Pure Year 1 Flashcards
How do you get from e to ln(x)
e^ln(x)=x
a^x x a^y
a^(x+y)
(a^x)^y
a^xy
(a^x)/(a^y)
a^(x-y)
(ab)^x
a^x x b^x
a^-x
1/a^x
(a/b)^-x
(b/a)^x = b^x/a^x
a^(p/q)
(qroota)^p
a^0
1
root ab =
roota x rootb
root a/b =
roota/rootb
Coordinate geometry length
root [(y2-y1)²+(x2-x1)²]
Coordinate geometry gradient
(y2-y1)/(x2-x1)
Coordinate geometry midpoint
(x1+x2)/2 , (y1+y2)/2
Perpendicular
m1xm2=-1
Perpendicular bisector
A line that passes through the midpoint of two points and is 90° to the original line
Linear modelling
Y=ax+b
Why may a linear model not be suitable
May not account for other factors (named)
Why may a linear model be suitable if drawn on a graph
The data/points lie close to a straight line
Interpret the meaning of a and b in y=ax+b
a= the increase in y as x increases by
The value of (named y) increases/decreases per (x)
Completed square form
y = a(x-b)^2 + c
How to solve a quadratic
Calculator
Completing the square
Quadratic formula
Graph of 0<a></a>
Exponential decay
Passes through y=1
Asymptote y=0
Vertical flip of exponential growth
Graph of a>1, a^x
Exponential growth
Passes through y=1
Asymptote y=0
Never negative y
9^x in terms of 3^x
3^2x
Caution with logs
Can’t be negative inside
List for day
Read Q properly Signs Mental maths vs calculator Reread question Signs in logs Will a graph help Something common for something simple (x/y) Dividing by negative logs with inequalities
Natural logarithm of x
ln(x)
Interpret the meaning of a and b in P=ab^x
The population at the start when x is 0
The proportional increase in … per year after …
Dealing with multiple translations
Y:
R
S
T
X:
T
S
R
N/Z+
Natural numbers
Positive integers
Z
Integers
Q
Rational numbers
Numbers that can be written as p/q where p and q are integers
R
Real numbers
All numbers on a number line
Proof by deduction
Direct proof
Complete the square, odd vs even etc for everything
Proof by exhaustion
Try it for all values necessary
Important for proof with trig
Trig graphs repeat, can prove in small range
Proof by counter example
Give 1 example only
Sub numbers in to disprove
Always, sometimes and never truth
If you think it’s always true or never true, prove it
If you think it’s sometimes, sub numbers in and get one for true and one for not true
i vs j
i is x
j is y
Should underline i and j
|v| for vectors
Square root of x^2 + y^2
Unit vector
(1/|v|) x v
How do you write a position vector or vector
Either with i, j and/or k or as a column vector
Vector AB
OB-OA
Trapezium in vectors
When AB and DC are parallel/scalar multiples
How to find the angle vector xi+yj+zk makes with the positive x axis for angle x, y and z
Cos(x)=x/(|v|)
Cos(y)=y/(|v|)
Cos(z)=z/(|z|)
Approximation for a number from binomial
Equate the two, cancel powers and get x
X should be small
Find h in trapezium rule
h=(b-a)/n
b=max x
a=min x
n=number of strips
Basically h is the width of the rectangles
Binomial expansion for positive integer powers
Use long NCR method
Binomial expansion for negative or fraction
Complicated formula
Start with ~ or end with …
Range of values for valid binomial notation
lxl<1
or
-1
How do you find the range of valid substitutions in binomial expansions
Take what is inside the bracket and solve for x
Modulus of value
How do you remove a negative in integrating in terms of the limits
Removing negative means the limits flip
How do you show a polynomial has no further linear factors
Factorise and solve equal to zero
Use b²-4ac and if its less than 0 for a quadratic then it has no roots and no other linear factors
