INEQUALITIES K1

 

If 12x + 8 ≥ 26 + 3x ≥ 9x – 10; find x.

 

Solution

 

12x + 8 ≥ 26 + 3x and  26 + 3x ≥ 9x - 10

 

12x – 3x ≤ 26 - 8 and  26 +10 ≤ 9x - 3x

 

9x ≤ 18 and  36 ≤ 6x

 

9x ≤ 18  and  36 ≤ 6x

9      9            6      6

 

x ≤ 2 and   6 ≤ x

 

x ≤ 2 and  x  ≥  6

 

TRY THIS…………….

 

 

If 12x + 34 ≥ 7 + 3x ≥ 9x – 22; find x

LOGARITHMS K3

 

If Log_ax = 18, find Log_a(¹/x⁵).

 

Solution

 

= Log_a(¹/x⁵)

 

= Log_ax^-5   [Since 1/aⁿ = a^-n]

 

= -5 x Log_ax   [Since Log_mp^q   = q x Log_mp]   

 

= -5 x 18

 

= -90

 

 

TRY THIS………..

 

If Log_ac= 40, find Log_a(¹/c⁶)

PERIMETERS K1

 

Area of rectangle is 68m². Find its perimeter if width is 4m. 

 

Solution

 

A = L x w

 

68 = L x 4

 

68 = 4L                 (Dividing by 4 both sides)

4      4

 

17 = L

 

So, L=17, w=4, P=?

 

Now; P =2(L + w)

 

= 2(17 + 4)

 

= 2 x 21            [because 17 + 4 = 21]

 

= 42m

 

Hence perimeter is 42m

 

TRY THIS………..

 

Area of rectangle is 96m². Find its perimeter if width is 4m.  

PROBABILITY K1

 Given the interval of 1-16 inclusive, Find the probability of having an even number or a prime number.

 

Solution

 

This is a non-mutually exclusive event.

n(S) = number of sample space = 14

 

Let P(E)= probability of even numbers = ⁸/14

Let P(R)= probability of prime  numbers = ⁶/14

P(EnR) = probability of having both even and prime numbers= ¹/14

 

∴ P(EuR) = P(E) + P(R)  - P(EnR).

 

 P(EuR) =  8  +  6 – 1

                14    14   14

 

P(EuR) =  14  -   1 

                14     14  

 

P(EuR) =  13   

                14         

 

Hence probability of an even number or a prime number is ^13/14  .

 

TRY THIS........

 

 

Given the interval of 1-18 inclusive, find the probability of having an even number or a prime number

EXPONENTIALS K2

 

Simplify (7a)² - 9a² - 8a².     

Solution

= (7a)² - 9a²   - 8a².        

= 7².a² - 9a²  - 8a².                    Since (mn)² =m² . n²

= 49a² - 9a²  - 8a².    {since the square number of 7 is 49}

= 46a²  - 17a².

= 29a²        

TRY THIS………………

Simplify (9C)² – 20C² – 11C²        

LOGARITHMS K3

 

Simplify        Log0.000001

                        Log(1/1000)

 

Solution

 

=    Log0.000001

       Log(1/1000)

 

=    Log10^-6

       Log1000^-1

 

=    Log10^-6

       Log(10³)^-1

 

=    Log10^-6

       Log10^-3

 

=    -6Log10

       -3Log10

 

=    -6                        [since Log 10 and Log 10 cancel out]

      -3

 

=    2

  

Hence           Log0.000001   =   2

                      Log(1/1000)

 

TRY THIS………..

 

Simplify        Log0.00000001

                    Log(¹/100000)

G.P. K1

 The 1st term of a geometric progression is 7 and the 5th term is 112. Find i) the common ratio. ii) 10^th term

 

Solution

 

G₁=7, G₅=112, n=5

 

i) G_n = G₁r^n-1;

 

G₅ = G₁r^5-1;

 

G₅ = G₁r⁴;

 

112 = 7 x r⁴;

 

112 = 7r⁴;

 7       7

 

16 = r⁴;  Finding the fourth root of 16,

 

r = 2

 

 

 

Hence the common ratio is 2.

 

ii)  10^th term

 

G_n = G₁r^n-1;

 

G₁₀ = G₁r^10-1;

 

G₁₀ = G₁r⁹;

 

G₁₀ = 7 x 2⁹;

 

G₈ = 7 x 512;

 

G₁₀ = 3584

 

Hence the tenth term is 3584

 

TRY THIS………………………….. 

 

The 1st term of a geometric progression is 2 and the 6th term is 486. Find i) the common ratio. ii) 10^th term

MATRIX K1

 

FACTORIZE K1

 

Factorize 4x² - 7x – 15.

 

Solution

 

= 4x² - 7x – 15. 

 

We split the middle term (-x)

 

We look for two items such that

mxn =  4x² x (-15)= -60x²and

m+n = -7x

The correct choice is -12x and 5x:

 

= 4x² - 12x + 5x – 15            Since    (-7x) = -12x + 5x.

 

= (4x² - 12x) + (5x - 15)    grouping the factors

 

= 4x(x - 3) + 5(x - 3)  factoring out common terms

 

= (4x + 5) (x - 3)  answer

 

Hence 2x² - x – 15 = (2x + 5) (x - 3) answer.

 

TRY THIS…..

 

 Factorize 3x² - 16x – 35.

ALGEBRA K2