EXPONENTIALS 5

Simplify 200

               8^-2/3

Solution

=  200

    8^-2/3

= 200 x 1

           8^-2/3

=  200 x 8^2/3        [Since 1/a^-n = aⁿ]

=  200 x (8^1/3)²        [Since 8^1/3 = cube root of 8 = 2]

=  200 x (2)²        

=  200 x 4

= 800       

          

Hence   200        = 800

              8^-2/3

TRY THIS…..

Simplify 100

             125^-2/3

LOGARITHMS 7

If Log_ax = 4, find Log_a(¹/x⁵)

Solution

= Log_a(¹/x⁵)

= Log_ax^-5          [since 1/aⁿ = a^-n]

= -5 x Log_ax    [since Log_axⁿ   = nLog_ax ]

= -5 x 4

= -20 answer

TRY THIS………..

If Log_ab= 33, find Log_a(¹/b⁴)

ARITHMETIC PROGRESSION 2

The 1st term of an A.P. is 80 and the common difference is 97. Find the nth term.

Solution

A₁= 80, d = 97

A_n = A₁ + (n-1)d

A_n = 80 + (n-1)97

A_n = 80 + 97n – 97

A_n = 97n - 17

Hence the nth term is 97n - 17.

TRY THIS……………

The 1st term of an A.P. is 124 and the common difference is 39. Find the nth term.

LOGARITHMS 6

If Log_ax = -413, find Log_a(¹/x)

Solution

= Log_a(¹/x)

= Log_ax^-1

= -1 x Log_ax

= -1 x -413

= 413 answer

TRY THIS………..

If Log_av= -526, find Log_a(¹/v)

VARIATION 1

x is inversely proportional to y. x=12 while y=5. Find y when x is 3.

Solution

x ⍺ k

      y

x = k

      y

12 = k

        5

k = 12 x 5

k = 60

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

k = 60, y= ?, x = 3.

x = k

      y

3 = 60

       y

y x 3 = 60  x  y   (multiply by y on both sides)

             y

3y = 60

3y = 60        (dividing by 3 on both sides)

3      3

y = 20

TRY THIS…………….

x is inversely proportional to y. x=20 while y=4. Find y when x is 16.

PERCENTAGE PROFIT 1

A man got a profit of 600/= after selling an item. Find the buying price if the percentage profit was 3%.

Solution

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

                         B.P

3% = 600   X  100   

            B.P.

B.P. x 3% = 60,000   x B.P.

                       B.P.

B.P. x 3 = 60,000   

        

B.P. x 3¹ =   60,000 ^20,000  

  ₁3                  3₁

B.P. = 20,000

                      

Hence Buying Price was 20,000/=

TRY THIS………………

A man got a profit of 34,100/= after selling an item. Find the buying price if the percentage profit was 10%.

FUNCTIONS 3

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.

GEOMETRIC PROGRESSION 1

Find the sum of the 1st eleven terms of the geometrical progression 2+6+18+54+…….

solution

G₁ = 2, r=3, n=11

    

S_n = G₁(rⁿ-1)/r-1

               

S₁₁ = 2(3¹¹-1)/3-1    [data substitution]

             

S₁₁ = ¹2(3¹¹-1)/2₁

              

S₁₁ = (3¹¹-1)

S₁₁ = 177147 - 1

S₁₁ = 177146

Hence the sum of the 1st 11 terms is 177146.

TRY THIS………………

Find the sum of the 1st 8 terms of the geometrical progression 7+14+28+…..

EXPAND 1

Expand 8w(5w – 7 + w)

solution

= 8w(5w – 7 + w)

= (8w x 5w) – (8w x 7) + (8w x w)

= 40w² – 56w + 8w²

= (40w² + 8w²)  – 56w

= 48w² – 56w answer

  

TRY THIS………..

Expand 3a(7a+ 30 - a)

MIDPOINT 1

Find a if the midpoint of a line from (a, 6) to (8, 10) is (7, 7)

Solution

x1=a, x2=8, y1=6, y2=10.

(7, 7)= (x1 + x2, y1 + y2)

                 2           2

(7, 7)= (a + 8,  6 + 10)

               2         2

(7, 7)= (a + 8, 16)

               2      2

Equating values of x;

7= a + 8

        2      

2 x 7= (a + 8)  x 2 ¹

               2 ₁     

14 = a + 8

14 - 8 = a

a = 6.

Hence a=6 answer

TRY THIS……………………..

Find c if the midpoint of a line from (c, 18) to (21, 14) is (3, 11)