Core

Question 1

Method of differences

Answer 1

If Uₙ = f(n) - f(n+1)

then sum to n from r=1 = f(1) + f(n+1)

Question 2

Sums of cubes for quadratic and cubic eq

Answer 2

a^3+b^3 = (a+b)^3-3ab(a+b)

a^3+b^3+y^3
= Σa^3 - 3(Σa)(Σab)+Σaby

Question 3

Conditions for a transformation to be linear

Answer 3

• Maps origin onto itself
• Can be represented by a matrix
• |ax+by|
  |cx+dy|

Question 4

Properties of linear transformations

Answer 4

T|kx| kT|x|
|ky| = |y|

T((x+x0)(y+y0) = T(x,y) + T(x0,y0)

Question 5

Properties of stretches (linear transformations)

Answer 5

|a 0|
|0 b| stretch sf a parallel to x axis and b to y axis
stretch parallel to x axis - points on y axis (x=0) invariant n vice versa

reflections self inverse P^2 = I

detM = area scale factor - negative if shape is reflected, 1 if shape is rotated

Question 6

Reflections about x,ynz axes

Answer 6

x |1 0 0| y |c 0 s|
|0 c -s| |0 1 0|
|0 c s| |-s 0 c|

z |c -s 0|
|s c 0 |
|0 0 1| s=sin etc angle is measured in anticlockwise direction from positive axis

Question 7

Order of applying multiple transformations

Answer 7

AB (x) = B then A

Question 8

f(n) = 3^4n + 2^(4n+2)

show f(k+1)-f(k) div by 15

then prove f(n) is div by 5

Answer 8

Write out subtraction, factor 3 n 2 terms, note that 3 coefficient (80) is 5x16, now there’s a factor of 3^k x 5, which will always be a multiple of 15 :)

Assume that f(k) is divisible by 5.
It would follow that f(k)=5m = where
m is an integer

f(k+1) = f(k) +5(16)(3^4k) + 60(2^4k)
take out factor of 1/3
15(16)(3^4k-1) + 60(2^4k)

also non negative integers means prove for 0 azwel

Question 9

1/i

Answer 9

-i (i/i^2)

Question 10

argand straight line

Answer 10

Re(z) = k

Question 11

meth odf differences simplifications for sum of f(r)-f(r+1) n f(r+2)

Answer 11

sum (fr - fr+1 = f(1)-f(n+1)
fr - fr+2 = f1 + f2 - f(n+1) - f(n+2)

Question 11

What makes a function improper
+ convergent/divergent

Answer 11

If one of the limits is infinite, or if the function is undefined at one of the limits, or in between the interval [a,b]

Convergent - if improper integral exists
Divergent - if integral doesn’t exist

Dont write integral from -inf to inf as integral from -t to t, instead from -t to 0 then 0 to t
both must converge to say function converges

Question 12

When to STOP using D,I meth

Answer 12

When there’s a 0 in the d row, or where you can integrate product of a row

Question 13

Mean value transformations

Answer 13

f(x)+k — f+k
kf(x) —– kf
-f(x) —— -f*

mean of f(x) + g(x) = sum of means on interval

where f* is the mean value

Question 14

Geometric reasoning to explain why mean value of sinx^5 is 0 on interval [0,2pi]

Answer 14

Integral on interval [0,π] is negative of integral on interval [π,2π], so summing them gives 0

Question 15

Definition of mean value of a function

Answer 15

Average value of a function over a given domain

Question 16

Don’t forget to square in volumes of rev

Answer 16

Yes king

Question 17

Polar integration tip

Answer 17

Integrating in 3rd n 4th quadrants doesn’t give negatives
Find values that give beginning and end of loop by solving r=0 e.g. r=asin4θ (0 n π/4)

Question 18

general solution to an equation with auxiliary solution sqrtk(k^2-n^2) where n>k and both are positive

Answer 18

Acos(sqrtn^2-k^2)…
as roots = isqrt(n^2-k^2) since n>k inside will be neg

Question 19

Invariant point Vs invariant line Vs line of invariant points

Answer 19

Point - point remains in same spot after matrix transformation (0,0) a solution for all linear transformations

Line - points that start on the line remain on the line (not necessarily remaining fixed)

Line of points - points that remain fixed, on a line.

Question 20

Centre of a polygon definition

Answer 20

The centre of a circle passing through all of the polygon’s vertices

Question 21

Conjugate of a complex number don’t get caught on lack

Answer 21

a-bi - IMAGINARY PART NEGATED

Question 22

Finding eq of tangent polar edition

Answer 22

E.g parallel to og line, calculate theta then sub in r=y/sintheta to find eq in terms of y, then after that sub back into

Question 23

Conditions on certain integrals

Answer 23

For arcsin(x/a) mod a needs to be less than x (cant arcsin smth more than one)

For arctan a needs to be greater than 0 and above

Question 24

Integrating smth to minus one

Answer 24

For LN, MAKE SURE U CHECK IF X COEFFICIENT IS NEG - THEN DIVIDE BY -1 FOR REVERSE CHEN RUL MAYOWA PLEASE

Question 25

Damped oscillations

Answer 25

discriminant>0 two roots heavy damping. No oscillations as resistive force large compared to restoring force.
Discriminant = 0 critical damping no oscillations
Discriminant<0 light damping oscillations decrease in magnitude exponentially

Heavy and critical motion determined from initial conditions. Light from period of oscillations

Question 26

Particular integral

Answer 26

Equation which satisfies differential eqs

constant a
linear ax+b
quad ax^2…
pe^kx = ae^kx
pcoswx+qsinwx = acoswx + bsinwx

Question 27

Ways of shifting summations

Answer 27

Sum from 0 to n of f(k) = Sum from 1 to n+1 of f(k-1)

Question 28

When one of the auxilliary roots is zero

Answer 28

Multiply PI by x

Same as cf and both contain a constant term

Question 29

Matrix properties

Answer 29

Non commutative - AB not BA
If AB exists doesn’t mean BA exists
Associative - (BC)A = B(CA)
Distributive A(B+C)=AB+AC

Question 30

Orientation reversed after linear t if

Answer 30

Determinant less than one

Question 31

Period of trig function

Answer 31

2pi/coefficient

for tan piover

Question 32

Damped vs forced harmonic motion

Answer 32

Damped - homogeneous, forced is non homogeneous (=f(t))

Question 33

Arcosh

Answer 33

positive root, valid for x>/1

x+sqrt(x^2-1), multiply by conjugate for diff of squares, rearrange for one then ln both sides - lnu = -ln(1/u)

each solution represents each x value that can produce y value

Question 34

Convex vs dimple polar curves

Answer 34

If a curve isn’t convex there will be more than two tangents perpendicular to the initial line

Question 35

Hyp functions

Answer 35

Sinh looks like a cubic, cosh a quadratic, tanh like arctan

Sinh odd, cosh even, tanh odd

Question 36

Parametric integration

Answer 36

y(t)dx/dt dt

for vs of rev, piy^2dx/dt dt