MATRIX 1B

EXPONENTIALS 1C

If b⁴ + 20 = 36; find b.

Solution

b⁴ + 20 = 36

b⁴ = 36 – 20  [taking 20 in the right hand side]

b⁴ = 16

b = 2 [since the 4^th root of 16 is 2]

Hence b = 2

TRY THIS…………………………

If m⁴ + 200 = 825; find m.

VECTORS 1A

Write (5,6) in terms of i and j.

Solution

= (5,6)

= (5,0) +  (0,6)

= 5(1,0) +  6(0,1)

= 5i + 6j   [since (1,0)=i and (0,1=j)

Hence (5,6) = 5i +  6j  

TRY THIS……………………..

Write (8,16) in terms of i and j.

MID-POINT 1A

Find a if the the midpoint of a line from (a, 6) to (9, 10) is(11,8)

Solution

x1=a, x2=9, y1=6, y2=10.

Midpoint = (x1 + x2, y1 + y2)

                        2             2

Midpoint = (a + 9 , 6 + 10)

                       2          2

But Midpoint is(11,8)

(11,8) = (a + 9 , 16)

                2        2

Equating the values of x

11 = a + 9

          2 

      

2  x 11 = (a + 9) x 2 Multiplying by 2 both sides.

                   2       

22 = a + 9

         

22 – 9 = a

Hence a = 13

TRY THIS……………………..

Find w if the midpoint of a line from (w, 4) to (7, 10) is(14,8)

SIMILARITY 1A

Prove that the 2 triangles below are similar.  

Solution

<NMP = <PRM = 30⁰ ------- Given

<MPN = <RPM = 90⁰ ------- Given

<MNP = <RMP = 60⁰ ------- (3^rd angles of a triangle)

Hence êMPN = êRPM --- (By AA – Similarity theorem)

TRY THIS……………………..

Prove that the 2 triangles below are similar.

SIMPLIFICATION 1A

Simplify: 5x + 9y + 2y - x

solution

= 5x + 9y + 2y – x

= 5x– x + 9y + 2y

= 4x + 11y

TRY THIS……………………… 

Simplify: 11x + 14y + 5y – 4x.

ALGEBRA 1F

Work out the value of 6a + 2b when a = 7 and b = 3.

Solution

=6a + 2b

=(6xa) + (2xb)

=(6x7) + (2x3)

=42+ 6

=48

TRY THIS……………………… 

Work out the value of 12m + 2n when m = 7 and n = 3.

ARITHMETICS 1C

Evaluate 16 x 362 + 638 x 16.

Solution

= 16 x 362 + 638 x 16

= 16 x (362 + 638)     factoring out the common number

= 16 x 1000   

= 16000      [after multiplying 16 by 1 and adding three zeros on the answer]

Hence 16 x 362 + 638 x 16 = 16000

TRY THIS………….

Evaluate 141 x 518 + 482 x 141. 

ALGEBRA 1E

If 4x + 3y= 8; find the x intercept.

Solution

x-intercept is when y=0.

4x + 3(0)= 8.

4x + 0 = 8.

4x = 8

4x = 8    dividing by 4 both sides.

4     4

x = 2

Hence x-intercept is (2,0)

TRY THIS………………………   

If 4x -3y= 20; find the x intercept. 

ROUNDING OFF 1A

Write 5 916 821 correct to the

i) Nearest million.

ii) Nearest thousand.

iii) Nearest ten.

Solution

i) Nearest million = 6,000,000

ii) Nearest thousand = 5 917000

iii) Nearest ten =5 916 820

TRY THIS……………………… 

Write 7 728 473 correct to the

i) Nearest million.

ii) Nearest thousand.

iii) Nearest ten.

REGULAR POLYGONS 1A

An interior angle of a regular polygon is 70⁰ 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+70⁰ …………………(2)

Substitute (2) in (1) above.

e+70⁰   + e=180⁰

e+ e+70⁰   =180⁰

2e+ 70⁰   =180⁰

2e=180⁰ - 70⁰   

2e=110⁰

2e=110⁰          dividing by 2 both sides.

2      2

e = 55⁰

But from equation (2)

But i = e+70⁰

        = 55⁰+70⁰

        = 125⁰      

Hence interior angle = 125⁰      

TRY THIS………………………   

An interior angle of a regular polygon is 76⁰ greater than an exterior angle. Find the interior angle.

ALGEBRA 1D

If 4x + 3y= 8; find the x intercept.

Solution

x-intercept is when y=0.

4x + 3(0)= 8.

4x = 8

4x = 8    dividing by 4 both sides.

4      4

x = 2

Hence x-intercept is (2,0)

TRY THIS………………………   

If 4x -3y= 20; find the x intercept.

MATRIX 1A

LOGARITHMS 1D

If log 2= 0.3010; find the value of log 50,000 without using tables.

solution

=log50,000

=log(100,000÷ 2)

=log100,000 – log2

=log10⁵ – log2

=5log10 – log2

=(5x1) – (0.3010)

=5 – 0.3010)

=4.699

Hence log50,000=4.699

TRY THIS……………

If log 2= 0.3010; find the value of log 5,000 without using tables.

FUNCTIONS 1A

If F(x) = 5x + 10; Find F^-1(40).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 5x + 10

y – 10 = 5x

y – 10  = ¹5x

   5          ₁5

y – 10   = x

   5

x  =   y – 10     after rearranging

            5

F^-1(x) = x – 10     after interchanging x and y variables.

                5

Now we calculate F^-1(40) as hereunder;

F^-1(40) = 40 – 10     

                 5

F^-1(40) =  30      

                5

F^-1(40) =  6      [after dividing 30 by 5]

               

Hence, F^-1(40) =  6     

                             

TRY THIS……………………………

If F(x) = 7x - 20; Find F^-1(6).

ALGEBRA 1C

Simplify: 5x + 9y + 2y - x

solution

= 5x + 9y + 2y – x

= 5x– x + 9y + 2y

= 4x + 11y

TRY THIS……………………… 

Simplify: 11x + 14y + 5y – 4x.

STATISTICS 1A

In a survey of 60 pupils, 22 of them said their favourite sport was football. What angle in a pie chart would this represent?

Solution

We make a fraction of 22 ot of 60, then multiply by 360⁰.

= 22 x 360⁰.

   60

= 22 x 360⁰₆

   60₁

= 22 x 6

= 132⁰.

Hence the angle is 132⁰ 

TRY THIS………………………   

In a survey of 30 people, 13 of them said their favourite sport was netball. What angle in a pie chart will this represent?

EXPONENTIALS 1B

If 3^2w (40^w) = 1000000^w-10 ; Find w.

Solution

5^2w (40^w) = 1000000 ^w-10

(5²)^w (40^w) = 10^6(w-10)

(25)^w (40^w) = 10^6w-60

(25 x 40)^w = 10^6w-60

(1000)^w = 10^6w-60

(10³)^w = 10^6w-60

10^3w = 10^6w-60  (Bases are alike, so they cancel out)

3w = 6w-60

0= 6w-3w – 60  [collecting letters together]

0= 3w – 60  [collecting letters together]

60 = 3w

²⁰60= 3w       [dividing by 3 on both sides]

    3     3

w = 20

TRY THIS…………………………….

If 4^2t (4^t) = 128^t+5 ; Find t.

SLOPE OR GRADIENT 1B

Find the c if the slope of a line which passes through (-3, -4) and (c-1,-11) is ^-7/11.

Solution

x₁ = -3,  y₁ =-4,  x₂ = c-1,  y₂ = -11, m=^-7/11.

m = y2 –y1

      x2 – x1

-7 =   -11 –(-4)

11    (c-1) –(-3)

-7 =   -11 + 4            [since –(-4)  becomes  + 4]             

11     c-1 + 3            [since –(-3)  becomes  + 3]                            

-7 =     -7

11     c+ 2

-7(c+2) = 11x-7     [after cross multiplication]

-7c-14 = -77

-7c= -77+14

-7c= -63

-7c= -63

-7     -7

c = 9

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

Find the c if the slope of a line which passes through (7, 3) and (c-3,-9) is ¹²/5.