Pure Maths How-Tos
Pure Maths with some Statistics (as specified)
- Mathematics
- Statistics, averages and distributionsAlgebra and functionsGraphs and transformationsLogarithms and exponentialsProbabilitySequences and seriesVectors
- AS
- AQA
- Created by: Betsy_2018
- Created on: 12-12-16 21:07
How To Find Polynomials
(1) You are given an expression stating that a term X is part of this expression
(2) This X is therfore equal to the original expression when multiplied with p(X)
Example
x^3 + 12x^2 + 34 - 12 = (x + 6) x p(x)
- = (x+6) x (Ax^2 + Bx + C)
- = Ax^3 + Bx^2 + Cx + 6x^2 + 6Bx + 6C
- = x^3(A) + x^2(B + 6) + x(6B + C) +6C
Therefore
- A = 1
- B + 6 = 12 | B = 6
- 6B + C = C + 36 = 34 | C = -2 | (can also be applied to '-12')
P(x) = x^2 + 6x - 2
Vertex And The Line Of Symmetry
To do this, you must complete the square on an equation.
Example
{y = x^2 + 4x + 12} = {(x + 2)^2 - 2^2 + 12} = {(x + 2)^2 + 8}
Vertex
Of (x + 2)^2 + 8, the x value of the vertex is '-1x' what is inside the bracket. They y value is what is outside the bracket
The vertex is (-2, 8)
Line of symmetry
The line of symmetry is sumply what the x equals when on the graph to split the graph in two identical lines. In this case, x = -2
Intersections In Quadratic Graphs
There will be 1, 2 or 0 intersections.
(1) You will be given 2 expressions, which you will need to turn into equations
(2) You will need to find a solution to these simultaneous equations. It may give you co-ordinates: one set, two sets or no sets.
Example
{x + 2y = 3} = {x = 3 - 2y}
{x^2 + 3yx = 10}
Therefore, via substitution, (2y + 1)(y+1) = 0
- When (y= -1/2), ( x= 4)
- When (y= -1), (x= 5)
Quadratic Inequalities
If the graph's equation >0, you are looking for values above (y= 0)
If the graph's equation <0, you are looking for values below (y= 0)
You are advised to always draw a graph for these questions, as it is very easy to make mistakes.
(1) Factorise to give 2 values of x | use the quadratic formula | use discriminant (for algebraic)
(2) plot these on a graph, highlight the area of the line above or below the x axis, depending on whether the equation >0 or <0
Example
x^2 + 3x - 10 ≥ 0
(x + 5)(x - 2) ≥ 0
x= -5, 2
Equation is above the line and is a positive graph so the inequality is (x≤5), (x≥2)
Dividing a Polynomial
Some can easily be separated into their constituents and cancelled out.
Others will require more complex methods of division. 2 ways of dividing polynomials are:
- Long division
- The Box Method
Both will give you a quotient, divisor and a remainder
Example
{x^2 + 2x + 1} ÷ {x + 1)
____ x + 1
x+1 | x^2 + 2x + 1
_____- (x^2 + x)
______________0
Factor Theorem
Factor Theorem
P(x) = []x^(n) + []x^(n-1) +... []x + []
(1) substitute the divisor in (flip the symbol in the brackets) into p(x)
(2) if it equals 0, it is a factor
(3) find out the rest of the divisors (quotient) to find the rest of the factors for this equation.
Remainder Theorem
P(x) = (x - a)Q(x) + Remainder
This is the same concept as the factor theorm. (x-a) will have its symbol flipped in order for it to go into P(x). This makes the answer {P(a) = 0} and the {remainder = 0}
Laws of Indices
- Multiplying: ____________________(x^n)(x^m) = x^(n+m)
- Dividing: ______________________ x^n/x^m =_(x^n-m)
- To the Power of 0: _______________x^0 = 1
- Multiplying by another indice:______(x^n)^m = x^nm
- Indices as a fraction: x^n/m = ______(m√x)^n
- Negative indices: ________________x^-n = 1/x^n
- Base > xª
- _______ ^ index
Exponentials and Expressions as Logarithms
Exponentials
- graph
- where a > 0
- always passes thrpugh (0, 1)
- steepness increases with greater base
- never touches x axis
Expressions as Logarithms
LogaN = x
a^x = n
e.g Log101000 = 3
10^3 = 1000
Laws of Logarithms
- Loga(XY) = LogaX + LogaY
- Loga(X/Y) = LogaX - LogaY
- Loga(X)^n = N(LogaX)
- Loga(1/X) = -LogaX
- LogaA = 1
Changing the Base (Logarithms) and Taking Logs
LogaX = (LogbX) / Logba
bLogaX = (b) / Logxa
- always use base 10
- state that you are working to base ten (unless specified otherwise)
Taking Logs
take logs using the index, multiplictaion, division and negative laws
e.g 7^(x+1) = 3^(x+2)
(x+1)Log7 = (x+2)Log3
xLog7 + Log7 = xLog3 + 2Log3
x = (2Log2 - Log7) / (Log7 - Log3) [This can be put into a calculator]
Sketching Cubic Graphs
{y = ax^3 + bx^2 + cx + d} = (x+a)(x+b)(x+c)
- From here you can find all of the points where the graph crosses the x/y axis
- Can be called a 'repeated root' graph [e.g (-1, 0) (0, 1) (1,0)
- The graph is a 'squiggle' shape
- goes up for positive, and down for negative
Example
y = x^3 - 2x^2 - x + 2
_ = (x+1)(x-1)(x-2)
0 = (-1,0) (1,0) (2, 0)
= 2 [when x=0]
TOTAL CO-ORDINATES: = (-1,0) (1,0) (2, 0) (0, 2)
Plot on a graph
Translations
Move in the x axis
| a | = [when a>0 = move right] = [when a<0 = move left]
| b |
Move in the y axis
| a | = [ when b>0 = move up] = [when b<0 = move down]
| b |
Formula
y - b = f(x - a)^n
Example: y = (x-5)^2 by -4 | 0
y - 0 = (x - 5 - (-4))^2
y = (x-1)^2
Factorials
n! = n x (n-1) x (n-2)...(n-n+1)
= n factorial
- if you divide a factorial by another factorial, many terms can be cancelled out
- 0! = 1 = (0 - 0 + 1)
N choose R
- n = number of objects
- r = number of different objects that can be chosen from n
- represented by:
(n!) / r!(n - r)!
Example
| 4 | = 4! / (3!)(4-3)! = 24 / 6(1) = 4
| 3 |
Pascal's Triangle
(a + b)^n
- use the relevant row of pascal's traingle (n=x)
Example
The co-efficent of x^2 in the expansion of (2-cx)^3 is 294. Find the value of c.
[a = 2] [b = cx] [n = 3] look up n = 3
3ab^2
3(2)(cx)^2 = 6(c^2)(x^2)
6c^2 = co-efficient of x^2
6c^2 = 294
c =7
Binomial Expansion
(a + b)^2 = a^n + | n | a^(n-1)b + | n | a^(n-2)b^2... + b^n
___________________| 1 |_____________| 2 |
- in the absence of a calculator, you must do the formula on the previous card ('choosing')
Example
(2x + y)^3
[a= 2x] [b= y] [n= 3]
(2x)^3 + (3C1)(2x^2)(y) + (3C2)(2x)(y^2) + (y^3)
= 8x^3 + 6x^2(3y) + (12x)(6y^2) + (y^3)
Suffix Notation
- This is to do with terms of a sequence
- terms go: t1, t2, t3...
Example
Un = n(n + 2) - find U3
U3 = (3)(5) = 15
Example
Un-1 = (n-1)(n-1+2) - find an expression for Un - Un - 1
- imagine the expression without '-1'; it would be (n)(n+2) = Un
= n(n + 2) - (n - 1)(n-1+2)
Inductive Sequences
- Inductive sequences are ones that have terms that include the value of the previous term
Un = Un-1 ... U1 = U0...
Un+1 = Un+1-1... U2 = U1...
Example
U1 = 3 | Un = Un-1 + 2n+1
Find U2
U2 = (3) + (2(2)+1) = 8
Limit of a Sequence
U∞ = L [this is a lot like taking Logs]
Example
Un+1 = 2 - 1/3Un [U1 = 3]
Therefore, U2 = 2 - 1/3U1
____________U2 = 2 - 1/3(3) = 1
U∞ = L
L = 2 - 1/3L
4/3L = 2
L = 2/4/3 = 3/2
Un => 1.5 as n => ∞
Arithmetic Sequences/Series [nth term]
- Specific term Un = Un-1 + d [inductive]
- Nth term = a + (n-1)d
Example
Find the first number after 1000 in the sequence with [a = 6], [d = 2.5]
Un = a + (n-1)d
___= 6 + 5/2(n - 1)
5/2(n - 1) > 994
n - 1 > 397.6
n > 398.6
n = 399, U399 = 1001
Sum of an Arithmetic Series
Sum of the first n terms
- 1/2 (n)(a + L)
Sum of the first (IN) natural numbers
- 1/2(n)(n+1)
Sum of the sequence
- common difference or first term can be found
- use simultaneous equations
- (1/2)n [2a + (n-1)d]
Sigma Notation
_x
∑ n^z
n = y
[x = final value of n] [y = first value of n] [z = nth term]
Look at the sequence, find a common term for 'z'. This could be n^x, or dividing each term to make n^x, n^x+1...
Examples
2^2 + 3^2 + 4^2 + 5^2...10^2 [x = 10, y= 2, z= n^2]
100
∑n^2
n = 2
1 + 3 + 9 + 27 = 3^0 + 3^1 + 3^2 + 3^3
x = 27, y = 1, nth = n-1
Geometric Series
Nth term
- Un = ar^(n-1)
Sum
- Sn = a(1-r^n) / (1 - r)
Example
Find the least number of terms of the geometric series 16 + 20 + 25... required to give a sum greater than 25,000 [a + ar + ar^2...]
20/16 = 5/4
Sn = a(1-r^n)/1-r = (16(1 - 5/4^n) / 1 - 5/4) > 25000
Cancel out, take logs and solve to find n
n = 27
Convergent Geometric Series
- a geometric series with common ratio r, converges when | r | < 1
- conergent geometric series has a sum to infinity
- S∞ = a / (1 - r)
Example
Find the least number of terms of the geometric series 16 + 20 + 25... required to give a sum greater than 25,000
a + ar + ar^2...
20/16 =
S∞ = a/1-r = 16 / 1 -
Writing Equations in the Form ax + by + c = 0
- The equation may require cancelling down or scaling
- Rearrange the equation so that all terms are equalling 0
- a= the multiplier for x, and b= the multiplier for y
Example
y = -1/2x - 3
2y = -x - 3
0 = -x - 2y - 3
x + 2y + 3 = 0
Straight Line Geometry
- Line Length (Distance beteen 2 points): √(x2- x1) + (y2- y1)
- Midpoint: (x1 +x2)/2, (y1 + y2)/2
- Gradient: Change in y / change in x
- Gradient of [OP: a/b] is perpendicular to [OP': -a/b]
- Equation of lime passing through (x1, y1), gradient m: y - y1 = m(x - x1)
Circle Geometry
Equation of a circle with centre (0, 0), radius r
x^2 + y^2 = r2
The x and y are any set of co-ordinates that lay on the circle
Equation of a circle with centre (a, b), radius r
(x - a)^2 + (y - b)^2 = r^2
For when the circle does have the centre (0, 0) (has 'moved away')
Finding a point outside of a circle circumference
Use the (x-a)^2 + (x-b)^2 = r^2
The outer-circle point is a new 'radius', so just apply its co-ordinates to the formula
Applying Translations on Circles
Translation = moving a curve without altering its shape by vector | a |
______________________________________________________>.>.| b |
Apply | a | to x^2 + y^2 = r^2
.>.>.>.| b |
This will give you the equation (x - a)^2 + (y - b)^2 = r^2
In general, if a circle is already translated, you just apply the translation further.
(x-p)^2 + (y - q)^2 = r^2 by a translation a b
= (x - p - a)^2 + (y - q - b)^2 = r^2
r^2 does not change
Finding the Equation of a Circle Using its Propert
This may involve finding the midpoint (should it be the centre), length of the diameter etc
Example
Circle with traingle P(2, 12), Q(-6, 2) and R(12, -10)
(1) Find midpoint of diameter co-ordinates
(2) Find the distance between midpoint and circumference co-ordinate (radius)
(3) use the equation (x-a)^2 + (y-b)^2 = r^2
- midpoint = [(2+12)/2, (14+-10)/2] = (7, 2)
- distance = √(2-14)^2 + (12--10)^2 = √26
- equation = (x-7)^2 + (x-2)^2 = √26^2
Length of Tangents
- P = point of tangents meeting
- A = 1 point on the circumference
- B = 1 point on the circumference
- C = centre of the circle
B and A are interchangable
You will be given the co-ordinates
PA^2 = CP^2 - CA^2
Differentiation To Find Tangent and Normal Equatio
(1) Given the equation of a circle and the co-ordinate of contact
(2) differentiate the equation and apply the co-ordinates to it
(3) this is the gradient
(4) apply the orignal co-ordinates and the new gradient to y-y1 = m(x-x1)
The normal is the same as the tangent, except it is perpendicular, meaning its gradient is -1/m
Example
Curve y=x^2 + 1 on point (3, 10)
Differentiated = 2x [2(3) = 6]
Tangent equation = y = 6x - 8
Normal equation = y = -1/6x + 10.5
Triangles (simple)
Finding the line of symmetry of a triangle
- Find midpoint of line of equilateral traingle
- find gradient of line between midpoint and point opposite the line
- find equation using gradient and a point on that line
Showing that a triangle has a right angle
- find the gradients of 2 lines
- multiply together
- if it = -1, they are perpendicular, and so must have a right angle between them
Trigonometry
Remember: SOH CAH TOA
Sine Rule (length) = [a/SinA = b/SinB]
Sine Rule (angle) = [SinA/a = SinB/b]
Cosine Rule (length) = a^2 = b^2 + c^2 - 2bcCosA
Cosine Rule (angle) = CosA = b^2 + c^2 - a^2 / 2bc
- Sin30 = 1/2
- Sin45 = 1/(√2)
- Sin60 = (√3)/2
- Cos30 = (√3)/2
- Cos45 = 1/√2
- Cos60 = 1/2
- Tan30 = (√3)/3
- Tan45 = 1
- Tan60 = √3
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