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?