x + 4v intersects the x-axis at only one point.

(a) Find the value of v. (b) Find the x-intercept of the graph.

29. Let k be an integer. If the graph of y = –2x2 – 6x + 3k does not intersect the x-axis, find the maximum

value of k.

30. The graph of y = kx2 + 4x – 8 intersects the x-axis, where k ≠ 0. Can k be a negative integer? Explain

your answer.

95

Level 2

Find the value of the discriminant for each of the following quadratic equations and state the nature of the

roots of the equation. [Nos. 31–34]

31. 2x2 + x = 9x – 8 32. 3(2 – x) = 4x2 + 1

33. (x + 2)(x – 2) = –5 34. (x – 6)2 = –4(x – 5)

Find the number of x-intercepts of the graph of each of the following equations. [Nos. 35–38]

35. y = x2 – 4(2 – 5x) 36. y = 7x(7x – 2) + 1

37. y = 8 – (1 – 2x)(x + 3) 38. y = 2(3 – x)2 + 3

39. Each of the following quadratic equations has one double real root. Find the value(s) of k.

(a) –3kx2 + 8x – 12k = 0

(b) 3x2 – 2kx = k

(c) x2 – 3kx + 6k = 4

40. If the quadratic equation x2 – 2(3x – 4k) = 5 has two distinct real roots, find the range of values of k.

41. If the quadratic equation (x + 2)(2x – k) + k(x – 1) = 0 has no real roots, find the range of values of k.

42. If the quadratic equation in x: –x2 + 2kx – (k – 3)(k + 2) = 0 has real root(s), find the range of values of k.

43. The quadratic equation h(x + 1)2 – 2x – 7 = 0 has no real roots. If h is an integer, find the maximum

value of h.

44. The quadratic equation (c + 1)x2 + 4cx – 2(3 – 2c) = 0 has real root(s), where c ≠ –1.

(a) Find the range of values of c.

(b) If c is a negative integer, how many possible values of c are there? Explain your answer.

45. The quadratic equation (n – 1)x2 – 3nx + 2(n + 2) = 0 has two equal real roots.

(a) Find the value of n.

(b) Solve the equation.

46. The quadratic equation (k – 4)x2 + 2(kx + 9) = 0 has two equal real roots.

(a) Find the two possible values of k.

(b) Solve the equation corresponding to each value of k.

96

47. The quadratic equation (3 – b)x2 + 2bx – b = 1 has real root(s), where b ≠ 3.

(a) Find the range of values of b.

(b) If b takes the minimum integer in (a), solve the equation.

48. The graph of y = mx2 – 2mx + 3 touches the x-axis at only one point T, where m ≠ 0.


(b) Find the coordinates of T.

49. The graph of y = (k – 1)x2 + 4x – 2 intersects the x-axis at only one point P.

(a) Can the value of k be equal to 1? Explain your answer.

(b) Hence, find the values of k.

(c) For each value of k obtained in (b), find the coordinates of P.

50. The figure shows the graph of y = x2 + 4x – b + 3.

(a) Find the range of values of b.

(b) Find the number of points of intersection of the graph

of y = –x2 + 4x – (b + 4) and the x-axis.

51. The graph of y = 3x2 + 4x – (5u + 7) intersects the x-axis.

(a) Find the range of values of u.

(b) If u takes the minimum value in (a), solve the equation 2x2 – ux + 2u + 3 = 0.

52. The graph of y = (2v + 1)x2 + 8vx – 4(3 – 2v) does not intersect the x-axis, where v ≠2

1− .

(a) Find the range of values of v.

(b) If v takes the maximum integer in (a), solve the equation 5x2 + 2v(v – 2)x – v = 0.

53. The graph of y = p + 12x – 2x2 touches the x-axis at only one point Q(q , 0).


(b) The straight line L passes through Q and (–1 , –32). If L cuts the y-axis at the point S, find the

coordinates of S.

54. Consider the quadratic equation (*): ax2 + bx + c = 0.

(a) Find the discriminant of (*) and express the answer in terms of a, b and c.

(b) Show that the discriminant of the quadratic equation x2 – bx + ac = 0 is the same as that of (*).

(c) Anna claims that the discriminant of the quadratic equation ax2 + (2a + b)x + (a + b + c) = 0 is

the same as that of (*). Do you agree? Explain your answer.

O

y

x

y = x2 + 4x – b + 3

97

Answers

Consolidation Exercise 3D

1. 28, two distinct real roots

2. 0, one double real root

3. –24, no real roots






9. 2 10. 2

11. 0 12. 1

13. 2 14. 0

15. (a) 4 – 12m (b) 3

1

16. (a) k2 – 64 (b) –8, 8

17. (a) 49 + 8c (b) c >

8

49−

18. (a) 64 – 12n (b) n >

3

16

19. (a) –2 (b) 2

3

(c) –12, 12

20. (a) c <

20

9 (b) c >

3

2−

(c) c <

6

1 and c ≠ 0

21. (a) k < –4 (b) k > 8

(c) k <

2

7−

22. (a) d ≥ –25 (b) d ≤ 2

(c) d ≤

16

9 and d ≠ 0

23. (a) 9 (b) 2

3(repeated)

24. (a) –8, 8 (b) 24

121

25. (a) u < 9 (b) u <

5

16 and u ≠ 0

26. (a) t >

8

3 (b) t >

24

25

27. (a) w ≤ 4

(b) w ≥

2

25− and w ≠ 0

28. (a) –3 (b) 2

29. –2 30. no





35. 2 36. 1

37. 0 38. 0

39. (a) 3

2− ,

3

2 (b) 0, –3

(c) 3

4

40. k <

4

7 41. k <

3

2−

42. k ≥ –6 43. –1

44. (a) c ≥ –3 and c ≠ –1

(b) 2

45. (a) 4 (b) 2 (repeated)

46. (a) 6, 12

(b) when k = 6, x = –3 (repeated);

when k = 12, x =2

3− (repeated)

47. (a) b ≥

2

3− and b ≠ 3

(b) 0,

2

1

48. (a) 3 (b) (1 , 0)

49. (a) yes (b) –1, 1

(c) when k = –1, P(1 , 0);

when k = 1, P

0 ,

2

1

50. (a) b < –1 (b) 2

51. (a) u ≥

3

5− (b) –1,

6

1

52. (a) v <

4

3− (b) –1,

5

1−

53. (a) p = –18, q = 3 (b) (0 , –24)

54. (a) b2 – 4ac (c) yes

98

F4A: Chapter 3E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)



(Full Solution)

Maths Corner Exercise 3E Level 1



___________ ( )

Maths Corner Exercise 3E Level 2



___________ ( )

Maths Corner Exercise 3E Multiple Choice


___________ ( )



_________

99


Objective: To solve practical problems leading to quadratic equations.

Review: Solving Quadratic Equations


1. 3x2 – x – 2 = 0 2. 2x2 – 3x – 4 = 0

( )( ) = 0

( ) = 0 or ( ) = 0 =)(

)()( ±

Solving Practical Problems Leading to Quadratic Equations

Step 1: Use a letter (e.g. x) to represent one of the unknowns.

Step 2: Form an equation in x.

Step 3: Choose the most appropriate method to solve the equation.

Step 4: Check the roots obtained to see if they are acceptable and reject those which are unreasonable.

Step 5: Write the answer(s) clearly in a complete sentence.

In each of the following, set up a quadratic equation in x according to the given information. [Nos. 3–4]

3. The lengths of the two legs of a right-angled 4. There are x students in a class. Each student

triangle are (x + 3) cm and (2x + 1) cm prepares a Christmas gift for each classmate.

respectively. The area of the triangle is 9 cm2. The total number of gifts needed is 462.


The product of two consecutive positive even

numbers is 168. Find the smaller number.

Let x be the smaller number. �

Step 1

Larger number = x + 2

x(x + 2) = 168 � Step

2

x2 + 2x – 168 = 0

(x – 12)(x + 14) = 0 �

Step 3

x = 12 or –14 (rejected) �

Step 4 ∴ The smaller number is 12. �

The product of two consecutive positive odd

numbers is 195. Find the larger number.

Let x be . � Step 1

=

( )( ) = ( ) �

Step 2

=

∴ . � Step 5

x =a

acbb

2

42 −±−

3x ( )

x ( )

( )x ( )x = ( )x

Do steps 3 and 4.

x =)(2

))((4)()( 2 −±−

100

Step 5

5. Peter is 3 years younger than his sister. The 6. The walking speed of Mr Chan is x m/s. He

product of the present ages of Peter and his can walk 1 440 m in (40x – 2) min. Find the

sister is equal to 40. Find the present age of Peter. value of x. �Ex 3E: 5

Let x be .

Present age of his sister =

7. The following shows some patterns. The number 8. The figure shows a right-angled triangle

of dots in the nth pattern is2

)1( +nn. formed by three metal sticks. Find the

length of the longest stick. �Ex 3E: 8

If the number of dots in the mth pattern is 66,

find the value of m.

By Pythagoras’ theorem,


9. The area of a rectangular garden is 63 m2. The length of the garden is 2 m longer than its width.

(a) Find the width of the garden.

(b) Find the perimeter of the garden.

1 min = s

Distance

= speed × time

If ∠C = 90°,

then a2 + b2 = c2. (x + 9) m x m

(x + 1) m

3rd pattern 1st pattern

…

2nd pattern

101



Consolidation Exercise 3E

Level 1

1. The sum of two numbers is 5 and their product is –6. Find the larger number.

2. The difference of two numbers is 8. If 23 is added to their product, the result is 8. Find the smaller

number.

3. The product of two consecutive positive odd numbers is 143. Find the two numbers.

4. Alex is 4 years older than Stephy. If the product of their ages after 5 years is 396, find the present age

of Stephy.

5. 60 candies are equally shared among x students. Each student gets (x – 4) candies. Find the number of

candies that each student gets.

6. The figure shows a rectangle with area 72 cm2. Find the value of x.

7. The figure shows a right-angled triangle. Find the value of x.

8. The difference of the lengths of the two legs of a right-angled triangle is 4 cm. If the length of the

hypotenuse is 58 cm, find the length of the longer leg of the triangle.

9. The perimeter and the area of a rectangle are 20 cm and 16 cm2 respectively. Find the length of the

longer side of the rectangle.

(x + 3) cm

(x + 9) cm

(6 + x) cm (14 + x) cm

(10 – 2x) cm

102

10. If two straight lines with slopes (m – 1) and

−

31

m are perpendicular to each other, find the values of

m.

11. The width of a rectangle is x cm. The length is longer than twice the width by 2 cm.

(a) Express the area of the rectangle in terms of x.

(b) If the area of the rectangle is 7 cm2, use the graph of y = 2x2 + 2x – 7 to find the width of the

rectangle, correct to 1 decimal place.

12. The base radius of a solid cylinder is x cm. The height of the cylinder is 4 cm.

(a) Express the total surface area of the cylinder in terms of x and π.

(b) If the total surface area of the cylinder is 18π cm2, use the graph of y = x2 + 4x – 9 to find the base

radius of the cylinder, correct to 1 decimal place.

13. In the figure, a circular path with uniform width of x m is built around

a circular fountain of diameter 12 m.

(a) Express the area of the path in terms of x and π.

(b) If the area of the path is 28π m2, find the width of the path.

5

x

–5

–10

–5 –4 –3 –2 –1

y

0 1 2

y = x2 + 4x – 9

2

x

–2

–4

–6

–2 –1

y

0 1 2

y = 2x2 + 2x – 7

12 m

fountain

path

x m

103

Level 2

14. A box of biscuits is equally shared among a group of children. If there are n children, each child can

get (3n – 1) biscuits. If one more child joins the group, each child gets (n – 1) biscuits less. Find the

total number of biscuits in the box.

15. A stone is thrown vertically upward from a point P on the ground. The height H m of the stone from P

after t s can be found by the formula H = 10t – 5t2. Can the stone hit a target 6 m vertically above P?


16. The sum of two positive numbers is 7. Is it possible that the sum of the squares of the two numbers is

equal to 30? Explain your answer.

17. The perimeter of a right-angled triangle is 24 cm. If the longest side of the triangle is longer than the

shortest side by 4 cm,

(a) find the hypotenuse of the triangle,

(b) find the area of the triangle.

18. In the figure, ABCD is a trapezium, where ∠B = ∠C = 90°.

Find the possible areas of ABCD.

19. The slant height of a right circular cone is 8 cm longer than its base radius. If the total surface area of

the cone is 90π cm2,

(a) find the height of the cone,

(b) find the volume of the cone in terms of π.

20. A wire of length 16π cm is cut into two parts. Each part is bent into a circle. If the total area of the two

circles is 34π cm2,

(a) find the radius of the larger circle,

(b) find the difference in the areas of the two circles in terms of π.

length = 16π cm

6 cm

(4 + x) cm A B

C D

(5 + x) cm 5 cm

104

21. In the figure, B and C are points on AD and AE respectively such that BC // DE. Find the values of x.

22. In the figure, a solid metal sphere of radius (r – 1) cm is put into an empty cylindrical container of

base radius r cm and height

+ 3

3

4r cm.

3

157π cm3 of water is then poured into the container and

the container is fully filled without any overflow.

(a) Find the value of r.

(b) If only

6

157π cm3 of water is poured into the empty cylindrical container, can the sphere in the

container be totally immersed in the water? Explain your answer.

23. The figure shows a rectangular field ABCD, where AB = 20 m and AD = 50 m. In a competition,

participants start from A and go to a point P on BC, and then reach D. Let BP = x m.

(a) If Connie chooses the point P such that she goes from P to D in 15 s with a speed of 3 m/s, find

the value of x.

(b) Samson chooses the point P such that ∠APD = 90°.

(i) Show that △ABP ~ △PCD.

(ii) Hence, find the value(s) of x and the total distance travelled by Samson.

(Give the answers correct to 3 significant figures if necessary.)

20 m

B C

D A 50 m

P

4 cm

x cm

Area = 12 cm2

Area = (x2 + 3x + 5) cm2

A

B C

D E

105

24. The following shows some patterns formed by identical squares. The number of squares in the nth

pattern is 3n2 + 4n – 4.

(a) If the number of squares in the pth pattern is 220, find the value of p.

(b) Is there any pattern with exactly 300 squares? Explain your answer.

25. The following shows some patterns. The number of dots in the nth pattern is

2

3n2 +

2

3n.

(a) If the number of dots in the kth pattern is 135, find the value of k.

(b) Can the 8th pattern be formed using all the dots in two consecutive patterns? Explain your answer.

26. The cost of manufacturing a car is $80 000. The marked price of the car is x% higher than its cost and

it is sold at a discount of

2

x%.

(a) Express the selling price of the car in terms of x.

(b) If a profit of $10 000 is made from selling the car, find the value of x.

27. (a) Solve x2 + 600x – 1 809 = 0.

(b) At the beginning of the first year, Mr Chan deposits $10 000 in a bank at an interest rate r%

compounded yearly. At the beginning of the second year, Mr Chan deposits $20 000 in another

bank at an interest rate 2r% compounded yearly. If the total amount received at the end of the

second year is $31 809, find the value of r.

1st pattern 2nd pattern 3rd pattern

…

1st pattern 2nd pattern 3rd pattern

…

106

Answers

Consolidation Exercise 3E

1. 6 2. –5, –3

3. 11, 13 4. 13

5. 6 6. 3

7. –1 8. 7 cm

9. 8 cm 10. 0, 4

11. (a) (2x2 + 2x) cm2

(b) 1.4 cm

12. (a) (2x2 + 8x)π cm2

(b) 1.6 cm

13. (a) (x2 + 12x)π m2

(b) 2 m

14. 24

15. no

16. yes

17. (a) 10 cm (b) 24 cm2

18. 12 cm2, 18 cm2

19. (a) 12 cm (b) 100π cm3

20. (a) 5 cm (b) 16π cm2

21. 2, 10

22. (a) 3 (b) yes

23. (a) 9.69

(b) (ii) 10, 40; 67.1 m

24. (a) 8 (b) no

25. (a) 9 (b) yes

26. (a) $(–4x2 + 400x + 80 000)

(b) 50

27. (a) –603, 3 (b) 3

107

F4A: Chapter 3F

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 24


(Video Teaching)

Book Example 25


(Video Teaching)



(Full Solution)

Maths Corner Exercise 3F Level 1



___________ ( )

Maths Corner Exercise 3F Level 2



___________ ( )

Maths Corner Exercise 3F Multiple Choice


___________ ( )



_________

108


Objective: To form quadratic equations from given roots.

Forming Quadratic Equations from Given Roots

The quadratic equation in x formed from the given roots α and β is:

x = α or x = β

x – α = 0 or x – β = 0

(x – α)(x – β) = 0 ∴ x2 – (α + β)x + αβ = 0


Form a quadratic equation in x with roots –2

and 4.

x = –2 or x = 4

x + 2 = 0 or x – 4 = 0 ∴ The required equation is

(x + 2)(x – 4) = 0

x2 – 2x – 8 = 0

Form a quadratic equation in x with roots 3

and –1.

x = 3 or x = –1

x – 3 = 0 or ( ) = 0 ∴ The required equation is

( )( ) = 0

In each of the following, form a quadratic equation in x from the given roots. [Nos. 1–10]

1. 0, 6 2. –2, 8 �Ex 3F: 1–13

x = 0 or x = 6 x = ( ) or x = ( )

x = 0 or ( ) = 0 ( ) = 0 or ( ) = 0

∴ The required equation is ∴ The required equation is

( )( ) = 0 ( )( ) = 0

= =

3. 8, –5 4. –7, –9

109

5. 2

1, 6 6.

3

2, 1

7. 4,4

1− 8. –6,

5

3−

9. 6

1,

6

1 10.

3

2− ,

4

3−


11. (a) Form a quadratic equation in x with roots5

1− and

5

1.

(b) Hence, solve the quadratic equation 25(y – 1)2 = 1.

110


Consolidation Exercise 3F

[In this exercise, write the quadratic equations in the answers in the form ax2 + bx + c = 0, where a, b and

c

are integers.]

Level 1


1. 3, 6 2. 5, –2 3. 1, –7 4. –6, 6

5. –2, –4 6. 0, –3 7. 5, 5 8. 3,

2

1

9. 4

1− , 2 10. –1,

5

2 11. –2,

4

3− 12. 0,

3

2

Level 2


13. 6

1,

4

1 14.

5

3,

7

1− 15.

2

3− ,

9

2 16.

7

6,

7

6−

17. 5

1− ,

3

2− 18.

3

4− ,

3

4− 19. a, –2a 20. c, c + 2

21. (a) Solve z2 – 6z + 8 = 0.

(b) Form a quadratic equation in x whose two roots are three times the roots of z2 – 6z + 8 = 0

respectively.

22. (a) Solve 3m2 + 7m + 2 = 0.

(b) Form a quadratic equation in n whose two roots are the squares of the roots of

3m2 + 7m + 2 = 0 respectively.

23. (a) Solve 8p2 + 10p – 7 = 0.

(b) Form a quadratic equation in q whose two roots are the reciprocals of the roots of

8p2 + 10p – 7 = 0 respectively.

111

Answers

Consolidation Exercise 3F

1. x2 – 9x + 18 = 0

2. x2 – 3x – 10 = 0

3. x2 + 6x – 7 = 0

4. x2 – 36 = 0

5. x2 + 6x + 8 = 0

6. x2 + 3x = 0

7. x2 – 10x + 25 = 0

8. 2x2 – 7x + 3 = 0

9. 4x2 – 7x – 2 = 0

10. 5x2 + 3x – 2 = 0

11. 4x2 + 11x + 6 = 0

12. 3x2 – 2x = 0

13. 24x2 – 10x + 1 = 0

14. 35x2 – 16x – 3 = 0

15. 18x2 + 23x – 6 = 0

16. 49x2 – 36 = 0

17. 15x2 + 13x + 2 = 0

18. 9x2 + 24x + 16 = 0

19. x2 + ax – 2a2 = 0

20. x2 – 2(2c + 2)x + c2 + 2c = 0

21. (a) 2, 4 (b) x2 – 18x + 72 = 0

22. (a) 3

1− , –2 (b) 9n2 – 37n + 4 = 0

23. (a) 2

1,

4

7− (b) 7q2 – 10q – 8 = 0

112

F4A: Chapter 3G

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 26


(Video Teaching)

Book Example 27


(Video Teaching)

Book Example 28


(Video Teaching)

Book Example 29


(Video Teaching)

Book Example 30


(Video Teaching)

Book Example 31


(Video Teaching)



(Full Solution)

Maths Corner Exercise 3G Level 1



___________ ( )

Maths Corner Exercise 3G Level 2



___________

113

○ Skipped ( )

Maths Corner Exercise 3G Multiple Choice


___________ ( )



_________

114


Objective: To understand the relations between the roots and the coefficients of a quadratic equation, and

to form quadratic equations using these relations.

Sum and Product of Two Roots

Let α and β be the roots of the quadratic equation ax2 + bx + c = 0 (where a ≠ 0).

Sum of roots = α + β =a

b− , product of roots = αβ =

a

c

In each of the following, write down the sum of roots and the product of roots. [Nos. 1–2] �Ex 3G: 1

1. x2 – 2x – 8 = 0 2. 4x2 + 3x – 5 = 0

Sum of roots =) (

) (− = Sum of roots =

Product of roots =) (

) (= Product of roots =


If the sum of roots of the equation 2x2 + kx – 11

= 0 is 5, find the value of k.

Sum of roots = 5

2

k− = 5

k = –10

If the product of roots of the equation

kx2 – 6x + 9 = 0 (where k ≠ 0) is –3, find the value

of k.

Product of roots = ( )

) (

) ( = ( )

k =

3. If the sum of roots of the equation 4. If the product of roots of the equation

3x2 + (k + 1)x + 7 = 0 is –2, find the kx2 – 3x + k + 8 = 0 (where k ≠ 0) is 5,

value of k. find the value of k. �Ex 3G: 2–5


If α and β are the roots of the equation

x2 + 4x – 6 = 0, find the value of (α + 1)(β + 1).

α + β =1

4− = –4

αβ =1

6−= –6

(α + 1)(β + 1) = αβ + β + α + 1

= (–6) + (–4) + 1

= –9

If α and β are the roots of the equation

x2 – 5x + 2 = 0, find the value of α2β + αβ 2.

α + β =) (

) (− =

αβ =) (

) (=

α2β + αβ 2 = αβ( )

=

a = b =

c =

�

a = b =

c =

115

5. If α and β are the roots of the equation x2 – 4x – 3 = 0, find the values of the following expressions.

(a) α + β (b) αβ �Ex 3G: 14–16

(c) α2 + β 2 (d) (β – 1)(α – 1)

x2 – (sum of roots)x + (product of roots) = 0

In each of the following, form a quadratic equation in x from the given sum and product of roots. [Nos. 6–7]

6. Sum of roots = 3, product of roots = 1 7. Sum of roots = –4, product of roots = 6


x2 – ( )x + ( ) = 0. x2 – ( )x + ( ) = 0

= 0


8. –2, 4 9. 73 + , 73 − �Ex 3G: 17

Sum of roots = ( ) + ( )

= ( )

Product of roots = ( )( )

= ( )

∴ The required equation is


10. α and β are the roots of the equation x2 + 3kx – (k + 2) = 0.

(a) Express α + β and αβ in terms of k. (b) If 111

=+βα

, find the value of k.

116



� Consolidation Exercise 3G

Level 1

In each of the following, write down

(a) the sum of roots,

(b) the product of roots. [Nos. 1–9]

1. x2 + 7x + 2 = 0 2. x2 – 6x + 4 = 0 3. 3x2 – 2x – 6 = 0

4. 4x2 – 4x + 1 = 0 5. 5x2 + 8x – 3 = 0 6. 3x2 – 8 = 0

7. 2x2 = 4x + 7 8. –5x2 + 1 = 7x 9. 3

2x2 –

6

1x = 0

10. If the sum of roots of the equation 3x2 – (2k + 1)x + 4 = 0 is –3, find the value of k.

11. If the product of roots of the equation 2ax2 – ax + (3 – a) = 0 (where a ≠ 0) is –2, find the value of a.

12. If the roots of the equation x2 – bx + c = 0 are –2 and 6, find the values of b and c.

13. If the roots of the equation 7x2 + mx + n = 0 are –3

1 and 2, find the values of m and n.

14. The sum of roots of the equation –2x2 + 4tx + 2 – t = 0 is 6.

(a) Find the value of t.

(b) Find the product of roots of the equation.

15. The product of roots of the equation hx2 + (h – 3)x – h + 6 = 0 (where h ≠ 0) is 2.

(a) Find the value of h.

(b) Find the sum of roots of the equation.

16. One root of the equation x2 – 7x + 6s = 0 is 3. Find the other root and the value of s.

17. One root of the equation x2 – 3tx + 14 = 0 is 2. Find the other root and the value of t.

117

18. If the sum of roots is equal to the product of roots of the equation rx2 – 2rx + 6 = 0 (where r ≠ 0), find

the value of r.

19. If the sum of roots is twice the product of roots of the equation 4qx2 – 3x + 1 – 2q = 0 (where q ≠ 0),

find the value of q.

20. If the sum of roots is greater than the product of roots of the equation 2x2 – px – 3p + 5 = 0 by 4, find

the value of p.

21. If the sum of roots is equal to the positive square root of the product of roots of the equation

nx2 – (1 + 2n)x + 9n = 0 (where n ≠ 0), find the value of n.

22. If α and β are the roots of the equation x2 + 3x + 5 = 0, find the values of the following expressions.

(a) 3α + 3β (b) 2αβ

(c) –2α – 2β (d) (–3α)(4β)

23. If α and β are the roots of the equation 4x2 – 7x + 1 = 0, find the values of the following expressions.

(a) αβ

8 (b)

βα +

7

(c) (2 + α)(β + 2) (d) (3α – 1)(1 – 3β)

24. If α and β are the roots of the equation 2x2 + 5x – 8 = 0, find the values of the following expressions.

(a) 3α(αβ + β2) (b) βα

11+

(c) βαβα

121++ (d)

+

+ 2

12

1

βα

25. In each of the following, form a quadratic equation in x from the given roots.

(a) 2, 9 (b) – 7 , 7

(c) 1 + 6 , 1 – 6 (d) –4 + 5 , –4 – 5

26. α and β are the roots of the equation 2x2 + 5x – 9 = 0. Form a quadratic equation in x from the roots in

each of the following.

(a) –α, –β (b) 4α, 4β (c) 5

α− ,

5

β−

27. α and β are the roots of the equation x2 – 6x – 10 = 0. Form a quadratic equation in x from the roots in


(a) α + 1, β + 1 (b) 2 – α, 2 – β (c) α

3,

β

3

118

Level 2

28. α and 2α + 2 are the roots of the equation 3x2 – 8x + 3k = k.

(a) Find the value of α.

(b) Find the value of k.

29. α and 2α are the roots of the equation 2x2 – 3mx + 16 = x.

(a) Find the two possible values of α.

(b) For each value of α obtained in (a), find the value of m.

30. α and β are the roots of the equation mx2 + 3x – (5 – 2m) = 0 (where m ≠ 0).

(a) Express α + β and αβ in terms of m.

(b) If (2α – 1)(2β – 1) = 2, find the value of m.

31. α and β are the roots of the equation 2x2 + (3n – 5)x – 6n = 0. If α2 + β2 = 10, find the values of n.

32. If α and β are the roots of the equation x2 – 6x – 1 = 0, find the values of the following expressions.

(a) α2 – αβ + β2 (b) α3 + β3

(c) (2α + 3β)(3α + 2β) (d) (α – β)

−

βα

11

(e) 12α + 2β2 + 2 (f) α3 + 37β

33. α and β are the roots of the equation 2x2 + 4x – 3 = 0, where α > β. Find the values of the following

expressions.


(a) α – β (b) βα

11−

(c) α4 – β4 (d) –2α – β2

34. If α and β are the roots of the equation x2 + x – 5 = 0, find the values of the following expressions.

(a) 11 −

+− β

β

α

α (b)

1

1

1

1

+

−+

+

−

α

β

β

α

(c) (α2 – β)(α – β2) (d) 2α2 – 2β

35. –3 + 4i and –3 – 4i are the roots of the equation x2 – 2ax + (3b + 1) = 0.

(a) Find the value of (–3 + 4i)(–3 – 4i).

(b) Find the values of a and b.

36. 2

1 +

2

3i and

2

1 –

2

3i are the roots of the equation 2x2 + cx + d = 0.

(a) Find the value of

−

+ ii

2

3

2

1

2

3

2

1.

(b) Find the values of c and d.

119

37. (a) Express

i23

26

+ in the form a + bi, where a and b are real numbers.

(b) It is given that the real part and the imaginary part of

i23

26

+ are the roots of the quadratic

equation x2 + 2px + 3q = 0.

(i) Find the values of p and q.

(ii) Is there a negative number n such that the quadratic equation x2 – px – q = n has real root(s)?


38. In the figure, the graph of y = –x2 + kx + 6 intersects the x-axis at two

points M(m , 0) and N(n , 0), where m < n.

(a) Express k in terms of m and n.

(b) Find the value of mn.

(c) If k < 0 and MN = 5, find the values of k, m and n.

39. In the figure, the graph of y = ax2 + bx – 10 intersects the x-axis at two

points A(α , 0) and B(β , 0).

(a) Express α + β in terms of a and b.

(b) Is it true that OA + OB = α + β? Explain your answer.

(c) It is given that αβ = –5 and M(2 , 0) is the mid-point of AB. Find

the values of a and b.

40. α and β are the roots of the equation 3x2 – 5x + 9 = 0.

(a) Find the value of

βα

11+ .

(b) Form a quadratic equation in x with roots

+

βα

1 and

+

αβ

1.

41. α and β are the roots of the equation x2 – 6x – 8 = 0.

(a) Find the value of α2 + β2.


−

2

βα and

−

2

αβ .

42. α and β are the roots of the equation 2x2 – 6x – 7 = 0.

(a) Find the value of (α – β)2.


βα −

1 and

αβ −

1.

x

y

O

y = –x2 + kx + 6

M(m , 0) N(n , 0)

x

y

O B(β , 0) A(α , 0)

y = ax2 + bx – 10

120

43. α and β are the roots of the equation x2 + 4x – 3 = 0.

(a) Find the value of α2 – αβ + β2.

(b) Form a quadratic equation in x with roots (α3 + β3) and α3β3.

44. α and β are the roots of the equation x2 – 4x + 7 = 0.

(a) Find the value of (α2 + β2)2.

(b) Form a quadratic equation in x with roots α4 and β4.

45. α and β are the roots of the quadratic equation ax2 + bx + c = 0, where a, b and c are non-zero

constants. Form a quadratic equation in x from the roots in each of the following.

(a) –α, –β (b) α

1,

β

1 (c) 3α, 3β

46. –2α and –2β are the roots of the equation 4x2 – 6x + 9 = 0.

(a) Find the values of α + β and αβ.

(b) Form a quadratic equation in x with roots α and β.

47.

+ 1

1

α and

+ 1

1

β are the roots of the equation x2 – 3x – 8 = 0.

(a) Find the values of α + β and αβ.

(b) Form a quadratic equation in x with roots (α + 1) and (β + 1).

48. It is given that

+=

+=

16

16

2

2

ββ

αα, where α ≠ β. Without finding the values of α and β, find the value of


(a) α2 + 6β

(b) α3β + α2

49. It is given that

−=

−=

64

64

2

2

ββ

αα.

(a) If α ≠ β, find the values of α + β and αβ.

(b) If α = β, find the values of α.


50. If the roots of the quadratic equation 3x2 + nx + 36 = 0 are integers, find all the possible values of n.

51. It is given that the roots of the quadratic equation x2 – 5x + p = 0 are non-negative integers. Fanny

claims that there are 6 different possible values of p. Do you agree? Explain your answer.

121

Answers

Consolidation Exercise 3G

1. (a) –7 (b) 2

2. (a) 6 (b) 4

3. (a) 3

2 (b) –2

4. (a) 1 (b) 4

1

5. (a) 5

8− (b)

5

3−

6. (a) 0 (b) 3

8−

7. (a) 2 (b) 2

7−

8. (a) 5

7− (b)

5

1−

9. (a) 4

1 (b) 0

10. –5 11. –1

12. b = 4, c = –12 13. m =3

35− , n =

3

14−

14. (a) 3 (b) 2

1

15. (a) 2 (b) 2

1

16. the other root = 4, s = 2

17. the other root = 7, t = 3

18. 3 19. 4

1−

20. 4

13 21. 1

22. (a) –9 (b) 10

(c) 6 (d) –60

23. (a) 32 (b) 4

(c) 4

31 (d) 2

24. (a) 30 (b) 8

5

(c) 8

1 (d) 5

25. (a) x2 – 11x + 18 = 0

(b) x2 – 7 = 0

(c) x2 – 2x – 5 = 0

(d) x2 + 8x + 11 = 0

26. (a) 2x2 – 5x – 9 = 0

(b) x2 + 10x – 72 = 0

(c) 50x2 – 25x – 9 = 0

27. (a) x2 – 8x – 3 = 0

(b) x2 + 2x – 18 = 0

(c) 10x2 + 18x – 9 = 0

28. (a) 9

2

(b) 27

22

29. (a) –2, 2

(b) when α = –2, m =3

13− ;

when α = 2, m =3

11

30. (a) α + β =m

3− , αβ =

m

m 52 −

(b) 2

31. –1,3

5

32. (a) 39 (b) 234

(c) 215 (d) 40

(e) 76 (f) 228

33. (a) 10 (b) 3

102

(c) 1014− (d) 1022

3−−

34. (a) 3 (b) 5

9−

(c) –36 (d) 12

35. (a) 25 (b) a = –3, b = 8

36. (a) 2

5 (b) c = –2, d = 5

37. (a) 6 – 4i

(b) (i) p = –1, q = –8

(ii) no

38. (a) k = m + n (b) –6

(c) k = –1, m = –3, n = 2

122

39. (a) α + β =a

b− (b) no

(c) a = 2, b = –8

40. (a) 9

5 (b) 9x2 – 20x + 48 = 0

41. (a) 52 (b) x2 – 3x – 36 = 0

42. (a) 23 (b) 23x2 – 1 = 0

43. (a) 25 (b) x2 + 127x + 2 700 = 0

44. (a) 4 (b) x2 + 94x + 2 401 = 0

45. (a) ax2 – bx + c = 0

(b) cx2 + bx + a = 0

(c) ax2 + 3bx + 9c = 0

46. (a) α + β =4

3− , αβ =

16

9

(b) 16x2 + 12x + 9 = 0

47. (a) α + β =10

1− , αβ =

10

1−

(b) 10x2 – 19x + 8 = 0

48. (a) 37

(b) 0

49. (a) α + β = 4, αβ = –6

(b) 2

404 ±(or 102 ± )

50. –39, –24, –21, 21, 24, 39

51. no

chapter 1 number systems 1a p.2 chapter 2 equations of

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