Cambridge Prelim MX1 Textbook Marathon/Q&A (2 Viewers)

axwe7

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Sorry for the late response,

The question is;

Use mathematical induction to prove that for all positive integers 'n':
(a) 1^2 + 2^2 + 3^2 + ... ... .. ... + n^2 = 1/6*n(n+1)(2n+1)


Cheers.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Sorry for the late response,

The question is;

Use mathematical induction to prove that for all positive integers 'n':
(a) 1^2 + 2^2 + 3^2 + ... ... .. ... + n^2 = 1/6*n(n+1)(2n+1)


Cheers.
 
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appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

The variable complex number z satisfies |z - z0 | = r . Find the max and min values of: |z|

I got the answer:

</= represents less than or equal to

|z0| - r </= |z| </= |z0| + r

But the answer has something different.

Instead of my |z0| - r

they have

| |z0| - r |

Can someone explain why there is an extra modulus sign around my entire answer that I got.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

The variable complex number z satisfies |z - z0 | = r . Find the max and min values of: |z|

I got the answer:

</= represents less than or equal to

|z0| - r </= |z| </= |z0| + r

But the answer has something different.

Instead of my |z0| - r

they have

| |z0| - r |

Can someone explain why there is an extra modulus sign around my entire answer that I got.
You need the absolute value sign to make sure answer is non-negative, since |z| is non-negative. (Draw a diagram.)
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

find the locus of z if the value of z - 1 / z - i is a) real, b) imaginary

for a) I got x + y = 1
b) I got (x -1/2)^2 + (y - 1/2)^2 = 1/2

The solutions are

a) The line through 1 and i, omitting i

b) The circle with diameter joining 1 and i, omitting these two points.

Could someone explain the answer please.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

find the locus of z if the value of z - 1 / z - i is a) real, b) imaginary

for a) I got x + y = 1
b) I got (x -1/2)^2 + (y - 1/2)^2 = 1/2

The solutions are

a) The line through 1 and i, omitting i

b) The circle with diameter joining 1 and i, omitting these two points.

Could someone explain the answer please.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

find the locus of z if the value of z - 1 / z - i is a) real, b) imaginary

for a) I got x + y = 1
b) I got (x -1/2)^2 + (y - 1/2)^2 = 1/2

The solutions are

a) The line through 1 and i, omitting i

b) The circle with diameter joining 1 and i, omitting these two points.

Could someone explain the answer please.
I haven't bothered doing the question, but notice that your answers match the answers given apart from some omissions of points in the answers. E.g. your x + y = 1 line is precisely the line through 1 (the point (1,0)) and i (the point (0,1)). To see why some points should be omitted, try subbing them in to the expression and see what happens.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

find the locus of z if the value of z - 1 / z - i is a) real, b) imaginary

for a) I got x + y = 1
b) I got (x -1/2)^2 + (y - 1/2)^2 = 1/2

The solutions are

a) The line through 1 and i, omitting i

b) The circle with diameter joining 1 and i, omitting these two points.

Could someone explain the answer please.
Not sure why they omitted those points for b) though, since they do give imaginary results if you sub. them in to the expression.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

THe function f(x) is defined by f(x) = x - 1/x , for x > 0

a) by considering the graphs of y = x and y = 1/x for x > 0, sketch y = f(x)
 

InteGrand

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appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I now need to sketch the inverse function:

I plugged it into wolfram and this is what it shows:

http://www.wolframalpha.com/input/?i=x+=+y+-+1/y+for+y+>+0+

I get how when you sub large y values it tends to negative infinity and so approaches as well the asymptote.

But shouldn't the inverse function pass through the point ( 0, 1)

Why does this not occur?

Also why does it as y approaches infinity approach the y axis and not y = x

If the inverse is a reflection across y = x, shouldn't the inverse approach y = x ??

Thanks
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I now need to sketch the inverse function:

I plugged it into wolfram and this is what it shows:

http://www.wolframalpha.com/input/?i=x+=+y+-+1/y+for+y+>+0+

I get how when you sub large y values it tends to negative infinity and so approaches as well the asymptote.

But shouldn't the inverse function pass through the point ( 0, 1)

Why does this not occur?

Also why does it as y approaches infinity approach the y axis and not y = x

If the inverse is a reflection across y = x, shouldn't the inverse approach y = x ??

Thanks
It actually does do all these things. Observe that in the WolframAlpha plot, the "x"-axis is actually the vertical axis. So they simply plotted the same graph but switched the axes around. So as you can see, the horizontal intercept on the graph is 1. This corresponds to the point where x = 0 and y = 1 (since the ''y''-axis here is actually horizontal). So it does go through the point (0, 1), since by (0, 1), we mean x = 0, y = 1.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Now by conpleting the square or using the quadratic formula, show that

f^-1 x = 1/2 ( x + ( 4 + x^2)^1/2 )

I used the quadratic formula and got the result

But instead of x + (4 + ....

I got x +/ - ( 4 + ....

Why does the answer disregard the negative option??
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Now by conpleting the square or using the quadratic formula, show that

f^-1 x = 1/2 ( x + ( 4 + x^2)^1/2 )

I used the quadratic formula and got the result

But instead of x + (4 + ....

I got x +/ - ( 4 + ....

Why does the answer disregard the negative option??


 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Screen Shot 2016-01-08 at 1.16.20 pm.png

Can you please explain part e) .
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

You lost me at solving for the inverse.......
 

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