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 Log_ax = ¹/₂  Log_a16+ ¹/₃ Log_a 343 find x.

Solution

Log_ax = ¹/₂  Log_a16+ ¹/₃ Log_a343

Log_ax = Log_a16^1/2  +  Log_a343^1/3   

Log_ax = Log_a 4 +  Log_a7

Log_ax = Log_a (4 x 7)

Log_ax = Log_a 28

x = 28

Hence x = 28

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  Log_aa^c = cLog_aa )

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

= 2 + 4

= 6

hence Log₂(4 x 16) = 6 

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)

Hence ∆PRU 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%

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)

Hence ∆ACF 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=?

A_n = 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  Log_aa^c = cLog_aa )

= (7 x 1) +  (5 x 1)          ( remember  Log_aa = 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 = 30⁰          

Hence y = 30⁰          

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)