GEOMETRY J1

 

An interior angle of a regular polygon is 76⁰ greater than an exterior angle. Find the interior angle.

 

Solution

 

Let i = interior angle, e = exterior angle.

 

Now i  + e=180⁰…………………(1)

 

But i = e+76⁰ …………………(2)

 

Substitute (2) in (1) above.

 

e+76⁰   + e=180⁰

 

e+ e+76⁰   =180⁰

 

2e+ 76⁰   =180⁰

 

2e=180⁰ - 76⁰   

 

2e=104⁰

 

2e=104⁰          dividing by 2 both sides.

2     2

 

e = 52⁰

 

From (1),

 

i  + e=180⁰

 

i  + 52⁰=180⁰

 

i  =180⁰ - 52⁰

 

i  =128⁰ answer

 

Hence interior angle is 128⁰

 

TRY THIS………………………   

 

An interior angle of a regular polygon is 48⁰ greater than an exterior angle. Find the interior angle.

FACTORIZE J3

 

Factorize 225 - m²

 

Solution

 

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

 

225-m² = 15²-m²       

 

          = (15 - m)(15 + m)

 

Hence 225 - m² = (15 - m)(15 + m)

 

 

TRY THIS…………….

 

 

Factorize 121 - H²

ALGEBRA J1

 

If  x+6y = 1            find  ^x/y             

    2x-3y   4

 

solution

 

x+6y =  1           

2x-3y   4

 

1(2x-3y) = 4(x+6y)           [after cross multiplying]

 

2x-3y = 4x+24y

 

2x - 4x = 3y + 24y   (Collecting like terms)

 

-2x = 27y

 

¹-2x =   27y                (dividing by -2 both sides)       

₁-2       -2  

 

 x = -13.5y             

     

x = -13.5y                  (dividing by y both sides)       

y          y

 

x = -13.5

y    

 

Hence ^x/_y = -13.5  answer.          

 

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

 

If   x+8y = 1            find  ^x/y             

    x- 6y     9

LOGARITHMS J2

 

Evaluate Log10000 -  log 0.001 - log 0.0001+ log 0.00001

 

Solution

 

= Log10000 -  log 0.001 -  log 0.0001+ log 0.00001

 

= Log10⁴ -  log 10^-3 – log 10^-4 + log 10^-5

 

= 4Log10 - (-3 log 10) – (-4log 10) + (-5log 10)

 

= (4x1) - (-3x1) - (-8x1) + (-5 x 1)

 

= 4 - (-3) – (-4) -5

 

= 4 + 3 + 4 - 5

 

= 11 – 5

 

= 6 ans.

 

  

Hence Log10000 -  log 0.001-  log 0.0001+ log 0.00001 = 6

 

 

TRY THIS……………………… 

 

 

Evaluate Log1000 -  log 0.001 - log 0.00000001+ log 0.00001

MIDPOINT J1

 

Find the midpoint of a line from (33, 12) to (5, 20)

 

Solution

 

x1=33, x2=5, y1=12,y2=20.

 

Mid point = (x1 + x2, y1 + y2)

                          2             2

 

Mid point = (33 + 5,  12 + 20)

                         2          2

 

Mid point = (38, 32)

                     2     2

 

Hence midpoint = (19, 16)

 

TRY THIS……………………..

 

 

Find the midpoint of a line from (-9, 24) to (9, -4)

FACTORIZE J2

 

Evaluate using factors: 996² – 986²

 

Solution

 

We apply difference of two squares: a² - b² = (a+b)(a-b)

 

996² – 986²  = (996 + 986)( 996 – 986)

 

                   = (1982)( 10)

 

                   = 19820   [only add a zero at 1982]

 

996² – 986² = 19820

 

TRY THIS…………….

 

Evaluate using factors: 708² – 608²

VARIATION J1

 

x is directly proportional to y. x=32 while y=4. Find y when x is 816.

 

Solution

 

x ⍺ y

x = ky

 

32 = k x 4

 

32 = 4k    [dividing by 4 both sides]

4      4

 

k = 8

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

 

k = 8, y= ? x = 816.

 

x = ky

 

816 = 8 x y

 

816 = 8y

8        8

 

y = 104.

 

TRY THIS…………….

 

x is directly proportional to y. x=40 while y=4. Find y when x is 370.

LOG J1

 

If log 2= 0.3010; find the value of log 1600000 without using tables.

 

solution

 

=log1600000

 

=log(16 x 100,000)

 

=log16 + log100,000

 

=log2⁴ + log10⁵

 

=4log2 + 5log10

 

=4(0.3010) + (5 x 1)  [since log10=1 and Log3=0.3010]

 

=(0.6020) +  5

 

=5.6020

 

Hence log1600,000=5.6020 answer.

 

 

TRY THIS……………

 

 

If log 2= 0.3010; find the value of log 40,000,000 without using tables.

 

FACTORIZE J1

 

Factorize completely tc + tr - cr - c².  

 

Solution

 

= tc + tr - cr - c².

 

= (tc + tr) – (cr - c²).      [Grouping the factors]

 

= t(c + r) - c(r + c).   [after factoring out]

 

 = (c + r)(t - c) answer

 

  Hence tc + tr - cr - c². =  (c + r)(t - c).   

 

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

 

Factorize completely yw + yr - rw - w². 

SETS J1

 If n(A)= 85 , n(B)= 96 and n(AuB)= 129, find n(AnB).

 

Solution

 

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

 

129 = 85 + 96 - n(AnB)

 

129 = 181 - n(AnB)

 

129 - 181 = - n(AnB)

 

-58 = -n(AnB)

 

n(AnB) = 58 [after dividing by -1 both sides]

 

Hence n(AnB) = 58 answer

 

 

TRY THIS………….

 

 

If n(A)= 91 , n(B)= 94 and n(AuB)= 127, find n(AnB).

G.PROGRESSION J1

 The 1st term of a geometric progression is 7 and the 5th term is 112. Find i)the common ratio. ii) 12^th term

 

Solution

 

G₁=7, G₅=112, n=5

 

i) G_n = G₁r^n-1;

 

G₅ = G₁r^5-1;

 

G₅ = G₁r⁴;

 

112 = 7 x r⁴;

 

112 = 7r⁴;

 7       7

 

16 = r⁴;  Finding the fourth root of 16,

 

r = 2

 

 

 

Hence the common ratio is 2.

 

ii)  12^th term

 

G_n = G₁r^n-1;

 

G₁₂ = G₁r^12-1;

 

G₁₂ = G₁r¹¹;

 

G₁₂ = 7 x 2¹¹;

 

G₁₂ = 7 x 2048;

 

G₁₂ = 14336

 

Hence the 12th term is 14336

 

TRY THIS………………………….. 

 

The 1st term of a geometric progression is 2 and the 6th term is 486. Find i) the common ratio. ii) 10^th term