FP1 COPY

Question 1

what are the steps for proof by induction?

Answer 1

prove it works for n=1
assume it works for n=k
prove it works for n=k+1

Question 2

what is the complex conjugate of z?

Answer 2

Z with the sign of the imaginary part reversed.

Question 3

If given a fraction with a complex denominator, how can it be written in the form a+bi?

Answer 3

Act as if you were rationalising the denominator. Multiply the bottom by a-bi and it should remove imaginary parts from the bottom.

Question 4

what does z* mean?

Answer 4

the complex conjugate of z

Question 5

find a and b such that (15+8i)^0.5=a+bi

Answer 5

square both sides.
15+8i=(a+bi)(a+bi)
expand the right side to get 
15+8i=a²-b²+2abi
thus 8=2ab and 15=a²-b². Solve simultaneously.

Question 6

what does [z] mean on an argand diagram?

Answer 6

the distance from the point z, to the origin.

Question 7

how would the area 3 ≥[z] be represented?

Answer 7

A shaded in circle of radius 3 centre of (0,0)

Question 8

how would you represent [z-(6+8i)]=10 ?

Answer 8

a circle of centre 6,8 and radius 10.

Question 9

how would you represent [z=2+4i]=0

Answer 9

a point at -2,4

Question 10

how would you represent Re(z)=-2

Answer 10

A infinite vertical line going through -2 on the x axis.

Question 11

how would you represent [z]=[z-4]?

Answer 11

a vertical line going though 2 on the x axis.

Question 12

how would you represent [z]≥[z-2i]

Answer 12

a horizontal line goes through 2 on the y axis and everything above it is shaded.

Question 13

what is arg(z)?

Answer 13

the angle between the line real axis and the line of z.

Question 14

how would you draw ? arg(z-6)=pi/4

Answer 14

A line goes from 6 =0i and makes an angle of pi/4 with the real axis.

Question 15

write -12 + 5i in modulus argument form.

Answer 15

first find the modulus. In this case, it is 13.
then find the angle between the line going from the origin to the point -12, 5. It is 2.75
z=13(cos(2.75)+isin(2.75))

Question 16

how would you draw arg(z-4i)=0?

Answer 16

a horizontal line going though 4 on the imaginary axis.

Question 17

what do complex roots of equations always come in?

Answer 17

Pairs. The sign of the imaginary part can be changed.

Question 18

if given a cubic and one imaginary root, (1+2i)how are the other 2 found?

Answer 18

the other complex root simply has the sign changed.
multiply. (z-1+2i)(z-1-2i). This gives (z²-2z+5). Then think of the thing you multiply this by to get the equation. Put this equal to 0 and solve.
This is the other root.

Question 19

If you know the real root to an equation, how can the other 2 be found?

Answer 19

if it were 7, divide the equation by (x-7). Then use the quadratic formula on the result to find the other roots.

Question 20

if you have an equation with unknown coefficients and you know one of the imaginary roots, how do you find the coefficients?

Answer 20

make x= the imaginary root in your equation. Then bring the real and imaginary parts of the equation together and both must equal.

Question 21

what is an identity matrix?

Answer 21

one where the leading diagonal is 1s and all the others are 0s.

Question 22

what must be true when you are multiplying matrices?

Answer 22

The number of columns is equal to the number of rows.

Question 23

if A and B are roots of a quadratic equation

Answer 23

A + B = -b/a

AB=c/a

Question 24

AB and C are the roots of a cubic equation.

Answer 24

A+B+C=-b/a
AB+BC+CA=c/a
ABC=-d/a

Question 25

If ABC and D are roots of a qurtic equation

Answer 25

A+B+C+D=-b/a
AB+AC+BD+BC+CD+DA=c/a
ABC+BCD+CDA+ABD=-d/a
ABCD=e/a

Question 26

what does ΣA mean?

Answer 26

A+B+C+D

Question 27

ΣAB

Answer 27

AB+AC+BD+BC+CD+DA

Question 28

ΣABC

Answer 28

ABC+BCD+CDA+ABD

Question 29

|5-6i-z|

How would this point be represented?

Answer 29

As the z is negative it would be (5,6) if the diagram was Cartesian.

Question 30

What does the determinent represent?

Answer 30

The ratio of the change of area when a matrix is multiplied by a shape.

Question 31

If matrix A provides the transformation B and matrix Y has the transformation Z, what matrix would transform by B and then Z?

Answer 31

Matrix Y* matrix A

Question 32

If you know all the roots of an equation but not the coefficients, how would you find them ?

Answer 32

Multiply all the coefficient together in the form (x-a-bi)

If a root was 2+2i you would write it (x-2-2i)

Question 33

If you have an equation where the denominatior and numerator are both polynomials of the same order, how is the vertical horizontal asymptote found?

Answer 33

Divide the coefficients of the highest degree terms and y= that value

Question 34

If the polynomial in the numerator is a higher degree than the denominator, how is the horizontal asymptote found?

Answer 34

There is no horizontal asymptote

Question 35

If the polynomial in the numerator is a lower degree than the denominator, how do you find the horizontal asymptote?

Answer 35

Y=0 is the asymptote