RADICALS 1

Radicals are among the challenging concepts in maths.

With this video below you can learn one of them and reach to the solution.





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ARITHMETICS 1

Evaluate 13 x 369 + 331 x 13.

Solution

= 13 x 369 + 331 x 13.

= 13 x (369 + 331)     factoring out the common number

= 13 x 700   

= 9100      [after multiplying 16 and 7 and adding two zeros on the answer]

Hence 13 x 369 + 331 x 13= 9100

TRY THIS………….

Evaluate 127 x 516 + 484 x 127.

SUM OF A GP 1

Find the sum of the 1st ten terms of the geometrical progression 2+6+18+54+…….

solution

G₁ = 2, r=3, n=10

S_n = G₁(rⁿ-1)/r-1

               

S₁₀ = 2(3¹⁰-1)/3-1            

S₁₀ = ¹2(3¹⁰-1)/2₁

               

S₁₀ = (3¹⁰-1)

S₁₀ = 59049 - 1

S₁₀ = 59048

Hence the sum of the 1st seven terms is 19682.

TRY THIS………………

Find the sum of the 1st eight terms of the geometrical progression 3+9+27+81+…..

ALGEBRA 2

If │4x – 7 │= 15; Find  x

Solution

±(4x – 11 )= 17

4x -11= 15   OR   –(4x-11) = 15

4x – 11 = 15    OR   -4x+11=15

4x = 15 + 11    OR   -4x= 15-11

4x = 26     OR   -4x  =   4

4      4               -4        -4

x = ¹³/2  OR  x = -1

Hence x = ¹³/2  OR  x = -1

TRY THIS………………………. 

If │2x – 7 │= 25; Find  x

FUNCTIONS 3

If F(x) = log₂x, Find F(32)

Solution

F(x) = log₂x

F(32) = log₂(32)

F(32) = log₂32^-1

F(32) = log₂(2⁵)^-1

F(32) = log₂2^{(5 x -1)}

F(32) = log₂2^-5

F(32) = -5log₂2

F(32) = -5 x 1

F(32) = -5

Therefore F(32) = -5

TRY THIS…………………

If F(x) = log₂x, Find F(512)

EXPAND 3

Expand 8vw(5w – 3)

Solution

= 8vw(5w – 3)

= (8vw x 5w) – (8vw x 3)

= 40vw² – 24vw answer

TRY THIS………..

Expand 4ab(7a+ 20)

FACTORIZE 1

Factorize 625x² - 9y²

Solution

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

625x²- 9y²  = 25²x² - 3²y²

                   = (25x)² - (3y)²

                   = (25x - 3y)( 25x + 3y)

Hence 625x² - 9y² = (25x - 3y)(25x + 3y)

TRY THIS………………………….

Factorize 225c²- 36d²

FUNCTIONS 2

Find a linear function f(x) with gradient -10 which is such that f(7)=16.

        Solution.

m=-10, points = [7,16] and [x, f(x)]

m = y₂-y₁/x₂-x₁

-10 = f(x) – 16/x-7

  

f(x)-16=-10(x-7) [after cross multiply]

f(x)-16=-10x+70

f(x)=-10x+70+16

f(x)=-10x+86

A  linear function is f(x) = -10x+86.

TRY THIS……….

Give out a linear function f(x) with gradient 6 and f(8)=-4

LOGARITHMS 4

Evaluate Log₂18 + Log₂8 - Log₂9

Solution

= Log₂18 + Log₂8 - Log₂9

= Log₂(18 x 8 ÷ 9)

= Log₂16

= Log₂2⁴

= 4Log₂2

= 4 x 1     [since Log₂2=1]

= 4 answer

TRY THIS………………….

Evaluate Log₂28 + Log₂16 - Log₂7

A. P. 1

The 1st term of arithmetic progression is 37 and the common difference is 40. Find the nth term

solution

A₁=37, d= 40, n=?

A_n =A1 + (n-1) d

A_n=37 + (n-1)40

A_n=37 + 40n -40

A_n=40n + 37-40

A_n=40n – 3

Hence the nth term is An=40n – 7  

TRY THIS………………….. 

The 1st term of arithmetic progression is 48 and the common difference is 98. Find the nth term.

SETS 1

If A={e, f, g, h, i} and B={a, h, b, c, g} Find n(AuB)

Solution

n(AuB) means everything found in A and B. If repetition occurs we take one element to represent others.

Hence n(AuB) = {a, b, c, e, f, g, h, i} answer

TRY THIS………………….

If R={e, f, g, h, I, p, q, r} and V={a, h, b, c, g, p} Find n(RuV)

LOGARITHMS 3

If Log₃(x²+7x+7)=0; find x.

Solution

Log₃(x²+7x+7)=0

 (x²+7x+7)=3⁰

x²+7x+7=1    [ Since n⁰=1 ]

x²+7x+7-1=0

x²+7x+6=0

We solve it by factorization;

x²+7x+6=0

x²+6x + x+6=0

x(x+6) + 1(x+6)=0

(x+6)(x+1)=0

x+6=0 OR x+1=0

x= -6 OR x= -1 answer

TRY THIS………………….

If Log₇(x² + 17x + 101)=2; find x.

ALGEBRA 1

LOGARITHMS 2

If log₉(2x + 11)=2; find x.

   Solution

Log₉(2x + 11)=2

(2x + 11)=9²           

                                            

2x + 11=81

2x =81 – 11

2x = 70

2x =   70   

2        2

x = 35

Hence x = 35

TRY THIS………………

If log₈(2t - 16)=2; find t

EXPAND 2

Expand (y – 7) (y – 16)

solution

= (y – 7) (y – 16)

= y (y – 16) – 7 (y – 16)

= y² – 16y – 7y + 112 

= y² – 23y + 112  since (– 16y – 7y) = – 23y

= y² – 23y + 112 answer

TRY THIS………..

Expand (c – 6) (c – 11)

UNITS OF DISTANCE 1

Change 91260m into Km.

Solution

1Km = 1000m

?  =  91260m

= 1  x  91260

       1000

=     91260

       1000

= 91.26Km

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

Change 65027m into Km.

LOGARITHMS 1

If Log_ax = 44, find Log_a(¹/x⁵)

Solution

= Log_a(¹/x⁵)

= Log_ax^-5

= -5 x Log_ax

= -5 x 44

= -220

TRY THIS………..

If Log_aw= 72, find Log_a(¹/w³)

MATRIX 1

FUNCTIONS 1

Given that F(x) = 3x  +  11. Find F(-2)

Solution

F(x) = 3x  +  11

F(-2) = 3(-2)  +  11

F(-2) = -6  +  11

F(-2) = 5

Hence  F(-2) = 5

TRY THIS………………..

Given that F(x) = 12x  +  78. Find F(-4)

UNITS OF VOLUME 1

Change 42.5 liters of water into milliliters

Solution

1L  =  1000mL

42.5L = ?

___________________________  

 = 42.5 x  1000

           1

= 42.5 x 1000

= 42500mL. Answer

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

Change 15.3 liters of milk into milliliters

PERFECT SQUARES 1

If 16x² + 24x + h is a perfect square, find h.

Solution

a = 16, b = 24, c = h.

b² = 4ac

(24)² = 4 x 16 x h

576 = 64h

576  =   64h

 64       64

           

h = 9

Hence h = 9

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

If 4x² + 12x + w is a perfect square, find w.

EXPAND 1

Expand 8w(5w + 5 - w)

Solution

= 8w(5w + 5 - w)

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

= 40w² + 40w - 8w²

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

= 32w² + 40w answer

TRY THIS………..

Expand 4a(9a+ 11 - a)

TRANSPOSE 1

BODMAS 1

Evaluate  70 + 6 – 5 – 7 – 9 + 15.

Solution

= 70 + 6 – 5 – 7 – 9 + 15

= 70 + 6 + 15 – 5 – 7 – 9

= 70 + 6 + 15 – ( 5 + 7 + 9)

= 91 – 21

= 70

TRY THIS………

Evaluate  55 + 6 – 7 – 3 – 13 + 6

EXPONENTIALS 1

Simplify (d⁴)⁸

solution

= (d⁴)⁸

=  d^4x8

=  d³²

TRY THIS........

Simplify  (k¹⁸)³

INEQUALITIES 1

Find x if 5x – 21 ≤ x + 15 ≤ 6x.

Solution

5x – 21 ≤ x + 15 and  x + 15 ≤ 6x

5x – x ≤ 21+ 15 and  15 ≤ 6x - x

4x ≤ 36 (divide by 4 both sides) and  15 ≤ 5x (divide by 5 both sides)

x ≤ 9 and  3 ≤ x

x ≤ 9 and  x ≥ 3 answer

TRY THIS……..

Find x if 6x – 35 ≤ x + 15 ≤ 2x.

SLOPE 1

Find the slope of a line which passes through (17, 13) and (6, 8)

Solution

x₁ = 17,  y₁ =13,  x₂ = 6,  y₂ = 8

m = y2 –y1

        x2 – x1

m =   8 –13

          6 –17

m =   -5

         -11

  

Hence the slope is ⁵/11

TRY THIS……..

Find the slope of a line which passes through (6, 13) and (9, -2)