Wednesday, 31 December 2014

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 = 36        

Hence y = 36        

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 = 180x  2               
       ∏     90

Ө =  2 180    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  Logaac = cLogaa )

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





Tuesday, 30 December 2014

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

Monday, 29 December 2014

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

= 4w70 - 10 

= 4w60

TRY THIS……………….

Simplify the following by writing in power form:
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 49x- 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 + 512 + 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 Logax = 44, find Loga(1/x).

Solution

= Loga(1/x)

= Logax-1

= -1 x Logax    

= -1 x 44    (But Logax = 44)

= -44


TRY THIS………..


If Logaw= 63, find Loga(1/w)



EXPONENTS 4


Simplify the following by writing in power form:
w70
w10

Solution

= w70 - 10 

=w60

TRY THIS……………….

Simplify the following by writing in power form:
h50

h20 


QUADRATIC FORMULA 1





SIMPLE INTEREST 1


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 = 5%

I  = PRT
      100

I  = 9000 x 5 x 2
            100

I  = 9000 x 5 x 2
            100

I  = 90 x 5 x 2
          
I  = 900/=

Hence the simple interest was Tsh 900/=


TRY THIS……………………


Dikshi deposited the amount of 9000/= in a bank which gives an interest rate of 6% for 7 years. Find the simple interest she got?