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# Hard maths questions for higher maths GCSE watch

1. (Original post by atsruser)
All of the questions that I've posted can be done with GCSE knowledge + insight. It's the insight that makes them hard, of course.
When I said the thread had gone off topic, I was referring to the posts above me e.g. your post #72.
2. A computer password for a top secret system comprises 50 randomly chosen digits, each in the range 0-9. There are distinct such passwords.

A computer cracker builds a powerful computer to break into the system. It tries each password in turn until the correct one is found. It can try a million million million passwords per second, and it is left running for a million million million years.

To the nearest one millionth of one percent, what percentage of passwords are still left to be tried, after this period of time?
3. Circle A is inscribed inside a square of side 2 so that it is tangent to all four sides. A smaller circle, B, is inscribed in the gap between circle A and the two sides of the square at one of the corners. Circle B is tangent to circle A and to the two sides of the square.

Find the radius of circle B.
4. Solve
5. 15 - 6 = 11

I'm sorry. It just is. No, it just is.
6. (Original post by atsruser)
Circle A is inscribed inside a square of side 2 so that it is tangent to all four sides. A smaller circle, B, is inscribed in the gap between circle A and the two sides of the square at one of the corners. Circle B is tangent to circle A and to the two sides of the square.

Find the radius of circle B.
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0.21?
(Original post by atsruser)
Solve
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Is n just 7? Since sqrt(7^2 - 4*45) gives you an irrational number?

(Original post by atsruser)
A computer password for a top secret system comprises 50 randomly chosen digits, each in the range 0-9. There are distinct such passwords.

A computer cracker builds a powerful computer to break into the system. It tries each password in turn until the correct one is found. It can try a million million million passwords per second, and it is left running for a million million million years.

To the nearest one millionth of one percent, what percentage of passwords are still left to be tried, after this period of time?
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I got 0.000032%
7. (Original post by TheOtherSide.)
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Is n just 7? Since sqrt(7^2 - 4*45) gives you an irrational number?
I would presume so, a small note however that the other roots are actually complex, not irrational.
8. (Original post by joostan)
I would presume so, a small note however that the other roots are actually complex, not irrational.
My bad! I'll keep that in mind when doing something like this again...
9. (Original post by TheOtherSide.)
My bad! I'll keep that in mind when doing something like this again...
It's not really an issue for GCSE level, I just thought I'd mention it for interest's sake, by definition an irrational number is by definition real.
10. (Original post by joostan)
It's not really an issue for GCSE level, I just thought I'd mention it for interest's sake, by definition an irrational number is by definition real.
If I'm being honest, I still don't get the difference between a complex number and an irrational number - do complex numbers only show up when working out quadratics?
11. (Original post by joostan)
It's not really an issue for GCSE level, I just thought I'd mention it for interest's sake, by definition an irrational number is by definition real.
To be fair, it's usually by convention that or so being irrational would be a sufficient condition to discard those solutions, I think.
12. (Original post by TheOtherSide.)
If I'm being honest, I still don't get the difference between a complex number and an irrational number - do complex numbers only show up when working out quadratics?
An irrational number is a number which cannot be expressed as the ratio, of two integers.
A complex number is a generalisation of the real numbers which gives every polynomial of order exactly solutions in the complex plane. Essentially one defines an imaginary number and proceed from there, though that is a discussion best suited to another thread.

(Original post by Zacken)
To be fair, it's usually by convention that or so being irrational would be a sufficient condition to discard those solutions, I think.
Agreed, though I tend to be a bit nit-picky, guess the pure courses have rubbed off on me :/
13. (Original post by TheOtherSide.)
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0.21?
That's not what I got. I'll check my result - it came to a nice exact answer, with surds.

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Is n just 7? Since sqrt(7^2 - 4*45) gives you an irrational number?
Yes but...
Hint:
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rewrite as then factorise both sides fully

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I got 0.000032%
I think that you misread the question. And that doesn't look quite right, even if you are trying to compute the % finished.

It would be easier to comment if you put up some working.
14. (Original post by joostan)
Essentially one defines an imaginary number and proceed from there, though that is a discussion best suited to another thread.
You can definitely ignore complex solutions here - it's a GCSE thread.
15. (Original post by atsruser)
That's not what I got. I'll check my result - it came to a nice exact answer, with surds.
I did get an exact answer with surds, except I wasn’t sure howto put it down: (-1 + sqrt2)/2

Yes but...
Hint:
Spoiler:
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rewrite as then factorise both sides fully
That’s what I did at the beginning, except I didn’t know whereto go from n(n+2)(n-2) = 315, so I ended up using the factor theorem to get(n-7) and then used long division to get a quadratic.
I think that you misread the question. And that doesn't look quite right, even if you are trying to compute the % finished.

It would be easier to comment if you put up some working.
I still haven’t familiarised myself with latex, so apologiesabout not being able to display everything in a way that’s easy to understand. Amillion million million passwords is the same as 10^17 passwords per second. Ithen multiplied this by 31536000, which is the number of seconds in a year, andthen multiplied by another 10^17 for the number of years. I divided all this by10^50 and multiplied by 100 to get a percentage. I’m not sure what else I could have done, unless I didmisread the question, or maybe made a stupid mistake.
16. (Original post by TheOtherSide.)
I did get an exact answer with surds, except I wasn’t sure howto put it down: (-1 + sqrt2)/2
I get

That’s what I did at the beginning, except I didn’t know whereto go from n(n+2)(n-2) = 315, so I ended up using the factor theorem to get(n-7) and then used long division to get a quadratic.
You didn't factorise 315 - do that, then use the fact that there is only one way to factorise both sides.

You can't use the factor theorem here - this is a GCSE thread, so you must restrict yourself to GCSE methods.

I still haven’t familiarised myself with latex, so apologiesabout not being able to display everything in a way that’s easy to understand. Amillion million million passwords is the same as 10^17 passwords per second. Ithen multiplied this by 31536000, which is the number of seconds in a year, andthen multiplied by another 10^17 for the number of years. I divided all this by10^50 and multiplied by 100 to get a percentage. I’m not sure what else I could have done, unless I didmisread the question, or maybe made a stupid mistake.
Actually I have noticed a typo in my question so it didn't quite work out as nice as I wanted it. But yes, you have badly misread the question. What does it ask you to find?

[edit: and you might want to check your powers - what is a million million million in powers of 10?]
17. This
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18. This question on counters
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19. This calculator paper question
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20. (Original post by OGGUS)
This
Is this correct? I skipped some steps as it isn't an exam.
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