LOGARITHMS B4

 

If log₃(10x + 7)=3; find x.

 

Solution

 

log₃(10x + 7)=3

 

 (10x + 7)=3³         

                                            

 10x + 7=27

 

10x = 27 – 7

 

10x =  20

 

10x =  20²   

10      10

 

x = 2

 

Hence x = 2

 

TRY THIS…………….

 

If log₃(5x + 1)=4; find x

ANGLES B1

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 + 20⁰ = 90⁰.

so,  5x = 90⁰ -  20⁰

so,  5x = 70⁰

 

so,  5x = 70¹⁴

       5       5

 

Hence x = 14

 

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

 

 If 3x - 22⁰ and 2x + 87⁰ are complementary angles, find the value of x. 

GEOMETRY B2

 

An interior angle of a regular polygon is 62⁰ 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+28⁰ …………………(2)

 

Substitute (2) in (1) above.

 

e+62⁰   + e=180⁰

 

e+ e+62⁰   =180⁰

 

2e+ 62⁰   =180⁰

 

2e=180⁰ - 62⁰   

 

2e=118⁰

 

2e=118⁰          dividing by 2 both sides.

2      2

 

e = 59⁰

 

But i  + e=180⁰…………………(1)

 

i  + 59⁰=180⁰.

 

i  =180⁰ - 59⁰

 

i = 121⁰

 

Hence i = 121⁰ 

  

TRY THIS………………………   

 

The interior angle of a regular polygon is 74⁰ greater than an exterior angle. Find the interior angle.

ALGEBRA B4

 

Factorize 225-9m²

 

Solution

 

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

 

225-9m² = 15² - 3²m²       

             = 15² - (3m)²     

             = (15 - 3m)(15 + 3m)

 

Hence 225 - 9m² = (15 - 3m)(15+ 3m)

 

TRY THIS…………….

 

Factorize 81 - 49c²

LOGARITHMS B3

 

If Log(30x-110/5+x) = 1. Find x.

Solution

Log(30x-110/5+x) = 1.

Log₁₀(30x-110/5+x) = 1.

 30x-110 = 10¹. After changing into exponential form

   5+x

 

30x-110 = 10    since [10¹=10]

   5+x

 

30x-110 = 10(5+x)    after cross multiply

  

30x-110 = 50+10x

 

30x-10x = 50+110 [collecting like terms]

 

20x = 160

 

x=8  [after dividing by 20 both sides]

Hence x=8.

TRY THIS……………

If Log(30x-50/7+x) = 1. Find x.

SETS B1

 

If n(A)= 83 , n(B)= 93 and n(AuB)= 130, find n(AnB).

Solution

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

130 = 83 + 93 - n(AnB)

130 = 176 - n(AnB)

130 - 176 = - n(AnB)

-46 = -n(AnB)

n(AnB) = 41       [after dividing by -1 both sides]

Hence n(AnB) = 41 answer

TRY THIS………….

If n(A)= 92 , n(B)= 98 and n(AuB)= 160, find n(AnB).

ALGEBRA B3

 

Simplify 21x-7(x-y)+5

 

Solution

 

=21x-7(x-y)+5

 

=21x-7x+7y+5        [since -7(x-y)= -7x+7y]

 

=14x+3y+5 answer

 

TRY THIS……..

 

Simplify 44x-9(x-y)+17

MAKING SUBJECT B1

 

QUADRATICS B2

 

What must be added to x²  + 8x to make the expression a perfect square?

 

Solution

 

a=1, b=8,c=?

 

We apply the formula: b² = 4ac

 

(8)² = 4 x 1 x c

 

64 = 4c

64 = 4c

4      4

 

16 = c

Hence number to be added is 16

 

TRY THIS……………………

What must be added to x²  + 12x to make the expression a perfect square?  

EXPONENTIALS B2

 

If 4¹⁰ = 32^a-2; find a

Solution

4¹⁰ = 32^a-2 

(2²)¹⁰ = (2⁵)^a-2 

2²⁰ = 2^5(a-2) 

2²⁰ = 2^5a-10

2²⁰ = 2^5a-10  [Same bases both sides cancels out]

20 = 5a – 10

20 + 10 = 5a

30 = 5a

30 = 5a

5      5

 

6 = a

Hence a=6

 

TRY THIS……………………….

 

If 4³ = 64^a-2; find a