Mathematics Lesson Notes SS3 First Term

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SCHEME OF WORK FOR SS3 MATHEMATICS

FIRST TERM

1. Revision: Indices and Logarithm
2. Surds
3. Surds in relation to Trigonometry
4. Matrices and Determinants
5. Linear and Quadratic Equations
6. Surface Area and Volume of Sphere and Hemispherical shapes
7. Mid-term Test
8. Longitude and Latitude
9. Longitude and Latitude
10. Arithmetic of Finance
11. Revision
12. Examination
13. Vacation

WEEK ONE                         

TOPIC: BASIC CONCEPT & APPLICATION OF LAWS OF INDICES

CONTENT:

• Basic Concept of Laws of Indices
• Application of Laws of Indices

Basic Concept of Laws of Indices

A number of the form a^m where a is a real number, a is multiplied by itself m times

The number a is called the base and the super script m is called the index (plural indices) or exponent.

1.            A ^m x A ⁿ = A^{m + n}    ——————–Multiplication law

Example:  X³ xX²  =( X x Xx X) x (X x X) = X ⁵

Or X³ x X² = X^{3 + 2} = X⁵

2.            A^m ÷ Aⁿ=A^m-n     ———————Division law

Example: X⁶ ÷ X⁴ = X^6-4 = X²

3.            (a ^m )ⁿ = a^mn   —————-Power law

Example:  (x³)² = X³ x X³ = X³⁺³ =X⁶

Or X^3X2 = X⁶

4.            a^m ÷ a^m = a^m-m = a⁰ =1

a^m ÷a^m = a^m/a^m = a^o = 1

a⁰………………………………….Zero Index

:. Any number raised to power zero is 1

Example: 3^o = 1, c^o = 1, y^o = 1

5.            (ab)^m =a^mb^m    ————-Product power law

e.g. (2xy)²= 4x²y²

6.            Negative index                

            a ^–m = 1/a^m        ————- Negative Index

Example: 2 ^-1 = ½, and 3 ^-2 = 1/3 ² = 1/9

7.            a^1/n  =ⁿ√a  ————–Root power law

Example : 9 ^½ =√9=3

27 ^1/3 =³√27 = 3 ie (3)³ = 3

8.            a ^m/n = (a ^1/n) ^m = (ⁿ√a)^m   ———–Fraction Index

or a ^m/n = (a^m) ^1/n =   (ⁿ√a)^m

Example : 27^2/3 = 3√27=3²=9.

EVALUATION

1.  27^5/3            2.   1000000000⁰

Application of Laws of Indices

Solve the following

(i)            32 ^3/5                                 (ii)           343 ^2/3   (iii) 64 ^2/3(iv) 0.001            (v) 14 ⁰

Solution:

i)             32 ^3/5 = (32 ^1/5) ³ = (⁵√32) ³

= 2 ³ = 8

ii)            343 ^2/3 = (343 ^1/3 )² = (³√343)²

 = (7 ³)^1/3)²

                = 7² = 49

iii)           64 ^2/3 = (64 ^1/3)² = (4 ³)^1/3)² = 4 ²

iv)           (0.001)³ = (1/100)³ = (1/10)³)³ = (10 ^-3)³

= 10 ^-9 = 1/10 ⁹

v)            14 ⁰ = 1

EVALUATION

1)            Simplify the following

a)            216 ^4/3   b) 25 ^1.5 c) (0.00001)²

d)            d 32 ^2/5        e) 81 ^¾

Reading Assignment: F/Maths project book 1(New third edition).Chapter 2 pg.4 – 6

WEEKEND ASSIGNMENT

1)            Evaluate 3 ^x  = 1/8 1 (a) 4 (b) -4 (c) -2 (d) 2

2)            Simplify                2r⁵ X 9r³                (a) P²     (b) 2p⁴ (c) P³   d)18r⁸

3)            Solve 3^-y = 243                   (a) -5     (b) 5                       (c) 3                       (d) -3

4)            Solve 25^-5n = 625               (a) 1/5   (b) 2/5  (c) 1 1/5                (d) – 2/5

5)            Simplify (0.0001)²=          (a) 10^-5  (b) 10 ^-3 (c) 10^-8  (d) 10^-10

Theory

1.   16^3/2 x 8^2/32.  3X² x 4X³

            32^1/56X⁷

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WEEK 2

TOPIC: INDICIAL/EXPONENTIAL EQUATION

CONTENT:

• Exponential Equation of Linear Form
• Exponential Equation of Quadratic Form

Under exponential equation, if the base numbers of any equation are equal, then the power will be equal&vice versa.

Exponential Equation of Linear Form

Solve the following exponential equations

a)            (1/2) ^x = 8                             b) (0.25) ^x+1 = 16

c)            3^x = 1/81                               d) 10 ^x = 1/0.001

e)            4/2^x = 64 ^x

Solution

a)            (1/2) ^X = 8

(2 ^-1) ^x = 2 ³

2 ^–x = 2 ³

-x = 3

X = – 3

b)            (0.25) ^x+1 = 16

                (25/100) ^x+1 = 16

                (1/4) ^x+1 = 4 ²

          (4 ^-1) ^x+1 = 4 ²

                – x – 1 = 2

                – x = 2 + 1

                – x = 3

X = – 3

c)            3^x = 1/81

                3^x = 1/3⁴

                3^x = 3 ^-4

                X = -4

d)            10^x = 1/0.001

                10 ^x = 1000

                10 ^x = 10 ³

                10^x = 10 ³

X = 3

e)            4/2^x = 64 ^x

                4÷2^x = 64 ^x

                2² ÷2^x = 64 ^x

                2 ^2-x = (2 ⁶) ^x

                2 ^2-x = 2 ^6x

2-x  = 6x

2=6x+x

2 = 7x

Divide both sides by 7

2/7 = 7x/7

X = 2/7

EVALUATION

Solve the following exponential equations

a)            2 ^x = 0.125            b) 25 ^(5x) = 625    c)            10 ^x = 1/100000

Exponential Equation of Quadratic Form

Some exponential equation can be reduced to quadratic form as can be seen below.

Example :  Solve the following equations.

a)            2 ^2x – 6 (2 ^x)  + 8 = 0

b)            5 ^2x + 4 X 5 ^x+1 – 125 = 0

c)        3 ^2x – 9 = 0

Solution

a)            2 ^2x – 6 (2 ^x) + 8 = 0

(2 ^x)² – 6 (2 ^x) + 8 = 0

Let 2 ^x = y.

Then y² – 6y + 8 = 0

Then factorize

Y ² – 4 y – 2y + 8 = 0

Y (y – 4) -2 (y -4) = 0

(y -2) (y – 4) = 0

Y – 2 = 0 or y – 4 = 0

Y = 2 or y= 4

Y = 2, 4

Since 2 ^x = y, and y = 2

                2 ^x = 2

                2 ^x = 2 ¹

                x = 1

Since 2 ^x = y and y = 4

                2 ^x = 4

                2 ^x = 2 ²

                N = 2

X = 1 and 2

b)            5 ^2x + 4 X 5 ^x+1 – 125 = 0

                (5 ^x) ² + 4 X (5 ^x X 5 ¹) – 125 = 0

                Let 5 ^x = p

                P ² + 4 X (p X 5) – 125 = 0

                P² + 4 (5p) – 125 = 0

                P² + 20p – 125 = 0

Then Factorise p² + 25p – 5p – 125 = 0

                P (p + 25) -5 (p + 25) = 0

                (p – 5) (p + 25) = 0

                P – 5 = 0 p + 25 = 0

                P = 5 or p = – 25

Since 5^x = p,        p = 5

                5 ^x  = 5 ¹

                X = 1

5x = -25 (Not simplified)               

1)            3 ^2x – 9 = 0

                (3 ^x) ² – 9 = 0

                Let 3 ^x  = a

                a ² – 9 = 0

                a ² = 9

                a = ±√9

                a = ± 3

                a = 3 or – 3

Since 3 ^x  = a,       when a = 3

                3 ^x  = 3 ¹

                X = 1

Since 3^x = a,        when a = -3

                3 ^x  = – 3 (Not a solution)          

EVALUATION

Solve the following exponential equations.

a)            2 ^{2x+ 1} – 5 (2 ^x) + 2 = 0

b)            3 ^2x – 4 (3 ^x+1) + 27 = 0

Reading Assignment : Further Maths Project Book 1(New third edition).Chapter 2 pg. 6- 10

WEEKEND ASSIGNMENT

1. Solve for X : (0.25) ^{X + 1} = 16 (a) -3 (b) 3 (d) 4 (d) -4
2. Solve for X : 3(3)^X = 27 (a) 3 (b) 4 (c) 2 (d) 5
3. Solve the exponential equation : 2^2X + 2^X+1 – 8 = 0 (a) 1 (b) 2 (c) 3 (d) 4
4. The second value of X in question 3 is (a) -1 (b) 1 (c) 2 (d) It has no solution
5. Solve for X : 10^-X = 0.000001 (a) 4 (b) 6 (c) -6 (d) 5

Theory

Solve the following exponential equations

(1.)  (3^X)² + 2(3^X )– 3 = 0    (2).  5^2X+1– 26(5^X) + 5 = 0

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                                                             WEEK THREE

TOPIC: LOGARITHM- SOLVING PROBLEMS BASED ON LAWS OF LOGARITHM

CONTENT:

• Logarithm of numbers (Index & Logarithmic Form)
• Laws of Logarithm
• Logarithmic Equation
• Change of Base

Logarithm of numbers (Index & Logarithmic Form)

The logarithm to base a of a number P, is the index x to which a must be raised to be equal to P.

Thus if P = a^x, then x is the logarithm to the base aof P. We write this as x = log a P. The relationship log_aP = x and a^x=P are equivalent to each other.

a^x=P is called the index form and log_aP = x is called the logarithm form

Conversion From Index to Logarithmic Form

Write each of the following index form in their logarithmic form

a)            2⁶ = 64   b)            25^1/2 = 5                                c)  4⁴= 1/256

Solution

a)            2⁶ = 64

                Log₂ 64 = 6

b)            25^1/2 = 5

                Log₂₅5=1/2

c)            4^-4= 1/256

Log₄1/256 = -4

Conversion From Logarithmic to Index form.

a)            Log₂128 = 7         b)            log₁₀ (0.01) = -2

c)            Log_1.5 2.25 = 2

Solution

a)            Log₂128 = 7

                2⁷ = 128

b)            Log₁₀(0.01) = -2

                10^-2= 0.01

c)            Log1.5 2.25 = 2

1.5² = 2.25

Laws of Logarithm

a)            let P = b^x, then log_bP = x

                Q = b^y, then log_bQ = y

                PQ = b^x X b^y = b^x+y (laws of indices)

                Log_bPQ = x + y

:.             Log_bPQ = log_bP + Log_bQ

b)            P÷Q = b^x÷by = b^x+y

                Log_bP/Q = x –y

:.             Log_bP/Q = logbP – logbQ

c)            Pⁿ= (b^x)ⁿ = b^xn

                Log_bpⁿ = nb^x

:.             LogPⁿ = logbP

d)            b = b¹

:.             Log_bb = 1

e)            1 = b⁰

                Logb1 = 0

EXAMPLE – Solve each of the following:.

a)            Log₃27 + 2log₃9 – log₃54

b)            Log₃13.5 – log₃10.5

c)            Log₂8  +  log₂3

d)            given that log₁₀2 = 0.3010 log₁₀3 = 0.4771 and log₁₀5 = 0.699 find the log₁₀64 + log₁₀27

Solution

a)            Log₃27 + 2 log₃9 – log₃54

= log₃ 27 + log₃ 9² –log₃54

= log₃ (27 X 9²/54)

= log₃ (27¹ X 81/54) = log₃ (81/2)               

= log₃ 3⁴/log32

= 4log₃ 3 – log₃ 2

4 X (1) – log₃ 2 = 4 – log₃ 2

= 4 – log₃ 2

b)            log₃ 13.5 – log₃ 10.5

                = log₃ (13.5)- Log310.5 = log₃ (135/105)

                = log₃ (27/21) = log₃ 27 – log₃ 21

                = log₃ 3 ³ – log₃ (3 X 7)

= 3log₃ 3  – log₃ 3 -log₃7

= 2 – Log₃ 7

c)            Log₂8 + Log₃3

                = log₂2³+ log₃3

                = 2log₂2 + log₃3

              2+1=3

d)            log₁₀ 64 + log₁₀ 27

                log₁₀ 2⁶ + log₁₀3³

                6 log₁₀ 2 + 3 log₁₀ 3

                6 (0.3010) + 3(0.4771)

                1.806 + 1.4314 = 3.2373.

EVALUATION

1.            Change the following index form into logarithmic form.

(a)          6³= 216 (b) 3³ = 1/27       (c) 9² = 81

2.            Change the following logarithm form into index form.

(a)          Log₈8 = 1              (b) log _½¼ = 2

3.            Simplify the following

a)            Log₅12.5 + log52

b)            ½ log₄8 + log₄32 – log₄2

c)            Log₃81

4.            Given that log 2 = 0.3010, log3 0.477

                Log₁₀5 = 0.699, find the log₁₀ 6.25 + log₁₀

Logarithmic Equation

Solve the following equation

a)            Log10 (x² – 4x + 7) = 2

b)            Log₈ (r² – 8r + 18) = 1/3

Solution

a)            Log₁₀ (x² – 4x + 7) = 2

                X² – 4x + 7 = 10² (index form)

                X² – 4x + 7 = 100

                X² – 4x + 7 – 100 = 0

                X² – 4x – 93 = 0

Using quadratic formular

                = – b ±√b²– 4ac

                   2a                       

                a = 1, b = -4, c = – 93

x = – (- 4) ± √(- 4) ² – 4 X 1 X (- 93)

                                2 X 1

                = + 4 ± √16 + 372

                             2 

                = + 4 ± √388/2

                = x = 4 +√ 388/2 or 4 – √388/2

                 x =  11.84 or x = – 7.85

2)            Log₈ (x² – 8x + 18) =8^1/3

                X² – 8x + 18 = 8^1/3

                X² – 8x + 18 = (2)^3X1/3

                X² – 8x + 18 =2

                X² – 8x 18 – 2 = 0

                X² – 8x + 16 = 0

                X² – 4x – 4x + 16 = 0

                X(x – 4) -4 (x – 4) = 0

                (x – 4) (x – 4) = 0

                (x – 4) twice

                X = + 4 twice

Change of Base

                Let log_bP = x and this means P = b^x

                Log_cP = log_cb^x = x log_cb

                If x log_cb = log_cP

                X = log_cP

                      Log_c b

:.             Log_cP = log_cP

                                     Log_cb

Example :  Shows that log_ab X log_ba = 1

                Log_ab = log_cb

                                     Log_ca

                Log_ba = log_ca

                                      Log_cb

:.             Log_ab X log_ba = log_cb  X log_ca

                                                                  Log_ca  +  log_cb= 1

EVALUATION

Solve the following logarithm equation.

                Log₃ (x² + 7x + 21) = 2

                Log₁₀ (x² – 3x + 12) = 1

Reading Assignment : Further Maths Project Book 1(New third edition).Chapter 2 pg. 8- 10                        

ASSIGNMENT

1. If log81/64 = x, find the value of x (a) 2 (b) 1 (c) -3 (d) -4.

2)            Solve 9^{(1 – x)} = (1/27) ^x+1 (a) -5 (b) -1 (c) 1 (d) ½

3)            Simplify log₇ 49 (a) 1/7   (b) 2  (c) 7  (d) log 1/7

4)            Solve the equation log₂16 = x  (a) 8  (b)  4 (c)  2   (d) 2

5)         Convert 5² = 25 into logarithm form (a) log₅25 = 2 (b) log ₂₅5 = 2 (c) log₂25 = 5 (d) None of the above

Theory

(1)          Find the value of x for which log₁₀ (4x² + 1) -2 log₁₀ x – log₁₀ 2 = 1 is valid.

(2)          Solve the logarithmic equation: Log₄ (x² + 6x + 11) = ½

TOPIC: LOGARITHM OF NUMBERS LESS THAN ONE

CONTENTS

• Standard forms
• Logarithm of numbers greater than one
• Multiplication and divisions of numbers greater than one using logarithm
• Using logarithm to solve problems with roots and powers (no > 1)
• Logarithm of numbers less than one.
• Multiplication and division of numbers less than one using logarithm
• Roots and powers of numbers less than one using logarithm

Standard Forms

Numbers such as 1000 can be converted to its power of ten in the form 10ⁿ where n can be term as the number of times the decimal point is shifted to the front of the first significant figure i.e. 10000 = 10⁴

Number                                                 Power of 10

1. 10²
2. 10¹
3. 10⁰
   1. 10^-3
   1. 10^-1

Note: One tenth; one hundredth, etc are expressed as negative powers of 10 because the decimal point is shifted to the right while that of whole numbers are shifted to the left to be after the first significant figure.

A number in the form A x 10ⁿ, where A is a number between 1 and 10 i.e. 1<A<10 and n is an integer is said to be in standard form e.g. 3.835 x 10³ and 8.2 x 10^-5 are numbers in standard form.

Examples : Express the following in standard form

1. 7853
2. 382
3. 0.387
4. 0.00104

Solutions

1. 7853 = 7.853 x 10³
2. 382 = 3.82 x 10²
3. 0.387 = 3.87 x 10^-1
4. 0.00104 = 1.04 x 10^-3

Logarithm of numbers greater than one   

Base ten logarithm of a number is the power to which 10 is raised to give that number e.g.

628000 = 6.28 x10⁵

628000 = 10^0.7980 x 10⁵

             = 10^{0.7980+ 5}

             = 10^5.7980

Log 628000 = 5.7980

          IntegerFraction (mantissa)

If a number is in its standard form, its power is its integer i.e. the integer of its logarithm e.g. log 7853 has integer 3 because 7853 = 7.853 x 10³

Examples: Use tables (log) to find the complete logarithm of the following numbers.

(a)  80030        (b) 8      (c) 135.80

(a)  80030    =             4.9033

(b)  8              =             0.9031

(c)   13580    =             2.1329

Multiplication and Division of number greater than one using logarithm

To multiply and divide numbers using logarithms, first express the number as logarithm and then apply the addition and subtraction laws of indices to the logarithms. Add the logarithm when multiplying and subtract when dividing.

Examples: Evaluate using logarithm.

1.    4627 x 29.3

2.    8198 ÷ 3.905

3.    48.63  x  8.53

              15.39

Solutions

1.    4627 x 29.3                         

                                        No            Log

                                        4627       3.6653

                                        X 29.3      + 1.4669

Antilog →   1356005.1322

                                                                 .     

                                \   4627 x 29.3 = 135600

To find the Antilog of the log 5.1322 use the antilogarithm table:

Check 13 under 2 diff 2 (add the value of the difference) the number is 0.1356. To place the decimal point at the appropriate place, add one to the integer of the log i.e. 5 + 1 = 6 then shift the decimal point of the antilog figure to the right (positive) in 6 places.

                                                       =    135600

2.            819.8 x 3.905                     

                                        No            Log

                                        819.8       2.9137

                                        3.905       0.5916

antilog →        209.9     2.3221

                                \   819.8 ÷ 3.905   =  209.9

3.            48.63 X 8.8.53

                        15.39

                                        No            Log

                                        48.63     1.6869

                         8.53       + 0.9309

                                                            2.6178

                         ÷ 15.39     – 1.1872

antilog →        26.95     1.4306

                                \48.63 ÷ 8.53   = 26.96

                                                  15.39

Evaluation:

1.    Use table to find the complete logarithm of the following:

(a)  183      (b) 89500     (c) 10.1300      (d) 7

2    Use logarithm to calculate.

3612 x 750.9

  113.2 x 9.98

Using logarithm to solve problems with powers and root

(nos. greater than one).

Examples:

Evaluate

(a)          3.53³                      (b)    4    40000        (c)    94100 x 38.2

                                                                                                5.68³ x 8.14 (2s.f)

Solution

No.                         Log_____

3.53³                      0.5478 x 3

44.00                     1.6434

\ 3.53³ = 44.00

(b)          4    40000

No.                         Log_____

4  4000  3.6021  ÷ 4

7.952                     0.9005

\ 4    4000   =   7.952

(c)           94100   x 38.2

                 5.683³   x 8.14

Find the single logarithm representing the numerator and the single logarithm representing the denominator, subtract the logarithm then find the anti log.

(Numerator – Denominator).

                No                                          Log

94100                                    4.9736 ÷ 2   = 2.4868

38.2                                        1.5821

Numerator                                                         4.0689 → 4.0689

5.68³                                      0.7543 x 3   = 2.2629        

8.14                                                                        0.9106

Denominator                                     3.1735  → 3.1735

7.859                                                                                                     0.8954

\94100  x  38.2   =  7.859              

                        5.68³ x 8.14

                                                                ~ 7.9 (2.sf)

Logarithm of number less than one.

To find the logarithm of number less than one, use negative power of 10 e.g.

                0.037     =             3.7 x 10^-2

                                                10 ^{0.5682 x 10-2}

                                                10 ^{0.5682 + (-2)}

                                                10^{-2 5682}

Log 0.037  =  2 . 5682

2     .          5682

Integer                 decimal fraction (mantissa)

Example: Find the complete log of the following.

(a)          0.004863     (b)  0.853    (c)  0.293

Solution

Log 0.004863      =             3.6369

Log 0.0853                           =             2.9309

Log 0.293                             =             1.4669

Evaluation

1.            Find the logarithm of the following:

(a)    0.064       (b)   0.002     (c)  0.802

2.            Evaluate using logarithm.

                95.3 x    318.4

                 1.29⁵ x 2.03                                                       

Using logarithm to evaluate problems of Multiplication, Division, Powers and roots with numbers less than one.

Examples:

1.            0.6735 x 0.928

2.            0.005692 ¸ 0.0943

3.            0.6104³

4.            4   0.000

5.            3  0.06642

Solution

1.            0.6735 x 0.928

                No.                         Log.___

                0.6735   1.8283

                0.928                     1.9675

                0.6248   1.7958

                \ 0.6735 x 0.928              =             0.6248

2.            0.005692 ¸  0.0943

No                          Log

                0.005692              3.7553

                ¸ 0.0943               2.9745

0.06037 2.7808

3.            0.6104³

                No                          Log_____

                0.6104³ 1.7856 x 3

                0.2274   1.3568

                \  0.6104³ =  0.2274

                \  0.005692  ¸   0.943  =  0.6037

4.            4   0.00083

                No.                                         Log._____

4  0.00083                            4.9191  ¸ 4

                0.1697                   1.2298

\ 4  0.06642  =  0.1697

5.            3  0.6642

                No.                                         Log.____________

3  0.6642                               2.8223 ¸ 3

                                                                2.1 + 1 + 0.8223 ¸ 3

                                                                3 + 1 .8223 ¸ 3

                                                                1 + 0.6074

                0.405                                     1.6074

    ³    0.6642           =             0.405    

Note:  3 cannot divide 2 therefore subtract 1 from the negative integer and

add 1 to the positive decimal fraction so as to have 3 which is divisible

by 3 without remainder.           

Evaluation:

1.            Evaluate

                5  (0.1684)³

2.            6.28 x  304

                                  981

Reading Assignment :Further Maths Project Book 1(New third edition).Chapter 2 pg.10- 16

 ASSIGNMENT

Use table to find the log of the following:

1.            900                         (a) 3.9542    (b) 1.9542    (c) 2.9542     (d) 0.9542

2.            12.34                     (a) 3.0899            (b) 1.089              (c)   2.0913             (d) 1.0913

3.            0.000197              (a) 4.2945            (b) 4.2945            (c)   3.2945             (d) 3.2945

4.            0.8                          (a) 1.9031            (b) 1.9031            (c)   0.9031             (d) 2.9031

5.            Use antilog table to write down the number whose logarithms is 3.8226.

                (a) 0.6646      (b) 0.06646     (c) 0.006646    (d) 66.46

Theory

Evaluate using logarithm.

1.            23.97  x     0.7124

                3.877 x 52.18 

2.            3  69.5² – 30.5²

Mathematics Elearning Notes – Edudelight.com

WEEK TWO (2)                                  DATE:

TOPIC: SURDS

COTENT:

• Rules of surds
• Basic Form of Surds
• Similar Surds
• Conjugate Surds
• Simplification of Surds
• Additional & Subtraction of Surds
• Multiplication and Division of Surds
• Rationalization of Surds
• Equality of Surds

Rules of Surds

 Surds are irrational numbers. They are the root of rational numbers whose value can not be expressed as exact fractions. Examples of surds are: √2, √7, √12, √18, etc.

1. √(a X b ) = √a X √ b
2. √(a / b )  = √a  / √b
3. √(a + b ) ≠ √a +  √b
4. √(a – b ) ≠ √a –  √b 

Basic Forms of Surds

      √a is said to be in its basic form if A does not have a factor that is a perfect square. E.g.  √6, √5, √3, √2 etc.  √18 is not in its basic form because it can be broken into √ (9×2) = 3√2. Hence 3√2 is now in its basic form.

Similar Surds

     Surds are similar if their irrational part contains the same numerals e.g.

1. 3√n and 5√ n
2. 6√2 and 7√2

Conjugate Surds 

    Conjugate surds are two surds whose product result is a rational number.

 (i)The conjugate of √3 – √5  is √3 + √5

    The conjugate of -2√7 + √3 is -2√7 – √3

     In general, the conjugate of √x + √y is √x – √y

     The conjugate of √x – √y = √x + √y

Simplification of Surds 

   Surds can be simplified either in the basic form or as a single surd.

   Examples 1. Simplify the following in its basic form (a) √45 (b) √98

Solution

(a) √45 = √ (9 x 5) = √9 x √5 = 3√5

(b) √98 = √ (49 x 2) = √49 x √2 = 7√2

Example 2. Simplify the following as a single surd (a) 2√5 (b) 17√2

Solution

(a) 2√5 = √4 x √5 = √ (4 x 5) = √20

(b) 17√2 = √289 x √2 = √ (289 x 2) = √578

Addition and Subtraction of Surds

 Surds in their basic forms which are similar can be added or subtracted.

Example:  Evaluate the following

(a)√32 + 3√8       (b) 7√3 – √75      (c) 3√48 – √75 + 2√12

Solution

•  (√32  + 3√8

          =√ (16 x 2) + 3√ (4 x 2)

          =4√2 + 6√2

          = 10√2

     (b) 7√3 – √75

= 7√3 – √ (25 x 3)

=7√3 – 5√3

=2√2

     (c) 3√48 – √75 + 2√12

          = 3√ (16 x 3) – √ (25 x 3) + 2√ (4 x 3)

          = 12√3 – 5√3 + 4√3

          = 11√3

EVALUATION

1. Simplify the following (a)5 √ 12  –  3 √ 18  +  4 √72  + 2  √75 (b) 3√2  – √32  + √50 + √98

2.  Multiply the following by their conjugate (a) √3 -2√5 (b) 3√2 + 2√3

Multiplication and Division of Surds

Example: Evaluate the following (a) √45 x √28    (b) √24 / √50

Solution

•  

 √45 x √28

= √ (9 x 5) x √ (4 x 7)

= 3√5 x 2√7

= 3 x 2 x √ (5 x 7)

= 6√35

(b)√24 / √50

          = √ (24 / 50)

          = √ (12 / 25)

          = √12 / √25

          = √ (4 x 3) / 5

          = 2√3 / 5

Surds Rationalisation

Rationalisation of surds means multiplying the numerator and denominator by the denominator or by the conjugate of the denominator.

Example: Evaluate the following     (a) 6 / √3     (b)     3

√3 + √2

Solution

•  6 / √3                                                                                 (b)        3

= 6 x √3                                                     √3 + √2

√3 x √3                                                = 3 (√3 – √2)

  = 6√3                                                       (√3 + √2) (√3 – √2)

 3                                                        = 3√3 – 3√2

  = 2√3                                                        (√3)2 – (√2)2

                        =         3√3 – 3√2

             3 – 1

=    3√3 – 3√2

1

   =  3( √3 -√2)

Equality of Surds

Given two surds i.e P + m and q + n if  P + m = q + n, then

P – q =  n  –  m the L.H.S

Of the equation is a rational number while the L.H.S and R.H.S can only be equal of they are both equal to zero (0)

                P – q = 0

                :. P = q and  n –  m = 0 i.e.

                N  =  m

Examples –  Find the square root of the following?

a)            7 + 2  10                b)  14  – 4  6

Solution

(a) Let the square root of 7 + 2  10  be   m +     n

(  m   +    n) ² = 7  + 2     10

M  +2 m n+ n = 7 + 2  10

M + n = 7 _____ (1)

2  m n  = 2  10

M n  =   10

Squarely both surds we have

M n =  10  _______(ii)

M    +  n = 7 ______ (i)

M n = 10   _______ (ii)

From equation (1)    m =  7 – n

Put m in (ii)   we have

                (7 – n) n = 10

7n – n² = 10

In sum; n² – 7n + 10  = 0

                n² – 2n – 5n + 10 =0

                n (n – 2) – 5 (n – 2) = 0

                (n -5) (n – 2) = 0

n = 5 or 2

m = 7 – 2, where n = 2

                m = 5,

m = 7 – 5 , when n = 5

                m = 2

                m= 5 or 2

The square root of 7 +  10 are  5 +  2 twice.

(b) Let the square root of 14 – 4   6   be   p  –   Q

The (   p  –   Q)² = 14 – 4    6

P –    2   pQ  + Q = 14 – 4   6

P +Q =  14 ……………………………(1)

-2    pQ      = 4     6

     -2                 – 2

PQ =   2     6      (squaring both sides)

PQ =        =   (2     6   )²

PQ = 4 x 6 ……………………………….. (11)

P +Q = 14 ………………………………… (1)

PQ = 24 ……………………………………… (11)

From equation……………… (1)   p = 14 –  Q

Sub for p in equation ………………… (11)

(14 –Q ) Q = 24

14 Q – Q ² = 24

In turn we have :

Q² – 14 Q + 24 = 0

Q² – 12Q– 2Q  + 24 = 0

Q (Q -12) – 2 (Q – 12)  = 0

Q = 2 or 12

if  p = 14 – Q ,when Q= 12

p = 14 – 12

p = 2

if  p = 14 – Q, when q = 2

p = 14- 2                                  

  = 12

12           –            2              =             2     3      –              2                              and  

2              –              12           =             2             –             2    3

EVALUATION

1.Express  3√2 – √3in the form √mwhere m and n are whole number.

                  2√3 – √2                    √n

2.Express            1   in the form p√5 + q√3, where p and q are rational numbers.

                   √5 +√3

Reading Assignment :Further Maths Project Book 1(New third edition).Chapter 3 pg.19-27

 ASSIGNMENT

1.Expand (3√2 – 1)(3√2 + 1)   (a) 16 (b) 20 (c) 17 (d) 24

2.Simplify √200 in its basic form (a) 10√2 (b) 5√4 (c) 2√10 (d) 2√50

3.Simplify 9/√3 (a) 3√2 (b) 3√3 (c) 1/3 (d) 2√2

4.Express 3√5 as a single surd (a) √40 (b) √55 (c) √45 (d) √35implify

5.Simplify √`128 – 4√8 (a) 0 (b) 1 (c) 2 (d) 3

Theory

1.Express  3√2 – √3in the form √mwhere m and n are whole number.

                  2√3 – √2                    √n

2.Express            1   in the form p√5 + q√3, where p and q are rational numbers.

                   √5 +√3

1 . Evaluate the following (a) 32^3/5 (b) 25^1.5 (c) (0.000001)²  (d) 343^2/3  (e) 19⁰

2 . Solve the following exponential equations (a) 2^x = 0.125 (b) 3^-x = 243 (c) 25^x = 625           (d) 10^x = 1/0.001 (e) 4/2^x =64^x

3 . Solve the following exponential equations (a) 2^2x -6(2^x) + 8 = 0 (b) 2^2x+1 -5(2^x) + 2 = 0

(c) 3^2x – 4(3^x+1) +27 = 0 (d) 3^2x – 9 = 0 (e) 7^2x – 2 X 7^x + 1 = 0

4 . Change each of the following index form to their logarithmic form (a) 2⁶ = 64                 (b) 3^-3 =1/27 (c) 25^1/2 =5 (d) 3⁵ = 243 (e) (0.01)² = 0.0001

5 . Change the following logarithmic form into index form (a) log₂128 = 7 (b) log_1/2(1/4) = 2 (c) log₇49 = 2 (d) log₅1/125 = -3 (e) log₅1 = 0

6 . Simplify each of the following (a) log₃27 + 2log₃9 –log₃54 (b) 1/2log₄8 + log₄32 – log₄2 (c) log₂√8 + log₃√3 (d) log_xx⁹ (e) log₅12.5 + log₅2

7 . Solve the following logarithmic equations (a) log₁₀(x² – 4x + 7) = 2                                   (b) log₈(x² – 8x + 18) = 1/3 (c) log₅(x² – 9) = 0 (d) log₄(x² + 6x + 11) = ½

8 . Use logarithm table to evaluate the following (a) (3.68)² x 6.705  (b)√0.897 x 3.536

                                                                                   √0.3581                       0.00249

9 . Simplify each of the following (a) 2√12 + 3√48 + √75 (b) 4√8 – 2√98 + √128                      (c) (3√2 – 1) (3√2 + 1 )

10 . Express                      1   in the formm√5 + n√3where m and n are rational numbers

                             3√5 + 5√3

Mathematics Elearning Notes – Edudelight.com

  WEEK 4                                            DATE………..

Matrices and determinants: concept, the basic operations of matrices. Identity matrices and equal matrices                 

MATRICES

Matrix is a rectangular array of numbers or elements in a row or column.

e.g ( a   b ),          a  

                             b

Elements arranged along the horizontal are called ROW. While elements arranged along the vertical is called COLUMN.

E,g 

               5        7                          5      7    row 1             

               8       12                         8     12    row 2

Column   1       2

NOTATION: A matrix is denoted by capital letter and the elements by small letters with reference to the position of the element. The position is defined in terms of the number of rows and columns.The first indicating the row, the second the column, thus:

              a₁₁   a₁₂     a₁₃b₁₁   b₁₂

 A =       a₂₁   a₂₂    a₂₃B =  b₂₁b₂₂                   

              a₃₁   a₃₂     a₃₃

Hence, a₂₁indicates the element in the second row and first column.

EVALUATION: Given the matrix,C=  6   -5   1 -3   write out the elements occupying the following, 

Positions.C₁₁, C_21, C₃₂, C₄₂, C₄₄ andC₃₄  2   -4   8   3  

  4   -7 -6   5

 -2   9   7 -1

Order of a matrix: A matrix can be identified or described by its order. In describing a matrix, the number of rows is stated first before the number of columns.

E.g             6   2   8     is a 2 x 3 matrix, i.e. order 2 by 3.

 5   7   3

BASIC OPERATION OF MATRICES:

Addition and subtraction of matrices: Two or more matrices can be added or subtracted when they are of the same order e.g 2 x 2, 3 x 3 and so on. The sum or subtraction is then determined by adding or subtracting corresponding elements.

If    A =  a₁₁  a₁₂      B =    b₁₁ b₁₂A + B = a₁₁ + b₁₁     a₁₂ + b₁₂ A – B = a₁₁ –b₁₁   a₁₂ –b₁₂

 a₂₁  a₂₂ b₂₁ b₂₂                   a₂₁+ b₂₁     a₂₂ + b₂₂a₂₁– b₂₁   a₂₂ – b₂₂

Example: Given the matrices below, find I A + B        II A – B      III B – A .

         A =     7   6   5                                      B =   12    -6   4

                    9   4   8                                                2    10   1

    I  A + B =   19    0   9             II          A – B = -5   12   1       III   B – A = 5   -12   -1

                       11   14   9                                          7   -6    7                          -7     6    -7

Addition of matrices is commutative, i.e A + B = B + A but matrix subtraction is not commutative,

A – B ≠ B – A 

MULTIPLICATION OF MATRICES:

1. Scalar multiplication: This is the multiplication of a matrix by a single number and it is done by multiplying each element in the matrix by the scalar.

e.g If C =  3 -8   12             find I 2C      II -3C

     4   5   7

   I  2C =  6    -16   24                     II – 3C =      -9    24     -36

  8      10   14                                         -12   -15   -21

2. Multiplication of two matrices: Two matrices can be multiplied together only when the number of columns in the first is equal to the number of rows in the second matrix.

If   A= a₁₁   a₁₂                 B =   b₁then A. B =   a₁₁b₁ + a₁₂b₂

 a₂₁   a₂₂                         b₂a₂₁b₁  + a₂₂b₂

 Matrix by matrix multiplication is not commutative, A.B ≠ BA

Example: Given the matrices A =    2    3     B =   4   1    find AB.

 5   7              8    9

                                   2   3        4   1

                       AB =             x

5   7       8     9

                     AB =     2×4 + 3×8   2 x1 + 3×9             8+24    2 +27

   = 

                                  5×4 + 7×8    5×1 + 7×9             20 +56   5+63

                    AB =   32     29

• 69

EVALUATION: 1. Find AC given the matrices A = -3    2    5          C =    4    7   2

0    1     8                    3   -5   1

 9    0    4

2. Given that A =    12     -8           B =   1     4                and C = 9    -5

 3        6                   2      8                            15   10

    Find  I.  A + B         II.   B – C      III.  2A – B + 3C

TYPES OF MATRICES:

Equal matrices: Two matrices are said to be equal if corresponding elements are equal and the matrices are of the same order.

Square matrix: Is a matrix having the same number of rows and columns. E.g 2 x 2, 3 x 3,and so on

Diagonal matrix: This is a square matrix with all elements zero except those on the main diagonal.

                                    2     0      0

                                    0     5      0

                                    0     0      8

Identity matrix: It is also called unit matrix and is a diagonal matrix in which the elements on the main diagonal are equal to one (1). It is denoted by I.

2 x 2 ,I =1       0                                      3 x 3,      I =   1    0  0

0      1                                                            0    1    0

       0    0   1

Null matrix: Is a matrix whose elements are zero. It is denoted by o. i.e 

                                                           0  0   0

                                                           0  0   0

                                                           0  0   0 

TRANSPOSE OF A MATRIX: The matrix obtained by interchanging the rows and columns of a matrix is called the transpose matrix. If B is the original matrix, its transpose is denoted by B^T.

    If    A =       2    5   ,     then A^T =     2   -8    7

                      -8   3                              5     3    4

                        7   4

EVALUATION: If A =    1   2    3    and   B = 7   10      find   I A.B and (A.B)^T

4   5    6                   8    11

9    12 

WEEKEND  ASSIGNMENT

Given that    A =    4     2   3                          B =   1    8   9

 5     7   6                                   3   5    4 

1.  Find A + B.     A.   5     10   12             B.      3    6    12

   8      12   10               2    -2   2

2.  Find A – B            A.       -3     -6   -6          B.  3     -6    -6

  -2     2    -2                2      2     2

3. Find (A + 2B) ^T                       A.   6     18    21      B.     6   11              C.       6     18

                      11   17   14              18   17                        21    11

                           21    14                       17    14

4. Find   A² – 4I

5. Find BA.

THEORY

1. Given that      P =    1     5                   find 2p² – 3p + 5I

                         -4    2

2.                            1      3     2                               2      4    – 3                             3      1   6

 If P =   8     -4    4                 Q  =        3      8      4                  R =      4      3   2

                7       3    5                               -1     3      6                              2   – 1    1

 Find (a) 5P + 2Q    (b) 4Q – 2R    (iii) 2P + Q + 3R      (iv) PR

Completing the square

To make a given expression a perfect square, the quantity to be added is the square of half of the  coefficient  of x ( or whatever letter is involved).

Examples: 

In each of the following, add the term that makes the given expression into a perfect square , then write the  result as the square of a bracketed expression.

1.  g²   – 4 ²/3 g

2. k²  – ¹¹/3 k

3.   m²  + 3mn.

Solutions.

g² – 4 ²/₃ g

the coefficient  of g is – ⁴²/3 = –¹⁴/3

half of –¹⁴/₃  = ½  x  –¹⁴/₃   =-⁷/₃

Square of half of coefficient of g = (-⁷/3)² = 49 (  + 5⁴/₉)

                                                                         9

:. 49 must be added to the given expression to make it a perfect square

     9

:. g² -4 ²/₃  + ⁴⁹/₉  = ( n – ⁷/₃)

k²  – 1 ¹/₃ k. the coefficient of k  is ……..¹¹/3 – –⁴/₃

Half of –⁴/₃  = ½  x –⁴/₃  = –²/₃

Square of half of coefficient of k =  ( –²/₃)²  = + ⁴/₉.

:. ⁴/9 must be added to the given expression to make it a perfect square 

:.  K²  – 1 ¹/₃ k  + ⁴/₉  =  ( k – ²/₃)²

1. m ^{c xx}  + 3mn 

the coefficient  of m is 3n .

half of + 3n =¹/₂ x + 3n  = + 3n

                                                2.

EVALUATION.

In each of the following add the term that makes the given expression into a perfect square . Write the result as the square of a bracketed expression

1b²  – ⁴/5b 

2.u²  – 1 ³/5 u

WEEKEND ASSIGNMENT

1.Find the value of k such that x2 + 5x +k is a perfect square.

   (a) 2 ½                                (b) 4 ¼   ( c) 6 ¼         ( d) 25  (e) 100.

2.  What must be added to n² + 1 1/s n to make it  a perfect square?

     (a)  ³/₅                              (b)  1 ¹/₅    ( c)  ¹⁶/ ₉              ( d)  ⁹/₁₆    ( e ) 1 ¼

3.  Solve the equation

   ( x + 1 ¼ )²  = 1 ⁹/₁₆

(  a)  2 ½, 0 )   (b) (0,2) ( c )  ( 0, –  )       ( d )  ( 0, -2 ½ )    (e) ( 3, 1 ½ ).

4.Solve the equation

  ( x +¹/₃ )²  =⁴/₉

( – ¹/₃, 1)                  (b) (1,  ¹/₃ )         ( c) (2, ²/₃ )     (d ) (-²/₃, 3)   (e )(¹/₃, -1).

5.  What must be added to v2 – 3/4 v to make it a perfect square?

                (a) ⁹/₆₄                   (b) ³/₈                    ( c) –³/₈  (d) 7 ¹/₉    (e)  ²/₉

Theory

In each of the following, add the term that makes the given expression into a perfect square.  Then write the result as the  square of a bracketed expression:

ii.   u² – 1 ³/5u

2.  a² – 6ad

Mathematics Elearning Notes – Edudelight.com

TOPIC: SOLUTION OF QUADRATIC EQUATION & SYMMETRIC PROPERTIES OF THE ROOT OF QUATION EQUATION

CONTENT

• Method of Factorization
• Completing the square method.
• Quadratic formula
• Sum & Product of Roots of a Quadratic Equation
• Symmetric Properties of Roots

Method of Factorization

A quadratic equation is an expression of the form ax² + bx + c = 0 in which a, b & c are numerals; and also the highest power of x is 2 & that the power of x will neither be fractions nor negatives. Quadratic equations can be solved using the method of factorization, completing the square, quadratic formula& graphical method

Steps in solving quadratic equation: (1)examine the middle term whose power of x is 1. (2) Find the product of the first & last term. (3) Find two terms whose sum is equal to the middle term & product is equal to the value of the product of the first & last term (4) Replace the middle term by two the two terms in step 3. (5) Factorize the first two & last two terms (6) equate the linear factors to zero to find the value of x.

Example – Solve by factorization: X² + 7X + 10 = 0 

Solution

X² + 7X + 10 = 0 

X² +2X + 5X + 10 = 0

X(X  + 2) + 5(X + 2) = 0

(X+2) (X + 5) = 0

X + 2 = 0

X = -2        OR

X + 5 = 0

X = -5

Hence X = -2 or -5

WEEK 5                                                                                  

Factorization of Quadratic Expressions.

Factorize the following:

1. 6a² + 15a + 9
2. 6a² – 19ax  – 36x²
3. (5x – 1) ( x – 3)  – ( x – 5)  ( x – 3)
4. 35 – 2b  – b²
5. x² – y² + ( x + y ) ²
6. 25a²  – 4  ( a – 2b ) ².

Solutions.

6a2 + 15x  + 9.

Since 3 is a common factor to all the terms, first take out 3 as the common factor:

6a² + 15a  + 9  = 3 (2a2+ 5a + 3)

 = 3 (2a² + 2a + 3a + 3)

 = 3 (2a (a+1)  + 3 (a+1)

= 3 (  (a+1)   (2a+ 3)).

Hence.

6a² + 15a + 9 = 3   (a+ 1)  (2a + 3)

  = 3 (a+1)  ( 2a + 3)

2. 6a² – 19ax – 36x²

1^st step:  Find the product of the first and last terms.

                6a² x -36x² = -216a²x²

2^nd step:  Find two terms such that their products is – 216a2x and their sum is -19ax ( the middle term).

Factors of -216a²x² sum of factors .

a.  +27ax and -8ax   + 19ax

b.  -27ax and +8ax  – 19ax

c.   +9ac  and -24ax    – 15ax

d.  -9x  and 24ax  + 15ax.

Of these only b gives the required result.

3^rd step:  Replace  -19ax in the given expression by -27ax and 8ax then factorize by grouping:

6a² – 19ax – 36x²

= 6a² – 27ax  + 8ax – 36x²

= 3a (2a – 9x)  + 4x (2a-9x)

=(2a – 9x ) ( 3a + 4x)

hence,

6a² – 19ax – 36x² = (2a -9x) (3a + 4x)

• (5x – 1 (x -3)  – (x -5)(x – 3)

= (x – 3)   ( 5x -1)  – (x – 5)           

= ( x -3)    ( 5x – 1  – x + 5 )

=  ( x – 3)  ( 5x  – x + 5 – 1)

=  ( x – 3)   ( 4x + 4)

=  ( x – 3)  + ( x + 1)

= ( x -3)  4(x+1)

=  4(x-3) (x+1)

     4.  35 – 2b –b²

  or  – b² – 2b + 35

   = -b2 – 7b  + 5b  + 35

  = -b (b+7)  + 5 (b + 7)

  =  (b+7)  ( -b + 5)

  =  (b+7)  ( 5-b)

or  (7+b)   ( 5 – b)

5.  x² – y² + (x + y)²

Since

X² – y²  = (x)² – (y)²

= ( x + y)  ( x –y)

then x² – y² + (x + y)²  = (x + y)(x-y) + (x+ y)²

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

= (x + y  ( x –y  + x + y)

= ( x + y ) ( x + x

= (x + y)  (2x)

(x +y ) (2x)  = 2x ( x +y)

EVALUATION.

Factorize the following

1. m² – 15mm – 54n²
2. 8a² – 18b²
3. If 17x = 37 5²  – 356²

Find the value of x

ASSIGNMENT

Factorize the following :

1. a² + 5ab – 36b²
2. r² – 25
3. 10p² – 41p  – 45
4. 12m² – 4mn -5n2
5. 12a²b²  + 11ab – 5.

Theory.

Factorize the following

1. 8c² – 14c – 9
2. x²y² – xy – 30.

If I is a root of the equation 5x² + kx – 3 = 0

Find the other root.

3.  Given that ½ and -3 are the roots of the equation 0 = ax² + bx + C, find a, b,c, where a, b and c are the least possible integers.

Solutions

1. Since 2 ½ is a root of the equation, then (x – 2 ½ ) is a factor, similarly, if -1 is a root of the equation then (x – (1-1) is a factor. i.e (x + 1) is a factor.

Thus, the required equation is : (x – 2 ½ ) ( x + 1)  = 0

                                                           x (x + 1) – 2 ½  ( x + 1)  = 0

                                                         x² + x  – 2 ½ x  – 2 ½   = 0

                                                         x² – 1 ½ x  – 2 ½  = 0.

                                                         X² – ³/2 x  – ⁵/2 =0

                                                         2x² – 3x – 5 =0

2.  Since I is a root of the given equation

                                                 5x² + kx – 3 = 0

then it must satisfy the given equation.

Substitute I for x,

5(1)² + k(1)  – 3 = 0

5 x 1 + k – 3 = 0

5 + k – 3 = 0

\5-3 + k = 0

2 + k = 0

k = 0-2

k = -2.

Hence the given equation becomes

5x² – 2x – 3 = 0

by factorization method, this quadratic  equations can be solved as follows :

5x² – 2x – 3 = 0

5x² – 5x + 3x – 3 = 0

5x (x -1)  + 3  ( x -1)  = 0

(x-1) ( 5x + 3)  = 0

(x -1) = 0 or 5x + 3 = 0

i.e. x = 0+ 1 or 5x = 0-3

x = 1 or 5x = -3

x = 1  or x = –³/5.

Hence the other roots of the given quadratic equation is x = ³/₅.

2. Since the given roots of the equation 0 = ax² + bx + C are ½ and -3, then the factors are (x – ½ ) and (x – c – 3)

i.e. ( x – ½ ) and   ( x + 3)

Thus, the required equation is

(x – ½ ) ( x + 3)  = 0

x(x+ 3)  – ½ (x  + 3)  = 0

x² + 3x – ½ x  – 3/2  = 0

2x² + 6x – x  – 3  = 0

2x² + 5x – 3  = 0

Comparing this with the given equation ax² + bx + C = 0

Then

A= 2, b= 5, c = -3.

EVALUATION

I.  7 and -3 are the roots of the quadratic equation  x² +kx -21 = 0,what is the value of k?

2.  -5 is a root of the 

 ASSIGNMENT

1.Find the quadratic equation whose roots are 2 and -3.

    (a) (x² + x -6    (b) x² – x -6    (c  ) x² + 5x – 6   (d)  x² – 5x + 6   (e) x² + 3x – 6.

2.  (2x + 3) is a factor of 6x² +  – 12.  The other factor is ………….

     (a) (x + 6)   (b) (2x – 3)   (c)  (3x + 4)  (d) (3x – 4)  (e) (4x – 9).

3.  Find the roots of the equation x2 + 12x – 28  = 0  the

Examples

1. The electrical resistance R ohms of a wire varies directly as the length cm and inversely as the square roots of the diameter d cm3.
2. Express d in terms of l,R and the constant of variation k.
3. Find the value of d, correct to 2 decimal places, when 1 = 15cm, R = 0.13ohms and k = 1.25 x 10-3.
4. V varies jointly as the square of x and inversely as y, if V =18 when x = 3 and y =4, find V when x = 6 and y =9.

Solutions.

The greater of the two roots is …………………………………

(a) -14                   (b) -2                     (c ) 2        (d)   14

4.  What is the product of the roots of the equation x² -3x + 2 =0?

                (a) -3                     (b) -1                     (c) 2                       (d ) 3         (e)5

5.  If x² – 10x -24 =0, the sum of the roots of the equation is ………….

                (a) – 10         (b) 10      (c ) -24  (d) -2

THEORY

2. find the quadratic equation whose roots are (3 + √5).
3. If 1 is  root of  the equation 5×2 + kx – 3 =0, find the other root.

Quadratic equation n² – 8n – 65 =0, what is the other root of the equation?

Topic: Solution of Quadratic Equation by Factorization Method

Examples:

Solve the following quadratic equations by the factorization methods.

• m² = 11m + 42
• 2x² – x – 21  =0
• 2p² + 11p = 30.

Solutions.

(1 ) m² = 11m + 42

then m² – 11m – 42 =0

1^st step:  Find the product of the first and last  terms :

m² x – 4² =-42m²

2^nd step:find two terms such that their product is -42m2 and their sum is -11m ( the middle term).

Factors of -42m² sum of factor s

(a)  + 3m and – 14m                        – 11m

(b)  – 6m and +7m                            + 11m

( c) -6m and + 7m                             +m

(d) +6m and -7m                              -m

Out of these, only (a) gives the required result. 

3^rd step:  Replace -11 in the given expression by +3m and -14m.  then factorise by grouping:

m² -11m -42 = 0.

M² + 3m  – 14m – 42 =0

M(m+ 3) (m – 14) = 0

M + 3 = 0 or m – 14 =0

i.e m = 0 -3, or m = 0 + 14.

M  = -3 or m = 14.

2.  2x² –x -21 =0

1^st Step:  Product  of first and last terms = 2x² x -21

                = – 42x²

2^nd step:  Find two terms such that their product is 42x² and their sum is –x (the middle term).

Factors of-42x²                  sum of factors

(a) + 2x and -21x                               -19x

(b) -2x and 21x                  +19x

( c)-3x and + 14x                               + 11x

(d) + 3x and -14x                              -11x

( e) +6x and -7x                 -x

(f)   -6x and + 7x                                +x

out of these, only (e)gives the required result.

3^rd step:  Replace –x in the given expression by +6x and -7x. then factorize by grouping

2x²  – x -21  =0

2x²  +6x -7x – 21 = 0

x ( x + 3)  -7 ( x + 3 ) = 0

( x + 3) ( x-7)  =0

either x + 3  = 0 or 2x -7 =0

i.e x = 0-3  or 2x = 7

x = -3  or x =⁷/2

x = -3  or x = 3 ½ .

3.  2p²  + 11p  = 30.

    2p² + 11p – 30 = 0

1^st step: Product of the first and last terms

  2p² x -30  =- 60p²

2^nd step: find two terms such that their products is -60p² and their sum is +11p (the middle term )

Factors of +11                    sum of factors

(a) + 6p and – 10p            -4p

(b) -15p and +4p               +4p

( c) -15p and +4p              -11p

(d) +15p and -4p               +11p.

Out of these, only (d) gives the more required solution. Replace + 11p by +15p and -4p in the  given expression .

2p² + 15p -4p -30 = 0

p ( 2p + 15) – 2 (2p + 15) = 0

(2p + 15) (p – 2) = 0

either 2p+ 15  = 0 or p -2 = 0

2p  = 0 – 15 or p = 0 + 2

2p = -15 or p = 2

p = -15 or p = 2

p = –¹⁵/2 or p = 2

p = -7 ½  or p = 2.

EVALUATION.

Solve the following quadratic equations by factorization method.

• 4e²  – 20 e + 25 = 0
• 4a² – 11a = 3

 ASSIGNMENT.

1. If x²  – 10 x – 24  = 0, then x = 12 or ……………

                (a) -3                     (b) -2                     (c)  -1    (d) 1     (e) 2

2.  if x² + Kx  + 16/9 is a  perfect square, then K = ………………..

                (a) ¹/3     (b)  ²/3       (c)  ⁴/2            (d) ⁸/3            (e)  ¹⁶/3

3.   What is the sum of the roots of the equation

                x² -3x +2 =0?

                (a) – 3                   (b) -1   ( c) 2    (d )3    (e)   5

4. Find the roots of the equation

                x² + 12x – 28 = 0

The greater of the two roots is ………….

(a) -14         (b) -2        (c ) 2(d)7   (e ) 14.

5. What are the factors of 6×2 + x – 12?

A.(2x+ 3)( x + 6)              B. (2x +3) ( 2x – 3)        C. (2x +3)(3x + 4)   D. (2x + 3) (3x -4)

E. (2x + 3 ) ( 4x -9).

THEORY.

Solve the following equations”

1.  4y²  – 28y + 49  =0

2. 2x² + 11x  + 5 = 0.

Completing the square method

Given a quadratic equation,

a  +bx =0 (where a, b and c are constants)

a  +bx =-c (subtract c from both sides)

 +  (divide both sides by a)

The coefficient of x=

Divide  by 2 =

Square  and add to both sides i.e

 +  =  i.e.

(X +  =  or

X +

X +

X =

X = . Q.E.D.

4Y + 4 = 0

SOLUTION

4Y =  4  (Re-arrange the equation)

 (Divide through by coefficient of )

4Y+ (  = . Half Y, square it and add it to both sides.

 =

(Y  = 0  Y 2 =

Y =  0 +2  Y = +2 or 2

OR using method of difference of two squares

 (Y 2)(Y+2) = 0

Y  2 = 0 or Y + 2 = 0

Y = 2 or Y =  2.

Example –  Solve the equation using completing the square method.

(1)          _X² – 8x + 3 = 0

(2)          3x² – 5x – 7 = 0

Solution

1.            x² – 8x + 3 = 0

                Step 1: Take C to the other side x² – 8x = -3

Step 2: Divide through by the coefficient of x²

                x² – 8x = -3

                 1    1      1

Step 3: Divide the coefficient of x by 2, then square and add it to both sides.

Co efficient of x = 8

                :               8 = 4

                                2

                                4² = 16

=x² – 8x + (-8)² = -3 + (- 8)²

                                                2                              2

x² – 8x + (-3)² = -3 + (-4)²

(x – 4)² = -3 + 16

(x – 4)² = 13

x – 4 = +   13

x = +4 + 13

x = 4 + 3.61

x = 4 – 3.61          or            4 + 3.61

                0.39                        or            7.61

2.            3x² – 5x – 7 = 0

                3x² – 5x = +7

                3x² – 5x = 7

                 3             3   3       

                x² – 5x = 7

                       3          3

                x² – 5x + (-5)² = 7 +      (-5)²

                       3            6            3                6

                (x – 5)² = 7 + 25

                       6            3    36

(x – 5)² = 84 + 25

                                        36

                (x -5)²    = +109

                     6                    36

x = 5 + +109

                      6                6

                x – 5 = +10.44

                     6           6

x – 5 = +10.44

                     6           6

                x = 5 + 10.44   or                5 – 10.44

                                  6                                 6

                x = 15.44              or            -5.44

                                6                                 6

                x = 2.5.573           or            – 0.906

                   = 2.57 or            – 0.91

                m = 7 + 2.24

                                   2

                m = 7 + 2.24   or 7 – 2.24

                                   2                                2

                m = 9.24               or            5.24

                                2                                2

                m = 4.62 or 2.62

2.            3x² + 7x -3 = 0

                3x² + 7x = 3

                3x² + 7x = 3

                 3             3     3

                x² + 7x = 1

                       3

                x² + 7x + (7)² = 1 + (7)²

                       3              6                          6

                (x + 7)² = 1 + 49

                       6                          36

                (x + 7)² = 1 + 36 + 49

                       6                              36

                (x + 7)² = 85

                       6             36

                x + 7 = +85

                      6              36

x + 7 = +85

                      6             6

                x + 7 = +9.22

                      6                6

                x = -7 + 9.22 or -7 – 9.22

                                   6                             6

= 0.37 or 2.70

– 16 .22       6             6

or

X= 2.22         6

Examples

Solve the following equations by completing the square method.

1. n²  – 12n + 1  = 0
2. y²  +7y  – 30 =0
3. m² -7m  + 11  = 0

Solutions

1.  n²  – 12n  + 1  = 0

    n² – 12n  = 0 – 1

   n² – 12n = -1.

Add to both sides  the square of –¹²/₂   ( -6).        

n² – 12n  ( -6) ² = -1 +  (-6)²

(n – 6) ²  = -1 + (-6) ²

(n – 6) ² =+ 35.

Take square root of both sides.

         n-6  = ± √35.

i.e n = ± √35   + 6

  n = + √35  + 6 or n = -√35 + 6

   n = 6 + √35  or n = 6 – √35

 n = 6 + 5.916  or n = 6 – 5.916

 n = 11.916 or n = 0.084.

  i. e .  n = 11.92 or n = 0.08 to 2 decimal places.

2.y² +7y – 30  = 0

   y² + 7y  = 0 + 30

    y²  + 7y  = 30

Add to both side the square of ⁷/2

y²  +7y  + ( ⁷/2)²  = 30  + (⁷/2) ²

i.e. ( y +⁷/2) ²  = 30+ 49

                             1        4.

( y + ⁷/₂) ²  = 120 + 49

                              4

( y + ⁷/₂)²  =  169

                         4.

Take square root of both sides :

:.  Y + ⁷/₂  = ±  √169/4/

y + ⁷/₂  =  ± ¹³/₂

i.e y =  ± ¹³/₂ –⁷/₂

.e y = +¹³/2  – ⁷/2  or y = – 13- 7

                                               2.

y = +6   or y  = -20

         2                 2

y = +3  or y = -10.

3.m² – 7m + 11 = 0

m² – 7m  = 0 – 11

m² -7m  = -11

Add to both sides the square of –⁷/₂

M² – 7m ( –⁷/₂)² = -11 + (-⁷/₂)²

( m – ⁷/₂)²  = -11   +  49

                       4

( m – ⁷/₂)²  = -44 + 49

                             4.

( m – ⁷/₂)²  = ⁵/₄

Take square root of both sides:

m  –⁷/₂  = ± √⁵/₄

m= ±√⁵/₄  + ⁷/₂

m=± √+ ⁷/₂

i.e. m = + √5   + 7  or m = – √5 + 7

                  2       2                 2      2.

m = + 2.236 +7  or m = – 2.236  + 7

              2        2                  2         2

m=+1.118 + 3.5 or m = -1.118 + 3. 5

m = + 4.618 or m = 2.418

i.e m = 4.62orm = 2.42.

to 2 decimal places

EVALUATION.

Solve the following method of completing  the squares:

1. x² – 8x – 1 = 0
2. p² -10p  + 15 = 0

WEEKEND ASSIGNMENT

1.  The greater of the two roots of the equations (2x -5) (3x +10 ) = 0 is

                (a) – 50       (b) -10  ( c)  -3     (d)  2 ½   ( e) 5.

2. If one of the roots of the equation ( n-13)2 = 9 is 10, what is the other root?

    (a)   4 (b) 16  ( c) 22  (d) 68  ( e ) 94.

3.  Find the value of k such that x² + 5x +k is a perfect square.

                (a) 2 ½    (b) 4 ¼    ( c )  6 ¼    ( d)  25   ( e)  100.

4.  If x² – 10x = 24, what is the values of x ?

                (a) 12,-3)   (b)  ( -2, 12)    ( c)  -1, 12)   (d) ( 1, 12)   (e) (2,12).

5.  What must be added to a² + ³/5a to make it a perfect square?

                (a) 3/10          (b) 6/5    ( c) 9/100  (d) 100/9  ( e) 36/5

THEORY

Solve the following equations, giving answers correct to 1 decimal place.

• x² + 14x + 8 = 0
• 2p²  +9p  – 30 =0

Quadratic Formula

Quadratic formula is derived by using the method of completing the square. Considering the general form of quadratic equation:

                ax² + bx + c = 0                              X =   The required quadratic formula.

Example – Solve x² + 3x – 2 = 0 using formula method

+

a = 1                       6 = 3                       C = – 2

x =  -b       b² – 4ac

+

                         2a 

x = -3      3² – 4(i) (-2)

+

                          2(i)

x = -3     9 + 8

                   2

x = -3       17

+

                 2

x = – 3     4.12

            2

x = -3 + 4.12        or            -3 – 4.12

                  2                                   2

x = 0.56                 or            – 3.56

2.            m² – 7m + 11 = 0

                Solve using formula method.

                m² – 7m + 11 = 0

+

                a = 1                       b = – 7    c = 11

                m = – b          b² – 4ac

+

                                      2a

                m = – (-7)          7² – 4(i) (ii)

+

                                                2(i)

                m = 7        49 – 44

+

                                     2

                m = 7      5

+

                                2

                m = 7     2.23

                                    2

                m = 7 + 2.23        or            7 – 2.23

                                   2                             2

                m = 4.62                               or            2.62

EVALUATION

1.            3x² + 7x – 3 = 0  solve using formula method

2.            Using completing the square and formula method solve 3x² – 12x + 10 = 0

Sum & Product of Roots of a Quadratic Equation

The expression for sum & product of roots of quadratic equation is gotten from the general expression of quadratic equation. If the distinct roots are α and β, then

α + β = -b/a   (sum of roots)

αβ =c/a         (product of roots)

Example 1 – find the sum and products of 2x² + 3x –  1 = 0

Solution

2x² + 3x – 1 = 0

a =2, b = 3, c = -1

Let α and β be the roots of the equation, then

α+β= -b/a= -3/2

αβ = c/a = -1/2

Example 2 – find the sum and products of 3x² – 5x -2 = 0

Solution

3x² -5x -2 =0

a= 3, b= -5, c= -2

let α and β be the roots of the equation, then

α+β= -b/a = 5/3

αβ= c/a= -2/3

NB: The quadratic equation whose root are α and β  is

            (X – α )(X – β) = 0

X² – (α +β)X + αβ = 0

Example – Find the quadratic equation whose roots are 3 & -2

Solution

α=3 and β=-2

α+β = 3 + (-2) = 1

αβ = 3 x -2 = -6

X² – (α +β)x + αβ = 0

X² – (1)X + (-6) = 0

X² – X -6 = 0

Symmetric Properties of Roots

Certain relations involving α and β can also be determined from α+β andαβ even when we do not know α and β distinctly. Such relations are usually said to be symmetric. They are symmetric in the sense that if α and α are interchanged, either the relation remains the same or is multiplied by -1

Example – If  α ≠ β, determine whether or not each of the following is symmetric

(a) α+β (b) αβ (c) α² + β² (d) α² – β² (e) 3α +2β (f) α³ + β³

Solution

• α+β = β +α , therefore α+β is symmetric
  • αβ =βα, therefore αβ is symmetric
  • α² + β² = β² + α², therefore α² + β² is symmetric
  • α² – β² = – (β² – α²), thereforeα² – β² is symmetric
  • 3α +2β ≠ 3β + 2α since α ≠ β, therefore 3α +2β is not symmetric
  •  α³ + β³ = β³ + α^3,  therefore α³ + β³ is symmetric

EVALUATION

1. Find the quadratic equation whose roots are ½  and 5
2. Find the sum & product of roots of the equation 3x² – 5x – 2 = 0

 ASSIGNMENT

1. Solve for x ; x² – 9 = 0 (a) 3, 3 (b) 3, -3 (c) 9, 3 (d) -3, -3
2. Find the sum and products of 2x² + 3x –  1 = 0 (a) -3/2, ½ (b) 3/2 , -1/2 (c) -3/2, -1/2 (d) 3, 2
3. Solve for x : 6x² – 13x + 5 = 0 (a) 5/3, ½ (b) -5/3,1/3 (c) 5/3, -1/3 (d) 3/5, 1/3
4. Factorize 6x² + 5x -6 (a) (3x – 2) (2x + 3) (b) (3x + 2) (2x + 3) (c)(3x – 2) (2x – 3) (d) (2x -3) (2x+ 2)
5. Find the sum and products of x² – 4x – 3 = 0

Theory

1. Solve S = ut + ½ at²  using the completing the square method
2. Find the quadratic equation whose roots are ¾ and ½

WEEK 6

 MENSURATION – Surface Area and Volume of Spheres and Hemispherical Shapes

MENSURATION Is defined as a branch of Mathematics that deals with measurement, especially the derivation and use of algebraic formulae to measure the areas, volumes and different parameters of geometric. Examples are cylinder, cone, cuboid, rectangular prism, rectangular based pyramid, total surface area of cylinder, cone and their volume.

FORMULAE

SHAPES	AREA	SURFACE AREA (C.S.A)	TOTAL SURFACE AREA (T.S.A)	VOLUME
CONE	rl	rl	l + r)
CYLINDER		2 rh	2 rh+2
CUBOID		2(lb+lh+bh)		l.b.h
TRIANGULAR PRISM			Sum of areas of all surfaces	Area of cross section x height
RECTANGULAR PYRAMID			Sum of all four triangular faces + base area	x base area x height
CUBE			6
SPHERE		4πR2		πR3
HEMISPHERE		2πR2	3πR2	4πR3

SPHERE                                                                                    A HEMISPHERE

          Volume of a Sphere

A sphere is a solid in which all the points on the round surface are equidistant from afixed point, known as the center of the sphere. The distance from the center to the surface is the radius.

Volume of sphere =   where r is the radius.

How to find the volume of a sphere? What is the volume of air in the ball?

Volume of a hemisphere

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

Volume of hemisphere  where r is the radius

Spheres

What is a sphere?
A sphere is a solid with all its points the same distance from the center. The distance is known as the radius of the sphere. The maximum straight distance through the center of a sphere is known as the diameter of the sphere. The diameter is twice the radius.

How to find the volume of a sphere?

The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius.

The volume and surface area of a sphere are given by the formulas:

where r is the radius of the sphere.
Example:

Calculate the volume of sphere with radius 4 cm.

Solution:

Volume of sphere

We can also change the subject of the formula to obtain the radius given the volume.

Example:

The volume of a spherical ball is 5,000 cm³. What is the radius of the ball?

Solution:

Example: Find the volume of a sphere with a diameter of 14 cm.