LOGARITHMS 1

Simplify Log₂1024 - Log₃27

Solution

= Log₂1024 - Log₃27

= Log₂2¹⁰ - Log₃3³

= 10Log₂2 - 3Log₃3

= (10 x 1) - (3 x 1)

= 10 - 3

= 7

Hence Log₂1024 - Log₃27 = 7

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

Simplify Log₂256 – Log₅125

EXPONENTIALS 1

If 5^2w (40^w) = 0.000001 ; Find w.

Solution

5^2w (40^w) = 0.000001

(5²)^w (40^w) = 10^-6

(25)^w (40^w) = 10^-6

(25 x 40)^w = 10^-6

(1000)^w = 10^-6

(10³)^w = 10^-6

10^3w = 10^-6   (Bases are alike, so they cancel out)

3w = -6

3w = -6²

3       3

w = -2

TRY THIS…………………………   

If 4^2a (2^a) = 128 ; Find a.

FUNCTIONS 1

If f(x) = x⁴ + kx² + 6x + 7 has a remainder of 10 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 = 10

16 + 4k + (-12) + 7 = 10

16 + 4k - 12 + 7 = 10

16 + 4k - 5 = 10

4k + 16 - 5 = 10

4k + 11= 10

4k = 10 - 11

4k = -1

4k = -1

4      4

k = ^-1/4

hence k= ^-1/4

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

  

If f(x) = x⁴ - kx² + 2x - 1 has a remainder of 18 when divided by x-3; find k.

VARIATION 1

x is directly proportional to y. x=32 while y=4. Find y when x is 184.

Solution

x ⍺ y

x = ky

32 = k x 4

32 = 4k    [dividing by 4 both sides]

4      4

k = 8

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

k = 8, y= ? x = 184.

x = ky

184 = 8 x y

184 = 8y

8        8

y = 23.

TRY THIS…………….

x is directly proportional to y. x=40 while y=4. Find y when x is 170.

SLOPE/GRADIENT 1

Find the slope of a line which passes through (-11, -8) and (6,-9)

Solution

x₁ = -11,  y₁ =-8,  x₂ = 6,  y₂ = -9

m = y2 –y1

       x2 – x1

m =   -9 –(-8)

          6 –(-11)

m =   -9 + 8

          6 + 11

m =    - 1

            17

Hence the slope is -1/17

TRY THIS……………………… 

Find the slope of a line which passes through (-11, -2) and (9,-14)

POLYGONS 1

A regular polygon has 64 sides. Find the total degrees of that polygon.

Solution

n = 64

Total degrees = (n - 2)180⁰

                       = (64 – 2)180⁰

                       = 62 x 180⁰

                       = 11160⁰

Total degrees = 11160⁰

TRY THIS……………………….

A regular polygon has 76 sides. Find the total angles of that polygon.

ALGEBRA 1

Calculate the value of 4x + k + 80 + y , when x = 8, k = 10 and y =− 33 .

Solution

=4x + k + 80 + y

=4(8) + 10 + 80 + (-33)

=4(8) + 10 + 80 + (-33)

=32 + 10 + 80 – 33

= 58 – 33

= 25 answer

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

Calculate the value of 8x + k +- 30 + y, when x = 11, k = 17 and y =− 10 .

A. P. 2

The 1st term of an A.P. is 72 and the common difference is 80. Find the 10th term.

Solution

A₁= 72, d = 80

A_n = A₁ + (n-1)d

A₁₀ = A₁ + (10-1)d

A₁₀ = A₁ + 9d

A₁₀ = 72 + (9x80) [ after substituting A₁= 72, d = 80 as given above]

∴ A₁₀ = 72 + 720

A₁₀ = 792

Hence the 10th term is 792.

  

TRY THIS……………

The 1st term of an A.P. is 160 and the common difference is 219. Find the 5th term.

EXPONENTIALS 3

Simplify 9a⁶b⁹ × 6a⁵b¹⁶ .

Solution

= 9a⁶b⁹ × 6a⁵b¹⁶ .

= (9 x 6)( a⁶x a⁵)( b⁹ × b¹⁶)

= 54a⁶⁺⁵b⁹⁺¹⁶.

= 54a¹¹b²⁵.

TRY THIS……………………… 

 Simplify 11c⁹d¹⁷ × 5c⁷d¹³ .

LOG 1

If Log_ax = 4, find Log_a(¹/x¹⁵)

Solution

= Log_a(¹/x¹⁵)

= Log_ax^-15          [since 1/aⁿ = a^-n]

= -15 x Log_ax    [since Log_axⁿ   = nLog_ax ]

= -15 x 4

= -60 answer

TRY THIS………..

If Log_ab= 6, find Log_a(¹/b⁴⁰)