My first 4u HSC Assessment (1 Viewer)

[Mumma]

today was my first 4U assesment... very straightforward stuff, but strangely some people found it difficult

 

[Riviet]

Mine was 2 weeks ago LOL.

 

[KeypadSDM]

I don't like showing that the sum = 0, I prefer showing 0 = sum, it's nicer:

x⁴ + 1 = 0 ... {1}
:. x⁶ + x² = 0 ... {2}

{1} + {2} =>
0 = x⁶ + x⁴ + x² + 1

 

[pLuvia]

It's the same, but I like ~~~ = 0

 

[tarsus]

I did my 4U assessment today. It was really straight forward, but I haven't mastered locus (the reasoning part with reference to vectors. Hehe, I haven't touched that excerise 2.5 on Cambridge but I know the basics) and couldn't do it at all, so I skipped that question. There goes 4 marks down the drain.

Overall, I found the test to be good- but some people found it difficult...

BTW, good luck for other people who haven't done their 4U assessment. I wish you do well.

 

[.ben]

i had 4u today. have a pretty good feeling bout it until the last question

given that:

1+sinθ+icosθ
-------------- = sinθ + icosθ
1+sinθ-icosθ

hence deduce that:

[1+sin(∏/5)+cos(∏/5)]^5+[1+sin(∏/5)-cos(∏/5)]^5=0

i had an inkling of de moivre's theorem in the actual test but wasn't able to finish cos there wasn't any time left

 

[YuMchA]

this yrs assessment post up =]
Q1)
a) if z=2-i and w=1+2i find:
i) z+w (1)<-- lil number is the wat the question is worth in marks
ii) w-z (1)
iii) zw (1)
iv) z times conjugate w (1)
v) z/w (1)

b) find all pairs of integers x and y that satisfy (x+iy)^2=24+10i (3)

c) consider the equation z^2+az +(1+i)=0 (2)
find the complex a, given that i is a root of the equation

d) it is given that 2+i is a root of P(z) z^3+rz^2+sz+20 r and s are real numbers
i) state why 2-i is also a root of P(z) (1)
ii) factorise P(z) over the real numbers (2)

e) the diagram below show a complex plane with origin O
the points P and Q represent arbitrary non_zero complex numbers z and w repectively. thus the length of PQ is |z-w|

i) copy the digram, and use it show that |z-w| less than or equal to |z|+|w| (2)
ii) construct the point R representing z+w wat type of quadrilater is OPRQ? (2)
iii) if |z-w|=|z+w| wat can be said about the complex number w/z (2)

Q2)
a)given z=square root 6 -squareroot 2i, find
i) Re(z^2) (1)
ii) (Imz)^2 (1)
iii)|z| (1)
iv) arg z (2)
v) z^4 in the form x+iy (2)

b) alpha =1+squareroot 3i and beta=1+i
i) find alpha/beta in the form x+iy (2)
ii) express alpha in mod arg form (2)
iii) given that beta has mod arg form beta= squareroot 2(cos pie/4+ isin pie/4) find the mod arg of alpha/beta (2)
iv) hence find the exact value of sin pie/12 (2)

c) sketch the region in the complex plane where the inequalities
|z-1-i|<2 and 0<arg(z-1-i)< pie/4 hold simultaneously (3)

d) the digram below shows the locus of the points z in the complex plane such that arg(z-3)-arg(z+1)= pie/3 this locus is part of a circle
The angle between the lines from -1 to z and from 3 to z is theta, as shown explain why theta= pie/3 (2)

max mark for test 40/40 scored by two ppl i got 39 damn =/

 

[pLuvia]

Here is my one of the 4u Complex number test

1a) Define modulus and conjugate of a complex number z=x+iy, (x,y real).
b) Prove that |z|² = zz and that (z₁z₂)=z₁z₂ for any two complex numbers z₁z₂.
c) Deduce that |z₁z₂|=|z₁||z₂|

2. Let A=1+i, B=2-i. Draw neat sketches to show the loci specified on the Argand diagram by.

a)Arg(z-A)= pi/4
b)|z-A|=|z-B|

3. Calculate the modulus and the argument of the product of the roots of the equation (5+3i)z² - (1+4i)z + (8-2i) = 0

4. Draw neat labelled sketches to indicate each of the subsets of the Argand diagram described below.
a){z:|z|> and o< Argz< pi/3}
b){z:z+z > 0}
c){z:|z-1|<|z+1|}
d){z:|z²-z²|<4}

5. Show that the point representing cos(pi/3)+isin(pi/3) on the Argand diagram lies on the circle of radius one with centre at the point which represents 1.

6. The complex number z is given by z=-sqrt3 + i.

a) Write down the value of arg(z) and |z|.
b) Hence, or otherwise show that z⁷+64z=0

7. Prove by induction that (cos@+isin@)ⁿ = cos(n@) + isin(n@) for all integers n>1.

8. R is a positive number and z₁,z₂ are complex numbers. Show that the point A,B,C which represent respectively the numbers z₁,z₂, (z₁ - iRz₂) / (1-iR) form the vertices of a right angled triangle

 

[Mill]

Some of these assessments are quite interesting.

Some of the questions are quite common.

Some have been ripped straight from HSC papers.


In response to Riviet's assessment:

You could do Question 3 using sum of a geometric series btw and it is perhaps more intuitive to think of it that way.

We are used to prooving results like that using a geometric series for traditional roots of unity questions. This is just a slight variation on that.

Roots are z, z³, z⁵, z⁷

thus z + z³ + z⁵ + z⁷ = 0

but z is non-zero

therefore 1 + z² + z⁴ + z⁶ = 0

arguably this method is somewhat hampered by the fact that your question stipulates z to be any root (rather than it being the root with least positive argument as we usually see) but it is possible to get around that... i don't feel like writing it out but i'm sure you can see how

so while this particular solution may be slightly longer given the unfortunate wording of the question, i still think it is the more intuitive way to think about it and people should be aware of it!

 

[Rax]

My First actual Assessment is the Half yearly in week 9......which is dodge
But I got this non-assessable assignment sheet thing which is 2 pages long and has some complex numbers, Conics and some complex polynomials
I will post it up when I have the time lol
good luck all
(I need to get better at 4u damnit lol)