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Maths (Paper 3)

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diwash

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Hello every one,
Let's post here the graph works in paper 3 mathsmatics...i will also post the answers of some graphs questions from past papers and i will highly encourage all of u to post answers of Q questions from past papers..indicating clearly from which year it is from....Lets help each other :good:
 
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MukeshG93

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Would be glad to help. Some of the most annoying graphs are from complex numbers as many of the students find it difficult to wrangle with the Argand Diagram and vectors can be annoying at times as well. I have quite the base in both of these so I can help as far as possible. Amazingly, there are numerical methods of finding out the maximum argument of the locus of a circle in an Argand diagram and I know a MUCH MUCH simpler way of finding the perpendicular distance from a point to a line and such. Will look forward to posting here tonight as my exam's tomorrow, GMT + 5:45 and I would love to answer questions seen on here, would boost my confidence level and level of understanding to a larger degree, hopefully. Thanks for starting a thread on this.
 
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ahmed t

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well u can start by posting ur easier way of findin the perpendicular distance, and the numerical methods for finding the maximum and minimum argument
 
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MukeshG93

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Easier and comprehensible way of finding the perpendicular distance from a point to a line (both of known position vector and vector equation, respectively)

Alright, so lets assign some variables first
our point P will have position vector (a, b, c)
our vector l will have equation r = (f, g, h) + t(i, j, k) [the use of i, j and k is mere coincidence, do not confuse with the vector notation]

Now, let the foot of the perpendicular from P to l (i.e. the point where P overshadows the line) be point Q

Q lies on the vector, so it must follow the vector equation and its position vector (i.e. coordinates) will be (x, y, z) = (f, g, h) + t(i, j, k) for a unique value of t.

Now, we can find the direction vector of the vector from P to Q using (Q - P) = (f + ti - a, g + tj - b, h + tk - c) = PQ
(NOTE: in the above, the values for a, b, c, f, g, h, i, j and k are known, they are provided in the question itself or we are asked to find them out before proceeding to this part of the question; t is the only unknown variable)

Now, PQ is perpendicular to l. So, PQ.l = 0 or, (f + ti - a, g + tj - b, h + tk - c).(i, j, k) = 0

Equating the above gives the value of t, which can be substituted into (x, y, z) = (f, g, h) + t(i, j, k) to obtain the position vector of Q.

Mission accomplished.
 
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intel1993

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hey thnx for making this thread,................
ca u plzz post j2002 q9 iii) argand diagram...........
and Jun 2007 q8 ii) argand diagram

plzz post asap...............
thnx 4 the help...
 
ammarelahi

ammarelahi

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how can we check the range of answers tht we get as in N09 P31 Q1 ?
and how to solve the inequality in N08 Q5 ii ?
 
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intel1993

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intel1993 said:
hey thnx for making this thread,................
ca u plzz post j2002 q9 iii) argand diagram...........
and Jun 2007 q8 ii) argand diagram

plzz post asap...............
thnx 4 the help...
Click to expand...



hey plzz help anyone.....................?????
 
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MukeshG93

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Numerical way of calculating the maximum and minimum argument of a complex locus

Alright, so the locus we usually get is in the form of a circle. Wherever the circle is, it will have a set of complex numbers that follow its requirements. Similarly, the locus has a point with min argument and max argument. Geometrically, the points with min and max arguments are where lines from the origin are tangent to the circle. Remember that a tangent to a circle will always make an angle of 90 degrees with the radius of the circle at that point. The locus usually follows the format |z - u| < a, where a is the radius and u is the centre. We can construct a right-angled triangle using the radius of the circle at the point with min/max argument, the line from the origin to the centre of the circle (it's length = |u|) and the line from the origin tangent to the circle at the min/max point. We also know the angle that the line from the origin to the centre of the circle makes with the x-axis (arg(u)). So, we have the length of two sides (the radius and the line from the origin to the centre) and a right-angle to work with. Trigonometry, much? use sin(theta) = opp/hyp and cos(theta) = adj/hyp and you should get the angles you need to find the argument in each case. Just try, if you don't get it, add me on yahoo messenger and we'll doodle it out.
 
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MukeshG93

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Min |z| and max |z| on a circle

Alright, if you draw a circle, you will find that there is a point closest to the origin. There are no special cases, and the point closest to the origin will ALWAYS be the closer point of intersection of the line passing through the origin and the centre of the circle with the circle's boundary itself. Finding this modulus is relatively easier than other argand ploys. What you do is: find the modulus of the point that is the centre of the circle. That gives the length of the line going from the origin to that centre. Now, you know the radius. So, subtracting the radius from the modulus should give you the length of that section. Similarly, adding it should give you the max|z|.
 
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MukeshG93

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Shading

Alright, how many of you have done the inequality shading stuff while in school? When we made a parabola, we'd have to shade the parts for which y > 0 and what not. In the complex plane, this is not very different. There is this other type of graphical representation called the polar form (not sure how many of you know about it). It involves using the distance of a point from the origin and the angle it makes with the x-axis instead of using x and y coordinates. This is what complex numbers get at. What is the modulus and argument? Exactly my point. Now, when we say draw arg(u), some people get stuck because they don't know what to do. Arg (u) is an angle, an angle that is made by a line with the x-axis. So, if you are asked to draw arg (u), first find the argument of the point and draw a straight line that makes that angle with the x-axis (if you have already plotted u, just draw a straight line through the centre and u that starts at the bottom end of the argand diagram and ends at the top end. Now, modulus means a certain distance from the origin. When we say |z| = 2, it means this complex number z is at a fixed distance of 2 from the origin. There is no mention of the angle it makes with the circle, so it can be anywhere within a 2 unit radius (a circle of radius 2 with the origin as its centre). Alright, now lets take inequalities into hand. What does it mean when it says arg(z) < arg(u)? It means the angle z subtends against the x-axis HAS to be less than the angle subtended by u. So, it technically has to lie below the line that corresponds to arg(u) but it can be ANYWHERE. 3 is less than 4 but so is 2, and so is 1. So, you have to shade whatever you see that makes an angle SMALLER than that made by the respective complex number. So, say u is 1 + 2i (coordinate of (1,2)). So, taking the previous case, z can be (1,1), even (2,1) and even (4,1), anything that makess a smaller angle. Similarly, if it says: |z - u| < 1, it means the distance between z and u has to be less than 1. The distance is mentioned but the any angle is not mentioned, so the locus, again, is a circle of radius one having u as its centre. Now, if the distance has to be less than 1, then you have to shade INSIDE the circle because inside the circle is where values of z lie such that the distance between z and u is less than 1. Now, when put simultaneously, you have to take both the relationships into consideration and shade adequately. If it asks you to shade |z - u| > 1 but arg(z) > arg(u), this means you have to shade OUTSIDE the circle ABOVE the line arg(u) because the points in that area make a larger angle with the x-axis than u does. You shad outside because the distance is GREATER than 1 so z cannot lie within a 1 unit radius of the point u. Get it? Hopefully you did.
 
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MukeshG93

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Alright, anything else you'd like me to cover? I can do past paper questions but would prefer not to. Ask me questions that regard conceptual perspectives. Like: why this? why not this? how does this make sense? why do we have to do this? is there an easier way to do this?

Teach me how to do this is fair enough but it's difficult to explain with words in mathematics, hope you understand.

Cheers

--Yours truly
 
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crash250

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thank u mukesh for this incredible job...but i was wondering if u can explain to us using any attached diagrams which will be more understandable to us
like u said its somewhat difficult to explain in words
many thanks mukesh its just up to u...u already done nice work far
 
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