SETS H20

 

 If n(A)= 85 , n(B)= 96 and n(AuB)= 119, find n(AnB).

 

Solution

 

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

 

119 = 85 + 96 - n(AnB)

 

119 = 181 - n(AnB)

 

119 - 181 = - n(AnB)

 

-62 = -n(AnB)

 

n(AnB) = 62 [after dividing by -1 both sides]

 

Hence n(AnB) = 62 answer

 

 

TRY THIS………….

 

 

If n(A)= 91 , n(B)= 95 and n(AuB)= 127, find n(AnB).

SIMPLIFY D2A

 

Simplify the expression c² - t²/ (c + t)

 

Solution

 

from difference of two squares p² - q² = (p - q)(p + q)

 

= c² - t²/ (c + t)

 

= (c + t)(c - t)

      c + t

 

= (c + t)(c - t)

      c + t

 

=  c - t    answer

      

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

 

 

Simplify the expression 

 

u² - y²/ (u + y)

SIMPLE INTEREST 1E

 

Rayan deposited the amount of 8,000/= in a bank for 5 years and got a profit of 2800/=. Find the interest rate?

 

Solution

 

I = 2800/=, P = 8,000/=, T = 5 years, R = ?

 

I  = PRT

      100

 

Where P=Principal amount

            I = simple Interest

           T= Time

           R=Rate

 

2800  = 8,000 x R x 5

                      100

 

 

2800  = 8,000 x R x 5

                      100

 

2800  = 80 x R x 5

                      

2800  = 400R

          

⁷2800  = 400R

  400       400

 

Hence the interest rate was Tsh 7%

 

 

TRY THIS……………….

 

 

Kibonde deposited the amount of 15,000/= in a bank for 4 years and got a profit of 3000/=. Find the interest rate?

 

EXPAND 3C

Expand 8w(5w + 7 – w²)

 

Solution

 

= 8w(5w + 7 – w²)

 

= (8w x 5w) + (8w x 7) - (8w x w²)

 

= 40w² + 56w - 8w³ answer 

ARITHMETIC PROGRESSION A2

 

Find the sum of the 1st 22 terms of the following arithmetic progression: 5+11+17+…………………

 

Solution

 

d=6, A₁ = 5,n=22

 

S_n = n[2A₁ + (n-1)d]

       2

 

S₂₂ = 22[2(5) + (22-1)d]

        2

 

S₂₂ = 11[10 + 21d]

 

S₂₂ = 11[10 + 21(6)]       since d=6

 

S₂₂ = 11[10 + 126]  

         

S₂₂ = 110 + 1386

          

S₂₂ = 1496

 

 

Therefore the sum of 1st 22 terms is 1496.

 

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

 

Find the sum of the 1st 15 terms of the following arithmetic progression: 6+10+14+…………………

ALGEBRA C2D

Factorize 121-9m²

 

Solution

 

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

 

121-9m² = 11² - 3²m²       

             = 11² - (3m)²     

             = (11 - 3m)(11 + 3m)

 

Hence 121 - 9m² = (11 - 3m)(11 + 3m)

 

TRY THIS…………….

 

Factorize 225 - 36c² 

EXPONENTIALS K10

 

If 4¹⁰ = 32^a-5; find a

 

Solution

 

4¹⁰ = 32^a-5 

 

(2²)¹⁰ = (2⁵)^a-5 

 

2²⁰ = 2^5(a-5) 

 

2²⁰ = 2^5a-25

 

2²⁰ = 2^5a-25  [Same bases both sides cancels out]

 

20 = 5a – 25

 

20 + 25 = 5a

 

45 = 5a

 

45 = 5a

5      5

 

9 = a

 

Hence a=9

 

TRY THIS……………………….

 

 

If 4³ = 32^x-2; find x

PIE CHART A1

 

In a survey of 60 people, 28 people said their favourite sport was football. What angle in a pie chart would this represent?

Solution

We make a fraction of 28 ot of 60, then multiply by 360⁰.

= 28 x 360⁰.

   60

 

= 28 x 360⁰₆

   60₁

 

= 28 x 6

 

= 168⁰.

 

Hence the angle is 168⁰ 

 

TRY THIS………………………   

 

In a survey of 30 people, 23 people said their favourite sport was football. What angle in a pie chart would this represent?

LOGARITHMS K13

 Simplify Log₂4096 - Log₃27

 

Solution

 

= Log₂4096 - Log₃27

 

= Log₂2¹² - Log₃3³

 

= 12Log₂2 - 3Log₃3

 

= (12 x 1) - (3 x 1)

 

= 12 - 3

 

= 9

 

Hence Log₂4096 - Log₃27 = 9

 

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

 

 

Simplify Log₂256 – Log₅25

COORDINATES A2

If m(4,6), B(7,10) and C(a,2) are collinear, find a.

 

Solution

 

x1=4, y=6, x2=7, y2=10, x3=a, y3=2.

 

y2 – y1  =   y3 – y2

x2 – x1     x3 – x2

 

10 – 6  =   2 – 10        

7 – 4        a – 7

 

 

4    =   -8         Now we cross multiply

3       a - 7

 

 

 

4(a-7) = 3 x -8

4a – 28 = -24

4a   = -24 + 28

4a  = 4

a= ⁴/₄ =1

 

Hence a = 1.

 

TRY THIS………………..

 

If m(2,-3), B(4,5) and C(w,29) are collinear, find w.