EXPONENTIALS K3

 If 8^10+2a = 32^a-2; find a

Solution

8^10+2a  = 32^a-2 

(2³)^10+2a  = (2⁵)^a-2 

2^30+6a = 2^5(a-2) 

2^30+6a  = 2^5a-10

2^30+6a   = 2^5a-10      [Same bases both sides, cancel out]

30+6a = 5a – 10

30 + 6a+10 = 5a        [after shifting 10 to the left]

40 + 6a = 5a        

40 = 5a-6a        [after shifting 6a to the left]

40 =    -1a

-1       -1

 

-40 = a

Hence a=-40

TRY THIS……………………….

 

If 4^7+2y = 64^y-2; find y

QUADRATICS K1

 

What must be added to x²  + 60x to make the expression a perfect square?

Solution

a=1, b=60,c=?

b² = 4ac

(60)² = 4 x 1 x c

3600 = 4c

3600 = 4c

 4        4

 

900 = c

Hence number to be added is 900

 

TRY THIS……………………

What must be added to x²  + 24x to make the expression a perfect square?  

LOGARITHMS K4

 If log₃(10x - 29)=4; find x.

 

Solution

 

log₃(10x - 29)=4

 

 (10x - 29)=3⁴       

                                            

 10x - 29=81

 

10x = 81 + 29

 

10x =  110

 

10x =  110¹¹   

10      10

 

x = 11

 

Hence x = 11

 

TRY THIS…………….

 

If log₃(5u + 16)=4; find u

VARIATION K2

 x is inversely proportional to y. x=12 while y=20. Find y when x is 15.

 

Solution

 

x ⍺ k

     y

 

x = k

     y

 

12 = k

      20

 

k = 12 x 20

 

k = 240

 

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

 

k = 240, y= ?, x = 15.

 

x = k

     y

 

15 = 240

        y

 

y x 15 = 240  x  y   (multiply by y on both sides)

              y

 

15y = 240

15     15

 

y = 16 .

 

TRY THIS…………….

 

x is inversely proportional to y. x=12 while y=40. Find y when x is 15.

SETS K1

 In Mwangata district the number of people who speak Kiswahili or Lingala is 410. 200 of them speak Kiswahili and 300 of them speak Lingala. How many speak both languages?

 

solution

 

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

 

Let Kiswahili=n(K), Lingala= n(L).

 ---------------------------------------------------------------------- 

n(K)= 200 ,

n(L)= 300,

n(KuL) = 410,

n(KnL)=?

---------------------------------------------------------------------- 

 

 

n(KuL) = n(K) + n(L) - n(KnL)

 

 

410  =  200 + 300 – n(KnL)

 

 

410  =  500 – n(KnL)

 

 

n(KnL) =  500 – 410

 

 

n(EnF) =  90

 

 

Hence n(EnF) = 90 answer

 

TRY THIS……………………………….. 

 

In Kazilabatemi district the number of people who speak Kiswahili or Spanish is 390. 260 of them speak Kiswahili and 320 of them speak Spanish. How many speak both languages?

 

INEQUALITIES K1

 

If 12x + 8 ≥ 26 + 3x ≥ 9x – 10; find x.

 

Solution

 

12x + 8 ≥ 26 + 3x and  26 + 3x ≥ 9x - 10

 

12x – 3x ≤ 26 - 8 and  26 +10 ≤ 9x - 3x

 

9x ≤ 18 and  36 ≤ 6x

 

9x ≤ 18  and  36 ≤ 6x

9      9            6      6

 

x ≤ 2 and   6 ≤ x

 

x ≤ 2 and  x  ≥  6

 

TRY THIS…………….

 

 

If 12x + 34 ≥ 7 + 3x ≥ 9x – 22; find x

LOGARITHMS K3

 

If Log_ax = 18, find Log_a(¹/x⁵).

 

Solution

 

= Log_a(¹/x⁵)

 

= Log_ax^-5   [Since 1/aⁿ = a^-n]

 

= -5 x Log_ax   [Since Log_mp^q   = q x Log_mp]   

 

= -5 x 18

 

= -90

 

 

TRY THIS………..

 

If Log_ac= 40, find Log_a(¹/c⁶)

PERIMETERS K1

 

Area of rectangle is 68m². Find its perimeter if width is 4m. 

 

Solution

 

A = L x w

 

68 = L x 4

 

68 = 4L                 (Dividing by 4 both sides)

4      4

 

17 = L

 

So, L=17, w=4, P=?

 

Now; P =2(L + w)

 

= 2(17 + 4)

 

= 2 x 21            [because 17 + 4 = 21]

 

= 42m

 

Hence perimeter is 42m

 

TRY THIS………..

 

Area of rectangle is 96m². Find its perimeter if width is 4m.  

PROBABILITY K1

 Given the interval of 1-16 inclusive, Find the probability of having an even number or a prime number.

 

Solution

 

This is a non-mutually exclusive event.

n(S) = number of sample space = 14

 

Let P(E)= probability of even numbers = ⁸/14

Let P(R)= probability of prime  numbers = ⁶/14

P(EnR) = probability of having both even and prime numbers= ¹/14

 

∴ P(EuR) = P(E) + P(R)  - P(EnR).

 

 P(EuR) =  8  +  6 – 1

                14    14   14

 

P(EuR) =  14  -   1 

                14     14  

 

P(EuR) =  13   

                14         

 

Hence probability of an even number or a prime number is ^13/14  .

 

TRY THIS........

 

 

Given the interval of 1-18 inclusive, find the probability of having an even number or a prime number

EXPONENTIALS K2

 

Simplify (7a)² - 9a² - 8a².     

Solution

= (7a)² - 9a²   - 8a².        

= 7².a² - 9a²  - 8a².                    Since (mn)² =m² . n²

= 49a² - 9a²  - 8a².    {since the square number of 7 is 49}

= 46a²  - 17a².

= 29a²        

TRY THIS………………

Simplify (9C)² – 20C² – 11C²