ALGEBRA 1

Multiply 2x - 7y by -9a

solution

= -9a(2x - 7y)

= (-9a x 2x) + (-9a x -7y)

= -12ax + 63ay     [since -9a x -7y = +63ay ]

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

Multiply 12p - 7q by  -4a

LOGARITHMS 3

Simplify Log₂128 - Log₃27

solution

= Log₂128 - Log₃27

= Log₂2⁷ - Log₃3³

= 7Log₂2 - 3Log₃3

= (7 x 1) - (3 x 1)

= 7 - 3

= 4

Hence Log₂128 - Log₃27 = 4

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

Simplify Log₂256 - Log₃27

GRADIENT 1

Find the slope of a line which passes through (-9, -4) and (8,-11)

Solution

x₁ = -9,  y₁ =-4,  x₂ = 8,  y₂ = -11

m = y2 –y1

       x2 – x1

m =   -11 –(-4)

          8 –(-9)

m =   -11 + 4

          8 + 9

m =    - 7

          17

Hence the slope is -7/17

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

Find the slope of a line which passes through (-11, -4) and (4,-11)

ARC LENGTH 1

Find radius if the length of an arc of the circle is ^2∏/₁₅ cm.

Solution

L= ^2∏/₁₅  , r = ?

L = ∏r

    180

^2∏/15  = ∏r

             180

180  x ^2∏/15  = ∏r  x   180    [multiplying by 180 both sides]

                      180

360 ∏  = ∏r 

     15      

                    

360 ∏  = ∏r 

    15∏  ∏

360   =  r 

    15      

r = 24cm

Hence radius is 24 cm

TRY THIS………………..

Find radius if the length of an arc of the circle is ^7∏/₃₀ cm.

ANGLES 1

If x-15⁰ and 4x + 45⁰ are complementary angles, find the value of x.

solution

Complementary angles add up to 90⁰.

so,  x-15⁰ + 4x + 45⁰ = 90⁰.

so,  x + 4x + 45⁰ - 15⁰ = 90⁰.

so,  5x + 30⁰ = 90⁰.

so,  5x = 90⁰ -  30⁰

so,  5x = 60⁰

so,  5x = 60¹²

       5       5

Hence x = 12

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

If 3x - 12⁰ and 2x + 77⁰ are complementary angles, find the value of x.

POLYNOMIALS 2

If f(x) = x⁴ + kx² + 3x + 7 has a remainder of 22 when divided by x+2; find k.

solution

f(x) = x⁴ + kx² + 3x + 7

x + 2 = 0

x = -2

f(x) = (-2)⁴ + k(-2)² + 3(-2) + 7 = 22

16 + 4k + (-6) + 7 = 22

16 + 4k + 1 = 22

4k + 17 = 22

4k = 22 - 17

4k = 5

4k = 5

4      4

k = 5/4

hence k=5/4

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

If f(x) = x⁴ - kx² + 3x - 8 has a remainder of 20 when divided by x-3; find k.

LOGARITHMS 2

Evaluate Log100000 +  log 0.001 + log₃243

Solution

= Log100000 +  log 0.001 + log₃243

= Log10⁵ +  log 10^-3 + log₃3^-5

= 5Log10 +  (-3log 10) + (-5log₃3)    [since log_aaⁿ  = nlog_aa]

= (5x1) + (-3x1) + (-5x1)             [since log_aa = 1]

= 5 + (-3) + (-5)

= 5 - 8

= -3

Hence Log100000 +  log 0.001 + log₃243 = -3

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

Evaluate Log100,000 +  log 0.01 + log₃729

CIRCLES 1

Given the figure below, prove that an angle inscribed in a semi –circle is a right angle.

solution

Given:

        diameter AOB

        O as centre of circle

        C is any point in the circumference

Required to prove:   

<ACB =90⁰       

Proof:      

< at centre = 2(< at circumference)

<AOB=2(<ACB) ------------    (i)

But <AOB is a straight angle

Straight angle=180⁰

Hence, <AOB = 180⁰

Now, from (i)

<ACB = 2(<ACB)

180⁰=2(<ACB)

90⁰= <ACB     [after dividing by 2 both sides]

Hence ACB = 90⁰ proved!!!

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

Given the figure below, prove that an angle inscribed in a semi –circle is a right angle.

RELATIONS 1

Let R={(10,3), (11,1), (-8,-6), (10,-20)}. Find the domain and range of R.

solution

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

Domain={10, 11, -8}

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

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

Hence Domain={10, 11, -8} and Range = {3, 1, -6, -20}

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

Let R={(7,3), (-41,-8), (-6,-6), (17, 0)}. Find the domain and range of R.

ABSOLUTE VALUE 1

If │2x + 11│= 47; Find  x

Solution

±(2x + 11 )= 47

  

2x +11= 47    OR   –(2x+11) = 47

  

2x+ 11 = 47    OR   -2x - 11=47

2x = 47- 11    OR   -2x= 47 + 11

2x = 36     OR     -2x  =  58

2       2                 -2       -2

x = 18  OR  x = -58/2 = -29

Hence,  x = 18 OR  x = -29 

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

If │2x + 19│= 37; Find  x

SETS 1

If n(A)= 173 , n(B)= 160 and n(AnB)=120, find n(AuB).

Solution

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

             = 173 + 160 – 120

             = 333 – 120

             = 213

Hence n(AuB) = 213 answer

  

TRY THIS…………………………….

If n(A)= 157 , n(B)= 163 and n(AnB)=110, find n(AuB).

POLYGONS 1

A regular polygon has 37 sides. Find the total angles of that polygon.

Solution

n = 37

Total angles = (n - 2)180⁰

                   = (37 – 2)180⁰

                   = 35 x 180⁰

                   = 6300⁰

Total angles = 6300⁰

TRY THIS…………………

A regular polygon has 47 sides. Find the total angles of that polygon.

MIDPOINT 1

Find the midpoint of a line from (31, 4) to (17, 10)

Solution

x1=31, x2=17, y1=4, y2=10.

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

                         2             2

Mid point = (31 + 17,  4 + 10)

                        2            2

Mid point = (48, 14)

                     2    2

Hence midpoint = (24, 7)

TRY THIS……………………..

Find the midpoint of a line from (34, 14) to (22, 16)

APPROXIMATIONS 1

Estimate the value of 9.3  x  0.064.

solution

= 9.3  x  0.064

= 9.0 x 0.06      [ 9.3 ≈ 9.0 to ones and 0.064 ≈ 0.06 to hundredths]

= 0.54

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

Estimate the value of 6.2  x  0.071

LOGARITHMS 1

If log₇(3x + 19)=2; find x

   Solution

log₇(3x + 19)=2

 (3x + 19)=7²   

                      

 3x + 19=49

3x = 49 – 19

3x =  30

3x =  30         [dividing by 3 both sides]

3         3

x = 10

Hence x = 10

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

If log₂(3x - 28) = 5; find x

POLYNOMIALS 1

If f(x) = x⁴ + kx² + 2x -20  has a remainder of 22 when divided by x+2; find k.

solution

f(x) = x⁴ + kx² + 2x - 20 ----------------- (i)

x + 2 = 0

x = -2

_____________________________

Then substitute x = -2 in (i) above

f(x) = (-2)⁴ + k(-2)² + 2(-2) -20 = 22

16 + 4k + (-4) - 20 = 22

16 + 4k -24 = 22    [since -4 – 20 = -24]

4k - 8 = 22       [since 16 – 24 = -8]

4k = 22 + 8

4k = 30

4k = 30

4      4

k = 15/2 = 7.5

hence k=7.5

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

If f(x) = x⁴ - kx² + 6x - 10 has a remainder of 40 when divided by x+3; find k.

ALGEBRA 1

If  x+5y = 1            find  ^x/y             

    x-3y     4

solution

x+5y = 1           

x-3y     4

1(x-3y) = 4(x+5y)           [after cross multiplying]

x-3y = 4x+20y

x - 4x = 20y + 3y

-3x = 23y

¹-3x = 23y

₁-3      -3

 x = -23y/3

x/y = -23/3

x = 5y

     

x = 5y

y     y

x = 5

y    

Hence ^x/_y = 5  answer.          

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

If  x+2y = 1            find  ^x/y             

    x- 2y     5