Saturday 18 January 2014

LOGARITHMS B11

If log3(3x + 6)=3; find x

   Solution

log3(3x + 6)=3

 (3x + 6)=33   
                      
 3x + 6=27


3x = 27 – 6

3x =  21


3x =  21   
3        3

X = 7


Hence x = 7

FUNCTIONS B10

If F(x) = log3x, Find F(1/243)

Solution

F(x) = log2x

F(1/243) = log3(1/243)

F(1/243) = log3243-1

F(1/243) = log3(35)-1

F(1/243) = log33(5 x -1)

F(1/243) = log33-5

F(1/243) = -5log33

F(1/243) = -5 x 1

F(1/243) = -5


Therefore F(1/243) = -5

LOGARITHMS B10

If Logax = 1/2  Loga16+ 1/3 Loga 343 find x.

Solution

Logax = 1/2  Loga16+ 1/3 Loga343

Logax = Loga161/2  +  Loga3431/3   

Logax = Loga 4 +  Loga7

Logax = Loga (4 x 7)

Logax = Loga 28


x = 28

Hence x = 28

Friday 17 January 2014

RADICALS B7




LOGARITHMS B9

Evaluate Log2(4 x 16).

Solution

= Log2(4 x 16)

= Log24 +  Log216          (applying the product rule)

= Log222 +  Log224             ( 4= 22 and 16=24 )

= 2Log22 +  4Log22       ( remember  Logaac = cLogaa )

= (2 x 1) +  (4 x 1)          ( remember  Logaa = 1 )

= 2 + 4

= 6


hence Log2(4 x 16) = 6 

Wednesday 15 January 2014

FUNCTIONS B9

Find the axis of symmetry for F(x) = 6x2 + 5x + 8

Solution

a=6, b=5

Axis of symmetry = -b/2a

                               =  -(5)
                                    2 x 6

                               =    -5
                                     12



Hence the axis of symmetry is -5/12

PERCENTAGES B16

A man got a profit of 4000/= after selling an item for 20,000/=. Find the percentage profit.

Solution

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

%’ge profit = 4000   X  100   
                        20,000

%’ge profit = 20%   

                       

Hence Percentage Profit was 20%

CONGRUENCE OF SIMPLE POLYGONS B11


 If PQ = RS and PQRS is a straight line, Prove that <UPQ = <RST



Solution

Given: figure PQTSRU.
Required to prove: <UPQ and <RST are equal.
Proof: PQ= RS    (given)

TQ= RU (equal sides)  

PU = TS (equal sides)

PQ + QR = QR + RS  (equal sides)

HencePRU and TQS are congruent by SSS rule.

Therefore <UPQ = <RST Proved!

RADICALS B6


LOGARITHMS B8

If log5(3x + 1)=3; find x

   Solution

log5(3x + 1)=3

 (3x + 1)=53      
                   
 3x + 1=125

3x =125 – 1

3x = 124

3x = 124   
3        3

X = 124/3


Hence x = 124/3

FUNCTIONS B8

Find the axis of symmetry for F(x) = 3x2 - 24x + 3

Solution

a=3, b=-24

Axis of symmetry = -b/2a

                               =  -(-24)
                                    2 x 3

                               =    24
                                      6

                               =    4
                                      

Hence the axis of symmetry is 4


PERCENTAGES B15


A man got a profit of 2000/= after selling an item for 8000/=. Find the percentage profit.

Solution

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

%’ge profit = 2000   X  100   
                        8000

%’ge profit = 25%   
                       

Hence Percentage Profit was 25%

Tuesday 14 January 2014

FUNCTIONS B7

If F(x) = log2x, Find F(128)

Solution

F(x) = log2x

F(128)= log2(128)

F(128)= log227

F(128)= 7log22

F(128)= 7 x 1

F(1/32) = 7

Therefore F(128)= 7

PERCENTAGES B14


A man got a profit of 400/= after selling an item. Find the buying price if the percentage profit was 5%.

Solution

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

5% = 400   X  100   
          B.P.

B.P. x 5 = 40,000   

        
B.P. x 51 =   40,000   
  15                   5

B.P. = 8000

                       

Hence Buying Price was 8000/=

CONGRUENCE OF SIMPLE POLYGONS B10


If AB = CD and ABCD is a straight line, Prove that <BAF = <CDE


Solution

Given: figure ABEDCF.
Required to prove: <BAF and <CDE are equal.
Proof: AB= CD    (given)

EB= CF (equal sides)   

AF = ED (equal sides)

AB + BC = CD + BC  (equal sides)

HenceACF and EBD by SSS rule

Therefore <BAF = <CDE Proved!


RADICALS B5



LOGARITHMS B7


If log5(3x + 1)=2; find x

   Solution

log5(3x + 1)=2

 (3x + 1)=52      
                   
 3x + 1=25

3x =25 – 1

3x = 24

3x = 24   
3        3

X = 8


Hence x = 8

FUNCTIONS B6

Find the axis of symmetry for F(x) = 7x2 - 98x + 3

Solution

a=7, b=-98

Axis of symmetry = -b/2a

                               =  -(-98)
                                    2 x 7

                               =    98
                                     14

                               =    7
                                      

Hence the axis of symmetry is 7


PERCENTAGES B13

A man got a profit of 2000/= after selling an item for 8000/=. Find the percentage profit.

Solution

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

%’ge profit = 2000   X  100   
                      8000

%’ge profit = 25%   

                       

Hence Percentage Profit was 25%

SEQUENCE AND SERIES B4

Find the compound interest on sh 4000 invested at 5% per annum after 4 years.

Solution

P= 4000/=, R=5%, T=1, n=4, A4=?, I=?

An = P(1 + RT/100)n

A4 = 4000(1 + (5x1)/100)4

A4= 4000(1 + 5/100)4

A4 = 4000(100/100 + 5/100)4

A4 = 4000(105/100)4  

A4 = 4000(1.05)4  

We use logarithm tables to calculate A4

NO
         LOG
4000 = 4.0 X 103
1.05 = 1.05 X 100
3. 6021
0. 2119 x 4 = 0.8476
   3.6021
+ 0.8476
2.8616 x 104
Answer

   4.4497
We check 4.4497 in the table of antilogarithms.

A4 = 2.8616 x 104

A4 = 28616

Compound interest = Amount(A4) – Principal(P)

                                   = 28616 – 4000

                                   = 24616


Hence the compound interest is sh. 24616/=

CONGRUENCE OF SIMPLE POLYGONS B9


In the following figure, prove that UVY is congruent to WYX



solution


Given: figure UVYWX .

Required to prove: UVY WYX

Proof: VY = YX (given)

            UY = YW (given)

<UYV = <XYW (opposite angles)

Hence UVY WYX by SAS rule (proved)

LOGARITHMS B6

Evaluate Log2(128 x 32).

Solution

= Log2(128 x 32)

= Log2128 +  Log232          (applying the product rule)

= Log227 +  Log225             ( 128= 27 and 32=25 )

= 7Log22 +  5Log22       ( remember  Logaac = cLogaa )

= (7 x 1) +  (5 x 1)          ( remember  Logaa = 1 )

= 7 + 5

= 12


hence Log2(128 x 32) = 12 

RADICALS B4




CIRCLES B5

Find y in the figure below, O is the center of the circle;



solution


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

= 2 x (5y)

= 10y

Angle 10y and 2y are opposite angles.

Therefore 10y + 2y = 360

12y = 360

12y = 360 
12         12 

y = 30        


Hence y = 30        

Monday 13 January 2014

SETS B6



If n(A)= 60 , n(B)= 70 and n(AnB)=40, find n(AuB).

Solution

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

             = 60 + 70 – 40

             = 130 – 40

             = 90

Hence n(AuB) = 90 answer

CONGRUENCE OF SIMPLE POLYGONS B8

In the figure below prove that DEH is congruent to GHF.




solution



Given: figure DEHGF .
Required to prove: DEH GHF
Proof: EH = HF (given)

            DH = HG (given)

          <EHD = <FHG (opposite angles)

Hence DEH GHF by SAS rule (proved)