2 Unit Revising Marathon (4 Viewers)

[jet]

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]

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]

Re: 2 Unit Revising Marathon HSC '10

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

a = -1 and b =1

 

[Dragonmaster262]

Re: 2 Unit Revising Marathon HSC '10

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

LaTeX

 

[gurmies]

Re: 2 Unit Revising Marathon HSC '10

a = -1 and b =1

Please read the guidelines of this exercise, you must post a question upon answering one correctly (which you did).

 

[Dragonmaster262]

Re: 2 Unit Revising Marathon HSC '10

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

 

[addikaye03]

Re: 2 Unit Revising Marathon HSC '10

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

u=x^2, v=ln(x)

u'=2x, v'=1/x

y'=(vu'-uv')/v^2 (chain rule)

=[2xln(x)-x]/(lnx)^2

=x[2ln(x)-1]/(lnx)^2

Question; Solve: tanx=3cotx

 

[gurmies]

Re: 2 Unit Revising Marathon HSC '10

u=x^2, v=ln(x)



u'=2x, v'=1/x



y'=(vu'-uv')/v^2 (chain rule)



=[2xln(x)-x]/(lnx)^2



=x[2ln(x)-1]/(lnx)^2



Question; Solve: tanx=3cotx

Quotient rule you mean xD

 

[Dragonmaster262]

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.

 

[addikaye03]

Re: 2 Unit Revising Marathon HSC '10

Quotient rule you mean xD

ROFL! Yeah that's the one!
Also i posted a Q

 

[ninetypercent]

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

 

[addikaye03]

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]

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]

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]

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

 

[violentdelights]

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

 

[cutemouse]

Re: 2 Unit Revising Marathon HSC '10

x = - 4 and 6

Wrong.

 

[violentdelights]

Re: 2 Unit Revising Marathon HSC '10

Wrong.

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]

Re: 2 Unit Revising Marathon HSC '10

^ there is no solution
as -5 is negative

 

[ninetypercent]

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)