ALGEBRA-3

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        [after dividing by y on both sides]

x = -23/3

y    

Hence ^x/_y = -23/3  answer.           

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

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

   x- 2y     5

LOGARITHMS-3

Find x if Log_x0.00001 = -5

solution

Log_x0.00001 = -5

0.00001  = x^-5 .  [0.000001  = 10^-5  ]

10^-5 = x^-5 .

10⁵ = x⁵ .  { equal powers cancel out}

x = 10

Hence x = 10

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

Find x if Log_x256 = 8

FUNCTIONS-1

If f(x) = |x - 24| evaluate f(-27)

solution

f(x) = |x - 24|

f(-27) = |-27 - 24|

f(-27) = |-51| = 51 since any number out of absolute signs is +ve.

Hence f(-27) = 51

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

If f(x) = |x - 40| evaluate f(-15)

RATIOS-1

Given the ratios M:N = 16: 8 and N:P = 4:7 Find the ratio M:P.

solution

M = M x N

P      N     P

M =   16 x 4¹

P      ₂ 8     7

M =      16

P        (7x2)

M =       ⁸16  =  8

P           ₇14      7

Hence M:P=8:7

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

Given the ratios M:N = 20: 8 and N:P = 12:7 Find the ratio M:P.

ABSOLUTE VALUE EQUATIONS-1

If │2x – 13 │= 21; Find  x

Solution

±(2x – 13 )= 21

2x -13= 21    OR   –(2x-13) = 21

2x – 13 = 21    OR   -2x+13= 21

2x = 21 + 13    OR   -2x= 21-13

2x = 34     OR   -2x  =  8

2      2                2        2

x = 17  OR  x = 4

Hence x = 17 OR x = 4     

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

If │4x – 16 │= 32; Find  x

PERCENTAGE PROFIT-1

A man got a profit of 200/= after selling an item for 8000/=. Find the percentage profit.

Solution

%’ge profit = Profit   X  100    [where B. P. represents Buying Price.]

                      B.P.

%’ge profit = 200   X  100   

                    8000

%’ge profit = 2.5%   

                       

Hence Percentage Profit is 2.5%

TRY THIS……………………………

A man got a profit of 10,000/= after selling an item for 600,000/=. Find the percentage profit.

DIFFERENCE OF 2 SQUARES-2

Factorize 1-V²

Solution

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

1-V² = 1²-V²       

        = (1 - V)(1 + V)

Hence 1 - V² = (1 - V)(1 + V)

TRY THIS…………….

Factorize 25 - W²

LOGARITHMS-2

If log 3= 0.4771; find the value of log 90,000 without using tables.

solution

log90,000=log(10,000 x 9)

=log10,000 + log9

=log10⁴ – log3²

=4log10 – 2log3

=(4x1) –  2(0.4771)

=4– 0.9542

= 3.0458

Hence log 90,000 = 3.0458

TRY THIS……………

If log 3= 0.4771; find the value of log 8,100 without using tables.

EXPONENTIALS-1

If a² = 3; find the value of a⁶.

solution

a⁶ = (a²)³.

a⁶ = (3)³.   [ since a² = 3]

a⁶ = 3 x 3 x 3

a⁶ = 27  answer

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

If a² = 13; find the value of a⁴.

APPROXIMATION-2

Estimate the value of 4143  ÷ 52

Solution

= 4123     

     52

= 4000      [ since 4143 ≈ 4000   and   52 ≈ 50 ]

     50

= 80

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

Estimate the value of 7203  ÷ 19

GEOMETRY-1

If x-7⁰ and 4x + 62⁰ are complementary angles, find the value of x.

solution

Complementary angles add up to 90⁰.

so,  x-7⁰ + 4x + 62⁰ = 90⁰.

so,  x + 4x + 62⁰ - 7⁰ = 90⁰.

so,  5x + 55⁰ = 90⁰.

so,  5x = 90⁰ -  55⁰

so,  5x = 35⁰

so,  5x = 35⁷

       5       5

Hence x = 7

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

If 3x - 4⁰ and 2x + 54⁰ are complementary angles, find the value of x.

RELATIONS-1

Let R={(10,3), (14,1), (-2,-6), (11,-20)}. Find the domain and range of R.

solution

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

    Domain={10, 14, -2, 11}

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

    Range={3, 1, -6, -20}

Hence Domain={10, 14, -2, 11} and Range = {3, 1, -6, -20}

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

Let R={(8,3), (-11,-8), (-3,-6), (19, 0)}. Find the domain and range of R.

AXIS OF SYMMETRY-1

Find the axis of symmetry for F(x) = 4x² + 50x + 8

Solution

a=4, b=10

Axis of symmetry = -b/2a

                                 =  -(50)

                                      2 x 4

                                 =   -50

                                        8

                                 =   -25

                                        4

Hence the axis of symmetry is ^-25/4

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

Find the axis of symmetry for F(x) = 7x² + 28x + 3.

SIMPLE INTEREST-1

Priya deposited the amount of 90,000/= in a bank which gives an interest rate of 3% for 5 years. Find the simple interest she got.

Solution

I = ?,  P = 90,000/=, T = 5 years, R = 3%

I  = PRT

      100

I  = 90,000 x 3 x 5

            100

I  = 90,000 x 3 x 5

            100

I  = 900 x 3 x 5

          

I  = 13,500/=

Hence the simple interest was Tsh 13,500/=

TRY THIS………………….

Edmund deposited the amount of 50,000/= in a bank which gives an interest rate of 2% for 4 years. Find the simple interest he got?

SETS-1

If n(A)= 173 , n(B)= 160 and n(AnB)=140, find n(AuB).

Solution

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

             = 173 + 160 – 140

             = 333 – 140

             = 193

Hence n(AuB) = 193 answer

TRY THIS…………………………….

If n(A)= 146 , n(B)= 164 and n(AnB)=130, find n(AuB).

FACTORIZATION-1

Factorize 6x² + 13x - 5

Solution

= 6x² + 13x - 5.  We split the middle term (+13x) to be (-2x + 15x).

= 2x² - 2x + 15x – 5                (-3x) = -14x + 5x.

= (2x² -2x) + (15x - 5)

= 2x(x - 1) + 5(x - 1)

= (2x + 5) (x -1)

Hence 6x² + 13x – 5 = (2x + 8) (x - 1) answer.

TRY THIS…..

Factorize 4x² - 17x – 15.

MID POINT-1

Find the midpoint of a line from (11, 4) to (17, 10)

Solution

x1=11, x2=17, y1=4, y2=10.

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

                         2            2

Mid point = (11 + 17,  4 + 10)

                        2            2

Mid point = (28, 14)

                     2    2

Hence midpoint = (14, 7)

TRY THIS……………………..

Find the midpoint of a line from (14, 18) to (22, 12)

ALGEBRA-3

Find w in the following equation;

3w - w-3 = w

7       2       5

solution

3w - w-3 = w

7       2       5

The LCM of 5, 7 and 2 is 70. So, we multiply by 70 throughout

70x 3w - 70(w-3) = w x 70

      7              2            5

¹⁰70x 3w - ³⁵70(w-3) = w x 70¹⁴

         ₁7              ₁2         5₁

10 x 3w - 35(w-3) = w x 14

30w - 35w + 105 = 14w

-5w + 105 = 2w

105 = 2w + 5w

105 = 7w

105 = 7w

7       7

15 = w

Hence w = 15

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

Find a in the following equation:

4a  -  a-3 = 3a

5        2       5

WORD PROBLEMS-1

1/9 of the lessons is Allocated to mathematics and 2/5 is allocated to civics. What fraction is left for other subjects?

solution

Amount of lessons allocated = 1  +  2

                                                9      5

Amount of lessons allocated = 5  +  18

                                                     45

Amount of lessons allocated = 23

                                                45

Now, the total number of lessons  = 1 = 45

                                                               45

Fraction for other subjects = total number of lessons  -  amount of lessons allocated.

Fraction for other subjects = 45  -  23

                                             45     45

Fraction for other subjects = 22 

                                            45    

Hence fraction for other subjects is 31/45

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

⁷/10 of the lessons is Allocated to mathematics and ⁵/8 is allocated to Geography. What fraction is left for other subjects?

POLYGONS-2

A regular polygon has 30 sides. Find the total angles of that polygon.

Solution

n = 30

Total angles = (n - 2)180⁰

                   = (30 – 2)180⁰

                   = 28 x 180⁰

                   = 5040⁰

Total angles = 5040⁰

TRY THIS……………………….

A regular polygon has 44 sides. Find the total angles of that polygon.

ARC LENGTH-1

Find radius if the length of an arc of the circle is ^∏/₃₆ cm.

Solution

L= ^∏/₃₆  , r = ?

L = ∏r

      180

^∏/12  = ∏r

           180

180  x ^∏/12  = ∏r  x   180   

                      180

 180 ∏  = ∏r 

     36      

                    

 180 ∏  = ∏r 

    36∏  ∏

5   =  r 

1      

     

r = 5cm

Hence radius is 5cm

TRY THIS………………..

Find radius if the length of an arc of the circle is ^∏/₂₀ cm.