FACTORIZATION 1

Factorize 121-d²

Solution

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

121-d² = 11²-d²       

        = (11 - d)(11 + d)

Hence 121-d² = (11 - d)(11 + d)

TRY THIS…………….

Factorize 144 - c²

PROBABILITY 1

A card is picked at random from a pack of playing cards. Find the probability that it is a red card.

Solution

Total playing card is normally 52. Half of them(26) are red.

 n(E) = 26, n(S) = 52, P(E) = ?

P(E) = n(E)

           n(S)

P(E) = 26

           52

  P(E) = 1

              2

 Probability that it is a red card is ½

TRY THIS………

A card is picked at random from a pack of playing cards. Find the probability that it is a red card.

PERCENTAGE PROFIT 1

A man got a profit of 130/= after selling an item for 5200/=. Find the percentage profit.

Solution

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

                      B.P.

%’ge profit = 130   X  100   

                    5200

%’ge profit = 2.5%   

                       

Hence Percentage Profit is 2.5%

TRY THIS……………………………

A man got a profit of 10,000/= after selling an item for 250,000/=. Find the percentage profit.

SETS 1

If n(A)= 60 , n(AuB) = 200 and n(AnB)=20, find n(B)

Solution

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

  200     = 60 + n(B)  – 20

  200     = 60 -20 + n(B) 

  200     = 40 + n(B) 

  200 - 40        = n(B)

  160        = n(B)

Hence n(B) = 160 answer

TRY THIS…………………………….

If n(A)= 68 , n(AuB) = 127 and n(AnB) = 25, find n(B).

MONEY RATES 1

How many US Dollars are worth Tsh. 8,500,000/=?

solution

1 US Dollar = 1700 Tsh.

             ?      = 10,200,000/=

After cross multiply we have:

= 10,200,000x 1

        1700

= 102,000 ⁶⁰⁰⁰

        17

= 6,000

Hence 10,200,000 Tshs equals 6000 USD

TRY THIS...............

How many US Dollars are worth Tsh. 51,000,000/=?[ Assume 1USD=1700 Tsh.]

TRIGONOMETRY 1

Express each of the following as trigonometric ratios of positive acute angles: (i)  Sin(-50⁰)  (ii) Cos(-210⁰)   (iii) Tan( -170⁰)

Solution

i)   Sin(-50⁰)  = Sin (360⁰-50⁰)

                      = Sin 310⁰

Sin θ = -Sin(360⁰- θ)  [since it is in the 4^th quadrant]

Sin 310⁰ = -Sin(360⁰- 310⁰)

               =  -Sin 50⁰ 

ii)   Cos(-210⁰)  = Cos (360⁰-210⁰)

                           = Cos 150⁰

Cos θ = -Cos(180⁰- θ)  [since it is in the 2nd quadrant]

Cos 150⁰ = -Cos(180⁰- 150⁰)

                =  -Cos 30⁰ 

ii)   Tan(-170⁰)  = Tan (360⁰-170⁰)

                           = Tan 190⁰

Tan θ = -Tan(θ - 180⁰)  [since it is in the 3rd quadrant]  

Tan 190⁰ = -Tan(190⁰- 180⁰)

                =  -Tan 10⁰ 

TRY THIS………

Express each of the following as trigonometric ratios of positive acute angles.

(i)  Sin(-60⁰)  (ii) Cos(-280⁰)   (iii) Tan( -130⁰)

LOGARITHMS 6

Evaluate Log1000 +  log 0.00001 + log₃243

Solution

= Log1000 +  log 0.00001 + log₃243

= Log10³ +  log 10^-5 + log₃3^-5

= 3Log10 +  (-5log 10) + (-5log₃3)    [since log_aaⁿ  = nlog_aa]

= (3x1) + (-5x1) + (-5x1)             [since log_aa = 1]

= 3 + (-5) + (-5)

= 3 - 10

= -7

Hence Log1000 +  log 0.00001 + log₃243 = -5

TRY THIS...............................

Evaluate Log100,000 +  log 0.01 + log₃729

FUNCTIONS 1

If F(x) = 13x + 14; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 14x + 10

y – 10 = 14x

y – 10  = 14x

   14        14

y – 10   = x

   14

x  =   y – 10     after rearranging

            14

 y^-1 = x – 10     after interchanging x and y variables.

             14

F^-1(x) = x – 10     after interchanging x and y variables.

                14

Hence, F^-1(x) = x – 10     

                             14

TRY THIS……………………………

If F(x) = 7x + 8; Find F^-1(x).

WORD PROBLEMS 1

In a box of 500 books, 160 are red in colour while the rest are bluish in color. Give out the percentage of blue books.

solution

1st we have to get the number of blue books.

 blue books = 500 - 160 = 340

then,=  340 x 100

             500

then,=  340

               5

   = 68%

Hence percentage of blue books is 68%

TRY THIS...............

In a box of 800 books, 300 are yellow in colour while the rest are bluish in color. Give out the percentage of blue books.

LOGARITHMS 5

Evaluate Log₂(512 ÷ 16).

Solution

= Log₂(512 ÷ 16)

= Log₂512 -  Log₂16          (applying the quotient rule)

= Log₂2⁹ -  Log₂2⁴             ( 512= 2⁹ and 16=2⁴ )

= 9Log₂2 -  4Log₂2       ( remember  Log_aa^c = cLog_aa )

= (9 x 1) - (4 x 1)          ( remember  Log_aa = 1 )

= 9 - 4

= 5

hence Log₂(512 ÷ 16) = 5

TRY THIS...............

Evaluate Log₂(1024 ÷ 32). 

DIFFERENCE OF 2 SQUARES 1

Evaluate 352² – 342²

Solution

352² – 342² = (352 + 342)( 352 – 342)

                     = (694)( 10)

                     = 6940

352² – 342² = 6940

TRY THIS…………….

Evaluate 905² – 805²

SIMPLE INTEREST 2

Dikshi deposited the amount of 9000/= in a bank which gives an interest rate of 4% for 4 years. Find the simple interest she got.

Solution

I = ?,  P = 9,000/=, T = 4 years, R = 4%

I  = PRT

      100

I  = 9000 x 4 x 4

            100

I  = 9000 x 4 x 4

            100

I  = 90 x 4 x 4

          

I  = 1440/=

Hence the simple interest was Tsh 1080/=

TRY THIS………………….

Edmund deposited the amount of 55,000/= in a bank which gives an interest rate of 6% for 6 years. Find the simple interest she got?

LOGARITHMS 4

If Log_ay = Log_a6 +  Log_a11 ; find y.

solution

Log_ay = Log_a6 +  Log_a11

Log_ay = Log_a(6 x 11)    [since Log_am +  Log_an = Log_a(m x n)] 

Log_ay = Log_a66

Hence y = 66    [since Log_a cancels out]

TRY THIS..............................

NECTA 1997 QN 13a

If Log_ax = Log_a5 +  Log_a3 ; find x.

MID-POINT 1

Find the midpoint of a line from (31, 12) to (17, 10)

Solution

x1=31, x2=17, y1=12, y2=10.

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

                         2             2

Mid point = (31 + 17,  12 + 10)

                        2            2

Mid point = (48, 22)

                     2    2

Hence midpoint = (24, 11)

TRY THIS……………………..

Find the midpoint of a line from (24, 14) to (22, 18)

INEQUALITIES 1

Find x in the following inequality 2x+50 ≤ 72

Solution

2x+50 ≤ 72

2x ≤ 72 – 50

2x≤ 22

2x  ≤  22         [ dividing by 2 on both sides]

2        2

x ≤ 11

Hence x ≤ 11

TRY THIS...............................

Find x in the following inequality 3x + 36 ≤ 87

RATIOS 1

Given the ratios U:V = 11: 2 and V:W = 20:7 Find the ratio U:W.

solution

U = U x V

W   V   W

U =   11 x 20⁵

W   ₁ 2     7

U =   11 x 5

W     1 x 7

U =   55

W     7

Hence U:W = 55:7

TRY THIS...............................

Given the ratios D:E = 5:7 and E:F = 3:35;  Find the ratio D:F.

SIMPLE INTEREST 1

Priya deposited the amount of 210,000/= in a bank which gives an interest rate of 3% for 5 years. Find the simple interest she got.

Solution

I = ?,  P = 210,000/=, T = 5 years, R = 3%

I  = PRT

      100

I  = 210,000 x 3 x 5

            100

I  = 210,000 x 3 x 5

            100

I  = 2100 x 3 x 5

          

I  = 31,500/=

Hence the simple interest was Tsh 31,500/=

TRY THIS………………….

Edmund deposited the amount of 70,000/= in a bank which gives an interest rate of 2% for 4 years. Find the simple interest he got?