LOGARITHMS-1

Simplify Log₂256 - Log₃729

Solution

= Log₂256 - Log₃729

= Log₂2⁸ - Log₃3⁵    [Since 256=2⁸ and 729=3⁶].

= 8Log₂2 - 6Log₃3    [since Log_aa = 1]

= (8 x 1) - (6 x 1)

= 8 - 6

= 2

Hence Log₂256 - Log₃729 = 2

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

Simplify Log₂2048 – Log₅625 

ALGEBRA 1

If 27x + 5y -9 = 0; find x-intercept.

Solution

x-intercept is when y=0.

27x + 5y -9 = 0

27x + 5(0) - 9 = 0

27x - 9 = 0

27x = 0+ 9

27x = 9

27      27

x= ⁹/27

Hence x= ⁹/27

TRY THIS………..

If 11x - 5y - 55 = 0; find x-intercept.

FUNCTIONS 3

If f(x) = x⁴ - kx² - 6x + 7 has a remainder of 22 when divided by x+2; find k.

solution

f(x) = x⁴ - kx² - 6x + 7

x + 2 = 0

x = -2

f(x) = (-2)⁴ - k(-2)² - 6(-2) + 7 = 22

16 - 4k + 12 + 7 = 22

16 - 4k + 19 = 22

16 + 19 = 22 + 4k

16 + 19 - 22=4k

35 - 22=4k

13 = 4k

k = ¹³/4    after dividing by 4 both sides.

hence k=¹³/4

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

If f(x) = x⁴ - kx² - 5x - 31 has a remainder of 18 when divided by x-2; find k.

PERCENTAGE PROFIT1

A man got a profit of 7400/= after selling an item. Find the buying price if the percentage profit was 5%.

Solution

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

                        B.P

5% = 7400   X  100   

           B.P.

B.P. x 5% = 740,000   x B.P.  [multiplying by B.P. both sides]

                       B.P.

B.P. x 5 = 740,000   

B.P. x 5¹ =   740,000  ¹⁴⁸⁰⁰⁰ 

  ₁5                   5₁

B.P. = 148000

                      

Hence Buying Price was 148000/=

  

TRY THIS………………

A man got a profit of 27,500/= after selling an item. Find the buying price if the percentage profit was 20%.

VARIATION 2

x is directly proportional to y. x=24 while y=4. Find y when x is 60.

Solution

x ⍺ y

x = ky

24 = k x 4

24 = 4k

4      4

k = 6

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

k = 6, y= ? x = 60.

x = ky

60 = 6 x y

60 = 6y

6      6

y = 10.

TRY THIS…………….

x is directly proportional to y. x=48 while y=12. Find y when x is 70.

LOGARITHMS 3

If log 2= 0.3010; find the value of log 400,000 without using tables.

solution

= log 400,000

=log(100,000 x 4)

=log100,000 + log4

=log10⁵ + log2²

=5log10 + 2log2

=(5x1) + 2(0.3010)

=5– 2(0.3010)

=5 – 0.6020

=4.398

Hence log2500=3.398

TRY THIS……………

If log 2= 0.3010; find the value of log 80,000 without using tables.

EXPONENTIALS 3

Simplify the following by writing in power form:

(m⁵⁵ x m¹⁵)/(m⁴⁶ x m⁶)

  

Solution

= (m⁵⁵ x m¹⁵)/(m⁴⁶ x m⁶)

= m⁵⁵⁺¹⁵

   m⁴⁶⁺⁶ 

= m⁷⁰

    m⁵² 

= m^70-52

= m¹⁸

TRY THIS……………….

Simplify the following by writing in power form:

(a⁶⁰ x a²⁰)/(a⁴⁶ x a¹⁴)

A. P. 1

The 1st term of arithmetic progression is 24 and the common difference is 40. Find the nth term

solution

A₁=24, d= 40, n=?

A_n =A₁ + (n-1) d

A_n=24 + (n-1)40

A_n=24 + 40n - 40

A_n=40n + 24 - 40

A_n=40n – 16

Hence the nth term is A_n=40n – 16  

TRY THIS………………….. 

The 1st term of arithmetic progression is 60 and the common difference is 122. Find the nth term.

AREAS 1

Area of a rectangle is 84 cm². Find its perimeter if its width is 7cm.

Solution

First we find length

A =  l  x w

84 =  l  x 7

84 = 7l

7       7

12 =  l

Now length is 12cm.

We can find perimeter since we have both length and width.     

Let l = length and w = width;

P = 2(l + w)

   = 2(12 + 7)

   = 2 x 19

   = 38

Hence perimeter is 38 cm. 

TRY THIS………………….. 

Area of a rectangle is 100 cm². Find its perimeter if its width is 4cm.

FUNCTIONS 2

If F(x) = 4x + 16; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 4x + 16

y – 16 = 4x

y – 16  = 4x

   4          4

y – 16   = x

   4

x  =   y – 16     after rearranging

           4

F^-1(x) = x – 16    after interchanging x and y variables.

                4

Hence, F^-1(x) = x – 16     

                              4

TRY THIS………………   

If F(x) = 11x + 30; Find F^-1(x).