ARITHMETIC PROGRESSION H1

 

The 1st term of an A.P. is 78 and the common difference is 45. Find the 10th term.

 

Solution

 

A₁= 78, d = 45

 

A_n = A₁ + (n-1)d

 

A₁₀ = A₁ + (10-1)d

 

A₁₀ = A₁ + 9d

 

A₁₀ = 78 + (9x45) [ after substituting A₁= 78, d = 45 as given above]

 

∴ A₁₀ = 78 + 405

 

A₁₀ = 483

 

Hence the 10th term is 483.

 

  

TRY THIS……………

 

 

The 1st term of an A.P. is 140 and the common difference is 23. Find the 18th term.

ALGEBRA H1

 

Simplify the expression r² - t²/ (r + t)

 

Solution

 

from difference of two squares p² - q² = (p - q)(p + q)

 

= r² - t²/ (r + t)

 

= (r + t)(r - t)

      r + t

 

= (r + t)(r - t)

      r + t

 

=  r - t    answer

      

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

 

 

Simplify the expression 

 

w² - y²/ (w + y)

EXPONENTIALS H1

 

If y⁵ = 3, find y¹⁰ – y²⁰.

 

Solution

 

= y¹⁰ – y²⁰

 

= (y⁵)² - (y⁵)⁴

 

= (3)²  - (3)⁴          remember y² = 3

 

= 9  -  (3x3x3x3)

 

= 9 – 81

 

= -72

 

Hence y¹⁰ – y²⁰ = -72.

 

TRY THIS……………….

 

If  y⁹ = 10, find y¹⁸ – y²⁷.

ARITHMETIC H1

 Evaluate 22 x 237 + 463 x 16.

 

Solution

 

= 222 x 237 + 463 x 222

 

= 222 x (237 + 463)     factoring out the common number

 

= 222 x 700   

 

= 155400      [after multiplying 222 and 7 and adding two zeros on the answer]

 

Hence 222 x 237 + 463 x 222 = 155400

 

 

TRY THIS………….

 

 

Evaluate 124 x 516 + 484 x 124.

COMPOUND INTEREST H1

 

Salva invested a certain amount of money in a bank which gives an interest rate of 10% compounded annually. How much did she invest at the start if she got 300,000 sh at the end of 4 years?

 

solution

 

n=4, T=1, R=10%, A₄=300,000, P=?

 

A_n = P(1 + ^RT/100)ⁿ

 

A₄ = P(1 + ^(10x1)/100)⁴

 

A₄ = P(1 + ¹⁰/100)⁴

 

A₄ = P(¹⁰⁰/100 + ¹⁰/100)⁴

 

A₄ = P(¹¹⁰/100)⁴  But A₄=8000,

 

300,000 = P(1.1)⁴  

 

300,000 = 1.4641P

 

300,000  =  1.4641P

1.4641        1.4641

 

300,000  =  P    [use math tables to divide if you like]

1.4641

 

P = 204,904.04 [to 2 d.p]

 

 

Hence at the start she invested Tsh 204,904.04

 

TRY THIS……………………..

 

Sylvia invested a certain amount of money in a bank which gives an interest rate of 10% compounded annually. How much did she invest at the start if she got 150,000 sh at the end of 5 years?

 

PRIME FACTORS H1

 

Write 1080 as a product of prime factors.

 

Solution

 

2	1080
2	540
2	270
3	135
3	45
3	15
5	5
	1

 

 Then, 540 = 2 x 2 x 2 x 3 x 3 x 3 x 5

 

Hence   1080 =2³ x 3³ x 5.

 

 

TRY THIS........

 

Write 840 as a product of prime factors.

VARIATION H1

 x is inversely proportional to y. x=18 while y=5. Find y when x is 30.

 

Solution

 

x ⍺ k

      y

 

x = k

      y

 

18 = k

        5

 

k = 18 x 5

 

k = 90

 

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

 

k = 90, y= ?, x = 30.

 

x = k

      y

 

30 = 90

         y

 

y x 30 = 90  x  y   (multiply by y on both sides)

                y

 

 

30y = 90

30      30

 

Hence y = 3  .

 

 

TRY THIS…………….

 

 

x is inversely proportional to y. x=120 while y=4. Find y when x is 10. 

LOGARITHMS H1

 If Log_ax = 0.9, find Log_a(¹/x³)

 

Solution

 

= Log_a(¹/x³)

 

= Log_ax^-3

 

= -3 x Log_ax

 

= -3 x 0.9

 

= -2.7

 

 

TRY THIS………..

 

If Log_aw= 1.2, find Log_a(¹/w⁴)

ANGLES H1

 

GRADIENT H1

 

Find the slope of a line which passes through (-1, -8) and (16,-9)

 

Solution

 

x₁ = -1,  y₁ =-8,  x₂ = 16,  y₂ = -9

 

m = y2 –y1

         x2 – x1

 

m =   -9 –(-8)

         16 –(-1)

 

m =   -9 + 8

         16 + 1

 

m =    - 1

            17

 

Hence the slope is -1/17

 

 

TRY THIS……………………… 

 

Find the slope of a line which passes through (-1, -2) and (9,-12)