Mathematics Applicants 2012

What method did you use to work out that $x^4+12x^3+44x^2+48x+16=(x^2+6x+4)^2$? I didn't have a clue how to do that bit.

(Think this is probably a really stupid question ).

hey does anyone know any step questions that don't need C3 trig or C4 knowledge (not done it yet)
thanks

The question implies that it can be written in the form [quadratic]^2. It is also fairly obvious that the coefficients are going to be integers so right away we know that the first term is x^2 and the last is 4. We are left with one other coefficient to find, I was feeling lazy at the time so I just called it a, expanded the expression, equated coefficients to give me a=6. Some people can just factorise things like that in their heads.

I'm defiantly applying to do maths finished my personal statement today . Currently thinking kings ucl royal holloway city and Kingston

A small goat is tethered by a rope to a fixed point at ground level on a square barn which stands in a horizontal field of grass. The sides of the barn are of length 2a and the rope is of length 4a. Let A be the area of the grass that the goat can graze. Prove that A <(or equal to) 14π(a^2) (π=pi) and determine the minimum value of A.

Here's a hint if you are struggling.



The answer will require a diagram so I can't really type it up clearly, but if anyone wants my answer I will happily provide although it may not make much sense if you aren't thinking along the right lines.

Gcse's: 3A*s 3As 2Bs
A levels: Maths A*(finished a year early), Further Maths, Physics, chemistry
Applying to: Cambridge, Bath, Durham, Southampton, Lancaster

The question implies that it can be written in the form [quadratic]^2. It is also fairly obvious that the coefficients are going to be integers so right away we know that the first term is x^2 and the last is 4. We are left with one other coefficient to find, I was feeling lazy at the time so I just called it a, expanded the expression, equated coefficients to give me a=6. Some people can just factorise things like that in their heads.

That was obviously too easy

This is one of my favourites: Prove
$\int^{\infty}_{-\infty} e^{-((x^2)/2)} \ dx = \sqrt{2\pi}$

That was obviously too easy

This is one of my favourites: Prove
$\int^{\infty}_{-\infty} e^{-((x^2)/2)} \ dx = \sqrt{2\pi}$

I'm defiantly applying to do maths finished my personal statement today . Currently thinking kings ucl royal holloway city and Kingston

Meh, bookwork. You turn it into a double integral and convert to polar coordinates. That result is too famous to ask as a brain teaser.
Beautiful result though.

Ok then. Prove it without any change of variable or transform.

I'd have to be Gauss to do that