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

ganesh
Registered: 2005-06-28
Posts: 23,082

### 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?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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

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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 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

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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) 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

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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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

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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

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

Fruityloop
Member
Registered: 2009-05-18
Posts: 135

### 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...

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.

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

The eclipses from Algol 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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