MID-POINT 1

Find the midpoint of a line from (11, 14) to (17, 10)

Solution

x₁=11, x₂=17, y₁=14, y₂=10.

Mid point = (x1 + x2, y1 + y2)

                        2             2

Mid point = (11 + 17, 14 + 10)

                       2               2

Mid point = (28, 24)

                     2    2

Hence midpoint = (14, 12)

TRY THIS……………………..

Find the midpoint of a line from (6, 12) to (26, 6)

SIMULTANEOUS EQUATIONS 1

Solve the following equations by substitution method.

3x - y = 11

x + 3y = 27

Solution

3x - y = 11 ----------- (i)

x + 3y = 27 ---------- (ii)

From equation (i)

3x - y = 11

3x - 11 = y

∴ y = 3x - 11 ----------(iii)

substitute equation (iii) in (ii) above,

x + 3(3x – 11 ) = 27

x + 9x – 33  = 27

x  + 9x  = 27 + 33

 10x  = 60                  [since x+9x=10x and 27+33=60]

10x  = 60

                  

10x = 60

10      10

x = 6

From equation (iii)

y = 3x - 11 ----------(iii)

y = 3(6) – 11

y = 18 – 11

y = 7

Hence x=6 and y=7.

TRY THIS………………………..

Solve the following equations by substitution method.

x + y = 11

2x - y = 4

FACTORIZATION 2

Factorize 5x² + 4x – 12.

Solution

= 5x² + 4x - 12.  We split the middle term (+4x) to be (-6x + 10x).

= 5x² - 6x + 10x – 12                (+4x) = -6x + 10x.

= 5x² - 6x + 10x – 12                grouping the factors.       

= x(5x - 6) + 2(5x - 6)   factoring out

= (x + 2) (5x - 6)

Hence 5x² + 4x - 12= (x + 2) (5x -6)answer.

TRY THIS…..

Factorize 2x² + 7x – 15.

LOGARITHMS 3

If log₇(3x + 19)=2; find x

   Solution

log₇(3x + 19)=2

 (3x + 19)=7²   

                      

 3x + 19=49

3x = 49 – 19

3x =  30

3x =  30   

3         3

x = 10

Hence x = 10

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

If log₂(3x - 28) = 5; find x

FUNCTIONS-2

If f(x) = x⁴ + kx² + 2x -20  has a remainder of 22 when divided by x+2; find k.

solution

f(x) = x⁴ + kx² + 2x - 20 ----------------- (i)

x + 2 = 0

x = -2

_____________________________

Then substitute x = -2 in (i) above

f(x) = (-2)⁴ + k(-2)² + 2(-2) -20 = 22

16 + 4k + (-4) - 20 = 22

16 + 4k -24 = 22    [since -4 – 20 = -24]

4k - 8 = 22       [since 16 – 24 = -8]

4k = 22 + 8

4k = 30

4k = 30

4      4

k = 15/2 = 7.5

hence k=7.5

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

If f(x) = x⁴ - kx² + 6x - 10 has a remainder of 40 when divided by x+3; find k.

ALGEBRA 1

If  x+5y = 1            find  ^x/y             

    x-3y     4

solution

x+5y = 1           

x-3y     4

1(x-3y) = 4(x+5y)           [after cross multiplying]

x-3y = 4x+20y

x - 4x = 20y + 3y

-3x = 23y

¹-3x = 23y

₁-3      -3

 X = -23y/3

x/y = -23/3

x = 5y

     

x = 5y

y     y

x = 5

y    

Hence ^x/_y = 5  answer.          

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

If  x+2y = 1            find  ^x/y             

    x- 2y     5

LOGARITHMS 2

Find x if Log_x0.00001 = -5

solution

Log_x0.00001 = -5

0.00001  = x^-5 .  [0.000001  = 10^-5  ]

10^-5 = x^-5 .

10⁵ = x⁵ .  { equal powers cancel out}

x = 10

Hence x = 10

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

Find x if Log_x256 = 8

FUNCTIONS 1

If f(x) = |x - 24| evaluate f(-27)

solution

f(x) = |x - 24|

f(-27) = |-27 - 24|

f(-27) = |-51| = 51 since any number out of absolute signs is +ve.

Hence f(-27) = 51

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

If f(x) = |x - 40| evaluate f(-15)

RATIOS 1

Given the ratios M:N = 16: 8 and N:P = 4:7 Find the ratio M:P.

solution

M = M x N

P      N     P

M =   16 x 4¹

P    ₂ 8     7

M =      16

P        (7x2)

M =       ⁸16  =  8

P           ₇14      7

Hence M:P=8:7

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

Given the ratios M:N = 20: 8 and N:P = 12:7 Find the ratio M:P.

PERCENTAGE PROFIT 1

A man got a profit of 200/= after selling an item for 8000/=. Find the percentage profit.

Solution

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

                      B.P.

%’ge profit = 200   X  100   

                    8000

%’ge profit = 2.5%   

                       

Hence Percentage Profit is 2.5%

TRY THIS……………………

A man got a profit of 10,000/= after selling an item for 600,000/=. Find the percentage profit.

FACTORIZATION 1

Factorize 1-V²

Solution

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

1-V² = 1²-V²       

        = (1 - V)(1 + V)

Hence 1 - V² = (1 - V)(1 + V)

TRY THIS…………….

Factorize 25 - W²

INEQUALITIES 1

If │2x – 13 │= 21; Find  x

Solution

±(2x – 13 )= 21

2x -13= 21    OR   –(2x-13) = 21

2x – 13 = 21    OR   -2x+13= 21

2x = 21 + 13    OR   -2x= 21-13

2x = 34     OR   -2x  =  8

2      2                2        2

x = 17  OR  x = 4

Hence x = 17 OR x = 4     

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

If │4x – 16 │= 32; Find  x

LOGARITHMS 1

If log 3= 0.4771; find the value of log 90,000 without using tables.

solution

log90,000=log(10,000 x 9)

=log10,000 + log9

=log10⁴ – log3²

=4log10 – 2log3

=(4x1) –  2(0.4771)

=4– 0.9542

= 3.0458

Hence log 90,000 = 3.0458

TRY THIS……………

If log 3= 0.4771; find the value of log 8,100 without using tables.

EXPONENTIAL EQUATIONS 1

If a² = 3; find the value of a⁶.

solution

a⁶ = (a²)³.

a⁶ = (3)³.   [ since a² = 3]

a⁴ = 3 x 3 x 3

a⁴ = 27  answer

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

If a² = 13; find the value of a⁴.

APPROXIMATIONS 1

Estimate the value of 4143  ÷ 52

Solution

= 4123     

     52

= 4000      [ since 4143 ≈ 4000   and   52 ≈ 50 ]

     50

= 80

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

Estimate the value of 7203  ÷ 19

GEOMETRY 1

If x-7⁰ and 4x + 62⁰ are complementary angles, find the value of x.

solution

Complementary angles add up to 90⁰.

so,  x-7⁰ + 4x + 62⁰ = 90⁰.

so,  x + 4x + 62⁰ - 7⁰ = 90⁰.

so,  5x + 55⁰ = 90⁰.

so,  5x = 90⁰ -  55⁰

so,  5x = 35⁰

so,  5x = 35⁷

       5       5

Hence x = 7

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

If 3x - 4⁰ and 2x + 54⁰ are complementary angles, find the value of x.