MATRIX 6

LOGARITHMS 7

Evaluate Log100 -  log 0.001 - log 0.0001+ log 0.00001

Solution

= Log100 -  log 0.001 -  log 0.0001+ log 0.00001

= Log10² -  log 10^-3 – log 10^-4 + log 10^-5

= 2Log10 - (-3 log 10) – (-4log 10) + (-5log 10)

= (2x1) - (-3x1) - (-8x1) + (-5 x 1)

= 2 - (-3) – (-4) -5

= 2 + 3 + 4 - 5

= 9 - 5

  

Hence Log100 -  log 0.001-  log 0.0001+ log 0.00001 = 4

TRY THIS……………………… 

Evaluate Log10,000 -  log 0.001 - log 0.00001+ log 0.00001

MATRIX 5

FACTORIZING 1

Factorize 81-m²

Solution

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

81-m² = 9²-m²       

        = (9 - m)(9 + m)

Hence 81 - m² = (9 - m)(9 + m)

TRY THIS…………….

Factorize 169 - c²

MATRIX 4

SIMPLE INTEREST 1

Asad deposited the amount of 200,000/= in a bank for 3 years and got a profit of 72000/=. Find the interest rate?

Solution

I = 72000/=, P = 200,000/=, T = 3 years, R = ?

I  = PRT

      100

72,000  = 200,000 x R x 3

                      100

72,000  = 200,000 x R x 3

                      100

72,000  = 2000 x R x 3

                     

72,000  = 6000R

          

¹²72,000  = 6000R

   6000       6000

Hence the interest rate was 12%

TRY THIS…………………..

Sonali deposited the amount of 300,000/= in a bank for 6 years and got a profit of 27,000/=. Find the interest rate?

MATRIX 3

LOGARITHMS 6

If Log_ax = 0.9, find Log_a(x²)

Solution

= Log_a(x²)

= Log_ax²

= 2 x Log_ax

= 2 x 0.9

= 1.8

TRY THIS………..

If Log_aw= 3.2, find Log_a(w²)

MATRIX 2

LOGARITHMS 5

Evaluate Log100 -  log 0.001 -  log 0.0001

Solution

= Log100 -  log 0.001 -  log 0.0001

= Log10² -  log 10^-3 – log 10^-4

= 2Log10 - (-3 log 10) – (-4log 10)

= (2x1) - (-3x1) - (-8x1)

=2 - (-3) – (-4)

= 2 + 3 + 4

=9

  

Hence Log100 -  log 0.001-  log 0.0001 = 9

TRY THIS……………………… 

Evaluate Log10000 -  log 0.000001 -  log 0.00000001

ALGEBRA 1

VARIATIONS 1

x is inversely proportional to y. x=18 while y=5. Find y when x is 15.

Solution

x ⍺ k

      y

x = k

      y

18 = k

        5

k = 18 x 5

k = 90

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

k = 90, y= ?, x = 15.

x = k

      y

15 = 90

         y

y x 15 = 90  x  y   (multiply by y on both sides)

                y

15y = 90

15      15

Hence y = 6  .

TRY THIS…………….

x is inversely proportional to y. x=80 while y=4. Find y when x is 10. 

MATRIX 1

LOGARITHMS 4

If log₉(2x + 33)=2; find x.

   Solution

Log₉(2x + 33)=2

(2x + 33)=9²           

                                            

2x + 33=81

2x =81 – 33

2x = 48

2x =   48   

2        2

x = 24

Hence x = 24

TRY THIS………………

If log₇(2t - 39)=2; find t

ARITHMETIC PROGRESSION 1

The first term of an AP is 41 and the last term is 97.If the arithmetic progression consists of 20 terms, calculate the sum of all the terms. 

Solution

A₁ = 41, n=20, A_n = 97

S_n = n(A₁ + A_n)

       2

S₂₀ = 20(41 + 97)

         2

S₂₀ = 10 x 138

S₂₀ = 1380

Hence the sum of all 20 terms is 1380.

TRYTHIS………………….

The first term of an AP is 11 and the last term is 53.If the arithmetic progression consists of 40 terms, calculate the sum of all the terms. 

EXPONENTIALS 1

Simplify the following by writing in power form:

w⁹⁰

w⁴⁰

Solution

= w^{90 - 40} 

=w⁵⁰

TRY THIS……………….

Simplify the following by writing in power form:

h⁵⁹

h²¹ 

LOGARITHMS 3

If Log_ax = 0.2, find Log_a(¹/x²)

Solution

= Log_a(¹/x²)

= Log_ax^-2

= -2 x Log_ax

= -2 x 0.2

= -0.4

TRY THIS………..

If Log_aw= 1.6, find Log_a(¹/w²)

ARITHMETIC 1

Evaluate 396² – 386²

Solution

We apply difference of two squares: a² - b² = (a+b)(a-b)

396² – 386²  = (396 + 386)( 396 – 386)

                   = (782)( 10)

                   = 7820   [only add a zero at 782]

396² – 386² = 13820

TRY THIS…………….

  

Evaluate 508² – 408²

LOGARITHMS 2

If Log_ax = 0.9, find Log_a(¹/x)

Solution

= Log_a(¹/x)

= Log_ax^-1

= -1 x Log_ax

= -1 x 0.9

= -0.9

TRY THIS………..

If Log_aw= 3.6, find Log_a(¹/w)

SLOPE OR GRADIENT 1

Find the slope of a line which passes through (-1, -8) and (6,-9)

Solution

x₁ = -1,  y₁ =-8,  x₂ = 6,  y₂ = -9

m = y2 –y1

       x2 – x1

m =   -9 –(-8)

         6 –(-1)

m =   -9 + 8

         6 + 1

m =    - 1

            7

Hence the slope is -1/7

TRY THIS……………………… 

Find the slope of a line which passes through (-11, -2) and (9,-12).

BINARIES-1

If p*k = 4pk + p – 2k; find 3*2.

Solution

p*k = (4 x p x k) + p – (2xk)   [rewriting the given expression more clearly]

3*2 = (4 x 3 x 2) + 7 – (2x2)   [substituting 7 for p and 2 for k]

3*2 = 24 + 7 – 4

3*2 = 31 – 4   [we add first before subtracting]

3*2 = 27.

Hence   (3*2) = 27.

 TRY THIS........

 If p*k = 2p - k – 5pk; find (-3*4).

QUADRATICS-1

Find the maximum value of the quadratic equation:5-6t-8t².

Solution

a=-8, b=-6, c=5

Maximum = 4ac-b²

                        4a

Maximum = (4 x -8 x 5) - (-6)²

                          4(-8)

Maximum = (-160)- 36

                         32

Maximum= -196                         since (-160)- 36=-196

                       32

Maximum = -98 =   -49

                      16         8

Hence maximum value is -49/8

TRY THIS………………………………..

NECTA  2003 QN. 10c

Find the maximum value of the quadratic equation:3+30t-5t².

POLYGONS 2

A regular polygon has 62 sides. Find the total degrees of that polygon.

Solution

n = 62

Total degrees = (n - 2)180⁰

                       = (62 – 2)180⁰

                       = 60 x 180⁰

                       = 10800⁰

Total degrees = 10800⁰

TRY THIS……………………….

A regular polygon has 53 sides. Find the total angles of that polygon.

LOGARITHMS 1

Evaluate Log10000 +  log 0.001 -  log 0.00000001

Solution

= Log10000 +  log 0.001 -  log 0.00000001

= Log10⁴ +  log 10^-3 – log 10^-8

= 4Log10 + (-3 log 10) – (-8log 10)

= (4x1) + (-3x1) - (-8x1)

=4 + (-3) – (-8)

= 4 – 3 + 8

=9

Hence Log10000 +  log 0.001-  log 0.00000001 = 9

TRY THIS……………………… 

Evaluate Log100 +  log 0.0000001 -  log 0.00000001

MID-POINT 1

Find the midpoint of a line from (23, 12) to (5, 20)

Solution

x1=23, x2=5, y1=12,y2=20.

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

                         2           2

Mid point = (23 + 5,  12 + 20)

                         2          2

Mid point = (28, 32)

                    2     2

Hence midpoint = (14, 16)

TRY THIS……………………..

Find the midpoint of a line from (-6, 14) to (6, -4)

PIE CHARTS 1

In a survey of 60 people, 28 people said their favourite sport was football. What angle in a pie chart would this represent?

Solution

We make a fraction of 28 ot of 60, then multiply by 360⁰.

= 28 x 360⁰.

   60

= 28 x 360⁰₆

   60₁

= 28 x 6

= 168⁰.

Hence the angle is 168⁰ 

TRY THIS………………………   

In a survey of 30 people, 23 people said their favourite sport was football. What angle in a pie chart would this represent?

POLYGONS 1

An interior angle of a regular polygon is 74⁰ greater than an exterior angle. Find the interior angle.

Solution

Let i = interior angle, e = exterior angle.

Now i  + e=180⁰…………………(1)

But i = e+74⁰ …………………(2)

Substitute (2) in (1) above.

e+74⁰   + e=180⁰

e+ e+74⁰   =180⁰

2e+ 74⁰   =180⁰

2e=180⁰ - 74⁰   

2e=106⁰

2e=106⁰          dividing by 2 both sides.

2      2

e = 53⁰

But i  + e=180⁰…………………(1)

i  + 53⁰=180⁰.

i  =180⁰ - 53⁰

i = 127⁰

Hence i = 127⁰ 

TRY THIS………………………   

An interior angle of a regular polygon is 78⁰ greater than an exterior angle. Find the interior angle.