Unit 2

Question 1

What is i equal to

Answer 1

the root of -1

Question 2

What is the form of complex numbers

Answer 2

a+bi

Question 3

How to add or subtract complex numbers

Answer 3

To add or subtract complex numbers gather the real parts together and the imaginary parts together

Question 4

How to multiply complex numbers

Answer 4

To multiply complex numbers multiply out brackets

Question 5

How to divide complex numbers

Answer 5

To divide times the division by the complex conjugate of the denominator

Question 6

How to find the square roots of complex numbers

Answer 6

let a + bi = the root square a+bi to remove the root on the other side multiply the left side out correlate the left to right in terms of complex and nornaml numers Then solve to find both a and b the answer is thus the a and b values in the form a + bi

Question 7

How to find the roots of a cubic or quartic when one complex root is given

Answer 7

If the root given is non-real then its conjugate is also a root of the polynomial we now know too roots and thus the factors are (z-(the first )) and (z-(the second)) Multiply them together Divide the original cubic by the product using long division the result of such can then be manipulated to find the final root meaning what is z equal too for the quartic the roots for the result will have to be find via a different method as there will be two

Question 8

What is the quadratic formula

Answer 8

x= -b+- the root of b^2 - 4ac divided by 2a

Question 9

Formula to find the nth term of an arthimetic sequence

Answer 9

un = a + (n-1)d

Question 10

Formula to find the nth term of a geometric sequence

Answer 10

un = ar^n-1

Question 11

How to find the sum to infinity of a geometric series

Answer 11

is r >1 then the sum to infinity is undefined if r<1 then the sum to infinity can be found using the equation s(infinity sign) = a divided by 1-r

Question 12

What is a

Answer 12

The first term of a series

Question 13

How to find the maclaurian expension of sin2x

Answer 13

find the expansion of sinx and substiutie the end answer x with 2x

Question 14

How to expand composite functions such as e^xcos3x

Answer 14

find the expansion of e^x find the expansion of cos3x multiply them together

Question 15

What is the expansion of e^x

Answer 15

1 + x + x^2/2! + x^3/3! + x^4/4! + …

Question 16

How to find The sum of (ar + b) when r =1

Answer 16

an/2(n+1)+ bn

Question 17

How to find the sum when r = k+1

Answer 17

find the sum when r =1 - the sum when n =k

Question 18

How to prove a function is even

Answer 18

show that f(-x) = f(x)

Question 19

How to prove a function is odd

Answer 19

show that f(-x)= -f(x)

Question 20

How to find stationary points

Answer 20

The first derivative equal to 0

Question 21

What order to sketch curve

Answer 21

1. Mark roots 2. Turning points 3. Vertical asymptotes 4. y-intercept

Question 22

How to prove that a function is even

Answer 22

show that f(-x) = f(x)

Question 23

How to prove that a function is odd

Answer 23

f(-x) = -f(x)

Question 24

What does -f(x) do

Answer 24

reflection of the graph f(x) in the x-axis

Question 25

graph of f(-x)

Answer 25

reflection of the graph of f(x) in the y-axis

Question 26

graph of f(x) +- a

Answer 26

+ a shifts the graph up a units - a shifts the graph down a units

Question 27

graph of kf(x)

Answer 27

k\>1 then the graph is strectched vertically k\<1 the graph will be compressed vertically

Question 28

f(x +- b)

Answer 28

f(x + b) will shift the graph left b units f(x - b) will shift the graph right b units

Question 29

f(cx)

Answer 29

c\>1 then the graph f(x) will be stretched horizontally c\<1 then the graph will be compressed horizontally

Question 30

What is the formula for volume of revolution around the x-axis

Answer 30

integral of b a limit pi x y^2 dx

Question 31

What is the formula for volume of revoltuion around the y-axis

Answer 31

integral of b a pi \* x^2 dy

Question 32

What must be true for matrix multiplcation to take place

Answer 32

for A x B to be possible, the number of columns of A must match the number of rows of B

Question 33

what makes a matrix orthogonal

Answer 33

if A'A= I

Question 34

what makes a matrix symmetric

Answer 34

if A' = A, the matrix will be symmetrical about the leading diagonal

Question 35

What makes a matrix skew-symmetric

Answer 35

If A' = -A there can only be zeros in the leading diagonal

Question 36

what makes matrix B the inverse of matrix A

Answer 36

if AB = BA = I thus only square matrixes can be inverse

Question 37

the determinant is denoted by det (A) or |A|, how is this calculated

Answer 37

is defined as |A| = ad - bc

Question 38

What makes a matrix non-singular and invertible

Answer 38

if |A| doesnt equal 0

Question 39

What makes a matrix a singular

Answer 39

if |A| = 0

Question 40

What is the General term of the binomail expansion

Answer 40

Question 41

How do you expand (n r)

Answer 41

Question 42

What are the four forms of auxillary equations

Answer 42

Distinct real roots: y = Ae^m1x + Be^m2x Repeated real roots: Y = (A + Bx)e^mx Imaginary roots: Y = Acosβx + Bsinβx (For imaginary roots in the form +/- βi) Complex roots: Y = e^αx(Acosβx + Bsinβx) (For imaginary roots in the form α +/- βi)