Download Mathematics (Extended) Flashcards (1)

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Natural numbers:
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Used for counting purposes
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Made up off all possible rational and irrational numbers
Integer:
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Prime numbers:
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Divisible only by itself and one
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1 is not a prime number
Rational numbers:
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A whole number
Can be written as a fraction
Irrational numbers:
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Cannot be written as a fraction e.g. 𝜋
Page 2
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Highest Common Factor and Lowest Common Multiple:
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HCF = product of common factors of x and y
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LCM = product of all items in Venn diagram
Prime Factorization: finding which prime numbers
multiply together to make the original number
Page 3
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Limits of accuracy:
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The degree of rounding of a number
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E.g. 2.1 to 1 d.p.
2.05 ≤ 𝑥 < 2.15
Standard form:
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104 = 10000
10−1 = 0.1
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103 = 1000
10−2 = 0.01
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102 = 100
10−3 = 0.001
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101 = 10
10−4 = 0.0001
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100 = 1
10−5 = 0.00001
Page 4
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Ratio
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Used to describe a fraction
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E.g. 3 : 1
Foreign exchange
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Money changed from one currency to another using proportion
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E.g.
Convert $22.50 to Dinars
$1 : 0.30KD
$22.50 : 6.75KD
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Map scales
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Using proportion to work out map scales
1km = 1000m
1m = 100cm
1cm = 10mm
Page 5
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Direct variation:
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𝑦 is proportional to 𝑥
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𝑦∝𝑥
𝑦 = 𝑘𝑥
Inverse variation:
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𝑦 is inversely proportional to 𝑥
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𝑦∝
1
𝑥
𝑦=
𝑘
𝑥
Page 6
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Percentage:
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Convenient way of expressing fractions
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Percent means per 100
Percentage increase or decrease:
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𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 =
𝐴𝑐𝑡𝑢𝑎𝑙 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐴𝑚𝑜𝑢𝑛𝑡
Simple interest:
𝑃𝑅𝑇
100
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𝐼=
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𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝑅 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑇 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑇𝑖𝑚𝑒
𝑅 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑛 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑇𝑖𝑚𝑒
Compound interest:
𝑅 𝑛
100
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𝐴 = 𝑃 1+
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𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
Page 7
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑖𝑚𝑒
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𝑆𝑝𝑒𝑒𝑑 =
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𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 =
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Units of speed:
km/hr
m/s
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Units of distance:
km
m
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Units of time:
hr
sec
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𝑘𝑚/ℎ𝑟 ×
5
18
= 𝑚/𝑠𝑒𝑐
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𝑚/𝑠𝑒𝑐 ×
18
5
= 𝑘𝑚/ℎ𝑟
𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒
Page 8
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Function notation:
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𝑓: 𝑥 → 2𝑥 − 1
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Function 𝑓 such that 𝑥 maps onto 2𝑥 − 1
Composite function
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Given two functions 𝑓 𝑥 and 𝑔 𝑥 , the composite function of 𝑓 and 𝑔 is the function
which maps 𝑥 onto 𝑓 𝑔 𝑥
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𝑓 2
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Substitute 𝑥 = 2 and solve for 𝑓 𝑥
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𝑓𝑔(𝑥)
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𝑓 −1 (𝑥)
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Substitute 𝑥 = 𝑔 𝑥
Let 𝑦 = 𝑓(𝑥) and make 𝑥 the subject
Page 9
Page 10
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General equation
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𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Solve quadratics by
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Trinomial factorization
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Quadratic formula
• 𝑥=
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−𝑏± 𝑏 2 −4𝑎𝑐
2𝑎
IMPORTANT!
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When question says “give your answer to two decimal places”, USE FORMULA!
Page 11
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Gradient of a Straight Line:
𝑦 −𝑦
• 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 2 1
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Equation of Line:
𝑥2 −𝑥1
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𝑦 = 𝑚𝑥 + 𝑐
• Find the gradient, 𝑚
• Find the 𝑦-intercept, 𝑐
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Midpoint of Graph:
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𝑥1 +𝑥2 𝑦1 +𝑦2
,
2
2
Length between two points:
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𝑥1 − 𝑥2
2
+ 𝑦1 − 𝑦2
2
Page 12
𝑓 𝑥 =1
𝑓 𝑥 =𝑥
𝑓 𝑥 = 𝑥3
𝑓 𝑥 =
1
𝑥
𝑓 𝑥 = 𝑥2
𝑓 𝑥 =
1
𝑥2
Page 13
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From O to A : Uniform speed
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From B to C : Uniform speed (return journey)
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From A to B : Stationery (speed = 0)
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Gradient = speed
Page 14
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From O to A : Uniform speed
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From A to B : Constant speed (acceleration = 0)
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From B to C : Uniform deceleration / retardation
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Area under a graph = distance travelled.
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Gradient = acceleration.
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If the acceleration is negative, it is called deceleration or retardation. (The moving body is
slowing down.)
Page 15
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Sum of angles at a point =360
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Angles on a straight line = 180
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Sum of angles in a triangle =180
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For regular polygon
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360
𝑛
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External angles =
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Internal angles = 180 −
Corresponding angles
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Alternate angles
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Co-interior angles
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Exterior angle=sum of interior opposite ∠
360
𝑛
For irregular polygon:
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Sum of exterior angles =360
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Sum of interior angles =180(n-2)
Vertically opposite angles
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Page 16
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A line of symmetry divides a two-dimensional shape into two congruent (identical) shapes.
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A plane of symmetry divides a three-dimensional shape into two congruent solid shapes.
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The number of times shape fits its outline during a complete revolution is called the order of
rotational symmetry.
Shape
Number of Lines of Symmetry
Order of Rotational Symmetry
Square
4
4
Rectangle
2
2
Parallelogram
0
2
Rhombus
2
2
Trapezium
0
1
Kite
1
1
Equilateral triangle
3
3
Regular hexagon
6
6
Page 17
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Rectangle: Opposite sides parallel and equal,
all angles 90°, diagonals bisect each other.
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Parallelogram : Opposite sides parallel and
equal, opposite angles equal, diagonals
bisect each other
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Rhombus: A parallelogram with all sides
equal, opposite angles equal, diagonals
bisect each other
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Trapezium: One pair of sides parallel
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Kite: Two pairs of adjacent sides equal,
diagonals meet at right angles bisecting one
of them
Page 18
= universal set
∉ = does not belongs to
∪ (union) = all the elements
⊆ = Subset
∩ (intersection) = common elements
𝐴′ = compliment of A
Ø or { } = empty set
n(A) = the number of elements in A.
∈ = belongs to
𝐴 ∩ 𝐵 is shaded
𝐴 ∪ 𝐵 is shaded
⊂ ‘is a subset of’
𝑏∈𝑋
= {a, b, c, d, e}
A’ is shaded
Page 19
Angle at centre = twice
Angle subtended by same arc
Angles in semicircle
angle on circumference
at circumference are equal
are 90°
Opposite angles in a cyclic
quadrilateral = 180°
Tangents from one point are equal
Alternate segment
∠ between tangent and radius is 90°
theorem
Page 20
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Right angled triangles:
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sin 𝑥 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
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cos 𝑥 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
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tan 𝑥 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
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For any other triangle:
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Sine rule:
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𝑎
sin 𝑎
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=
𝑏
sin 𝑏
=
𝑐
sin 𝑐
Cosine rule
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To find the angle given 3 sides
• cos 𝑎 =
One pair of information needed
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𝑏 2 +𝑐 2 −𝑎2
2𝑏𝑐
To find side given angle and two sides
• 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝑎
Page 21
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Pythagoras theorem
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To find hypotenuse
• 𝑎2 + 𝑏 2 = 𝑐 2
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To find one of the shorter sides
• 𝑎2 = 𝑐 2 − 𝑏 2
• 𝑏 2 = 𝑐 2 − 𝑎2
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Angle of elevation:
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Angle of depression:
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Angle above the horizontal line.
Angle below the horizontal line.
1
2
Area of a triangle: 𝑎𝑏 sin 𝑐
Page 22
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The bearing of a point B from another point A is:
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An angle measured from the north at A.
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In a clockwise direction.
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Written as three-figure number (i.e. from 000 ° to 360°)
Eg: The bearing of B from A is 050°
Page 23
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Area:
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Parallelogram = 𝑏 × ℎ
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1
Triangle= 𝑏 × ℎ
2
1
Trapezium= 𝑎 +
2
2
Circle= 𝜋𝑟
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Sector= 𝜋𝑟 2 ×
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OR
𝑎𝑏 sin 𝜃
𝑏 ℎ
𝜃
360
Volume and surface area:
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Cylinder
Sphere
• 𝐶𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 2𝜋𝑟ℎ
• 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 4𝜋𝑟 2
• 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ
• 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3
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Cone
• 𝐶𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 𝜋𝑟𝑙
• 𝑉𝑜𝑙𝑢𝑚𝑒 =
4
3
1
(𝜋𝑟 2 ℎ)
3
Hemisphere
• 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 2𝜋𝑟 2
2
3
• 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3
Page 24
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Volume:
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Capacity and Mass:
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Connecting volume and capacity:
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1𝑚𝑙 = 1𝑐𝑚3
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1𝑘𝑙 = 1𝑚3
Density =
𝑀𝑎𝑠𝑠
𝑉𝑜𝑙𝑢𝑚𝑒
Page 25
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𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
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𝑎 𝑚
𝑏
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𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛
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𝑛
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𝑎𝑚
𝑛
= 𝑎𝑚𝑛
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𝑎0 = 1
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𝑎−𝑛 =
1
𝑎𝑛
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𝑎×𝑏
𝑚
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𝑚
𝑎
𝑎𝑚
𝑏𝑚
=
𝑚
= 𝑎𝑛
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𝑎× 𝑏= 𝑎×𝑏
•
𝑎
𝑏
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=
𝑎
2
𝑎
𝑏
=𝑎
= 𝑎𝑚 × 𝑏 𝑚
Exponential equations:
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Equations involving unknown indices
Page 26
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Condition 1: Given distance from a point
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Condition 2: Given distance from a straight
line
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Condition 3: Equi-distant from two given
points
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Condition 4: Equi-distant from two
intersecting lines
Page 27
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A vector quantity has both magnitude and direction.
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E.g. Vectors a and b represented by the line segments can be added using the
parallelogram rule or the nose-to-tail method.
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Multiplication by a scalar:
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A scalar quantity has a magnitude but no direction
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The negative sign reverses the direction of the vector
Column vector:
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Top number is the horizontal component and bottom number is the vertical component
Parallel vectors:
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Vectors are parallel if they have the same direction
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In general the vector 𝑘
𝑎
𝑏
is parallel to
𝑎
𝑏
Modulus of a vector:
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In general, if 𝑥 =
𝑚
𝑛
, 𝑥 =
(𝑚2 + 𝑛2
Page 28
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Addition:
𝑝 𝑞
𝑎+𝑝 𝑏+𝑞
𝑎 𝑏
•
+
=
𝑟 𝑠
𝑐 𝑑
𝑐+𝑟 𝑑+𝑠
Multiplication by scalar
𝑎 𝑏
𝑘𝑎 𝑘𝑏
• 𝑘
=
𝑐 𝑑
𝑘𝑐 𝑘𝑑
Multiplication by vector:
𝑎𝑝 + 𝑏𝑟 𝑎𝑞 + 𝑏𝑠
𝑝 𝑞
𝑎 𝑏
•
×
=
𝑟 𝑠
𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠
𝑐 𝑑
• You can only multiply if no. of columns in left equals to no. of rows in right
Determinant:
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Determinant = leading diagonal – secondary diagonal
𝑎 𝑏
• A =
𝐴 = 𝑎𝑑 − (𝑏𝑐)
𝑐 𝑑
Inverse:
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•
1
To work out inverse, switch leading diagonal, negate secondary diagonal, multiply by
𝑎
1
𝑎 𝑏
𝑑 −𝑏
A =
𝐴−1 =
(𝑎𝑑−𝑏𝑐) −𝑐
𝑐 𝑑
𝑎
Page 29
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Reflection:
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When describing a reflection, the position of the mirror line is essential.
Rotation:
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To describe a rotation, the centre of rotation, the angle of rotation and the direction of
rotation are required.
•
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Translation:
•
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A clockwise rotation is negative and an anticlockwise rotation is positive.
When describing a translation it is necessary to give the translation vector
Enlargement:
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To describe an enlargement, state the scale factor, K and the centre of enlargement
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𝑆𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 =
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If K > 0, both object and image lie on same side of the centre of enlargement.
•
If K < 0, object and image lie on opposite side of the centre of enlargement.
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝐾 2 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
Page 30
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Shear:
•
To describe a shear, state; the shear factor, the invariant line and the direction of the
shear
•
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑚𝑜𝑣𝑒𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑠ℎ𝑒𝑎𝑟
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑓𝑖𝑥𝑒𝑑 𝑙𝑖𝑛𝑒
•
𝑆ℎ𝑒𝑎𝑟 𝑓𝑎𝑐𝑡𝑜𝑟 =
•
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
=
𝑎
𝑏
Stretch:
•
To describe a stretch, state; the stretch factor, the invariant line and the direction of the
stretch
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 ′ 𝑓𝑟𝑜𝑚 𝐴𝐵
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 𝑓𝑟𝑜𝑚 𝐴𝐵
•
𝑆𝑡𝑟𝑒𝑡𝑐ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 =
•
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝑆𝑡𝑟𝑒𝑡𝑐ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
Page 31
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•
Reflection:
1 0
•
0 −1
•
−1
0
•
0 1
1 0
•
0
−1
0
1
Reflection in the 𝑥 − 𝑎𝑥𝑖𝑠
Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠
Reflection in the line 𝑦 = 𝑥
−1
0
Enlargement:
𝑘 0
•
0 𝑘
Reflection in the line 𝑦 = −𝑥
where k=scale factor and centre of enlargement = (0,0)
Page 32
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•
Rotation:
0 −1
•
1 0
•
0
−1
1
0
•
−1 0
0 −1
Stretch:
1 0
•
0 𝑘
•
𝑘
0
0
1
Rotation 90° anticlockwise, centre (0,0)
Rotation 90° clockwise, centre (0,0)
Rotation 180° clockwise/anticlockwise, centre (0,0)
Stretch factor k, invariant line x-axis & parallel to y-axis
Stretch factor k, invariant line y-axis & parallel to y-axis
Page 33
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Shear:
1
•
0
𝑘
1
Shear factor k, invariant line x-axis & parallel to x-axis
1
𝑘
0
1
Shear factor k, invariant line y-axis & parallel to y-axis
•
Page 34
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Histograms:
•
A histogram displays the frequency of either continuous or grouped discrete data in the
form of bars.
•
The bars are joined together.
•
The bars can be of varying width.
•
The frequency of the data is represented by the area of the bar and not the height.
•
When class intervals are different it is the area of the bar which represents the
frequency not the
•
height
•
Instead of frequency being plotted on the vertical axis, frequency density is plotted.
•
Class width = Interval
•
Frequency density = Height
•
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝐶𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ × 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦
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•
Mean:
•
•
•
Median:
•
The middle value when the data has been written in ascending or descending order
•
Odd no. of values
•
Even no. of
(add two values divide by 2)
Most frequently occurring value
Range:
•
•
5+1
= 3𝑟𝑑 𝑣𝑎𝑙𝑢𝑒
2
6+1
values
= 3.5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
Mode:
•
•
𝑆𝑢𝑚 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
Difference between highest and lowest values
Estimated mean of grouped data:
•
Work out midpoints of each group and multiply by frequency
•
Divide by number of values
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•
Cumulative frequency is the total frequency up to a given point.
•
Inter-quartile range = 𝑢𝑝𝑝𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 − 𝑙𝑜𝑤𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒
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•
Probability is the study of chance, or the likelihood of an event happening.
•
Probability of an event =
•
If probability = 0, the event is impossible and if probability =1, the event is certain to happen
•
All probabilities lie between 0 and 1.
•
Exclusive events:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
•
Two events are exclusive if they cannot occur at the same time.
•
The OR Rule:
• For exclusive events A and B
• p(A or B) = p(A) + p(B)
•
Independent events:
•
Two events are independent if occurrence of one is unaffected by occurrence of other.
•
The AND Rule:
•
p(A and B) = p(A) × p(B)
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•
Linear sequences:
•
•
Find common difference e.g. 3 then multiply by 𝑛 and work out what needs to be added
Quadratic sequences:
•
Format: 𝑎𝑛2 + 𝑏𝑛 + 𝑐
𝑎+𝑏+𝑐 =
3𝑎 + 𝑏 =
2𝑎 =
•
•
Work out the values and then place into formula to work out nth term formula
Geometric progression:
•
Sequence where term has been multiplied by a constant to form next term
•
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑜𝑓 𝐺. 𝑃. = 𝑎𝑟 (𝑛−1)
• a = 1st term
r = common difference
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