Thursday 31 August 2017

EXPONENTIALS 4


Simplify 11
              64-2/3

Solution

11
  64-2/3

=  11 x 1
          64-2/3

=  11 x 642/3        [Since 1/a-n = an]

=  11 x (641/3)2        [Since 641/3 = 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)

Tuesday 29 August 2017

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  Logaac = cLogaa )

= (11 x 1) +  (4 x 1)          ( remember  Logaa = 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 = 129


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)


Thursday 10 August 2017

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.

Wednesday 9 August 2017

FACTORIZE 2


Factorize 289x- 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 52w (40w) = 10w+40 ; Find w.

Solution

52w (40w) = 10w+40

(52)w (40w) = 10w+40

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

(25 x 40)w = 10w+40

(1000)w = 10w+40

(103)w = 10w+40

103w = 10w+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 32t (4t) = 6t+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.


Sunday 6 August 2017

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 = 32a-2; find a

Solution

810 = 32a-2 

(23)10 = (25)a-2 

230 = 25(a-2) 

230 = 25a-10

230 = 25a-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 = 625h-2; find h


QUADRATICS 1


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


Saturday 5 August 2017

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

Friday 4 August 2017

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

10x505   
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.