ALGEBRA K2

 

GRADIENT K1

 

Find the slope of a line which passes through (-3, -4) and (8,-11)

 

Solution

 

x₁ = -3,  y₁ =-4,  x₂ = 8,  y₂ = -11

 

m = y2 –y1

          x2 – x1

 

m =   -11 –(-4)

          8 –(-3)

 

m =   -11 + 4

          8 + 3

 

m =    - 7

          11

 

 

Hence the slope is -7/11

 

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

 

Find the slope of a line which passes through (-11, -4) and (4,-15)

ALGEBRA K1

 

Multiply 2x - 7y by -5a

 

solution

 

= -5a(2x - 7y)

 

= (-5a x 2x) + (-5a x -7y)

 

= -10ax + 35ay     [since -5a x -7y = +35ay ]

 

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

 

Multiply 14p - 3q by  -4a

 

FUNCTIONS K1

 

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 - 12 + 7 = 22

 

16 + 4k - 5 = 22

 

4k + 16 - 5 = 22

 

4k + 11= 22

 

4k = 22 - 11

 

4k = 11

 

4k = 11

4      4

 

k = ¹¹/4

 

hence k=¹¹/4

 

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

 

 If f(x) = x⁴ - kx² + 3x - 11 has a remainder of 16 when divided by x-3; find k.

LOGARITHMS K2

 

Evaluate Log100000 +  log 0.00001 + log₃243

 

Solution

 

= Log100000 +  log 0.001 + log₃243

 

= Log10⁵ +  log 10^-3 + log₃3⁵

 

= 5Log10 +  (-3log 10) + (5log₃3)    [since log_aaⁿ  = nlog_aa]

 

= (5x1) + (-3x1) + (5x1)             [since log_aa = 1]

 

= 5 + (-3) + (5)

 

= 7

 

Hence Log100,000 +  log 0.001 + log₃243 = 7

 

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

 

Evaluate Log10,000,000 +  log 0.01 + log₃729

 

EXPONENTIALS K1

 

If 5^2w (40^w) = 1000000 ; Find w.

Solution

5^2w (40^w) = 1000000

(5²)^w (40^w) = 10⁶

(25)^w (40^w) = 10⁶

(25 x 40)^w = 10⁶

(1000)^w = 10⁶

(10³)^w = 10⁶

10^3w = 10⁶   (Bases are alike, so they cancel out)

3w = 6

3w = 6²

3       3

 

w = 3

 

TRY THIS…………………………….

 

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

FACTORS K1

 Evaluate using factors 1168² – 832²

Solution

We apply the difference of two squares: a² - b² = (a-b)(a+b).

1168² – 832² = (1168 + 832)( 1168 - 832)

                     = (2000)( 336)

                     = 672000   [only multiply 336 and 2, then add three zeros]

Hence 1168² – 832² = 672000

TRY THIS……………………

 

Evaluate 1253² – 747²

EXPONENTIALS K1

 

If 5^2w (40^w) = (0.000001)²⁰ ; Find w.

 

Solution

 

5^2w (40^w) = (0.000001)²⁰ ;

 

(5²)^w (40^w) = (10^-6)²⁰ ;

 

(25)^w (40^w) = 10^{-6 x 20} ;

 

(25 x 40)^w = 10^-120

 

(1000)^w = 10^-120

 

(10³)^w = 10^-120

 

10^3w = 10^-120   (Bases are alike, so they cancel out)

 

3w = -120

 

3w = -120^-40

3         3

 

w = -40 answer

 

TRY THIS…………………………   

 

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

VARIATION K1

 

x is inversely proportional to y. x=54 while y=5. Find y when x is 9.

 

Solution

 

x ⍺ k

      y

 

x = k

      y

 

54 = k

        5

 

k = 54 x 5

 

k = 270

 

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

 

k = 270, y= ?, x = 9.

 

x = k

      y

 

9  = 270

        y

 

y x 9 = 270  x  y   (multiply by y on both sides)

             y

 

 

9y=270

 

9y = 270           [Dividing by 10 both sides]

9      9

 

Hence y = 30.

 

 

TRY THIS…………….

 

 

x is inversely proportional to y. x=80 while y=4. Find y when x is 10. 

LOG J3

 

If Log_ax = 56, find Log_a(¹/c)

 

Solution

 

= Log_a(¹/c)

 

= Log_ac^-1

 

= -1 x Log_ac

 

= -1 x 56

 

= -56 answer

 

TRY THIS………..

 

If Log_am= 34.6, find Log_a(¹/m)