DIRECT VARIATION F1

x is directly proportional to y. x=12 while y=4. Find y when x is 93.

 

Solution

 

x ⍺ y

x = ky

 

12 = k x 4

 

12 = 4k

4      4

 

k = 3

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

 

k = 3, y= ? x = 93.

 

93 = 3 x y

93 = 3y

3      3

 

y = 31.

 

TRY THIS…………….

 

x is directly proportional to y. x=20 while y=5. Find y when x is 100.

 

Ans. y=25 

FACTORIZE F1

Factorize 144x² - 169y²

 

Solution

 

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

 

144x²- 169y²  = 12²x² - 13²y²

 

                   = (12x)² - (13y)²

 

                   = (12x - 13y)( 12x + 13y)

 

Hence 144x² - 169y² = (12x - 13y)(12x + 13y)

 

TRY THIS………………………….

 

Factorize 196e²- 49f²

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