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    (Original post by IrrationalRoot)
    I've done virtually every pure one from 1991 to 2013 and have never seen this? Where did you learn this idea?
    STEP I, 2002, Q2. Makes it a c. 8 minute question if you use logarithmic differentiation.

    I learnt it from random experimenting. :dontknow:
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    (Original post by Zacken)
    STEP I, 2002, Q2. Makes it a c. 8 minute question if you use logarithmic differentiation.

    I learnt it from random experimenting. :dontknow:
    That idea iz just so damn cool. OMFG
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    This is quickly turning into the 'Zain shares all his wisdom' thread :lol:
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    (Original post by Kvothe the arcane)
    Duke Glacia, Zacken

    Lovely idea. perhaps hyperlink the OP with relevant post urls?
    Yes! Brilliant, done that now.
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    (Original post by Zacken)
    Use the fact that some trigonometric functions are shifts of another, common examples include...
    Spoiler:
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    This is going to have people wondering why it has rep.
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    Here's one idea which I found particularly useful.(Tho it may be very obvious)

    Use of Determinants to show that the y>0 -or- y<0 (depending on the coefficient of x^2 )for all values of x in a quadratic.

    b^2 -4ac<0

    Here's one Question where this may come handy.

    Whats the minimum angles to the vertical for a projectile(ball) to be realeased with speed v such that at any point of time the distance to the ball is increasing.
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    (Original post by EricPiphany)
    ..
    Embarrassing; thanks and fixed now.
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    (Original post by EricPiphany)
    ..
    I noticed that too lol.
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    (Original post by Duke Glacia)
    Here's one idea which I found particularly useful.(Tho it may be very obvious)

    Use of Determinants to show that the y>0 for all values of x in a quadratic.
    Careful now, if the coefficient of x^2 is negative, then you're showing that y < 0 for all x.
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    (Original post by IrrationalRoot)
    Never came across the idea of logarithmic differentiation for products. Was this featured in any STEP questions?
    The best use of this is when you want to find stationary points of a function; those for f(x) are the same as log f(x) (or indeed, any strictly monotonic function, but logs simplify things for the reasons Zacken has already said). In uni stats you'll see it all the time in maximum likelihood questions (or I believe OCR MEI).

    For integrals, don't forget the obvious such as Weierstrass's substitutions (https://en.m.wikipedia.org/wiki/Tang...e_substitution)
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    (Original post by Zacken)
    Careful now, if the coefficient of x^2 is negative, then you're showing that y < 0 for all x.
    yikes
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    (Original post by Kvothe the arcane)
    Duke Glacia, Zacken

    Lovely idea. perhaps hyperlink the OP with relevant post urls?

    He edits very quickly .
    Didnt get what u mean but amazing idea kvoth(saw the edit on OP). Thnx !
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    Given any quadratic polynomial:
    f(x) = ax^2+bx+c

    The sum of the roots is -\frac{b}{a}.
    The product of the roots is \frac{c}{a}.

    I can't remember a particular question, but there are a few questions which by using this allows you to skip a lot of algebra.
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    Zacken is such a lad
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    (Original post by Alex:)
    Given any quadratic polynomial:
    f(x) = ax^2+bx+c

    The sum of the roots is -\frac{b}{a}.
    The product of the roots is \frac{c}{a}.

    I can't remember a particular question, but there are a few questions which by using this allows you to skip a lot of algebra.
    Isn't this just standard FP1?
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    (Original post by zetamcfc)
    Isn't this just standard FP1?
    Not that I can recall.
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    Inequalities:

    One key thing that many people seem to fail to grasp (or at least that I've noticed) is that you can use multiple inequalities to show one inequality, especially when your intuition tells you that the inequality is very loose.

    So, for example - if I needed to show that n! \geq n (stupid example, but I can't think of any good ones right now) then you could say that n! > 2^n (let's say it was in a question that was asking you to prove Sterling or some ****) then you could say n! > 2^n > n. I'll see if I can think up some good example of this later.

    Another thing to be aware of in STEP is to be very careful vis whether you're applying monotone functions to both sides of an inequality. So for example, it is true that (over the reals) x^3 > y^3 \Rightarrow x > y but it is not true that (over the reals) x^2 > y^2 \Rightarrow x > y since the x \mapsto x^3 function is injective (but more importantly: increasing) over the reals and hence so is its inverse, but the x \mapsto x^2 function doesn't have that same privilege.

    If you're ever making a claim like x^3 > y^3 \Rightarrow x > y in STEP, then make sure to say something like x^3 is increasing or such; or x^2 > y^2 \Rightarrow x > y then say something like because x,y > 0 or x^2 is increasing over \mathbb{R}^{+}.

    A good way of establishing inequalities in STEP is to find a minimum or maximum and then show that the function is strictly decreasing or increasing from that point by considering the derivative over the interval. This comes in useful quite a bit as well.
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    (Original post by Alex:)
    Not that I can recall.
    It is, very standard FP1 - it's also called Vieta's forumla but you're not taught the name at A-Level.
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    (Original post by Alex:)
    Not that I can recall.
    It is for AQA anyway, just looked at the spec.
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    Summing arctans is useful:

    \arctan a + \arctan b = \arctan \left( \frac{a + b}{1-ab} \right) + n\pi

    Which can be derived from the tangent compound angle formula.
 
 
 
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