GRADIENT 1

A line passes through (9a, 7) and (8, 5a). If its slope is 2, find a.

Solution

x1=9a, y1 = 7, x2 = 8, y2 = 5a

Slope(m) =    y2 – y1

                     x2 – x1

               2 =  5a – 7

                      8 – 29a

    2 =    5a – 7     [cross multiplying]

   1       8 – 9a

2(8 – 9a) = 5a – 7

16 – 18a = 5a – 7

16 – 18a + 7 = 5a

16 + 7 = 5a+ 18a

23 = 23a

23 = 23a

23    23

1 = a

Hence a = 1

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

A line passes through (2a, 5) and (4, 6a). If its slope is 8, find a.

LOGARITHMS 2

If log 2= 0.3010; find the value of log 25,000 without using tables.

solution

log25000=log(100,000÷ 4)

=log100,000 - log4

=log10⁵ – log2²

=5log10 – 2log2

=(5x1) – 2(0.3010)

=5– 2(0.3010)

=5 – 0.6020

= 4.398

Hence log25 = 4.398

TRY THIS……………

If log 2= 0.3010; find the value of log 5000 without using tables.

FUNCTIONS 1

If F(x) = 12x + 12; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 12x + 12

y – 12 = 12x

y – 12  = 12x

   12        12

y – 12   = x

   12

x  =   y – 12     after rearranging

            12

 y^-1 = x – 12     after interchanging x and y variables.

             12

F^-1(x) = x – 12     after interchanging x and y variables.

                12

Hence, F^-1(x) = x – 12     

                              12

TRY THIS……………………………

If F(x) = 3x + 16; Find F^-1(x).

INEQUALITIES 1

If 3x – 1 ≤ x + 5 ≤ 21x; find x

solution

3x – 1 ≤ x + 5 and  x + 5 ≤ 21x

3x – x ≤ 1+ 5 and  5 ≤ 21x - x

2x ≤ 6(divide by 5 both sides) and  5 ≤ 20x (divide by 20 both sides)

x ≤ 3 and  ¹/4 ≤ x

x ≤ 3 and  x ≥ ¹/4 

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

If 14x – 1 ≤ x + 25 ≤ 6x; find x