2 Unit Revising Marathon (1 Viewer)

jet

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OK, now that 2010 is here I am reviving the old Revision Marathon game.

Rules:
The OP posts a question, first person to get it correct asks the next question, and so on.

First Question
 

blackratpoo

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Re: 2 Unit Revising Marathon HSC '10

how do you get that font? that wikipedia one that allows you to do (divide by signs)?
 

Dragonmaster262

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Re: 2 Unit Revising Marathon HSC '10

Sorry; my net got disconnected and it took ages to reconnect.

 
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Dragonmaster262

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Re: 2 Unit Revising Marathon HSC '10

I'm having troubles reading your answer but I believe it is correct. Please post the next question.
 

ninetypercent

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Re: 2 Unit Revising Marathon HSC '10

tanx = 3cotx
tanx = 3/tanx
tan^2x = 3
tanx = +/- sqrt 3
x = pi(n) + pi/3
where n is an integer

Find the equation of the tangent(s) to x^2 + y^2 = 4, which are parallel to y= x + 3
 
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addikaye03

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Re: 2 Unit Revising Marathon HSC '10

where n is an integer

Find the equation of the tangent(s) to x^2 + y^2 = 4, which are parallel to y= x + 3
This Q can be solved geometrically. There is also an algebraic method. If not solved by a 2010 student by tomoz, i will show both
 

cutemouse

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Re: 2 Unit Revising Marathon HSC '10

Find the equation of the tangent(s) to x^2 + y^2 = 4, which are parallel to y= x + 3
2U (ie. long) method:

All lines parallel to y=x+3 are of the form y=x+k, where k is a constant.

Thus if y=x+k is a tangent to x^2+y^2=4 the quadratic formed by subsituting y=x+k into x^2+y^2=4 will have a discriminant of zero.

Now sub y=x+kinto x^2+y^2=4

Thus we have x^2+(x+k)^2=4
x^2+x^2+2xk+k^2-4=0
2x^2+2xk+k^2-4=0

Δ=(2k)^2-4.2.(k^2-4)
=4k^2 - 8k^2 + 32
= -4k^2 + 32

But Δ=0
Thus 4k^2=32
k^2=16
k=+/- 4

Thus the required equation of tangent is y=x+4 or y=x-4
 

Trebla

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Re: 2 Unit Revising Marathon HSC '10

If a(x) and b(x) are even functions while c(x) and d(x) are odd functions, prove that:

(i) a(x).b(x) is an even function

(ii) a(x).c(x) is an odd function

(iii) c(x).d(x) is an even function
 

cutemouse

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Re: 2 Unit Revising Marathon HSC '10

If a(x) and b(x) are even functions while c(x) and d(x) are odd functions, prove that:

(i) a(x).b(x) is an even function

(ii) a(x).c(x) is an odd function

(iii) c(x).d(x) is an even function
If a(x) and b(x) are even functions, a(-x)=a(x) and b(-x)=b(x)
If c(x) and d(x) are odd functions, c(-x)=-c(x) and d(-x)=-d(x)

(i) Let f(x)=a(x).b(x)
Thus f(-x)=a(-x).b(-x) = a(x).b(x) = f(x)

(ii) Let f(x) = a(x).c(x)
Thus f(-x)=a(-x).c(-x) = a(x).-c(x) = -a(x).c(x) = -f(x)

(iii) Let f(x)=c(x).d(x)
Thus f(-x)=c(-x).d(-x) = -c(x).-d(x) = c(x).d(x) = f(x)

Question: Solve for x: |x - 1| = -5
 
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Re: 2 Unit Revising Marathon HSC '10

x = - 4 and 6

Find the eqn of the locus of the point so that it is equidistant from points 3, 2 and -1, 5
 
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Re: 2 Unit Revising Marathon HSC '10

Really? I'm not thinking well...
Our teacher taught us that you must find
x - 1 = -5 normally without absolute values
and then find
-(x - 1) = -5

Hmm. How'd you do it?
 

ninetypercent

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Re: 2 Unit Revising Marathon HSC '10

^ there is no solution
as -5 is negative
 

ninetypercent

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Re: 2 Unit Revising Marathon HSC '10

Find the eqn of the locus of the point so that it is equidistant from points 3, 2 and -1, 5


Find the locus of the points that move equidistant from the line y = 3 and the point (3,4)
 

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