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FORM 1 Maths
Nitaanza na Secondary Mathematics
Tutanzia mwanzo kabisa, yaani pale unapoingia form 1....."Numbers"
Mnakaribishwa kuchangia hii topic, au kuendeleza kutokea hapa..

Numbers​

Types of Numbers
Integers are whole numbers (both positive and negative, including zero). So they are ..., -2, -1, 0, 1, 2, .... So a negative integer is a negative whole number, such as -3, -10 or -23. Natural numbers are positive integers.

A rational number is a number which can be written as a fraction where numerator and denominator are integers (where the top and bottom of the fraction are whole numbers). For example 1/2, 4, 1.75 (=7/4).

Irrational numbers are numbers which cannot be written as fractions, such as pi and √2. In decimal form these numbers go on forever and the same pattern of digits are not repeated.

Square numbers are numbers which can be obtained by multiplying another number by itself. E.g. 36 is a square number because it is 6 x 6 .

Surds are numbers left written as √n , where n is positive but not a square number. E.g. √2 (see 'surds').
In Mathematics, surds are the values in square root that cannot be further simplified into whole numbers or integers. Surds are irrational numbers. The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified.

Prime numbers are numbers above 1 which cannot be divided by anything (other than 1 and itself) to give an integer. The first 8 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

Real numbers are all the numbers which you will have come across (i.e. all the rational and irrational numbers). All real numbers can be written in decimal form (such as 3.165).

A factor (or divisor) of a number is a number which will divide into your number exactly. So you can divide a number by one of its factors and you won't be left with a remainder. For example, 3 is a factor of 6 because you can divide 6 by 3 and you won't be left with a remainder (you get 2).

Prime Factor Decomposition
An important fact is that any number can be written as the product (multiplication) of prime numbers in one way. For example, 20 = 5 x 2 x 2 . This is the only way of writing 20 as the product of prime numbers. Writing a number in this way is called prime factor decomposition.
Example
Find the prime factor decomposition of 36.
We look at 36 and try to find numbers which we can divide it by. We can see that it divides by 2.
36 = 18 × 2
2 is a prime number, but 18 isn't. So we need to split 18 up into prime numbers. We can also divide 18 by 2.
18 = 9 × 2
and so 36 = 18 × 2 = 9 × 2 × 2
But we haven't finished, because 9 is not a prime number. We know that 9 divides by 3.
9 = 3 x 3.
Hence 36 = 9 × 2 × 2 = 3 × 3 × 2 × 2.
This is the answer, because both 2 and 3 are prime numbers.
Example
a and b are prime numbers, ab3 = 54. Find the values of a and b.
So ab3 is the prime factor decomposition of 54.
54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3 = 2 × 33
So a = 2 and b = 3.

LCM and HCF
The least (or lowest) common multiple (LCM) of two or more numbers is the smallest number into which they evenly divide. For example, the LCM of 2, 3, 4, 6 and 9 is 36.
The highest common factor (HCF) of two or more numbers is the highest number which will divide into them both. Therefore the HCF of 6 and 9 is 3.

Approximations
If the side of a square field is given as 90m, correct to the nearest 10m:
The smallest value the actual length could be is 85m (since this is the lowest value which, to the nearest 10m, would be rounded up to 90m). The largest value is 95m.
Using inequalities, 85£ length <95.
Sometimes you will be asked the upper and lower bounds of the area. The area will be smallest when the side of the square is 85m. In this case, the area will be 7725m². The largest possible area is 9025m² (when the length of the sides are 95m).

BODMAS (/BIDMAS)
When simplifying an expression such as 3 + 4 × 5 - 4(3 + 2), remember to work it out in the following order: brackets, of (/indices), division, multiplication, addition, subtraction.
So do the thing in the brackets first, then any division, followed by multiplication and so on. The above is: 3 + 20 - 4 × 5 = 3 + 20 - 20 = 3

Tunaeendela...

Measurements​

Systems of Measurement
Most of the world now uses the metric system of measurement. In the UK metric measurements have been taught in schools since the early 1970’s; however imperial measurements are still used in some cases. We therefore have to know how to convert between metric and imperial measurements.

The Metric system
The metric system has almost universally been adopted as it is much easier to calculate accurately as all metric measurements are based on units of 10.
Milli – means one thousandth (or 1/1000)
Centi means one hundredth (or 1/100)
Kilo means one thousand (1,000)
Metric Measurements
Length is typically measured in:
Centimetre (cm) = 10mm
Metre (m) = 100cm
Kilometre (km) = 1,000m
Mass is measured in:
Kilogram (kg) = 1,000 grams
Tonne = 1,000kg
Volume is measured in:
Litre = 1,000ml
1,000 litres = 1m3

Imperial Measurements
Length is measured in:
1 foot = 12inches
1 yard = 3 feet
1 mile = 1760 yards
Mass is measured in:
1 pound = 16 ounces
1 stone = 14 pounds
1 ton = 160 stones
Volume is measured in:
1 gallon = 8 pints

Converting Metric and Imperial Units
As some imperial units are still in use today you do need to know how to convert them to metric and vice versa. The conversion rates below have been rounded to 2 decimal points.

Kilometres and Miles:
Miles to km - Multiply by 1.61
km to Miles - Multiply by 0.62

Kilograms and Pounds:
kg to Pounds - Multiply by 2.20
Pounds to kg - Multiply by 0.45

Litres and Gallons:
Litres to Gallons - Multiply by 0.22
Gallons to Litres - Multiply by 4.55

Metres and Centimetres to feet and inches:
Inch to cm - Multiply by 2.54
cm to Inch - Multiply by 0.39

Ratios​

If the ratio of one length to another is 1 : 2, this means that the second length is twice as large as the first. If a boy has 5 sweets and a girl has 3, the ratio of the boy's sweets to the girl's sweets is 5 : 3 . The boy has 5/3 times more sweets as the girl, and the girl has 3/5 as many sweets as the boy. Ratios behave like fractions and can be simplified.

Example
Simon made a scale model of a car on a scale of 1 to 12.5 . The height of the model car is 10cm.

(a) Work out the height of the real car.

The ratio of the lengths is 1 : 12.5 .

So for every 1 unit of length the small car is, the real car is 12.5 units. So if the small car is 10 units long, the real car is 125 units long. If the small car is 10cm high, the real car is 125cm high.


(b) The length of the real car is 500cm. Work out the length of the model car.

We know that model : real = 1 : 12.5 . However, the real car is 500cm, so 1 : 12.5 = x : 500 (the ratios have to remain the same). x is the length of the model car.

To work out the answer, we convert the ratios into fractions:

1/12.5 =x/500

multiply both sides by 500:

500/12.5 = x

so x = 40cm

Example
Alix and Chloe divide £40 in the ratio 3 : 5. How much do they each get?

First, add up the two numbers in the ratio to get 8. Next divide the total amount by 8, i.e. divide £40 by 8 to get £5. £5 is the amount of each 'unit' in the ratio.


To find out how much Alix gets, multiply £5 by 3 ('units') = £15. To find out how much Chloe gets, multiply £5 by 5 = £25.

Map Scales

If a map has a scale of 1 : 50 000, this means that 1 unit on the map is actually 50 000 units across the land. So 1cm on the map is 50 000cm along the ground (= 0.5km).
So 1cm on the map is equivalent to half a kilometre in real life.
For 1 : 25 000, 1 unit on the map is the same as 25 000 units on the land. So 1 inch on the map is 25 000 inches across the land, or 1cm on the map is 25 000 cm in real life.

Proportion​

Proportion
If a "is proportional" to b (which is the same as 'a is in direct proportion with b') then as b increases, a increases. In fact, there is a constant number k with a = kb. We write a ∝ b if a is proportional to b.
The value of k will be the same for all values of a and b and so it can be found by substituting in values for a and b.
Example
If a ∝ b, and b = 10 when a = 5, find an equation connecting a and b.
a = kb (1)
Substitute the values of 5 and 10 into the equation to find k:
5 = 10k
so k = 1/2
substitute this into (1)
a = ½b
In this example we might then be asked to find the value of a when b = 2. Now that we have a formula connecting a and b (a = ½ b) we can subsitute b=2 to get a = 1.
Similarly, if m is proportional to n2, then m = kn2 for some constant number k.
If x and y are in direct proportion then the graph of y against x will be a straight line.
Inverse Proportion (HIGHER TIER)
If a and b are inversely proportionally to one another,
a ∝ 1/b
therefore a = k/b
In these examples, k is known as the constant of variation.
Example
If b is inversely proportional to the square of a, and when a = 3, b = 1, find the constant of variation.
b = k/a2 when a = 3,
b = 1
therefore 1 = k/32
therefore k = 9

Rates of Change​

Calculating rates of change is an important part of the Maths curriculum for students studying the higher paper.
To calculate rates of change in your exam you will need to be able to interpret graphs.
The graph below shows the cost of three different mobile phone tariffs.

Line A shows a direct proportion. The gradient of the line represent the rate of change.
The formula is therefore the change in the y axis divided by the change in the x axis.
In this example that equals 10 ÷ 40 = 0.25. This represents a charge of 25p per minute and shows a constant proportion.
Line B shows a fixed charge of £5 regardless of how many calls are made. The gradient is 20 ÷ 100 = 0.2 or 20p per minute.
Line C shows a fixed charge of £12.50, with no call charges up to 60 minutes. After 60 minutes the gradient is 3 ÷ 60 = 0.05 0r 5p a minute.
You can use the graph to see which tariff represents best value for you. After about 55 minutes this would be tariff C, if you use fewer minutes than this it would be tariff A. Tariff B is therefore never the best value option.
Repetitive rate of change
With repetitive rates of change the percentage change is applied more than once. You therefore have to calculate one step at a time.
For example, if a business buys a computer for £1,000. In the first year it depreciates by 25%, the next year it loses 20% of its value and then 10% every year after that.
You may then be asked how much is the computer worth after 3 years? To calculate this you can use one of two methods.
Method 1: Step by Step:
Lose 25% so worth 75% after year one.
75% of 1,000 = 0.75 x 1000 = £750
Then 80% of 750 = 0.80 x 750 = £600
Then 90% of 600 = 0.90 x 600 = £540
So the computer is worth £540 after 3 years.
Method 2: Using multipliers:
Lose 25% so worth 75%
Lose 20% so worth 80%
Lose 10% so worth 90%
0.75 x 0.80 x 0.90 = 0.54
Then 0.54 x 1000 = £540

Gradients and Graphs​

Gradient is another word for "slope". The higher the gradient of a graph at a point, the steeper the line is at that point. A negative gradient means that the line slopes downwards.

Finding the gradient of a straight-line graph


It is often useful or necessary to find out what the gradient of a graph is. For a straight-line graph, pick two points on the graph. The gradient of the line = (change in y-coordinate)/(change in x-coordinate) .

In this graph, the gradient = (change in y-coordinate)/(change in x-coordinate) =(6,6)=(x1, y1); (10,8)=(x2,y2)
=(y2-y2) divide by (x2-x1)
= (8-6)/(10-6) = 2/4 = 1/2

We can, of course, use this to find the equation of the line. Since the line crosses the y-axis when y = 3, the equation of this graph is y = ½x + 3 .


Finding the gradient of a curve


To find the gradient of a curve, you must draw an accurate sketch of the curve. At the point where you need to know the gradient, draw a tangent to the curve. A tangent is a straight line which touches the curve at one point only. You then find the gradient of this tangent.

Example

Find the gradient of the curve y = x² at the point (3, 9).

Gradient of tangent = (change in y)/(change in x)
= (9 - 5)/ (3 - 2.3)
= 5.71


Note: this method only gives an approximate answer. The better your graph is, the closer your answer will be to the correct answer. If your graph is perfect, you should get an answer of 6 for the above question.


Parallel Lines


Two lines are parallel if they have the same gradent.


Example


The lines y = 2x + 1 and y = 2x + 3 are parallel, because both have a gradient of 2.

Perpendicular Lines (HIGHER TIER)
Two lines are perpendicular if one is at right angles to another- in other words, if the two lines cross and the angle between the lines is 90 degrees.
If two lines are perpendicular, then their gradients will multiply together to give -1.
Example
Find the equation of a line perpendicular to y = 3 - 5x.
This line has gradient -5. A perpendicular line will have to have a gradient of 1/5, because then (-5) × (1/5) = -1. Any line with gradient 1/5 will be perpendicular to our line, for example, y = (1/5)x.
S

Source:Revision Maths - Maths GCSE and A-Level Revision

 

[The last don]

Safi sana umepatia jibu mkuu ila kwenye njia kidogo sijakuelewa hasa hapo uliposema 11× = 400 hapa ukitafuta thamani ya x haiji 36 sijui ulipataje mkuu.

Safi sana umepatia jibu mkuu ila kwenye njia kidogo sijakuelewa hasa hapo uliposema 11× = 400 hapa ukitafuta thamani ya x haiji 36 sijui ulipataje mkuu.

Ni kweli chief,niliminus 4 from 400,nikasahu kuiandika.

 

[itakiamo]

Tuanze na format za mitihani, kujua format kutamuwezesha mwanafunzi kupata dira ya kinachoendelea katika assessments mbalimbali


FORM TWO NATIONAL ASSESSMENT EXAMINATION FORMAT
1 (a)L.C.M or G.CF
(b) Approximation
2 (a)Fraction
(b) Decimal and percentage
3. (a) Units
(b) Profit, loss and ratio
4. (a) Geometry
(b) Perimeter and Area
5. (a) Algebra ( simultaneously equation)
(b) Algebra ( Quadratic expression or equation)
6. (a) Coordinate geometry
(b) Transformation
7. (a) Exponents or logarithm
(b) Radical ( especially rationalization)
8. (a)Similarities
(b) Congruence
9. (a) Pythagoras theorem
(b) Trigonometric ratio
10.(a) Set
(b) Statistics

Itaendelea..........

 

[The last don]

Safi sana umepatia jibu mkuu ila kwenye njia kidogo sijakuelewa hasa hapo uliposema 11× = 400 hapa ukitafuta thamani ya x haiji 36 sijui ulipataje mkuu.

Niliminus 4 from 400 hundred kwa kichwa then nikasahau kuidocument as part of solution Boss,
edited.

 

[Mtoboa siri]

Niliminus 4 from 400 hundred kwa kichwa then nikasahau kuidocument as part of solution Boss,
edited.

Hapo sasa umesomeka maana nilikuwa gizani kuona jibu lipo sawa ila njia ni ya vumbi. Pamoja sana mkuu.

 

[Okoth p'Bitek]

Hesabu ni rahisi
Just imagine intergration ni unachukua huku kule na hapa ukifanya summation inakuwa imeisha

Opposite yake ni differentiation
Physics ndio kisanga mazee

 

[Teknocrat]

Tunaenedelea.....
ALGEBRA

Factorising​

Factorising - Expanding Brackets
This section shows you how to factorise and includes examples, sample questions and videos.
Brackets should be expanded in the following ways:
For an expression of the form a(b + c), the expanded version is ab + ac, i.e., multiply the term outside the bracket by everything inside the bracket (e.g. 2x(x + 3) = 2x² + 6x [remember x × x is x²]).
For an expression of the form (a + b)(c + d), the expanded version is ac + ad + bc + bd, in other words everything in the first bracket should be multiplied by everything in the second.
Example
Expand (2x + 3)(x - 1):
(2x + 3)(x - 1)
= 2x² - 2x + 3x - 3
= 2x² + x - 3
Factorising
Factorising is the reverse of expanding brackets, so it is, for example, putting 2x² + x - 3 into the form (2x + 3)(x - 1). This is an important way of solving quadratic equations.
The first step of factorising an expression is to 'take out' any common factors which the terms have. So if you were asked to factorise x² + x, since x goes into both terms, you would write x(x + 1) .

Factorising Quadratics
There is no simple method of factorising a quadratic expression, but with a little practise it becomes easier. One systematic method, however, is as follows:

Example

Factorise 12y² - 20y + 3
= 12y² - 18y - 2y + 3 [here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].
The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y.
6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y - 3) -1(2y - 3)
The answer is (2y - 3)(6y - 1)

Example
Factorise x² + 2x - 8
We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and -2.
x² + 4x - 2x - 8
x(x + 4) - 2x - 8
x(x + 4)- 2(x + 4)
(x + 4)(x - 2)

Once you work out what is going on, this method makes factorising any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorising is by trial and error.

The Difference of Two Squares

If you are asked to factorise an expression which is one square number minus another, you can factorise it immediately. This is because a² - b² = (a + b)(a - b) .

Example
Factorise 25 - x²
= (5 + x)(5 - x) [imagine that a = 5 and b = x]

Quadratic Equations​

Quadratic equations can be solved by factorising, completing the square and using a formula. In this section you will learn how to:

• solve quadratic equations by factorising
• solve quadratic equations by completing the square
• solve quadratic equations by using the formula
• solve simultaneous equations when one of them is quadratic

This animated video states that a quadratic is an expression featuring an unknown number which has been squared. Examples are used to show how to simplify quadratics by factorisation. Working with both positive and negative terms is shown. Plotting a quadratic equation on a graph is used to show why quadratics can have more than one value.

Solving quadratic equations by factorising
Unless a graphical method is asked for, quadratic equations on the non-calculator paper will probably involve factorising or completion of the square. Quadratic equations can have two different solutions or roots.
You may need a quick look at 'factorising' again to remind yourself how to factorise expressions such as:
x2 − x − 6
which factorises into (x − 3)(x + 2),
a2 − 3a
which factorises into a(a − 3)
and
b2 − 2b + 1
which will factorise into (b − 1)2.

Solving quadratic equations by completing the square

NOTE: Check by substituting both roots back into the original equation.

This following is a common way to lead into asking you to use completion of the square.

NOTE: Remember in, for example, (x + n)2 the number of xs (called the coefficient of x) is 2n. So the coefficient of x will be 6 in (x + 3)2.


Solving quadratic equations by using the formula

When using the quadratic formula, don’t forget the ‘2a’denominator. Also, be careful when dealing with negative numbers
inside the square root. State your values of a,b and c to be used in the formula.

 

[Teknocrat]

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Narudi.......