Core 2 Flashcards

0
Q

Remainder Theorem

A

When a polynomial f(x) is divided by (ax - b) the remainder is given by f(b/a)

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1
Q

Factor Theorem

A

If (x-a) is a factor of f(x) then:
> f(a) = 0
> x=a is a root/solution to the equation f(x) = 0

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2
Q

Sine Rule

A

a/sinA = b/sinB = c/sinC
OR
sinA/a = sinB/b = sinC/c

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3
Q

Cosine Rule

A

a² = b² + c² - 2bc cosA
OR
cosA = (b²+c²-a²) / (2bc)

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4
Q

Area of a Triangle

A

1/2ab*sinC

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5
Q

The Ambiguous Case

A

When the angle you are finding is bigger than the angle you are given there are two possible results as two triangles can be drawn with the given information ( two sides, one angle)

In general:
sin(180-x) = sinx

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6
Q

Logs

A
x = loga n
x equals log base a of n
Means
a^x = n
a to the power x equals n
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7
Q

Logs

Multiplication Law

A

loga x + loga y = loga xy

Log base a of x plus log base a of y equals log base a of xy

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8
Q

Logs

The Division Law

A

loga x - loga y = loga (x/y)

log a of x minus log a of y equals log a of x/y

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9
Q

Logs

Changing Bases

A

loga x = (logb x) / (logb a)

Log base a of x equals log b base of x divided by log base b of a

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10
Q

Logs
Changing Bases - Special Case
loga b

A

loga b = (logb b) / (logb a) = 1 / (logb b)

log base a of b equals 1 divisions sed by log base b of b

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11
Q

Midpoint of a Line

A

Line (x1,y1) -> (x2,y2)

Midpoint = (x1+x2 / 2, y1+y2 / 2)

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12
Q

Distance Between Two Points On A Line

A

√[(x1-x2)² + (y1-y2)²]

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13
Q

Equation of a Circle

A

(x-a)² + (y-b)² = r²

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14
Q

Equation of a Circle

Centre

A

(a,b)

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15
Q

Equation of a Circle

Radius

A

r

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16
Q

360° In Radians

A

2π radians

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17
Q

180° In Radians

A

π radians

18
Q

Converting from Radians to Degrees

A

multiply by 180/π

19
Q

Converting from Degrees to Radians

A

Divide by 180/π

20
Q

Length of an Arc When Angle is in Radians

A

l = rθ

l = length of arc
r = radius of circle
θ = angle in radians
21
Q

Geometric Series

General Term

A

a r^n-1

a= first term 
r = common ratio
n = term number
22
Q

Sum of a Geometric Series

A

Sn = a(1-r^n) / r-1

Sn = sum
a = first term
r = common ratio
n = number of terms
23
Q

Geometric Series

Definition

A

To get from one number to the next, multiply by the same number each time, the common ratio

24
Sum to Infinity of a Geometric Sequence
If -1< r >1 the series is convergent and the sum to infinity exists This means that r^n tends to 0 as n tends towards infinity In this case: SumToInfinity = a / 1-r Otherwise the series is divergent and the sum to infinity does not exist
25
Circle Triangle | sinθ
sinθ = y/r
26
Circle Triangle | cosθ
cosθ = x/r
27
Circle Triangle | tanθ
tanθ = y/x
28
Types of Stationary Point
Maximum Point Minimum Point Point of Inflection
29
d²y/dx² | Minimum Point
d²y/dx² > 0
30
d²y/dx² | Maximum Point
d²y/dx² < 0
31
d²y/dx² | Point of Inflection
d²y/dx² = 0
32
Stationary Points | Gradient
f'(x) = 0
33
Trigonometrical Identities
tanθ = sinθ / cosθ sin²θ + cos²θ = 1
34
sin 30
1 / 2
35
cos 30
√3 / 2
36
tan 30
√3 / 3
37
sin 60
√3 / 2
38
cos 60
1 / 2
39
tan 60
√3
40
sin 45
√2 / 2
41
cos 45
√2 / 2
42
tan 45
1
43
Trapezium Rule
Area Under Line | ≈ 1/2 h [ y0 + 2( y1 + y2 + ... + yn-1) + yn]