FACTORIZE E2

 Factorize completely qc + qr - cr - c².  

 

Solution

 

= qc + qr - cr - c².

 

= (qc + qr) – (cr - c²).      [Grouping the factors]

 

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

 

 = (c + r)(q - c).

 

  Hence qc + qr - cr - c². =  (c + r)(q - c).   

 

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

 

Factorize completely pe + pr - re - e². 

QUADRATICS E1

 

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

 

Solution

 

a=1, b=8,c=?

 

b² = 4ac

 

(8)² = 4 x 1 x c

 

64 = 4c

64 = 4c

4      4

 

16 = c

Hence number to be added is 16

 

TRY THIS……………………

What must be added to x²  + 12x to make the expression a perfect square?   

RELATIONS E1

 Let R={(7,23), (0,11), (-2,-5), (14,-20)}. Find the domain and range of R.

 

solution

 

For domain we check on the values of x in each point.

Domain={7, 0, -2, 14}

 

For range we check on the values of y in each point.

Range={23, 11, -5, -20}

 

Hence Domain={7, 0, -2, 14} and Range = {23, 11, -5, -20}

 

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

 

Let R={(12,3), (-33,8), (-18,-9), (29, -14)}. Find the domain and range of R.

LOGARITHMS E2

 Change the following into exponential form

i) Log₇ 49 = 2

ii) Log₂ 16 = 4

iii) Log₈ 1 = 0

iv) log₅ (¹/₁₂₅) = -3

Solution

 

i) 7² = 49 ⇔ log₇ 49 = 2

ii) 2⁴ = 16 ⇔ Log₂ 16 = 4

iii) 8⁰ = 1 ⇔ log₈ 1 = 0

iv) 5^-3 = ¹/₁₂₅ ⇔ log₅ (¹/₁₂₅) = -3

TRY THIS…………….

 

Change the following into exponential form

i) Log₁₁ 121 = 2

ii) Log₃ 81 = 4

iii) Log₂₆1 = 0

iv) log₄ (¹/₆₄) = -3

EXPAND E1

 

Expand 8w(5w – 7)

solution

= 8w(5w – 7)

= (8w x 5w) – (8w x 7)

= 40w² – 56w answer

TRY THIS………..

Expand 2a(7a+ 30)

LOGARITHMS E1

 Evaluate Log100 +  log 0.001 -  log 0.00000001

 

Solution

 

= Log1000 +  log 0.001 -  log 0.00000001

 

= Log10³ +  log 10^-3 – log 10^-8

 

= 3Log10 + (-3 log 10) – (-8log 10)

 

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

 

=3 + (-3) – (-8)

 

= 3 – 3 + 8

 

=8

 

 

Hence Log100 +  log 0.001-  log 0.00000001 = 7

 

 

TRY THIS……………………… 

 

 

Evaluate Log10000 +  log 0.00001 -  log 0.0000000000001

PROFIT AND LOSS E1

 

A man got a profit of 2800/= after selling an item. Find the buying price if the percentage profit was 10%.

 

Solution

 

%’ge profit = Profit   X  100    where B. P. represents Buying Price.

                         B.P

 

10% = 2800   X  100   

            B.P.

 

10% = 280,000   

               B.P.

 

 

B.P. x 10% = 280,000   x B.P.

                       B.P.

  

B.P. x 10 = 280,000   

 

        

B.P. x 10¹ =   280,000    [dividing by 10 both sides]

  ₁10                   10

 

B.P. = 28,000

                      

Hence Buying Price was 28,000/=

 

 

TRY THIS………………

 

A man got a profit of 2400/= after selling an item. Find the buying price if the percentage profit was 20%.

APPROXIMATIONS E1

 

Estimate the value of 7.2  x  0.033

 

solution

 

= 7.2  x  0.034

 

= 7.0 x 0.03      [ 7.2 ≈ 7.0 to ones and 0.033 ≈ 0.03 to hundredths]

 

= 0.21

 

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

  

Estimate the value of 8.4  x  0.053

FUNCTIONS E1

 

Find a linear function f(x) with gradient -6 which is such that f(8)=12.

 

       Solution.

 

m=-6, points = [8,12] and [x, f(x)]

 

m = y₂-y₁/x₂-x₁

 

-6 = [f(x) – 12]/x-8

 

 

f(x)-12=-6(x-8) [after cross multiply]

f(x)-12=-6x+48

f(x)=-6x+48+12

f(x)=-6x+60

 

A  linear function is f(x) = -6x+ 60.

 

TRY THIS……….

 

Give out a linear function f(x) with gradient -5 and f(6)=19

SETS E1

 

If n(A)= 55 , n(AuB) = 130 and n(AnB)=30, find n(B)

 

Solution

 

n(AuB) = n(A) + n(B) - n(AnB)

 

  130     = 55 + n(B)  – 30

 

  130     = 55 - 30 + n(B) 

 

  130     = 25 + n(B) 

 

  130 - 25        = n(B)

 

  105        = n(B)

 

Hence n(B) = 105 answer

TRY THIS…………………………….

 

 

If n(A)= 62 , n(AuB) = 138 and n(AnB) = 28, find n(B).

FACTORIZE E1

 

Factorize completely cd + cr - dr - d².  

 

Solution

 

= cd + cr - dr - d². [after rearranging]

 

= cr + cd - dr - d². [after rearranging]

 

= (cr + cd) – (dr - d²).      [Grouping the factors]

 

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

 

 = (r + d)(c - d).

 

  Hence cd + cr - dr - d².  . (r + d)(c - d).

 

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

 

NECTA 1995 QN. 20a

 

Factorize completely pq + pr - rq - q². 

 

ARITHMETICS E1

 

Write 1600 as a product of prime factors.

 

Solution

 

2	1600
2	800
2	400
2	200
2	100
2	50
5	25
5	5
	1

 

 Then, 1600 = 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5

 

Hence   1600 = 2⁶ x 5².

 

 

TRY THIS........

 

Write 1200 as a product of prime factors.