WORD PROBLEMS J4

 

VECTORS J5

 

If u = 60i + 2j and v = 3i + 10j find 7u - 3v

 

solution

 

= 7u - 3v

 

= 7(60i + 2j) - 3(3i + 10j)

 

= 420i + 14j - 9i - 30j

 

= (420i - 9i) + (14j - 30j)

 

= 411i - 66j

 

Hence 7u - 3v = 411i - 66j

 

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

 

 

If u = 9i + 3j and v = 2i + 6j;  find 8u - 4v.

LOGARITHMS J7

 

Evaluate Log₂(¹/256 ÷ 8).

 

Solution

 

= Log₂(¹/256 ÷ 8)

 

= Log₂(¹/256) -  Log₂8          (applying the quotient rule)

 

= Log₂256^-1 -  Log₂2³             ( 256= 2⁸ and 8=2³ )

 

= -1Log₂2⁸ -  3Log₂2       ( remember  Log_aa^c = cLog_aa )

 

= -1 x 8Log₂2 -  3Log₂2      

 

= (-8 x 1) - (3 x 1)          ( remember  Log_aa = 1 )

 

= -8 - 3

 

= -11

 

 

Hence Log₂(¹/256 ÷ 8) = -11 

 

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

 

 

Evaluate Log₂(¹/1024 ÷ 32). 

LOGARITHMS J6

 

Find x if Log_x128 = 7

 

solution

 

Log_x128 = 7

 

128 = x⁷ .

 

2⁷ = x⁷ . [since 625 = 5⁴ by prime factorization]

 

2⁷ = x⁷ . [ powers ARE SAME, they cancel out]

 

x = 2

 

Hence x = 2

 

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

 

Find y if Log_y64 = 3

FACTORIZE J6

 

Factorize 1-121B²

 

Solution

 

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

1-121B² = 1²-121B²   

    

             = 1²-11²B²      

 

             = 1²-(11B)²       

 

 

             = (1 - 11B)(1 + 11B)

 

 

Hence 1 - 121B² = (1 - 11B)(1 + 11B)

 

 

TRY THIS…………….

 

 

 Factorize 16 – 4C²

EXPONENTIALS J3

 

If a² = 7; find the value of a⁶.

 

solution

 

a⁶ = (a²)³.

 

a⁶ = (7)³.   [ since a² = 30]

 

a⁶ = 7 x 7 x 7

 

a⁶ = 343  answer

 

 

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

 

 

NECTA FORM II - 2012 QN. 8

 

 

If a² = 12; find the value of a⁴.

PERCENTS J4

 

In a box of 500 books, 60 are red in colour, 140 are yellow in colour while the rest are bluish in color. Give out the percentage of blue books.

 

solution

 

1st we have to get the number of blue books.

 

 blue books = 500 – 60 - 140 = 300

 

then,=  300 x 100

            500

 

then,=  300

             5

 

   = 60%

 

Hence percentage of blue books is 60%

 

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

  

In a box of 800 books, 300 are yellow in colour, 200 are red, while the rest are greyishish in color. Give out the percentage of grey books.

SQUARES J2

 

Evaluate 20396² – 20386²

 

Solution

We use difference of two squares;

a² – b² = (a – b)(a + b)

 

20396² – 20386² = (20396 + 20386)(20396 – 20386)

 

                     = (40782)(10)

 

                     = 407820

 

20396² – 20386² = 407820

 

TRY THIS…………….

 

 

Evaluate 457308² – 457208²

FUNCTIONS J2

 

Find the axis of symmetry for F(x) = 4x² + 20x + 8

 

Solution

 

a=4, b=10

 

Axis of symmetry = -b/2a

 

                               =  -(20)

                                    2 x 4

 

                               =    -20

                                      8

 

 

Hence the axis of symmetry is -2.5

 

TRY THIS…………………

 

 

Find the axis of symmetry for F(x) = 6x² + 40x + 3

ALGEBRA J3

 

Solve the following equations by substitution method.

x + y = 3

x + 3y = -3

 

Solution

 

x + y = 3 ----------- (i)

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

 

from equation (i)

 

x + y = 3

x = 3 - y ----------(iii)

 

substitute equation (iii) in (ii) above,

 

x + 3y = -3

 

(3 - y) + 3y = -3

3 - y + 3y = -3  

3 + 2y = -3  

2y = -3 - 3 

 

2y = -6           [since –y+3y=2y]

 

 

 

2y = -6

2      2

 

y = -3

 

 

From equation (iii)

x = 3 - y

x = 3 – (-3) ----------(iii)

x = 3+3

x=6

Hence x=6 and y=-3.

 

 

TRY THIS………………………..

 

 

5x - 2y = 14

x  -  y = 4

EXPAND J2

 

Expand 7w(w – 60 - w²)

 

solution

 

= 8w(w – 60 - w²)

 

= (8w x w) – (8w x 60) + (8w x w²)

 

= 8w² – 480w +   8w³   answer

 

 

TRY THIS………..

 

 

Expand 4e(e + 70 – e²)

ALGEBRA J1

 

Expand 8w(w – 3 + w²)

 

solution

 

= 8w(w – 3 + w²)

 

= (8w x w) – (8w x 3) + (8w x w²)

 

= 8w² – 24w +   8w³   answer

 

 

TRY THIS………..

 

 

Expand 10p(p + 7 – p²)

LOGARITHMS J1

 

If Log_ax = 46, find Log_a(¹/x⁵)

 

Solution

 

= Log_a(¹/x⁵)

 

= Log_ax^-5

 

= -5 x Log_ax   [Since  Log_axⁿ = nLog_ax]

 

= -5 x 46

 

= -230

 

 

TRY THIS………..

 

 

If Log_aW= 60, find Log_a(¹/W⁷)