RADICALS 1

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Evaluate $\sqrt{6}\times \sqrt{8}$

solution

$=\sqrt{6}\times \sqrt{8}$

$=\sqrt{6\times 8}$

$=\sqrt{48}$

$=\sqrt{2\times 2\times 2\times 2\times 3}$

$=2\times 2\times \sqrt{3}$

$=4\sqrt{3}$
$$ TRY THIS............

Evaluate $$ 12 × 6

INEQUALITIES 2

Indicate |x| > 2 on a number line

Solution

±(x) > 2

+(x) > 2 or -(x) > 2

x > 2     or -(x) < 2

                  -1    -1

x > 2     or x <- 2

TRY THIS……………….

Indicate |x| > 3 on a number line

ALGEBRA 1

RATIOS 1

In a mixed school the number of boys is 95. If there are 165 students, find the ratio of boys to girls.

Solution

Number of girls = Total – number of boys

                          = 165 – 95

                          = 70

Ratio of boys to girls = 95:70

Find the HCF of 95 and 70.

Then we divide both numbers in our ratio by the HCF = 5.

Ratio of boys to girls = 95: 70

                                      5  :  5

                                  = 19:14

Hence the required ratio is 19:14

TRY THIS……………….

In a mixed school the number of boys is 120. If there are 620 students, find the ratio of boys to girls.

FUNCTIONS 1

If F(x) = 24x -5. Find F(-3)

Solution

F(x) = 24x -5

F(-3) = 24(-3) -5

F(-3) = (-72) -5

F(-3) = -77

Hence  F(-3) = -77

TRY THIS……………….

If F(x) = 2x + 10. Find F(-6). 

ARITHMETIC PROGRESSION 1

The 1st term of an A.P. is 16 and the common difference is 24. Find the 10th term.

Solution                                               

A₁= 26, d = 24

A_n = A₁ + (n-1)d

A₁₀ = A₁ + (10-1)d

A₁₀ = A₁ + 9d

A₁₀ = 26 + (9x24)

A₁₀ = 26 + 216

A₁₀ = 242

Hence the 10th term is 242.

TRY THIS……………….

The 1st term of an A.P. is 51 and the common difference is 13. Find the 10th term.

EXPONENTIALS 1

If 32^y-4 = 128^y ; find y

Solution

32^y-4 = 128^y

(2⁵)^y-4= (2⁷)^y

2^5y-20 = 2^7y

2^5y-20 = 2^7y

5y – 20 = 7y

5y – 7y = 20

-2y = 20

  - 2y   =  20

  - 2        -2

y = -10

Hence y = -10

TRY THIS……………….

If 16^w-6 = 128^w ; find w

DECIMAL PLACES 1

Write 0.0605049 correct to

i)  2 decimal places

ii)  3 decimal places

iiii)  4 decimal places

iv)  5 decimal places

Solution

i) 0.0605049 = 0.06

ii) 0.0605049 = 0.061

iii) 0.0605049 = 0.0605

iv) 0.0605049 = 0.06050

TRY THIS…………………….

Write 0.0807026 correct to

i)  2 decimal places

ii)  3 decimal places

iii)  4 decimal places

iv)  5 decimal places

UNITS OF WEIGHT 1

How many grams are there in 0.076Kg?

Solution

1Kg = 1000g

0.076Kg = ?

= 0.076 x 1000

            1

= 76grams

TRY THIS…………………….

How many grams are there in 0.0936Kg?

SETS 3

In Udumuka village the number of people who speak English or French is 300. 180 of them speak English and 170 of them speak French. How many speak both languages?

solution

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

Let French=n(F), English= n(E).

n(E)= 180,

n(F)= 170,

n(EuF) = 300,

n(EnF)=?

n(EuF) = n(E) + n(F) - n(EnF)

300  =  180 + 170 – n(EnF)

300  =  350 – n(EnF)

n(EnF) =  350 – 300

n(EnF) =  50

Hence n(EnF)=50 answer

∴  50 speak both languages   .

TRY THIS……..

In Ihomasa village the number of people who speak English or French is 300. 180 of them speak English and 170 of them speak French. How many speak both languages?

FACTORIZATION 2

Factorize 3x² + 10x + 7

solution

3x² + 7x + 3x + 7

(3x² + 7x) + (3x + 7)

3x(x + 7) +1(3x + 7)

(3x + 7) (x + 7)

Hence (3x + 7) (x + 7) answer

TRY THIS……..

Factorize 10x² + 13x – 3

GRADIENT 1

Find the slope of a line which passes through (10, 13) and (14, 8)

Solution

x₁ = 10,  y₁ =13,  x₂ = 14,  y₂ = 8

m = y2 –y1

       x2 – x1

m =   8 –13

        14 –10

m =   -5

          4

  

Hence the slope is -5/4

TRY THIS……………………

Find the slope of a line which passes through (4, 17) and (12, 4)

INEQUALITIES 1

Find x if 4x – 21 ≤ x + 45 ≤ 6x.

Solution

4x – 21 ≤ x + 45 and  x + 45 ≤ 6x

4x – x ≤ 21+ 45 and  45 ≤ 6x - x

3x ≤ 66 (divide by 3 both sides) and  45 ≤ 5x (divide by 5 both sides)

x ≤ 22 and  9 ≤ x

x ≤ 22 and  x ≥ 9

TRY THIS……..

Find x if 61x – 105 ≤ x + 15 ≤ 2x

SETS 2

If n(AuB)=437 , n(B)= 300 and n(AnB)=70, find n(A).

Solution

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

        437  =  n(A) + 300 – 70

        437  = n(A) +  230

       437 - 230   = n(A)

       n(A)= 207

hence n(A)= 207 answer

TRY THIS……..

If n(AuB)=480 , n(B)= 320 and n(AnB)=190, find n(A).

LOGARITHMS 4

If log₅ (2x + 3) =2; find x

   Solution

log₅ (2x + 3) =2

(2x + 3)=5²     

                    

2x + 3=25

2x =25 – 3

2x = 22

2x = 22   

2        2

X = 11

Hence x = 11

TRY THIS……..

If log₇ (2x + 11) =2; find x

FACTORIZATION 1

Factorize 2x² + 5x + 2

Solution

2x² + 4x + x + 2

(2x² + 4x) + (x + 2)

2x(x + 2) +1(x + 2)

(2x + 1) (x + 2)

Hence (2x + 1) (x + 2) answer

TRY THIS……..

Factorize 3x² + 3x - 18

SETS 1

If n(A)= 80 , n(B)= 70 and n(AnB)=60, find n(AuB).

Solution

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

             =  80 + 70 – 60

             = 150 – 60

             = 90

Hence n(AuB) = 90 answer

TRY THIS……..

If n(A)= 100 , n(B)= 125 and n(AnB)=80, find n(AuB).

REGULAR POLYGONS 2

Find the exterior angle of a regular polygon with 10 sides.

Solution

n = 10

e =   360

         n

e =   360

         10

e =   360

        10

e = 36⁰

Hence the interior angle is 36⁰

TRY THIS………………………………………..

Find the exterior angle of a regular polygon with 20 sides.

SIMULTANEOUS EQUATIONS 1

Solve the following equations by substitution method.

x + y = 9

x + 3y = 13

Solution

x + y = 9 ----------- (i)

x + 3y = 13 ---------- (ii)

from equation (i)

x + y = 9

x = 9- y ----------(iii)

substitute equation (iii) in (ii) above,

x + 3y = 13

(9- y)  + 3y = 13

9+2y = 13

2y = 13 - 9

2y = 4

2      2

y = 2

From equation (iii)

x = 9- 2 ----------(iii)

x = 9- 2

 x = 7

Hence x=7 and y=2.

TRY THIS………………………..

4x - y = 20

3x + 2y = 70

EXPONENTIAL EQUATIONS 1

Find x if 64^x = 0.25; find x.

Solution

64^x = 0.25

(2⁶)^x = 0.25          [since 64=2⁶ ]

2^6x = 0.25          

2^6x = ²⁵/₁₀₀     [converting 0.25 into fraction]       

2^6x = ¹/₄    

2^6x = 4¹           [since ¹/_a=a^-1]

2^6x = (2⁴)¹

2^6x = 2^4x1           

2^6x = 2⁴           [since (a^c)^d =a^cd]

2^6x = 2⁴           [same bases cancel out]

6x = 4         

6x = 4 ²        

6      6 ₃

x= ²/3

Hence x= ²/3

TRY THIS…………………….

Find x if 16^x = 0.125; find x.

LOGARITHMS 3

Evaluate Log₂16 + Log₂10 + Log₂40

Solution

= Log₂16 - Log₂10 + Log₂40

= Log₂(16x40)

               10

= Log₂(640)

              10

= Log₂(64)

            

= Log₂2⁶)

= 6Log₂2

= 6 x 1

= 6

Hence Log₂16 + Log₂10 + Log₂40 = 6 

TRY THIS…………………….

Evaluate Log₂8 - Log₂7 + Log₂ 224

DIFFERENCE OF 2 SQUARES 1

Evaluate 6257² – 6247²

Solution

6257² – 6247² = (6257 + 6247)( 6257 – 6247)

  

                   = (12504)( 10)

                     = 125040

Hence 6257² – 6247² = 125040.

TRY THIS…………………

Evaluate 8155² – 8145²

REGULAR POLYGONS 1

Find the interior angle of a regular polygon with 20 sides.

Solution

n = 20

i = (n-2)180

         n

i = (20-2)180

          20

i = (18) x 180⁹

          20₁

i = 18 x 9

i = 162

Hence the interior angle is 162⁰

TRY THIS………………………………………..

Find the interior angle of a regular polygon with 20 sides.

LOGARITHMS 2

Evaluate log₃243¹²

Solution

= log₃243¹²

= log₃(3⁵)¹²

= log₃3⁶⁰

= 60 x log₃3    [since log_aa = 1]

= 60 x 1

= 60

Hence log₃243¹² = 40

TRY THIS………………

Evaluate log₅3125¹⁰

DIFFERENCE OF 2 SQUARES 1

Factorize 9-h²

Solution

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

9-h² = 3²-h²

        = (3 - h)(3 + h)

Hence 9-h² = (3 - h)(3 + h)

TRY THIS…………….

Factorize 25-m²

LOGARITHMS 1

Evaluate Log₂(32 x 2).

Solution

= Log₂(32 x 2)

= Log₂32 +  Log₂2          (applying the product rule)

= Log₂2⁵ +  Log₂2¹             (32= 2⁵ and 2=2¹ )

= 5Log₂2 +  1Log₂2       ( remember  Log_aa^c = cLog_aa )

= (5 x 1) +  (1x 1)          ( remember  Log_aa = 1 )

= 5 + 1

= 6

hence Log₂(32 x 2) = 6

TRY THIS………………

Evaluate Log₂(128 x 4).