EXPONENTIALS 4

Simplify 11

              64^-2/3

Solution

=  11

  64^-2/3

=  11 x 1

          64^-2/3

=  11 x 64^2/3        [Since 1/a^-n = aⁿ]

=  11 x (64^1/3)²        [Since 64^1/3 = cube root of 64 = 4]

=  11 x (4)²        

=  11 x 16

= 176        

         

Hence   11        = 176

             64^-2/3

TRY THIS…………………………

simplify 7

             27^-2/3

EXPAND 1

Expand 8w(5w + 3 - w)

Solution

= 8w(5w + 3 - w)

= (8w x 5w) + (8w x 3) - (8w x w)

= 40w² + 24w - 8w²

= [40w² - 8w²] + 24w collecting like terms

= 32w² + 24w answer

TRY THIS………..

Expand 4a(7a+ 12 - a)

TRANSPOSITION 1

SIMPLIFY 2

Simplify: 2 + 6(6n ÷ ⁿ/2) - 30

Solution

=2 + 6(6n ÷ ⁿ/2) - 30

=2 + 6(6n x ²/n) – 30

=2 + 6(6n¹ x ²/n₁) – 30  cancelling n

=2 + 6(6 x  2) – 30  

=2 + 6(12) – 30  

=2 + 72 – 30  

=74 – 30  

=44  

=44 answer

TRY THIS……..

Simplify 20 + 4(10n ÷ ⁿ/2) - 70

LOGARITHMS 4

Evaluate Log₂(2048 x 16).

Solution

= Log₂(2048 x 16)

= Log₂2048 +  Log₂16          (applying the product rule)

= Log₂2¹¹ +  Log₂2⁴             ( 2048= 2¹¹ and 16=2⁴ )

= 11Log₂2 +  4Log₂2       ( remember  Log_aa^c = cLog_aa )

= (11 x 1) +  (4 x 1)          ( remember  Log_aa = 1 )

= 11 + 4

= 15

hence Log₂(2048 x 16) = 15

TRY THIS…………………

Evaluate Log₂(8 x 512)

SIMPLIFY 1

Simplify 11x-3(x-y)+5

Solution

=11x-3(x-y)+5

=11x-3x+3y+5

=8x+3y+5 answer

TRY THIS……..

Simplify 14x-9(x-y)+12

POLYGONS 3

An interior angle of a regular polygon is 78⁰ 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+78⁰ …………………(2)

Substitute (2) in (1) above.

e+78⁰   + e=180⁰

e+ e+78⁰   =180⁰

2e+ 78⁰   =180⁰

2e=180⁰ - 78⁰   

2e=102⁰

2e=102⁰          dividing by 2 both sides.

2      2

e = 51⁰

But i  + e=180⁰…………………(1)

i  + 51⁰=180⁰.

i  =180⁰ - 51⁰

i = 129⁰

Hence i = 129⁰ 

TRY THIS………………………   

An interior angle of a regular polygon is 82⁰ greater than an exterior angle. Find the interior angle.

ALGEBRA 5

If a² – 20 = 61; find a.

SOLUTION

a² – 20 = 61

a² = 61+ 20

a² = 81

√a² = √81    keep root sign on both sides

a = √81   

 a = 9     since the square root of 81 is 9.

Hence a = 9

TRY THIS………..

If m² – 23 = 98; find m.

INTERCEPT 1

If 2x + 5y -11 = 0; find x-intercept.

Solution

x-intercept is when y=0.

2x + 5y -11 = 0

2x + 5(0) - 11 = 0

2x - 11 = 0

2x = 0+ 11

2x = 11

2      2

x= ¹¹/2

Hence x= ¹¹/2

TRY THIS………..

If 8x - 3y - 17 = 0; find x-intercept.

MIDPOINT 1

Find a if the midpoint of a line from (a, 6) to (8, 10) is (10, 7)

Solution

x1=a, x2=8, y1=6, y2=10.

(10, 7)= (x1 + x2, y1 + y2)

                2             2

(10, 7)= (a + 8,  6 + 10)

                2          2

(10, 7)= (a + 8, 16)

                 2      2

Equating values of x;

10= a + 8

         2      

2 x 10= (a + 8)  x 2 ¹

               2 ₁     

20 = a + 8

20-8 = a

a = 12.

Hence a=12

TRY THIS……………………..

Find m if the midpoint of a line from (m, 19) to (13, 10) is (9, 11)

ALGEBRA 4

Expand 10w(5w – 3)

solution

= 10w(5w – 3)

= (10w x 5w) – (10w x 3)

= 50w² – 30w answer

TRY THIS………..

Expand 4y(9y+ 10)  

VECTORS 2

If u = 20i + 2j and v = 3i + 10j find 5u + 3v

solution

= 5u + 3v

= 5(20i + 2j) + 3(3i + 10j)

= 100i + 10j + 9i + 30j

= (100i + 9i) + (10j + 30j)

= 109i + 40j

Hence 5u + 3v = 109i + 40j .

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

NECTA 2001 QN. 12a (i)

If u = 4i + 3j and v = 2i + 4j;  find 2u + 3v.

FUNCTIONS 2

If F(x) = log₂x, Find F(¹/ ₈₁₉₂)

Solution

F(x) = log₂x

F(¹/8192) = log₂(¹/ ₈₁₉₂)

F(¹/8192) = log₂ 8192^-1

F(¹/8192) = log₂(2¹³)^-1

F(¹/8192) = log₂2^{(13 x -1)}

F(¹/8192) = log₂2^-13

F(¹/8192) = -13log₂2

F(¹/8192) = -13 x 1

F(¹/8192) = -13

Therefore F(¹/ 8192) = -13

TRY THIS…………………

If F(x) = log₅x, Find F(¹/₃₁₂₅)

PROBABILITY 1

A number is chosen at random from 1 – 15 inclusive. Find the probability that it is a multiple of five or an even number greater than 8.

Solution

THIS IS A NON-MUTUALLY EXCLUSIVE EVENT.

Let n(S) represent sample space

P(E) = probability of even number greater than 8

P(M) = probability of a multiple of 5

n(S) = {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15} = 15

n(E) = {10, 12, 14} = 3

n(M) = {5, 10, 15} = 3

But 10 appears on both categories.

P(E) = ³/15

P(M) = ³/15

P(EnM) = ¹/15

………………………….

P(EuM) = P(E) + P(M) - P(EnM)  

P(EuM) = ³/15 + ³/15 -¹/15

              = ⁵/15

              = ¹/3  after simplification

∴ P(EuM) = ¹/5

TRY THIS………………

A number is chosen at random from 17 – 30 inclusive. Find the probability that it is a multiple of 5 or an even number.

FACTORIZE 2

Factorize 289x² - 169y²

Solution

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

289x²- 169y²  = 17²x² - 13²y²

                   = (17x)² - (13y)²

                   = (17x - 13y)( 17x + 13y)

Hence 289x² - 169y² = (17x - 13y)(17x + 13y)

TRY THIS………………………….

factorize 121c²- 49d²

ALGEBRA 3

Multiply 2x - 4y by -5a

solution

= -5a(2x - 4y)

= (-5a x 2x) - (-5a x 4y)

= (-10ax) - (-20ay) [since -5a x 4y = -20ay ]

= -10ax + 20ay    

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

Multiply 40p - 3q by  -4m

EXPONENTIALS 3

If 5^2w (40^w) = 10^w+40 ; Find w.

Solution

5^2w (40^w) = 10^w+40

(5²)^w (40^w) = 10^w+40

(25)^w (40^w) = 10^w+40

(25 x 40)^w = 10^w+40

(1000)^w = 10^w+40

(10³)^w = 10^w+40

10^3w = 10^w+40  (Bases are alike, so they cancel out)

3w = w+40

3w - w = 40

2w = 40

2w = 40²⁰

2       2

w = 20

TRY THIS…………………………….

If 3^2t (4^t) = 6^t+10 ; Find t.

VARIATIONS 2

x is directly proportional to y. x=12 while y=4. Find y when x is 69.

Solution

x ⍺ y

x = ky

12 = k x 4

12 = 4k

4      4

k = 3

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

k = 3, y= ? x = 69.

69 = 3 x y

69 = 3y

3      3

y = 23.

TRY THIS…………….

x is directly proportional to y. x=20 while y=5. Find y when x is 68.

Ans. y=17

LOGARITHMS 3

If log 2= 0.3010; find the value of log 400,000 without using tables.

solution

=log200,000

=log(4 x 100,000)

=log4 + log100,000

=log2² + log10⁵

=2log2 + 5log10

=2(0.3010) + (5 x 1)  [since log10=1 and Log3=0.3010]

=(0.6020) +  5

=5.6020

Hence log400,000=5.6020

TRY THIS……………

If log 2= 0.3010; find the value of log 4,000,000 without using tables.

ALGEBRA 2

If 13w² = 325; find w

Solution

13w² = 325

13w² = 325               [Dividing by 13 both sides]

13          13

w² = 25

w = 5          Because the square root of 25 is 5.   

Hence w = 5 .          

TRY THIS……………………….

If 14y² = 5600; find y.

POLYGONS 2

A regular polygon has 27 sides. Find the total interior angles of that polygon.

Solution

n = 27

Total angles = (n - 2)180⁰

                   = (27 – 2)180⁰

                   = 25 x 180⁰

                   = 4500⁰

Total angles = 4500⁰

TRY THIS……………………….

A regular polygon has 38 sides. Find the total interior angles of that polygon.

EXPONENTIALS 2

If 8¹⁰ = 32^a-2; find a

Solution

8¹⁰ = 32^a-2 

(2³)¹⁰ = (2⁵)^a-2 

2³⁰ = 2^5(a-2) 

2³⁰ = 2^5a-10

2³⁰ = 2^5a-10  [Same bases both sides cancels out]

30 = 5a – 10

30 + 10 = 5a

40 = 5a

40 = 5a

5      5

8 = a

Hence a=8

TRY THIS……………………….

If 5¹⁰ = 625^h-2; find h

QUADRATICS 1

What must be added to x²  + 20x to make the expression a perfect square?

Solution

a=1, b=20,c=?

b² = 4ac

(20)² = 4 x 1 x c

400 = 4c

400 = 4c

4       4

100 = c

Hence number to be added is 100

TRY THIS……………………

What must be added to x²  + 10x to make the expression a perfect square?

FACTORIZE 1

Factorize completely mc + mr - cr - c².  

Solution

= mc + mr - cr - c².

= (mc + mr) – (cr - c²).      [Grouping the factors]

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

 = (c + r)(m - c) answer

  Hence mc + mr - cr - c². =  (c + r)(m - c).   

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

Factorize completely pw + pr - rw - w².

UNITS OF LENGTH 1

Convert 53734m into km.

Solution

1km  = 1000m

  ?     = 53734m

After cross-multiplying;

= 1 x 53734

      1000

=   53734

     1000

= 53.734 km

Hence 53734m = 53.734km

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

Convert 72682m into km.

VARIATIONS 1

x is directly proportional to y. x=32 while y=4. Find y when x is 736.

Solution

x ⍺ y

x = ky

32 = k x 4

32 = 4k    [dividing by 4 both sides]

4      4

k = 8

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

k = 8, y= ? x = 736.

x = ky

736 = 8 x y

736 = 8y

8       8

y = 92.

TRY THIS…………….

x is directly proportional to y. x=40 while y=4. Find y when x is 380.

APPROXIMATIONS 1

Estimate the value of 2.1  x  0.034

solution

= 2.1  x  0.034

= 2.0 x 0.03    [ 2.1 ≈ 2.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.06 

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

Estimate the value of 5.4  x  0.073

LOGARITHMS 2

If log₃(10x - 23)=3; find x.

Solution

log₃(10x - 23)=3

 (10x - 23)=3³       

                                            

 10x - 23=27

10x = 27 + 23

10x =  50

10x =  50⁵   

10      10

x = 5

Hence x = 5

TRY THIS…………….

If log₃(5x + 6)=4; find x. 

FUNCTIONS 1

Find a linear function f(x) with gradient -6 which is such that f(5)=14.

Solution.

m=-6, points = [5,14] and [x, f(x)]

m = y₂-y₁/x₂-x₁

-6 = [f(x) – 14]/x-5

f(x)-14=-6(x-5)  [after cross multiplying]

f(x)-14=-6x+30

f(x)=-6x+30+14

f(x)=-6x+44

A  linear function is f(x) = -6x + 44.

TRY THIS……….

Give out a linear function f(x) with gradient -4 and f(5)=11.