CIRCLES 1

Find the value of y in the figure below. [Diagram not to scale]

Solution

Angle at the centre = 2 x (angle at the circumference

= 2 x (y)

=2y

Angle 2y and 8y are opposite angles. They add up to 360⁰.

Therefore 2y + 8y = 360⁰

10y = 360⁰

10y = 360⁰ 

10        10 

y = 36⁰          

Hence y = 36⁰          

TRY THIS………….

Find the value of w in the figure below 

SIMULTANEOUS EQUATIONS 2

Solve the following equations by substitution method.

x + y = 10

x + 3y = 18

Solution

x + y = 10 ----------- (i)

x + 3y = 18 ---------- (ii)

from equation (i)

x + y = 10

x = 10 - y ----------(iii)

substitute equation (iii) in (ii) above,

x + 3y = 18

(10 - y)  + 3y = 18  

10+2y = 18            [since –y+3y=2y]

2y = 18 - 10

2y = 8

2      2

y = 4

From equation (iii)

x = 10 - 4 ----------(iii)

x = 6

Hence x=6 and y=4.

TRY THIS………………………..

5x - 2y = 17

x  -  y = 2

SIMPLE INTEREST 4

Rayan deposited the amount of 8,000/= in a bank for 5 years and got a profit of 2000/=. Find the interest rate?

Solution

I = 2000/=, P = 8,000/=, T = 5 years, R = ?

I  = PRT

      100

2000  = 8,000 x R x 5

                      100

2000  = 8,000 x R x 5

                      100

2000  = 80 x R x 5

                      

2000  = 400R

          

⁵2000  = 400R

  400       400

Hence the interest rate was Tsh 5%

TRY THIS……………….

Kibonde deposited the amount of 12,000/= in a bank for 4 years and got a profit of 3000/=. Find the interest rate?

RADIAN TO DEGREES 1

Change ^2∏/90 radians into degrees.

Solution

Ө = 180⁰ x s                but s = ^∏/90

       ∏

Ө = 180⁰  x  2∏               

       ∏     90

Ө =  ² 180   x   2∏ ¹             

          ₁∏     90₁

Ө =  2 x 2

Ө =  4⁰

Hence ^2∏/90 radian is equal to 4⁰.

TRY THIS……………

Change ^5∏/20 radian into degrees.

COORDINATE GEOMETRY 2

Find the midpoint of a line from (9, 4) to (5, 10)

Solution

x1=9, x2=5, y1=4, y2=10.

Mid point = (x1 + x2, y1 + y2)

                         2             2

Mid point = (9 + 5,  4 + 10)

                        2          2

Mid point = (14, 14)

                     2    2

Hence midpoint = (7, 7)

TRY THIS……………………..

Find the midpoint of a line from (18, 8) to (6, 12)

FACTORIZATION 3

Factorize 2x² - 3x - 20

Solution

= 2x² - 3x – 20.  We split the middle term (-3x) to be (-8x + 5x).

= 2x² - 8x + 5x – 20                (-3x) = -8x + 5x.

= (2x² -8x) + (5x - 20)

= 2x(x - 4) + 5(x - 4)

= (2x + 5) (x -4)

Hence 2x² - 3x – 20 = (2x + 5) (x - 4) answer.

TRY THIS…..

Factorize 2x² - x – 28.

LOGARITHMS 3

Evaluate Log₂(1024 x 16).

Solution

= Log₂(1024 x 16)

= Log₂1024 +  Log₂16          (applying the product rule)

= Log₂2¹⁰ +  Log₂2⁴             (1024= 2¹⁰ and 16=2⁴ )

= 10Log₂2 +  4Log₂2       ( remember  Log_aa^c = cLog_aa )

= (10 x 1) +  (4 x 1)          ( remember  Log_aa = 1 )

= 10 + 4

= 14

hence Log₂(1024 x 16) = 14

TRY THIS………………

Evaluate Log₂(32 x 64). 

DIFFERENCE OF 2 SQUARES 2

Factorize 1-w²

Solution

We use difference of two squares a² – b² = (a - b)(a + b)

1-w² = 1²-w²       

        = (1 - w)(1 + w)

Hence 1-w² = (1 - w)(1 + w)

TRY THIS…………….

Factorize 9 - c²

MATRIX 4

PERCENTAGE PROFIT 2

A man got a profit of 200/= after selling an item for 2500/=. Find the percentage profit.

Solution

%’ge profit = Profit   X  100    where B. P. represents Buying Price.

                         B.P

%’ge profit = 200   X  100   

                       2500

%’ge profit = 8%   

                       

Hence Percentage Profit is 8%

TRY THIS……………………………

A man got a profit of 600/= after selling an item for 9000/=. Find the percentage profit.

LENGTH OF AN ARC 1

Find the length of an arc if the radius of the circle is 6cm.

Solution

L = ∏r

      180⁰

L = ∏ x  6

             180⁰

L = ∏ cm

      30

Hence length of an arc is ^∏/30 cm.

TRY THIS……………

Find the length of an arc if the radius of the circle is 18cm.

SIMPLE INTEREST 3

Dikshi deposited the amount of 12,000/= in a bank which gives an interest rate of 3% for 4 years. Find the simple interest she got.

Solution

I = ?,  P = 12,000/=, T = 4 years, R = 3%

I  = PRT

      100

I  = 12,000 x 3 x 4

            100

I  = 12,000 x 3 x 4

            100

I  = 120 x 3 x 4

          

I  = 1440/=

Hence the simple interest was Tsh 1440/=

TRY THIS…………………………

Harshita deposited the amount of 70,000/= in a bank which gives an interest rate of 3% for 6 years. Find the simple interest she got.

MAKING THE SUBJECT 2

If A = HQE make E the subject of the formula.

Solution

A = HQE

A   =   HQE

HQ      HQ

A   =   E

HQ     

Hence   E =      A 

                      HQ     

TRY THIS………………………..

If K = CDG make G the subject of the formula.

BINARIES 1

SETS 2

If n(A)= 30 , n(AuB) = 100 and n(AnB)=10, find n(B)

Solution

n(AuB) = n(A) + n(B) - n(AnB)

  100     = 30 + n(B)  – 10

  100     = 30 -10 + n(B) 

  100     = 20 + n(B) 

  100 - 20        = n(B)

  80        = n(B)

Hence n(B) = 80 answer

TRY THIS…………………………….

If n(A)= 60 , n(AuB) = 110 and n(AnB) = 30, find n(B).

AXIS OF SYMMETRY 2

Find the axis of symmetry for F(x) = 6x² + 20x + 8

Solution

a=6, b=10

Axis of symmetry = -b/2a

                               =  -(20)

                                    2 x 6

                               =    -20

                                      12

Hence the axis of symmetry is ^-5/3

TRY THIS…………………

Find the axis of symmetry for F(x) = 2x² + 40x + 3

SETS 1

If n(A)= 100 , n(B)= 160 and n(AnB)=150, find n(AuB).

Solution

n(AuB) = n(A) + n(B) - n(AnB)

             = 100 + 160 – 150

             = 260 – 150

             = 110

Hence n(AuB) = 110 answer

TRY THIS…………………………….

If n(A)= 130 , n(B)= 170 and n(AnB)=150, find n(AuB).

SIMULTANEOUS EQUATIONS 1

Solve the following equations by substitution method.

2x - y = 1

x + y = 8

Solution

2x - y = 1----------- (i)

x + y = 8---------- (ii)

from equation (ii)

x + y = 8

x = 8- y ----------(iii)

substitute equation (iii) in (i) above,

2x - y = 1

2(8- y) - y = 1

16 - 2y - y = 1  

16 - 3y = 1 (since -2y – y = -3y)

16- 1 = 3y  [after collecting like terms]

15 = 3y

15 = 3y

 3     3

5 = y

From equation (iii)

x = 8- y ----------(iii)

x = 8- 5

 x = 3

Hence x=3 and y=5.

TRY THIS…………………….

Solve the following equations by substitution method.

2x - 3y = 4

x + y = 7

SIMPLE INTEREST 2

Dikshi deposited the amount of 9000/= in a bank which gives an interest rate of 3% for 2 years. Find the simple interest she got.

Solution

I = ?,  P = 9,000/=, T = 2 years, R = 3%

I  = PRT

      100

I  = 9000 x 3 x 2

            100

I  = 9000 x 3 x 2

            100

I  = 90 x 3 x 2

          

I  = 540/=

Hence the simple interest was Tsh 540/=

TRY THIS………………….

Edmund deposited the amount of 60,000/= in a bank which gives an interest rate of 6% for 4 years. Find the simple interest she got?

MAKING THE SUBJECT 1

If W = MPE make E the subject of the formula

solution

W = MPE

 W   =   MPE

MP      MP

 W   =   E

MN     

Hence   E =      W 

                       MP

TRY THIS………………….

If A = REP make P the subject of the formula

ALGEBRA 2

DIFFERENCE OF 2 SQUARES 1

Evaluate 396² – 386²

Solution

396² – 386² = (396 + 386)( 396 – 386)

                     = (782)( 10)

                     = 7820

396² – 386² = 7820

TRY THIS…………….

Evaluate 308² – 108²

INEQUALITIES 2

If 10x + 8 ≥ 7 + 3x ≥ 9x – 13; find x

Solution

10x + 8 ≥ 7 + 3x and  7 + 3x ≥ 9x - 13

10x – 3x ≤ 7 - 8 and  7 +13 ≤ 9x - 3x

7x ≤ 1 and  20 ≤ 6x

7x ≤ 1  and  20 ≤ 6x

7       7         6      6

x ≤ ¹/7 and  ¹⁰/3 ≤ x

x ≤ ¹/7 and  x  ≥ ¹⁰/3

TRY THIS…………….

If 14x + 8 ≥ 7 + 3x ≥ 11x – 20; find x

CONGRUENCE OF TRIANGLES 1

Prove that the following triangle is isosceles. 

Solution

Given: triangle SRT with R produced to M.

Required to prove: ∆SRT is isosceles

Proof: SM = MT …………….… (given)

<SMR = <RMT = 90⁰ ………..(given)

RM = MR…………………..……..(common line)

∆UVT ≡ ∆WVT by SAS rule

RS = RT (Equal corresponding sides)

Hence ∆ SRT is isosceles (proved)!

TRY THIS………………..

Prove that the following triangle is isosceles. 

EXPONENTS 7

MATRIX 3

EXPONENTS 6

Simplify the following by writing in power form:

4w⁷⁰

  w¹⁰

Solution

= 4w^{70 - 10} 

= 4w⁶⁰

TRY THIS……………….

Simplify the following by writing in power form:

8 h⁵⁰

   h⁴⁰ 

INVERSE OF A MATRIX 1

ALGEBRA 1

Expand 8w(w – 5)

solution

= 8w(w – 5)

= (8w x w) – (8w x 5)

= 8w² – 40w answer

TRY THIS………..

Expand 13a(a+ 5)

FACTORIZATION 2

Factorize 49x² - 81y²

Solution

we use difference of two squares a² – b² = (a - b)(a + b)

49x²- 81y²  = 7²x² - 9²y²

                   = (7x)² - (3y)²

                   = (7x - 3y)( 7x + 3y)

Hence 49x² - 81y² = (7x - 3y)(7x + 3y)

TRY THIS………………………….

factorize 49x²- 121y²

FACTORIZATION 1

EXPONENTS 5

Write in expanded form: 4⁵

Solution

4⁵ = 4 x 4 x 4 x 4 x 4

TRY THIS……………….

Write in expanded form: 6⁴

COORDINATE GEOMETRY 1

Find the midpoint of a line from (13, 12) to (5, 10)

Solution

x1=13, x2=5, y1=12,y2=10.

Mid point = (x1 + x2, y1 + y2)

                          2             2

Mid point = (13 + 5,  12 + 10)

                         2          2

Mid point = (18, 22)

                      2     2

Hence midpoint = (9, 11)

TRY THIS……………………..

Find the midpoint of a line from (-4, 8) to (1, -2)

MATRIX 1

LOGARITHMS 2

If Log_ax = 44, find Log_a(¹/x).

Solution

= Log_a(¹/x)

= Log_ax^-1

= -1 x Log_ax    

= -1 x 44    (But Log_ax = 44)

= -44

TRY THIS………..

If Log_aw= 63, find Log_a(¹/w)

EXPONENTS 4

Simplify the following by writing in power form:

w⁷⁰

w¹⁰

Solution

= w^{70 - 10} 

=w⁶⁰

TRY THIS……………….

Simplify the following by writing in power form:

h⁵⁰

h²⁰