Core Pure 2

Question 1

What is Eulers relation

Answer 1

e^iθ = cos θ + isin θ

Question 2

How to write a complex number in exponential form

Answer 2

z = re^iθ

θ = arg z

r = |z|

Question 3

What is de Moivre’s theorem

Answer 3

(r(cosθ + isinθ))ⁿ = rⁿ(cos nθ + i sin nθ)

Question 4

How to find the mean value of a function f(x) over the interval [a,b]

Answer 4

1/(b-a) ∫^b_a f(x) dx

Question 5

d/dx (arcsin x) =

Answer 5

= 1/ √(1-x²)

Question 6

d/dx (arccos x) =

Answer 6

-1 / √ (1-x²)

Question 7

d/dx (arctanx) =

Answer 7

= 1 / 1+x²

Question 8

i³ = ?

Answer 8

-i

Question 9

i⁴ = ?

Answer 9

1

Question 10

i² = ?

Answer 10

-1

Question 11

For a complex number z = a+bi, what is the complex conjugate

Answer 11

z* = a-bi

Question 12

If the roots of a quadratic equation are α and β, what are 2 ways you can write the equation in terms of α and β

What is the relationship between α and β

Answer 12

0 = (z-α)(z-β)

or

0 = z² - (α+β)z + αβ

α and β are a conjugate pair

Question 13

Where z = a+bi

z + z* = ?

Answer 13

2a

Question 14

Where z = a+bi

zz* = ?

Answer 14

a² + b²

Question 15

For a cubic equation, describe the possible combinations of real and imaginary roots

Answer 15

3 real roots

or

2 imaginary roots(complex conjugate pair) and 1 real root

Question 16

For a quartic equation, describe the possible combinations of real and imaginary roots

Answer 16

4 real roots

2 real roots, 2 imaginary roots (complex conjugate pair)

4 imaginary roots (2 complex conjugate pairs)

Question 17

What is an argand diagram

Answer 17

A diagram used to show complex numbers

y axis = Im

x axis = Re

Question 18

What is the argument of a complex number (arg z)

Answer 18

The angle made between the line joining the number to the origin and the x axis

It is measured in the anitclockwise direction

Question 19

For a complex no. z with |z| = r and arg z = θ, what is the modulus argument form

Answer 19

z = r(cos θ + isin θ)

Question 20

arg (z₁ / z₂) = ?

arg (z₁z₂) = ?

Answer 20

arg (z₁ / z₂) = arg z₁ - arg z₂

arg (z₁z₂) = arg z₁ + arg z₂

Question 21

Given that z₁ = x₁ + iy₁ , what is the locus of point z, such that |z-z₁| = r

Answer 21

A circle of centre (x₁,y₁) and radius r

Question 22

Given that z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ ,what is the locus of points z on an argand diagram such that

|z-z₁| = |z-z₂|

Answer 22

The perpendicular bisector of the line segment joining z₁ and z₂

Question 23

Given that z₁ = x₁ + iy₁ ,what is the locus of points z such that arg(z-z₁) = θ

Answer 23

A half line (line from a point that extends infinitely in 1 direction) from the fixed point z₁ ,making an angle θ with a line from z₁ ,parallel to the real axis