FP1 Flashcards
what are the steps for proof by induction?
prove it works for n=1
assume it works for n=k
prove it works for n=k+1
what is the complex conjugate of z?
Z with the sign of the imaginary part reversed.
If given a fraction with a complex denominator, how can it be written in the form a+bi?
Act as if you were rationalising the denominator. Multiply the bottom by a-bi and it should remove imaginary parts from the bottom.
what does z* mean?
the complex conjugate of z
find a and b such that (15+8i)^0.5=a+bi
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.
what does [z] mean on an argand diagram?
the distance from the point z, to the origin.
how would the area 3 ≥[z] be represented?
A shaded in circle of radius 3 centre of (0,0)
how would you represent [z-(6+8i)]=10 ?
a circle of centre 6,8 and radius 10.
how would you represent [z=2+4i]=0
a point at -2,4
how would you represent Re(z)=-2
A infinite vertical line going through -2 on the x axis.
how would you represent [z]=[z-4]?
a vertical line going though 2 on the x axis.
how would you represent [z]≥[z-2i]
a horizontal line goes through 2 on the y axis and everything above it is shaded.
what is arg(z)?
the angle between the line real axis and the line of z.
how would you draw ? arg(z-6)=pi/4
A line goes from 6 =0i and makes an angle of pi/4 with the real axis.
write -12 + 5i in modulus argument form.
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))
how would you draw arg(z-4i)=0?
a horizontal line going though 4 on the imaginary axis.
what do complex roots of equations always come in?
Pairs. The sign of the imaginary part can be changed.
if given a cubic and one imaginary root, (1+2i)how are the other 2 found?
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.
If you know the real root to an equation, how can the other 2 be found?
if it were 7, divide the equation by (x-7). Then use the quadratic formula on the result to find the other roots.
if you have an equation with unknown coefficients and you know one of the imaginary roots, how do you find the coefficients?
make x= the imaginary root in your equation. Then bring the real and imaginary parts of the equation together and both must equal.
what is an identity matrix?
one where the leading diagonal is 1s and all the others are 0s.
what must be true when you are multiplying matrices?
The number of columns is equal to the number of rows.
if A and B are roots of a quadratic equation
A + B = -b/a
AB=c/a
AB and C are the roots of a cubic equation.
A+B+C=-b/a
AB+BC+CA=c/a
ABC=-d/a
If ABC and D are roots of a qurtic equation
A+B+C+D=-b/a
AB+AC+BD+BC+CD+DA=c/a
ABC+BCD+CDA+ABD=-d/a
ABCD=e/a
what does ΣA mean?
A+B+C+D
ΣAB
AB+AC+BD+BC+CD+DA
ΣABC
ABC+BCD+CDA+ABD
|5-6i-z|
How would this point be represented?
As the z is negative it would be (5,6) if the diagram was Cartesian.
What does the determinent represent?
The ratio of the change of area when a matrix is multiplied by a shape.
If matrix A provides the transformation B and matrix Y has the transformation Z, what matrix would transform by B and then Z?
Matrix Y* matrix A
If you know all the roots of an equation but not the coefficients, how would you find them ?
Multiply all the coefficient together in the form (x-a-bi)
If a root was 2+2i you would write it (x-2-2i)
If you have an equation where the denominatior and numerator are both polynomials of the same order, how is the vertical horizontal asymptote found?
Divide the coefficients of the highest degree terms and y= that value
If the polynomial in the numerator is a higher degree than the denominator, how is the horizontal asymptote found?
There is no horizontal asymptote
If the polynomial in the numerator is a lower degree than the denominator, how do you find the horizontal asymptote?
Y=0 is the asymptote
