IIT JEE Complex Numbers (part 3)

IIT JEE Complex Numbers (part 3) 2010 IIT JEE Paper 1 Problem 39 Complex Numbers (part 3)

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IIT JEE Complex Numbers (part 3)

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so we're in the home stretch, let's see if part C is true. We're running out of real estate, so let me
copy and paste it. Actually maybe i can do it over here to the left, i have some doodles here, let me
clear those out. let's see if we can do it here to the left
Part C. So we have this determinate, and it's claiming it equals 0
Let's see if that's true, so we can actually try to multiply these out and see if anything interesting happens
but we could even better leverage some of the albegra we;ve already done for Part A
In part A, we figured out... we figured out that z minus z1 is equal to T times Z two minus Z one
So this thing right overe here... this is the same thing as T times Z two minus z one
The reason why that's interesting is I have z two minus z one
here, so I'm starting to have similar things
and then this thing over here... so if I take the conjugate of each of these... if I take the conjugate
of each of these... if I take the conjugate of z minus the conjugate of z one, this is going to be equal
to T times the conjugate of Z2 minus the conjugate of Z1, and i know that because when i do this, I'm
swapping all the signs on the imaginary parts of the complex numbers. So if you were to go through this
entire process, just swapping the imaginary parts of the complex number. The end result
would have its imaginary parts swapped, and that's exactly what we have over here. We have the imaginary
parts swapped over here.
so if you want to work it out, set this equal to A plus B I or set this equal to A one plus B one. you
can work it out
but i think it's a pretty intuitive idea, that we're just making all of the... We're just swapping the
signs of all the imaginary parts of these complex numbers.
Now with this said, this determinate becomers pretty simple. This becomes T times z two minus z1. this
becomes T times... T times the conjugate of Z2-Z1 this down here is Z2-Z1 and then this over here
is the conjugate of Z2-Z1. So what is this determinate, It's going to be this times this, so it's T
times Z2 minus Z1 times z2 minus z1, times the conjugate of those, so it;s that times that
minus this times this, so minus t Z2-Z1 times the conjugate of Z2 minus the conjugate of Z1. Now this
is exactly equal to this, these things are obviously going to cancel out, and we're clearly going to
get 0, so C is also true, so the correct answers to this original problem. The correct answers were

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IIT JEE Complex Numbers (part 3) 2010 IIT JEE Paper 1 Problem 39 Complex Numbers (part 3)

IIT JEE Complex Numbers (part 3)

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