MID-POINT 1

Find a if the midpoint of a line from (a, 6) to (4, 10) is (7, 7)

Solution

x1=a, x2=4, y1=6, y2=10.

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

                 2             2

(7, 7)= (a + 4,  6 + 10)

                2          2

(7, 7)= (a + 4, 16)

                2      2

Equating values of x;

7= a + 4

        2      

2 x 7= (a + 4)  x 2 ¹

              2 ₁     

14 = a + 4

14-4 = a

a = 10.

Hence a=10

TRY THIS……………………..

Find a if the midpoint of a line from (a, 12) to (7, 10) is (3, 11)

ALGEBRA 6

If a² – 20 = 61; find a.

solution

a² = 61+ 20

a² = 81

√a² = √81    keep root sign on both sides

a = √81 

  

 a=9

Hence a = 9

TRY THIS………..

If a² – 19 = 102; find a.

SETS 5

If n(A)= 80 , n(B)= 90 and n(AuB)= 166, find n(AnB).

Solution

n(AuB) = n(A) + n(B) - n(AnB)

166 = 80 + 90 - n(AnB)

166 = 170 - n(AnB)

166 - 170 = - n(AnB)

-4 = -n(AnB)

n(AnB) = 4 [after dividing by -1 both sides]

Hence n(AnB) = 4 answer

TRY THIS………….

If n(A)= 89 , n(B)= 87 and n(AuB)= 131, find n(AnB).

VECTORS 2

Write (8,6) in terms of i and j.

Solution

= (8,6)

= (8,0) +  (0,6)

= 8(1,0) +  6(0,1)

= 8i + 6j   [since (1,0)=i and (0,1=j)

Hence (8,6) = 8i +  6j  

TRY THIS……………………..

Write (2,14) in terms of i and j.

LOGARITHMS 21

Find x if Log₄x + Log₄(3x+8) =2

Solution

Log₄x + Log₄(3x+8) =2

Log₄x(3x+8) =2

x(3x+8) =4²

3x²+ 8x =4².

3x²+ 8x =16.

3x²+ 8x - 16=0

3x²+ 12x – 4x- 16=0       [Splitting +8x into 12x – 4x]

(3x²+ 12x) – (4x+16)=0     [grouping the factors] 

3x(x+ 4) – 4(x+ 4)=0     [taking out common factors] 

(3x- 4)(x+4)=0    

3x- 4=0 OR x+4=0    

3x = 4 OR x = -4

3x = 4 OR x = -4

3      3

x = ⁴/3 OR x = -4

Hence x = ⁴/3 OR x = -4

TRY THIS…………………………  

Find m if Log₉m + Log₉(m-24) =0

ALGEBRA 5

If k⁸ - 200 = 56; find k.

Solution

K⁸ - 200 = 56

K⁸ = 56 + 200 [taking 200 in the right hand side]

K⁸ = 256

k = 2 [since the 8^th root of 256 is 2]

Hence k = 2

TRY THIS…………………………

If c⁵ - 17 = 15; find c.

LOGARITHMS 20

If log_x 2401 – log₃729 = -4; find x

solution

log_x 2401 – log₃729 = -2

log_x 2401 – log₃3⁶ = -2

log_x 2401 – 6log₃3 = -2

log_x 2401 – 6 x 1 = -2

log_x 2401 – 6 = -2

log_x 2401 = -2 + 6

log_x 2401 = 4 

2401 = x⁴        2401=7x7x7x7=7⁴ by prime factors

7⁴ = x⁴      Powers cancel out

x = 7

Hence x = 7.

TRY THIS………….

If log_m2048 – log₆216 = 8; find m

FUNCTIONS 4

Find the inverse of {(9,7), (7,0), (-9,8), (-14,-6), (17,-4)}

Solution

Here, the inverse is the opposite of a given expression. Therefore, the values will be swapped. 

Hence, inverse is { (7,9), (0,7), (8,-9), (-6,-14), (-4,17)  }

TRY THIS………….

Find the inverse of {(4,6), (-1,3), (30,5), (-4, 6), (19,-7)}

SLOPE OR GRADIENT-1

A line passes through (3a, 27) and (8,5a). If its slope is 4, find a.

Solution

x1=3a, y1 = 27, x2 = 8, y2 = 5a

Slope(m) =    y2 – y1

                     x2 – x1

               4 =  5a – 27

                      8 – 3a

               4 =    5a – 27     [cross multiplying]

               1       8 – 3a

4(8 – 3a) = 5a – 27

32 – 12a = 5a – 27

32 – 12a + 27 = 5a

32 + 27 = 12a+ 8a

60 = 20a

60 = 20a

20    20

3 = a

Hence a = 3

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

A line passes through (2a, 7) and (8,5a). If its slope is 2, find a.

LOGARITHMS 19

Simplify        Log0.0000001

                        Log(1/100)

Solution

=    Log0.0000001

       Log(1/100)

=    Log10^-7

       Log100^-1

=    Log10^-7

       Log(10²)^-1

=    Log10^-7

       Log10^-2

=    -7Log10

       -2Log10

=    -7

       -2

=    ⁷/2

Hence           Log0.0000001         =  ⁷/2

                        Log(1/100)

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

Simplify        Log0.001

                    Log(1/1000)

INEQUALITIES 2

Find x in the following inequality. 46 – 6x ≤ 72

Solution

46 – 6x ≤ 72

46-46 – 6x ≤ 72-46    subtracting 46 on both sides.

0 – 6x ≤ 26

-6x ≤ 26

-6x ≥ 26

-6     -6

x  ≥ ^-13/3  The inequality sign changes because x has changed from negative to positive.

Hence x  ≥ -13/3

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

Find x in the following inequality. 56 – 7x ≤ 77