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#1 2009-02-06 03:19:29

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,534

Question Bank : Age Group 16-17 : I

1. Find the angle between the asymptotes to the hyperbola 3x² - 5xy -2y² + 17x + y + 14 = 0.

2. Two sides of a triangle are 4 meters and 5 meters in length and the angle between them is increasing at the rate of 0.06 radians/second. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length if

.

3. If

,
find
.

4. Solve:

.

5. Find the equation of the ellipse whose foci are (4,0) and (-4,0) and

.

6. For what values of x is the rate of increase of


twice the rate of increase of x?


Character is who you are when no one is looking.

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#2 2009-05-06 09:06:17

quittyqat
Member
Registered: 2009-04-08
Posts: 1,213

Re: Question Bank : Age Group 16-17 : I

Ganesh, can you make a question bank for younger kids?


I'll be here at least once every month. XP

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#3 2009-08-15 02:32:40

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Question Bank : Age Group 16-17 : I

Hi ganesh
For the second problem:
Assume that the angle between the  sides of fixed length is θ, so
A = ½(4)(5) sin θ=10 sin θ where  A is the area of triangle at any time ,  so 
dA/dt =10 cos θ  (dθ/dt) when θ=pi/3 and dθ/dt = 0.06
dA/dt =10 cos (pi/3) * 0.06
         =10 (½)(0.06)=0.3

Best Regards
Riad Zaidan

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#4 2009-08-15 22:03:58

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Question Bank : Age Group 16-17 : I

Hi ganesh
For the third problem:
w=x+2y+z^2  ,   x= cos(t) ,  y = sin(t)   ,   z=t

dw/dt sad dw/dx)(dx/dt)+ (dw/dy)(dy/dt)+( dw/dz)(dz/dt)
   
         =1(-sin (t) + 2(cos (t)  + 2 z (1)

          = -sin(t)+2 cos(t) + 2z 

Best Regards
Riad Zaidan

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#5 2009-08-15 22:11:24

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Question Bank : Age Group 16-17 : I

Hi ganesh
For the forth problem:
dy/dx  + xy =x
dy/dx=x-xy=x(1-y)
dy/(1-y )=x dx
∫dy/(1-y )=∫x dx
-ln(1-y) sad x²) /2 + c
ln(1/(1-y))=( x²) /2 + c ⇒
1/(1-y)=e^(( x²) /2 + c) ⇒
1-y = 1/(e^(( x²) /2 + c))
y=1-1/(e^(( x²) /2 + c)) and you can simplify more
Best Regards
Riad Zaidan

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#6 2009-08-15 22:24:04

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,238

Re: Question Bank : Age Group 16-17 : I

Hi ganesh;

We were working on this one at the same time but Riad was faster.

For #4;

Use separation of variables.

Integrate both sides:

Raise both sides to the power of e.

c= e^c

Invert both sides and solve for y:
Solution is:

Last edited by bobbym (2009-08-15 22:26:56)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2009-08-16 08:00:57

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Question Bank : Age Group 16-17 : I

Hi ganesh;

For the sixth problem:

Assume that

y= x^3-5x^2+5x+8 and differentiate both sides w.r.t  (t) we get:

dy/dt=(3 x^2 - 10 x + 5) (dx/dt)  ...........(1)

but  dy/dt  =2 *  (dx/dt)  so  substitute  in   (1)    we  have the following:

  2  (dx/dt) =(3 x^2 - 10 x + 5) (dx/dt)  so  if dx/dt ≠0 we  get 

2   =  3 x^2 - 10 x + 5  therfore

3 x^2 - 10 x + 3 = 0  so   

(3x-1)(x-3)=0   so   

either  x= 1/3   or    x= 3    Q.E.D 

Best Regards
Riad Zaidan

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#8 2009-08-16 08:09:07

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Question Bank : Age Group 16-17 : I

Hi ganesh;

For the fifth problem:


The  foci are (4,0) and (-4,0) and  e=1/3 the foci are on the x-axis with center on (0,0)

   c = 4  but   e=c/a   so   c/a= 1/3   and  we have 

4/a=1/3   so  a =12 but  c^2 = a^2 - b^2 or  b^2=a^2-c^2=144-16=128

so the requiered equation is

(x^2)/144 +  (y^2)/128   =    1


Best Regards
Riad Zaidan

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#9 2009-09-10 19:52:38

Fruityloop
Member
Registered: 2009-05-18
Posts: 120

Re: Question Bank : Age Group 16-17 : I

1. Find the angle between the asymptotes to the hyperbola 3x² - 5xy -2y² + 17x + y + 14 = 0.

This is honestly the most lengthy, involved problem I've ever worked on.  I never studied the general conic equation while in school, so this is new for me.  Hopefully I've done everything correctly.
First we need to get rid of the xy term by doing a rotation of axes..





using the formulae for the axes of rotation


We then substitute these into the given equation and after some work we get..

The x'y' terms cancel and after much, much more work we end up with...
(I changed x' and y' to simply x and y)

Now here I had a little help from the computer and amazingly...
eek
continuing further along we eventually end up with...

So the slope of the asymptotes are

The hyperbola is oriented parallel to the y-axis and to get the angle between the asymptote and x-axis we use

So the angle between the asymptotes is


or on the other side the angle is

I hope I did everything right.  I wouldn't wish this problem upon my worst enemy.dizzy

Last edited by Fruityloop (2009-09-10 21:36:40)


The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

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