a level further maths

conjugate of z

what does z* mean

a-bi

what is the conjugate of a+bi

natural numbers, 1,2,3

what does N stand for in the number system, give examples

Integers, -1,0,1

what does Z stand for in the number system, give examples

Rational numbers, 1/2,1/3,1/4

what does Q stand for in the number system, give examples

Irrational numbers, square root of 2, square root of 3, square root of 5

what does I stand for in the number system, give examples

Real numbers, 1,-1,1/2,square root of 2

what does R stand for in the number system, give examples

Imaginary/ complex numbers, i, 2i, 3i

what does C stand for in the number system, give examples

an array of figures providing some type of information

what is a matrix

down

is a row in a matrix across or down

across

is a column in a matrix across or down

3 rows, 2 columns

what does the matrix 3x2 mean

2x2

what is a square matrix

any number x 1

what is a column matrix

1 x any number

what is a row matrix

square root of -1

what is the value of i

-1

what is the value of i^2

-i

what is the value of i^3

1

what is the value of i^4

i

what is the value of i^5

-1

what is the value of i^6

-i

what is the value of i^7

1

what is the value of i^8

ad-bc

how to find the determinant of a matrix

the determinant equals 0, the two lines are either parallel or the same line

what is a singular matrix

rotations, reflections, enlargements, stretches

what can be represented as a transformation matrix

rotation

what does a matrix with a determinant of 1 show

reflection

what does a matrix with a determinant of -1 show

area scale factor

what does the determinant of a transformation matrix show

1, 3, 5, 7, 9

example of an arithmetic sequence

1, 2, 4, 8, 16

example of a geometric sequence

1, 4, 1, 4, 1

example of a periodic sequence (oscillating sequence)

a + (n-1)d, a is the first term, d is the common difference

how to generate an arithmetic sequence

1/2n(2a + (n-1)d)

how to generate an arithmetic series

ar^n-1, a is the first term, r is the common ratio

how to generate a geometric sequence

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

how to generate a geometric series

shorthand notation used for series

what is sigma notation

a finite sequence of step by step instructions carried out to solve a problem

what is an algorithm

1/2n(n+1)

what is sigma r

1/6n(n+1)(2n+1)

what is sigma r^2

1/4n^2(n+1)^2

what is sigma r^3

2

what is the base of 2^4

4

what is the logarithm of 2^4

log2 8

how can you write 2^3 as a logarithm

x=3.58

solve 2^x=12

x=1.76

solve 5^x=17

logAB

logA+logB

log(A/B)

logA-logB

log(AxA)= logA+logA = 2logA = log(A^n) = nlogA

log(A)^2

0

log1

log2 8 + log2(x^3) - log2 y = 3 + 3log2 x - log2 y

simplify log2(8x^3/y)

multiplying or dividing by a negative number

what causes the inequality to change

positive

the modulus function is always

ax^2 + bx + c = 0, x^2 + b/ax + c/a = 0, (x + b/2a)^2 + c/a - b^2-4a^2 = 0, (x + b/2a)^2 = b^2/4a^2 - c/a, (x + b/2a)^2 = b^2-4ac/4a^2, x + b/2a = +- square root of b^2-4ac / 2a, x = -b/2a +- square root of b^2-4ac / 2a, x = -b +- square root of b^2-4ac / 2a

Prove the quadratic formula

prove true if n=1, assume true for n=k, k+1=Sk+Uk+1, equivalent to 1/2n(n+1) with n=k+1, if true for n=k then true for n=k+1

prove by induction that n sigma r r=1 = 1/2n(n+1)

prove true if n=1, assume true for n=k, k+1=Sk+Uk+1, equivalent to 1/6n(n+1)(2n+1) with n=k+1, if true for n=k then true for n=k+1

prove by induction that n sigma r^2 r=1 = 1/6n(n+1)(2n+1)

prove true if n=1, assume true for n=k, k+1=Sk+Uk+1, equivalent to 1/4n^2(n+1)^2 with n=k+1, if true for n=k then true for n=k+1

prove by induction that n sigma r^3 r=1 = 1/4n^2(n+1)^2

prove true if n=1, assume true for n=k, M^k+1=MM^k, M^k+1, equivalent to ((top row) 1-3n 9n (bottom row) -n 3n+1) with n=k+1, if true for n=k then true for n=k+1

prove by induction that if M = ((top row) -2 9 (bottom row) -1 4) then M^n = ((top row) 1-3n 9n (bottom row) -n 3n+1)

let f(n)=n^3-7n+9, prove true if n=1, assume true for n=k, n=k+1, now f(k+1)-f(k)=3(k^2+k-2), f(k+1)=f(k)+3(k^2+k-2), which is a multiple of 3, if true for n=k then true for n=k+1

prove by induction that n^3-7n+9 is a multiple of 3

let f(n)=3^2n+11, prove true if n=1, assume true for n=k, n=k+1, now f(k+1)-f(k)=8(3^2k), f(k+1)=f(k)+8(3^2k), which is a multiple of 4, if true for n=k then true for n=k+1

prove by induction that 3^2n+11 is a multiple of 4

-b/a

ax^2+bx+c=0 has roots alpha and beta, what is alpha + beta

c/a

ax^2+bx+c=0 has roots alpha and beta, what is alpha x beta

alpha + beta/ alpha x beta = (-b/a)/(c/a)

ax^2+bx+c=0 has roots alpha and beta, what is 1/alpha + 1/beta

(alpha + beta)^2 - 2(alpha x beta) = (-b/a)^2 - 2(c/a)

ax^2+bx+c=0 has roots alpha and beta, what is alpha^2 + beta^2

(alpha x beta)^2 = (c/a)^2

ax^2+bx+c=0 has roots alpha and beta, what is alpha^2 x beta^2

-b/a

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is alpha + beta + gamma

c/a

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is (alpha x beta) + (alpha x gamma) + (beta x gamma)

-d/a

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is alpha x beta x gamma

(alpha x beta) + (alpha x gamma) + (beta x gamma)/ alpha x beta x gamma = (c/a)/(-d/a)

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is 1/alpha + 1/beta + 1/gamma

(alpha + beta + gamma)^2 - 2((alpha x beta) + (alpha x gamma) + (beta x gamma)) = -b/a - 2(c/a)

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is alpha^2 + beta^2 + gamma^2

(alpha x beta x gamma)^2 = (-d/a)^2

ax^3+bx^2+cx+d=0 has roots alpha, beta and gamma, what is alpha^2 x beta^2 x gamma^2

-b/a

ax^4+bx^3+cx^2+dx+e=0 has roots alpha, beta, gamma and delta, what is alpha + beta + gamma + delta

c/a

ax^4+bx^3+cx^2+dx+e=0 has roots alpha, beta, gamma and delta, what is (alpha x beta) + (alpha x gamma) + (alpha x delta) + (beta x gamma) + (beta x delta) + (gamma x delta)

-d/a

ax^4+bx^3+cx^2+dx+e=0 has roots alpha, beta, gamma and delta, what is (alpha x beta x gamma) + (alpha x beta x delta) + (alpha x gamma x delta) + (beta x gamma x delta)

e/a

ax^4+bx^3+cx^2+dx+e=0 has roots alpha, beta, gamma and delta, what is alpha x beta x gamma x delta

360°

2 pi radians =

180°/pi

1 radian =

90°

pi/2 radians =

60°

pi/3 radians =

45°

pi/4 radians =

30°

pi/6 radians =

feta/360 x 2 x pi x r

arc length =

feta/360 x pi x r^2

sector area =

r x feta

arc length using radians =

1/2 r^2 x feta

sector area using radians =

let z=x+yi, then z=x-yi, z+5iz=(x-yi)+5i(x+yi) = x-yi+5xi-5y, (x-5y)+(5x-y)i=2+34i, x-5y=2 (equation 1) 5x-y=34 (equation 2), (equation 1 x 5)=5x-25y=10 (equation 3), (equation 2 - equation 3) = 24y=24, y=1, y into (equation 1) gives, x-5=2, x=7, z=7+i

solve the equation z*+5iz=2+34i

square root of a^2 + b^2

what is the modulus of a+bi

tan^-1(b/a)

what is the argument of a+bi

r(cos(feta)+isin(feta))

double angle formula

cos n (feta) + isin n (feta)

demoivre's theorem

let z = (-1+i), modulus of z = square root of 2, modulus of z^10 = (square root of 2)^10 = 32, arg(z) = 135, arg(z^10) = 10x135 = 1350 = -pi/2, (-1+i)^10 = -32i

work out (-1+i)^10 using modulus argument form

|z|^n

what is |z^n|

n x arg(z)

what is arg(z^n)

(x-2)(x-(1-3i))(x-(1+3i)), (x-2)(x-1+3i)(x-1-3i), (x-2)(x^2-x-3xi-x+1+3i+3xi-3i+9), (x-2)(x^2-2x+10), (x^3-2x^2+10x-2x^2+4x-20) = x^3-4x^2+14x-20, p=-4 and q=14

f(x)=x^3+px^2+qx-20, given that f(2)=f(1-3i)=0, find the values of p and q

real numbers (Re)

what does the x-axis represent on an argand diagram

imaginary numbers (Im)

what does the y-axis represent on an argand diagram

modulus

what does the r represent on r(cos(feta)+isin(feta))

argument

what does the feta represent on r(cos(feta)+isin(feta))

modulus = square root of (-1)^2+4^2 = square root of 17, argument = tan^-1(4/-1) = -1.33, pi x -1.33 = 1.82, z = square root of 17 (cos(1.82)+isin(1.82))

write z=-1+4i in modulus-argument form

r1+r2(cos(feta1+feta2)+isin(feta1+feta2))

how do u write z1z2 in modulus-argument form