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#1 2012-03-23 18:13:21

Denominator
Member
Registered: 2009-11-23
Posts: 155

Stuck on previous exam questions!

Hey guys!

Well I've been trying to do previous exams as practice for the one I have next week.

I came across 2 questions, of which I have no idea how to solve.

If you guys could show working and explanations so that I understand as well, that'll be great.

Q 1
While attempting to solve a quadratic equation, Christobel inadvertently interchanged the coefficient of x² with the constant term, causing the equation to change.
She solved this different equation accurately.
One of the roots she got was 2 and the other was a root of the original equation.
What are the sum of the squares of the two roots of the original equation?

I tried letting the first equation be ax²+bx+c=0 and the other be cx²+bx+a=0 but then I didn't know what to do...

Q 2
A goods train leaves Southampton at 9.17 am and arrives in London at 12.02 pm.
On the same day, a passenger express train leaves Southampton at 9.56 am and arrives in London at 11.36 am.
At what time does the passenger train pass the goods train if each is travelling non-stop at constant speed?

Yeah... I didn't even bother trying this question, I have no clue as to what variables I need to make or such..

Any help is appreciated. Thanks guys.

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#2 2012-03-23 21:14:58

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

Re: Stuck on previous exam questions!

Hi;

I have a question are a,b,c integers?


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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#3 2012-03-23 21:27:26

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Stuck on previous exam questions!

Yeah, probably

I used the quadratic formula to solve for the two equations,
but I don't know which one to be 2 or to be equal with the other roots from the other equation

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#4 2012-03-23 21:31:30

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

Hi Denominator

You should probably not done that. Instead you should have x-2 as a factor of that second polynomial.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#5 2012-03-23 21:31:36

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,442

Re: Stuck on previous exam questions!

hi Denominator,

I'll have a go at these but it'll be in about half an hour if that's OK with you.

No problem whether a, b and c are integers or not.

Back shortly.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#6 2012-03-23 21:38:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

Hi Denominator


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#7 2012-03-23 21:41:30

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Stuck on previous exam questions!

Oh okay anonimnystefy,

so I make the equation: (x-2)(ax+b)=0
ax²+(b-2a)x-2b=0
And the other equation is -2bx²+(b-2a)x+a=0
Which factorises to (bx+a)(-2x+1)=0
And the solutions are -a/b and 1/2
Okay so either -b/a=-a/b or -b/a=1/2
Doesn't that mean theres still many solutions for a and b?

Also, I don't have the answers so I'm trusting you guys ;S

Last edited by Denominator (2012-03-23 21:42:17)

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#8 2012-03-23 21:42:45

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

Re: Stuck on previous exam questions!

Hi;


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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#9 2012-03-23 21:46:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

What are those?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#10 2012-03-23 21:49:39

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

Re: Stuck on previous exam questions!

The sums of the squares of the roots, I think.


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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#11 2012-03-23 21:59:13

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

Look at mine.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#12 2012-03-23 22:05:03

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,442

Re: Stuck on previous exam questions!

hi Denominator

This is the corrected version.

let the equations be

and

multiplying out

and

equating coefficients gives

from (ii) and (iii)

and putting (ii) into (i) gives

Putting (iv) into (v) gives

which looks a bit nasty.  But you can easily spot that R - 1 is a factor which leaves a quadratic, which factorises easily and it should be plain sailing from there.

Bob

ps.  Now to read number 2


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#13 2012-03-23 22:05:44

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

Re: Stuck on previous exam questions!

Hi anonimnystefy;

Who should be looking?


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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#14 2012-03-23 22:08:19

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

You, m.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#15 2012-03-23 22:10:46

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

Re: Stuck on previous exam questions!

There are other solutions if we relax the integer coefficients idea. What are your quadratic two equations?


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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#16 2012-03-23 22:14:07

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,039

Re: Stuck on previous exam questions!

They are both 2x^2-5x+2.

Last edited by anonimnystefy (2012-03-23 22:14:34)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#17 2012-03-23 22:15:24

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

Re: Stuck on previous exam questions!

That equation is illegal.

You also should have posted a new thread instead of tacking on to an active one.


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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#18 2012-03-23 22:20:30

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,442

Re: Stuck on previous exam questions!

hi Denominator

trains:

For questions like this I like to sketch a distance time graph so i get a good picture in my head of what is happening.  see below.

You have enough data to work out the gradients for the two lines.  And you know that at the moment of passing the 't' values and the 'd' values are equal.

So you could make two equations for d/t based on the gradients and solve for d and t.

Bob

View Image: Denominator.gif

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#19 2012-03-24 07:19:00

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Stuck on previous exam questions!

Oh okay thanks guys!

Yeah, after using your methods, I get 5/4 as well

I tried doing something different:
Let the solutions of the different equation be X and 2
The solutions of the original equation is 1/X and 1/2 (after playing around, I found this rule)
So 1/X=X
Which means X=1 or -1
Sum of squares (1)²+(1/2)²=5/4

Btw even though 2x²-5x+2 works, the question says 2 different equations, 1 original and the other with the coefficients swapped... this stays the same.
Thanks a lot guys!

As for Q2
Can you tell me what equations you formed, because I couldn't make some..

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#20 2012-03-24 07:31:29

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

Re: Stuck on previous exam questions!

Hi;

Isn't 5 also an answer?


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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#21 2012-03-24 09:29:24

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Stuck on previous exam questions!

I don't think so because the question asks for the sum of the roots of the ORIGINAL equation, not the different equation
That is if you're thinking the roots are 2 and 1, but that is from the different equation, not the original

Unless your saying the roots of the original equation are 1/2 and √19 / 2
Which I don't think is right

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#22 2012-03-24 09:36:31

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

Re: Stuck on previous exam questions!

Why can't you switch the equations calling  the second one the original one like this?

Original equation:

Different equation:


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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#23 2012-03-24 09:40:09

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Stuck on previous exam questions!

Because the question says that the different equation has to have 2 as a root

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#24 2012-03-24 09:44:46

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

Re: Stuck on previous exam questions!

Hi;

Okay, yes you are right, I did not see that.



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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#25 2012-03-24 12:07:49

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,442

Re: Stuck on previous exam questions!

hi Denominator

Let the distance from Southampton to London be D, and the distance to the overtaking point be d.

Let the time in minutes from 9.17 to the overtaking point be t

The goods train takes 165 minutes
The express train takes 100 minutes

So the gradient (velocity in miles / min) for the goods train is

and for the express train

You can eliminate both D and d by dividing these equations:

and then solve for t.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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