CO-ORDINATE GEOMETRY-5

Find the slope of a line which passes through (-5, -2) and (6,-9)

Solution

x₁ = -5,  y₁ =-2,  x₂ = 6,  y₂ = -9

m = y2 –y1

       x2 – x1

m =   -9 –(-2)

           6 –(-5)

m =   -9 + 2

           6 + 5

m =    - 7

           11

Hence the slope is -7/11

PYTHAGORAS - 3

Find the length AB

Solution

We will use the PYTHAGORAS theorem [ a² + b² = c²  ]

a² + b² = c²

AB² + BC² = AC²

AB² + 10² = 26²

AB² + 100 = 676

AB² = 676 - 100

AB² = 576    [Find the square root of 576]

Hence AB= 24cm      

MATRIX-3

MATRIX-2

SETS-6

In Bibanja district the number of people who speak Kiswahili or Lingala is 400. 200 of them speak Kiswahili and 300 of them speak Lingala. How many speak both languages?

solution

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

Let Kiswahili=n(K), Lingala= n(L).

n(K)= 200 ,

n(L)= 300,

n(KuL) = 400,

n(KnL)=?

n(KuL) = n(K) + n(L) - n(KnL)

400  =  200 + 300 – n(KnL)

400  =  500 – n(KnL)

n(KnL) =  500 – 400

n(EnF) =  100

Hence n(EnF)=100 answer

LOGARITHMS-6

Evaluate Log100000 +  log 0.001 -  log 0.00000001

Solution

= Log100000 +  log 0.001 -  log 0.00000001

= Log10⁵ +  log 10^-3 – log 10^-8

= 5Log10 + (-3 log 10) – (-8log 10)

= (5x1) + (-3x1) - (-8x1)

=5 + (-3) – (-8)

= 5 – 3 + 8

=10

Hence Log100000 +  log 0.001-  log 0.00000001 = 10

ALGEBRA-4

The sum of five consecutive numbers is 590. Find the 4th number.

Solution

Let the numbers be as shown in the table below

1st number	2nd number	3rd number	4th number	5th number	TOTAL
n	n+1	n+2	n+3	n+4	590

Then, n + (n+1) + (n+2) + (n+3)+ (n+4) = 590

5n + 1+2+3 +4= 590

5n + 10= 590

5n = 590 -10

5n = 580

5n = 580

 5        5

n = 116

4th number = n +3

                       = 116 + 3

                       = 119

Hence the 4th number is 119

QUADRATIC-5

If zx² + 12x + 4 = 0 is a perfect square, find the value of z .

Solution

In zx² + 12x + 4 = 0;   a=z, b=12 and c=4.

For the perfect square, b² = 4ac

(12)² = 4 x z x 4

144 = 16z

144 =  16z

 16       16

9 = z

Hence the value of z is 9

CO-ORDINATE GEOMETRY-4

Find the slope of a line which passes through (8, 6) and (-11,-3)

Solution

x₁ = 8,  y₁ =6,  x₂ =-11,  y₂ = -3

m = y2 –y1

       x2 – x1

m =    -3 –6

         -11 –8

m =     - 9

          - 19

m =  9

       19

Hence the slope is 9/19

FUNCTIONS-5

Given that F(x) = 13x  +  65. Find F(8)

Solution

F(x) = 13x  +  65

F(8) = 13(8)  +  65

F(8) = 104  +  65

F(8) = 169

Hence  F(8) = 169

SEQUENCE&SERIES-4

The 1st term of an A.P. is 10 and the common difference is 58. Find the 12th term.

Solution

A₁= 10, d = 58

A_n = A₁ + (n-1)d   [ formula for n terms ]

A₁₂ = A₁ + (12-1)d

A₁₂ = A₁ + 11d    [ formula for 12 terms ]

A₁₂ = 10 + (11 x 58)

A₁₂ = 10 + 638

A₁₂ = 648

Hence the 12th term is 648.

ABSOLUTE VALUE-3

If │4x + 7 │= 47; Find  x

Solution

±(4x + 7 )= 47

4x +7= 47    OR   –(4x+7) = 47

4x+ 7 = 47    OR   -4x - 7=47

4x = 47- 7    OR   -4x= 47 + 7

4x = 40     OR     -4x  =  54

4       4                   4        4

x = 10  OR  x = 27/2 = 13.5

Hence,  x =10  OR  x = 13.5