FACTORIZING 3

Factorize completely cm + cr - mr - m².  

Solution

= cm + cr - mr - m².

= (cm + cr) – (mr - m²).      [Grouping the factors]

= c(m + r) - m(r + m).   [after factoring out]

 = (m + r)(c - m).

  Hence cm + cr - mr - m². =  (m + r)(c - m).   

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

NECTA 1995 QN. 20a

Factorize completely pq + pr - rq - q². 

VARIATIONS 2

x is directly proportional to y. x=12 while y=4. Find y when x is 240.

Solution

x ⍺ y

x = ky

12 = k x 4

12 = 4k

4      4

k = 3

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

k = 3, y= ? x = 240.

x = ky

240 = 3 x y

240 = 3y

 3       3

y = 80.

TRY THIS…………….

x is directly proportional to y. x=20 while y=8. Find y when x is 200.

LOGARITHMS 10

Evaluate Log₂(2048 x 16).

Solution

= Log₂(2048 x 16)

= Log₂2048 +  Log₂16          (applying the product rule)

= Log₂2¹¹ +  Log₂2⁴             (2048= 2¹¹ and 16=2⁴ )

= 11Log₂2 +  4Log₂2       (remember  Log_aa^c = cLog_aa )

= (11 x 1) +  (4x 1)          (remember  Log_aa = 1 )

= 11 + 4

= 15

hence Log₂(2048 x 16) = 15

TRY THIS………………

Evaluate Log₂(2048 x 256). 

FUNCTIONS 6

Find a linear function f(x) with gradient -6 which is such that f(5)=12.

       Solution.

m=-6, points = [5,12] and [x, f(x)]

m = y₂-y₁/x₂-x₁

-6 = [f(x) – 12]/x-5

  

f(x)-12=-6(x-5) [after cross multiply]

f(x)-12=-6x+30

f(x)=-6x+30+12

f(x)=-6x+52

Hence A  linear function is f(x) = -6x+ 52.

TRY THIS……….

Give out a linear function f(x) with gradient -5 and f(6)=14

APPROXIMATIONS 2

Estimate the value of 7.3  x  0.034

solution

= 7.3  x  0.034

= 7.0 x 0.03      [ 7.3 ≈ 7.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.21

  

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

Estimate the value of 6.4  x  0.081

REGULAR POLYGONS 4

A regular polygon has 34 sides. Find the total angles of that polygon.

Solution

n = 34

Total angles = (n - 2)180⁰

                   = (34 – 2)180⁰

                   = 32 x 180⁰

                   = 5760⁰

Total angles = 5760⁰

TRY THIS……………………….

A regular polygon has 58 sides. Find the total angles of that polygon.  

FUNCTIONS 5

Find the maximum value of the quadratic equation:3-2t-8t².

Solution

a=-8, b=-2, c=3

Maximum = 4ac-b²

                        4a

Maximum = (4 x -8 x 3) - (-2)²

                          4(-8)

Maximum = (-96)- 4

                         32

Maximum= -100

                       32

Maximum = -25

                       8

Hence maximum value is -25/8

TRY THIS………………………………..

NECTA  2003 QN. 10c

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

PERFECT SQUARES 1

What must be added to x²  + 5x to make the expression a perfect square?

Solution

a=1, b=5,c=?

b² = 4ac

(5)² = 4 x 1 x c

25 = 4c

25 = 4c

4      4

²⁵/4 = c

Hence number to be added is ²⁵/4.

TRY THIS……………………

NECTA 2013,  QN. 10a

What must be added to x²  + 8x to make the expression a perfect square?   

LOGARITHMS 9

Log₄7x – Log₄(x+3)-Log₄5=0.

Solution

Log₄7x – Log₄(x+3)-Log₄5=0.

Log₄4x – [ Log₄(x+7) + Log₄5]=0.

Log₄{4x/(x+7)5} =0.

4x            =   4⁰   = 1

5(x+7)

4x          = 1

5x+35

5x + 35 = 4x

5x - 4x + 35 = 0

5x - 4x = - 35

x = -35

Hence x=-35

TRY THIS……………………….. 

NECTA 2011,  QN 2a

Log₄5x – Log₄(x+2) - Log₄3 = 0.

VARIATIONS 1

x is directly proportional to y. x=12 while y=4. Find y when x is 60.

Solution

x ⍺ y

x = ky

12 = k x 4

12 = 4k

4      4

k = 3

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

k = 3, y= ? x = 60.

60 = 3 x y

60 = 3y

3      3

y = 20.

TRY THIS…………….

x is directly proportional to y. x=20 while y=4. Find y when x is 70.