PRIME FACTORS 1

Write 420 as a product of prime factors.

Solution

2	420
2	210
3	105
5	35
7	7
	1

 Then, 420 = 2 x 2 x 3 x 5 x 7

Hence   420 = 2² x 5 x 7.

TRY THIS........

Write 540 as a product of prime factors.

GEOMETRY 2

If x-15⁰ and 4x + 35⁰ are complementary angles, find the value of x.

Solution

Complementary angles add up to 90⁰.

so,  x-15⁰ + 4x + 35⁰ = 90⁰.

so,  x + 4x + 35⁰ - 15⁰ = 90⁰.

so,  5x + 15⁰ = 90⁰.

so,  5x = 90⁰ -  15⁰

so,  5x = 75⁰

so,  5x = 75¹⁵

       5       5

Hence x = 15⁰

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

If 3x - 18⁰ and 2x + 43⁰ are complementary angles, find the value of x.

EXPONENTIAL EQUATIONS - 2

If 2^(6m-7) x 3⁽ⁿ⁺⁹⁾ = (3⁴⁴) x (2⁹³); Find m and n.

solution

equating equal bases;

2^(6m-7) =  2⁹³

2^(6m-7) =  2⁹³

6m - 7 = 93

6m  = 93 + 7

6m  = 100

¹6m  = 100⁵⁰

₁6           6₃

m = ⁵⁰/₃

again for n we equate equal bases.

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

n + 9 = 44

n = 44 - 9

n = 35

Hence m = 20 and n = 35

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

If 8^(2m-9) x 11⁽ⁿ⁺⁴⁰⁾ = (11⁶⁰) x (8²³); Find m and n.

FACTORIZATION - 8

Factorize completely ca + cr - ar - a².  

Solution

= ca + cr - ar - a².

= (ca + cr) – (ar - a²).      [Grouping the factors]

= c(a + r) - a(r + a).

 = (a + r)(c - a).

  Hence ca + cr - ar - a². =  (a + r)((c - a).   

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

NECTA 1995 QN. 20a

Factorize completely pq + pr - rq - q².   

LOGARITHMS-11

If Log_ay = Log_a22 +  Log_a3 ; find y.

solution

Log_ay = Log_a22 +  Log_a3

Log_ay = Log_a(22x 3)         [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.

UNITS 2

Convert 4073m into km.

Solution

1km  = 1000m

  ?     = 4073m

After cross-multiplying;

= 1 x 4073

      1000

=   4073

     1000

= 4.073 km

Hence 4073m = 4.073km

TRY THIS........

Convert 8023m into km.

FACTORIZATION - 7

Factorize completely   (3x-5)² - x²  

solution

= (3x-5)² - x²  

= [(3x-5) - x][ (3x-5) + x ]  --- since a² + b² = (a+b)(a-b) Here a is (x-2) and b is x.

= [3x-5 - x][ 3x-5 + x ] 

= [3x-x-5][ 3x+ x - 5  ] 

= [2x-5][ 4x - 5  ] 

= [2x-5][ 4x - 5 ] 

Hence  (2x-5)² - x²  = (2x-2)(4x - 5) 

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

Factorize completely   (6x-1)² - x²  

POLYGONS-2

A regular polygon has 88 sides. Find the total angles of that polygon.

Solution

n = 88

Total angles = (n - 2)180⁰

                   = (88 – 2)180⁰

                   = 86 x 180⁰

                   = 15480⁰

Total angles = 15480⁰

TRY THIS……………………….

A regular polygon has 68 sides. Find the total angles of that polygon.

LOGARITHMS-10

If log₅(3x + 101)=3; find x

   Solution

log₅(3x + 92)=3

 (3x + 92)=5³   

                      

 3x + 92=125

3x = 125 – 92

3x =  33

3x =  33   

3        3

X = 11

Hence x = 11

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

If log₂(2x - 22)=5; find x

APPROXIMATIONS-4

Estimate the value of 2.3  x  0.034

solution

= 2.3  x  0.034

= 2.0 x 0.03      [ 2.3 ≈ 2.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.06

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

Estimate the value of 6.2  x  0.083