Complex numbers - exam questions - answers. Question 1: Jan 2009. 1. 1. 2. 2. ) 4. 2 is the region inside the circle centre A(0,4) and radius 2. ) raw the two ...
Complex numbers ‐ exam questions Question 1: Jan 2009
Question 2: Jan 2007
Question 3: Jan 2008
Question 4: June 2010
Question 5: Jan 2010
Question 6: Jan 2006
in the polar form ( r , θ ), with r>0 and
Complex numbers ‐ exam questions ‐ answers Question 1: Jan 2009 a) z 4i 2 is the region inside the circle
centre A(0,4) and radius r 2. b) Draw the two tangents to the circle from the origin O. We call the points of contact P1 ( z1 ) and P2 ( z2 ). Use trig.properties to work out the argument of z1 and z 2 : In the right-angles triangle OAP1 ,sin
opp 2 1 hyp 4 2
1 so sin 1 2 6 arg( z1 )
2
6
3
and arg( z2 ) arg( z1 ) 2
arg( z )
3 Question 2: Jan 2007
2 3
2 3
a ) i ) Let z A 4 2i and A(4, 2) The point M represents z in the Argand diagram. z 4 2i 2 z z A 2 is equivalent to AM 2 The locus of M is the circle centre A(4, 2) radius r 2 ii ) Let zB 3 2i and B(3, 2) z z 3 2i z zo z z B is equivalent to OM= BM The locus of M is the prependicular bisector of OB. b) z 4 2i 2 is " inside " the circle z z 3 2i is the "half-plane" containing O. Question 3: Jan 2008
a ) i ) i 2 3 i 2 3 2i (2 3) 2 (2) 2 12 4 16 4 The circle C passes through the point where z i ii ) The centre of C is the point where z 2 3 i arg( z i ) arg(2 3 i i ) arg(2 3 2i ) 2 Tan 1 ( ) . 6 2 3 The half-line L passes through the centre of C. b) c)
Question 4: June 2010
z 2 2i and M ( z ) Does M belong to L1 ? z 1 3i 2 2i 1 3i 3 5i 9 25 34 z 5 7i 2 2i 5 7i 3 5i 9 25 34 M ( z 2 2i ) belongs to L1
Does M belong to L 2 ? 2 arg( z ) arg(2 2i ) tan 1 2 4 M ( z 2 2i ) belongs to L 2 M ( z ) is a point of the intersection between L1 and L2 b) L1 is the perpendicular bisector of the line AB with A(z A 1 3i ) and B( z B 5 7i ) L2 is the half line from O with gradient tan
4
1.
c)
Question 5: Jan 2010
a ) i ) z 4 2i 4 this is the circel centre A(z A ) with z A 4 2i and radius r 4 ii ) z z 2i This is the perpendicular bisector of the line OB with z B 2i and zO 0 b) The region is the intersection of the inside of the circle and the half-plane containing B.
The ability to manipulate complex numbers is very handy in circuit anal- ysis and in electrical engineering in general. Complex numbers are par- ticularly useful ...
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Normal distribution - Exam questions. Question 1: Jan 2006. Question 2: Jan 2008. Question 3: Jun 2008. Page 2. Question 4: Jan 2007. Question 5: Jun 2010.
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Question 4: Jan 2009. Question 5: Jan 2010. Question 6: June 2007. Page 3. Question 7: June 2006. Question 8: June 2008. Page 4. Question 9: June 2009 ...