|
 Ordinary
Level Paper 1
Complex
numbers
1999
Q4 Paper 1
1998 Q4
Paper 1
1997 Q4
Paper 1
1996
Q4 Paper 1
1995
Q4 Paper 1
1999
Q4 Paper 1 Complex numbers
(A)
Z = 5+4i where  .
Plot (1) z, (2) z - 4i on an Argand diagram
 
Z
= 5+ 4i, Z-4i = 5+4i-4i
= 5 +0i
(10 Marks)
(B)(i)
Let u =3 - 6i we are asked to find /3 - 6i / this the modulus
of the complex number
(10 Marks)
(B)(ii)
Show  .
Multiply everything by the common denominator i
this is the standard way to solve an equation where a bottom
line is involved, this gives
 True.
Another idea when you get a "solve" problem in complex numbers
is just to fill in the equation with the things that you
know and take it from there! Eg
 ,
By doing this you are guaranteed the attempt mark! You can
then multiply everything by i and tidy it up.
(5 Marks)
(B)(iii)
In this last part of part b we are asked to express in
the form p + q I.
Solution:
First
replace all the u's by 3 - 6i. This gives
 (10marks)
(C)
Here we are given w = i - 2.
Solution:
We
are asked to find  this
just means multiply w by w. this gives  (5Marks)
In
the second part we are asked to solve  for
real k and real t.
Solution:
Just
replace w and  in
the equation. This gives k (3-4i)
= 2(i - 2)+1+ ti
3k
- 4ki = 2i - 4 +1 + ti. Now set reals equal to the reals
and the i = i.
This
gives  and
(10
marks)
Comments:
Tasks involved in this question (1)plot on an Argand
diagram(a),(2)Solve a Complex Number equation(c)(3)Multiply(c)(4)Divide(b)Modulus(b)
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1998
Q4 Paper 1 Complex numbers
(A)
Given
w = 2i plot  on
an Argand diagram.
 (10
marks)
Very
easy same thing was asked in '99,98.'96,95 so you must be
able to answer this.
(B)
This part is based on the roots of a quadratic equation
.We are asked to show 4 - 3i is a root of  and
find the other root, probably the easiest way is to use
the roots formula 
(B)(ii)
Investigate if
here we are
asked to find the modulus of two complex numbers
Solution:
 ,
Now find the modulus of 10-10i = .
The results are equal!
(10 marks)
(C)(i)Given
u =2 - i. We are asked to write  in
the form a +bi.
Solution:
Again
very easy
 ,
First get  in
the form a + ib as follows 
(10
Marks)
(C)(ii)
In the last part of this question we use the information
found in part (i) to solve the equation ,
Solution:
Just
fill in the parts that you know 
 :
(10 marks)
Comment:
Tasks (1)Plot includes  (there
must have been lots of errors with this in'97)
(2)Modulus
(b)Divide(c)(3)Complex Number equation(c)(4)Roots of a quadratic(b)
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1997
Q4 Paper 1 Complex numbers
(A)
Here
we are asked to get rid of the brackets and tidy it up
Solution:
3(1
+5i)+ i (3 - 2i) = 3
+15i + 3i-2(-1)
= 5+ 18 i. This was worth 10 marks!
(B)(i)
Another question based on the modulus of a Complex number
In
this case we are asked to find the values of a for
which ?
Solution:
(10marks)
(ii)
Given w = 4i we are asked to verify 
Solution
just replace w in the equation by 4i
 true
(10
Marks)
©
In this part of the question we have to divide two
complex numbers and solve an equation .
We
are given z =1 +i . We are asked to find  (10marks)
Now
we are asked to use this result to Solve (very similar to
'98,)
Solution:
(10 marks)
Comments
This was a very short question ,You were not asked to plot
on an Argand diagram here but all the rest of the usual
tasks are here. (1)Multiply(a),(2)Conjugate(c) (3)complex
number equation (c)(4)modulus(b).
What
was a bit unusual was the second part of (b) it had not
been asked before but as you can see from above it really
was an exercise in multiplying out complex numbers.
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1996
Q4 Paper 1 Complex numbers
(A)
Given z = 1 -4i as usual we are asked to plot Complex numbers
on the Argand Diagram. The want us to plot z and 2 +z Z
= 1 - 4i, 2 + z = 2 + 1 - 4i=3-4i.
Part
(b) Again
consists of three parts this was a feature of this question
up to 1996 thereafter just 2 parts.
a)
w = (1-3i)(2+i). Here we are asked to multiply out
two complex numbers.
Solution
(10) marks
(2)
Based on the modulus and the conjugate of w.
Show
Solution.
Just sub in for w and it's (
) conjugate to get 
True
(10 marks)
(3)
We are asked to find a if
this is a Complex number equation.
Solution:
Just fill in for w and 
and cross multiply,
then set reals equal to reals and i's = to i's. This gives
(5 marks)
(C)
Based on a Quadratic Equation:
Given
z =2 - i is a root of
find p
and q.
Solution:
Since 2 - i is a root then 2 + i is also a root. We know
the sum of the roots is -p this gives
, and the product of the roots is q this gives
(15
Marks)
Comments
Contains all of the usual tasks .
(1)Plot
(a) (2)Multiply (b) (3)complex equation(b). (4)modulus and
conjugate (b)(5)Roots of a Quadratic(c)
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1995
Q4 Paper 1 Complex numbers
(a)
Given  .
We are asked to plot the following  this
is very straightforward since we know the first two numbers
already
To get
 just
add 5+4i + -3-5i = 2-i
for doing this you got 10 marks! (2 marks for axes)
b)
This
consists of three parts
(1)
We
must change w =  into
the form p+iq .
(2)
Solution:We
do this by multiplying above and below by the conjugate
of the bottom-line this gives
 =
W (10marks)
(2)
We are asked for the modulus of w = (5
marks)
(3)
We
are asked to show modulus squared is equal to w multiplied
by it's conjugate i.e. is
Which
is true! (5 marks)
(3)
Again three parts two of which
are Complex number equations.
(1)
Given u = 6 - 5i find a and b if u + ai = 2b
Solution
Just replace u by 6 - 5i and set reals equal to reals and
i's = to i's.
This
gives 6-5i + ai = 2b  10 marks
The
next part is just a slightly more complicated version of
the last part; again it's a Complex number equation.
(2)
Solve
for real s and t. s (2-i) +ti (4+2i) = 1 + s + ti
Solution
just
multiply it out and set reals equal to reals and i = i.
 (5marks)
A
bit long but not difficult, this type of question is a regular
on LCMaths paper 1.
(3) The
Last part of this question was a bit off the wall it falls
into the category of ask a quadratic at all costs. You are
told that a complex number Z = x + iy we are asked what
type of curve is represented by 
Solution:
 a
circle. (5 marks)
The
way the marks are allocated clearly shows how badly parts
(2) and (3) were attempted!
Comments
Last bit of part c was unusual but all of the regular tasks
were asked again
(1)Plot(a)(2)Divide(b)(3)Complex
Number equations(c)(4)Modulus (b)(5)Conjugate(b)
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