Friday 29 December 2017

LOGARITHMS 11


Change the following into Logarithmic form.

i)  53 = 125

ii)  33 = 27

iii)  110 = 1

Solution


53 = 125 log5 125 = 3

33 =27
log3 27 = 3

110 = 1
log11 1 = 0

TRY THIS………………….

Change the following into Logarithmic form.


i)  112 = 121

ii)  34 = 81

iii)  150 = 1 

FACTORIZING-3


Factorize 1 - 225m2

Solution

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

1 - 225m2 = 12 - 152m2       

                = 12 – (15m)2       

                = (1 - 15m)(1 + 15m)

Hence 1 - 225m2 = (1 - 15m)(1 + 15m) answer


TRY THIS…………….



Factorize 1 – 121c2

EXPONENTIALS 7


If 820 = 32a-2; find a.

Solution

820 = 32a-2 

(23)20 = (25)a-2 

260 = 25(a-2) 

260 = 25a-10

260 = 25a-10      [Same bases both sides, cancel out]

60 = 5a – 10

60 + 10 = 5a

70 = 5a

70 = 5a
5      5

14 = a

Hence a=14

TRY THIS……………………….


If 47 = 64y-2; find y.

LOGARITHMS 10


If log5(2x + 19)=2; find x.

   Solution

Log5(2x + 19)=2

(2x + 19)=52           
                                            
2x + 19=25

2x = 25 – 19

2x = 6

2x =   6   
2        2

x = 3

Hence x = 3

TRY THIS………………


If log7(2t - 23)=2; find t

Thursday 28 December 2017

WORD PROBLEMS 1


Ana, Joy and Fred shared 800 sweets such that Ana got 10 sweets less than Joy while Fred got three as much as what Ana got. Find those given to Fred.

Solution

The starting point is Joy. Let the sweets given to Joy be x. See the table below:

ANA
JOY
FRED
x-10
x
3(x-10)=3x-30
  
Then;

ANA + JOY + FRED = 800

x-10 + x + 3x-30 = 800

x + x + 3x - 30 - 10 = 800

5x - 40 = 800

5x = 800 + 40

5x = 840

5x = 840
5       5

x = 168

The sweets given to Fred

= 3x – 30

= 3(168) – 30

= 504 – 30

=474

Hence Fred got 474 sweets.


TRY THIS………………………


Kemi, Roy and Isaya shared 610 sweets such that Kemi got 10 sweets less than Roy while Isaya got four as much as what Kemi got. Find those given to Kemi.   [100 answer] 

RATIOS 1


Given the ratios U:V = 9: 40 and V:W = 20:7 Find the ratio U:W.

solution

U = U x V
W   V   W

U =    9  x  201
W   2 40    7

U =   9 x 1
W    2 x 7

U =   9
W    14

Hence U:W = 9:14

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



Given the ratios D:E = 20:7 and E:F = 28:15;  Find the ratio D:F.

PERCENTAGE PROFIT 2


Dikshi got a profit of 1,794/= after selling an item for 24,794/=. Find the percentage profit.

Solution

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

But; Profit = SP - BP;     [where S. P. represents Selling Price.]

So; B.P = SP – Profit

B.P = 24,794 – 1794

       = 23,000

%’ge profit = 1794   x  100    
                      23000

%’ge profit = 1794 
                      230

%’ge profit = 74/5%   
        
              
Hence Percentage Profit was 74/5%   


TRY THIS………………..


Janhwi got a profit of 220,000/= after selling an item for 266,200/=. Find the percentage profit.      Answer 21%


AXIS OF SYMMETRY 1


Find the axis of symmetry for F(x) = 7x2 - 50x + 7

Solution

a=7, b=-4

Axis of symmetry = -b/2a

                              =  -(-50)
                                  2 x 7

                              =    50
                                   14

                              =     25
                                     7


Hence the axis of symmetry is 25/7


TRY THIS…………


Find the axis of symmetry for F(x) = 8x2 - 15x + 7 


Wednesday 27 December 2017

VECTORS 1


If u = 6i + 2j and v = 7i + 10j find 5u - 3v

solution

= 5u - 3v

= 5(6i + 2j) - 3(7i + 10j)

= 30i + 10j - 21i - 30j

= (30i + 21i) + (10j - 30j)

= 51i - 20j

Hence 5u - 3v = 51i - 20j

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



If u = 7i + 3j and v = 4i + 4j;  find 2u + 6v.

LOGARITHMS 9


Evaluate Log2(1024 x 8).

Solution

= Log2(1024 x 8)

= Log21024 +  Log28          (applying the product rule)

= Log2210 +  Log223             ( 1024= 210 and 8=23 )

= 10Log22 +  3Log22       ( remember  Logaac = cLogaa )

= (10 x 1) +  (3 x 1)          ( remember  Logaa = 1 )

= 10 + 3

= 13

hence Log2(1024 x 8) = 13 

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


Evaluate Log2(1024 x 64). 

EXPONENTIALS 6


If k5 - 200 = 2925; find k.

Solution

k5 - 200 = 2925

k5 = 2925 + 200 [taking 200 in the right hand side]

k5 = 3125

k = 5 [since the 5th root of 3125 is 5]

Hence k = 3

TRY THIS…………………………


If w5 - 20 = 223; find w.


QUADRATICS 3


What must be added to x+ 40x to make the expression a perfect square?

Solution

a=1, b=12,c=?
b2 = 4ac
(40)2 = 4 x 1 x c
1600 = 4c

1600 = 4c
 4         4

400 = c

Hence number to be added is 400

TRY THIS……………………


What must be added to x+ 28x to make the expression a perfect square?  


LOGARITHMS 8


Simplify Log2128 – Log53125 – Log6216

Solution

= Log2128 – Log53125 – Log6216

= Log227 – Log55– Log663

= 7Log22 – 5Log55 – 3Log6because Logaan =  nLogaa

= (7 x 1) – (5 x 1) – (3 x 1)    

= 7 – 5 - 3

= -1

Hence Log2128 – Log53125 – Log6216 = -1 

TRY THIS…………..



Simplify Log22048 – Log39 – Log53125

Tuesday 26 December 2017

PROBABILITY 2


A number is chosen at random from 1 – 10 inclusive. Find the probability that it is a multiple of five or an even number.

Solution

THIS IS A MUTUALLY EXCLUSIVE EVENT.

Let n(S) represent sample space
P(E) = probability of even number
P(M) = probability of a multiple of 5

n(S) = {1, 2, 3, 4, 5, 6, 7, 8, 9,10} = 10
n(E) = {2, 4, 6, 8, 10} = 5
n(M) = {5,10} = 2
P(E) = 5/10
P(M) = 2/10
………………………….
P(EuM) = P(E) + P(M)

P(EuM) = 5/10 + 2/10

P(EuM) = 7/10

TRY THIS………………

A number is chosen at random from 17 – 24 inclusive. Find the probability that it is a prime or an even number.

VARIATION 2


x is directly proportional to y. x=12 while y=4. Find y when x is 600.

Solution

x y
x = ky

12 = k x 4

12 = 4k
4      4

k = 3
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 3, y= ? x = 600.

600 = 3 x y

600  = 3y       (dividing by 3 on both sides)
 3        3

Hence y = 200 answer.

TRY THIS…………….

x is directly proportional to y. x=20 while y=4. Find y when x is 70.


FUNCTIONS 4


If F(x) = 21x + 10; 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= 21x + 10

y – 10 = 21x

y – 10  = 21x
   21        21

y – 10   = x
   21

x  =   y – 10     after rearranging
           21

 y-1 = x – 10     after interchanging x and y variables.
             21

F-1(x) = x – 10     after interchanging x and y variables.
                21

Hence, F-1(x) = x – 10     
                             21

TRY THIS……………………………



If F(x) = 17x + 26; Find F-1(x).