PRIME FACTORS 1

Write 420 as a product of prime factors.

Solution

2	420
2	210
3	105
5	35
7	7
	1

 Then, 420 = 2 x 2 x 3 x 5 x 7

Hence   420 = 2² x 5 x 7.

TRY THIS........

Write 540 as a product of prime factors.

GEOMETRY 2

If x-15⁰ and 4x + 35⁰ are complementary angles, find the value of x.

Solution

Complementary angles add up to 90⁰.

so,  x-15⁰ + 4x + 35⁰ = 90⁰.

so,  x + 4x + 35⁰ - 15⁰ = 90⁰.

so,  5x + 15⁰ = 90⁰.

so,  5x = 90⁰ -  15⁰

so,  5x = 75⁰

so,  5x = 75¹⁵

       5       5

Hence x = 15⁰

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

If 3x - 18⁰ and 2x + 43⁰ are complementary angles, find the value of x.

EXPONENTIAL EQUATIONS - 2

If 2^(6m-7) x 3⁽ⁿ⁺⁹⁾ = (3⁴⁴) x (2⁹³); Find m and n.

solution

equating equal bases;

2^(6m-7) =  2⁹³

2^(6m-7) =  2⁹³

6m - 7 = 93

6m  = 93 + 7

6m  = 100

¹6m  = 100⁵⁰

₁6           6₃

m = ⁵⁰/₃

again for n we equate equal bases.

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

n + 9 = 44

n = 44 - 9

n = 35

Hence m = 20 and n = 35

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

If 8^(2m-9) x 11⁽ⁿ⁺⁴⁰⁾ = (11⁶⁰) x (8²³); Find m and n.

FACTORIZATION - 8

Factorize completely ca + cr - ar - a².  

Solution

= ca + cr - ar - a².

= (ca + cr) – (ar - a²).      [Grouping the factors]

= c(a + r) - a(r + a).

 = (a + r)(c - a).

  Hence ca + cr - ar - a². =  (a + r)((c - a).   

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

NECTA 1995 QN. 20a

Factorize completely pq + pr - rq - q².   

LOGARITHMS-11

If Log_ay = Log_a22 +  Log_a3 ; find y.

solution

Log_ay = Log_a22 +  Log_a3

Log_ay = Log_a(22x 3)         [since Log_am +  Log_an = Log_a(m x n)] 

Log_ay = Log_a66

Hence y = 66                  [since Log_a cancels out]

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

NECTA 1997 QN 13a

If Log_ax = Log_a5 +  Log_a3 ; find x.

UNITS 2

Convert 4073m into km.

Solution

1km  = 1000m

  ?     = 4073m

After cross-multiplying;

= 1 x 4073

      1000

=   4073

     1000

= 4.073 km

Hence 4073m = 4.073km

TRY THIS........

Convert 8023m into km.

FACTORIZATION - 7

Factorize completely   (3x-5)² - x²  

solution

= (3x-5)² - x²  

= [(3x-5) - x][ (3x-5) + x ]  --- since a² + b² = (a+b)(a-b) Here a is (x-2) and b is x.

= [3x-5 - x][ 3x-5 + x ] 

= [3x-x-5][ 3x+ x - 5  ] 

= [2x-5][ 4x - 5  ] 

= [2x-5][ 4x - 5 ] 

Hence  (2x-5)² - x²  = (2x-2)(4x - 5) 

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

Factorize completely   (6x-1)² - x²  

POLYGONS-2

A regular polygon has 88 sides. Find the total angles of that polygon.

Solution

n = 88

Total angles = (n - 2)180⁰

                   = (88 – 2)180⁰

                   = 86 x 180⁰

                   = 15480⁰

Total angles = 15480⁰

TRY THIS……………………….

A regular polygon has 68 sides. Find the total angles of that polygon.

LOGARITHMS-10

If log₅(3x + 101)=3; find x

   Solution

log₅(3x + 92)=3

 (3x + 92)=5³   

                      

 3x + 92=125

3x = 125 – 92

3x =  33

3x =  33   

3        3

X = 11

Hence x = 11

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

If log₂(2x - 22)=5; find x

APPROXIMATIONS-4

Estimate the value of 2.3  x  0.034

solution

= 2.3  x  0.034

= 2.0 x 0.03      [ 2.3 ≈ 2.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.06

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

Estimate the value of 6.2  x  0.083

FUNCTIONS - 2

If F(x) = 12x + 6; 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 + 6

y – 6 = 12x

y – 6  = 12x

 12        12

y – 6   = x

   12

x  =   y – 6     after rearranging

            12

 y^-1 = x – 6     after interchanging x and y variables.

            12

F^-1(x) = x – 6     after interchanging x and y variables.

               12

Hence, F^-1(x) = x – 6     

                           12

TRY THIS……………………………

If F(x) = 5x + 28; Find F^-1(x).

LOGARITHMS-9

Evaluate Log₂(512 ÷ 8).

Solution

= Log₂(512 ÷ 8)

= Log₂512 -  Log₂8          (applying the quotient rule)

= Log₂2⁹ -  Log₂2³             ( 512= 2⁸ and 8=2³ )

= 9Log₂2 -  3Log₂2       ( remember  Log_aa^c = cLog_aa )

= (9 x 1) - (3 x 1)          ( remember  Log_aa = 1 )

= 9 - 3

= 6

hence Log₂(512 ÷ 8) = 6

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

Evaluate Log₂(1024 ÷ 32). 

MATOKEO YA FORM IV - 2015

 BOFYA HAPA KUPATA MATOKEO YA 

KIDATO CHA NNE 2015

Chagua Herufi Ya Shule Yako.
A	B	C	D	E	F
G	H	I	J	K	L
M	N	O	P	Q	R
S	T	U	V	W	X
Y			Z

FRACTIONS - 2

Express as a single fraction m - 5 + m+1

                                               6          4

solution

= m - 5 + m+1

      6          4

= 2(m – 5) + 3(m+1)

            12         

= 2m - 10 + 3m+3

            12          

= 2m + 3m + 3- 10

            12         

= 5m +  -7

       12         

= 5m - 7

     12         

=  5m - 7                 answer

       12       

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

Express as a single fraction 4c - 1 + 2d+1

                                              4           2

RELATIONS - 2

Let R={(5,3), (6,1), (-2,-6), (14,-20)}. Find the domain and range of R.

solution

For domain we check on the values of x in each point.

Domain={5, 6, -2, 14}

For range we check on the values of y in each point.

Range={3, 1, -6, -20}

Hence Domain={5, 5, -2, 14} and Range = {3, 1, -6, -20}

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

Let R={(7,1), (-13,5), (-1,-2), (14, -9)}. Find the domain and range of R.

DIFFERENCE OF 2 SQUARES - 2

Evaluate 2797² – 2787²

Solution

2797² – 2787² = (2797 + 2787)( 2797 – 2787)

                     = (5584)( 10)

                     = 55840

Hence 2797² – 2787² = 55840

TRY THIS…………….

Evaluate 936² – 836²

SETS - 4

If n(A)= 176 , n(B)= 160 and n(AnB)=150, find n(AuB).

Solution

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

             = 176 + 160 – 150

             = 336 – 150

             = 186

Hence n(AuB) = 186 answer

TRY THIS…………………………….

If n(A)= 146 , n(B)= 164 and n(AnB)=100, find n(AuB).

SIMPLE INTEREST-3

Rayan deposited the amount of 80,000/= in a bank for 2 years and got a profit of 2500/=. Find the interest rate?

Solution

I = 1200/=, P = 80,000/=, T = 2 years, R = ?

I  = PRT

      100

2500  = 80,000 x R x 2

                      100

2500  = 80,000 x R x 2

                      100

2500  = 800 x R x 2

                      

2500  = 4000R

           

⁵2500  = ¹4000R

₈4000      ₁4000

Hence the interest rate was  1.6%

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

Rayan deposited the amount of 8,000/= in a bank for 6 years and got a profit of 2100/=. Find the interest rate?