STEP Tricks and Hacks Thread

OP

Original post by Alex:

Fair enough. It's not mentioned at all in Edexcel.

'tis the easiest topic in Edexcel FP1.

Reply 42

zetamcfc

19

Idk if this is any use for STEP but

\ \frac{d}{dx}( \arctan(f(x))) = \frac{d}{dx}( -\arctan(\frac{1}{f(x)}))

Reply 43

username1162972

Original post by Zacken

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.

Illdo complex numbers/ geometry :
:wink:

Reply 44

DylanJ42

Original post by 16Characters....

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.

III, 2002, Q2 IIRC (done this question yesterday after seeing it mentioned in the main STEP thread)

Reply 45

joostan

13

Since no one's mentioned it yet, L'Hôpital's rule is a sometimes convenient way of computing the limit of a function as it tends to a specific value and I'm pretty sure can be quoted without proof.

Reply 46

16Characters....

13

Original post by DylanJ42

III, 2002, Q2 IIRC (done this question yesterday after seeing it mentioned in the main STEP thread)

That's the question I first met it on (also, I loved III 2002). It's come up a few times though.

Reply 47

Ayman!

15

Original post by Alex:

Fair enough. It's not mentioned at all in Edexcel.

It's actually in the book in the complex numbers chapter. The knowledge is compulsory in Edexcel IAL FP1 and was also a part of my IGCSE course. Very useful technique for polynomials.

Reply 48

Ayman!

15

Not a STEP-er but I'm joining this thread to pick up on useful techniques. :biggrin:

Here's an easy one that might be helpful:

Factorising

x^{4} + 1

and polynomials of the like -

It's fairly easy to write out

\displaystyle x^{4} + 1 \equiv \left ( x^{2} + \alpha x +1 \right )\left( x^{2}+\beta x +1 \right )

and then compare coefficients using C3 techniques, but in case you're feeling adventurous,

Original post by Zacken

\displaystyle x^4 + 1 = x^4 + 2x^2 + 1 - 2x^2 = (x^2 + 1)^2 - (\sqrt{2}x)^2 = (x^2 + \sqrt{2}x +1)(x^2 - \sqrt{2}x + 1)

This could come in handy in certain integrals. Try the classic

\displaystyle \int \sqrt{\tan x} \ \mathrm{d}x

using this. :h:

(edited 8 years ago)

Reply 49

shamika

16

How about strong induction? (Look it up, can't be bothered to explain it properly :tongue:

)

Oh: and modular arithmetic. I'll let Zacken do the honours and explain both :smile:

Reply 50

username1162972

Original post by aymanzayedmannan

Not a STEP-er but I'm joining this thread to pick up on useful techniques. :biggrin:

Here's an easy one that might be helpful:

Factorising

x^{4} + 1

and polynomials of the like -

It's fairly easy to write out

\displaystyle x^{4} + 1 \equiv \left ( x^{2} + \alpha x +1 \right )\left( x^{2}+\beta x +1 \right )

and then compare coefficients using C3 techniques, but in case you're feeling adventurous,

This could come in handy in certain integrals. Try the classic

\displaystyle \int \sqrt{\tan x} \ \mathrm{d}x

using this. :h:

Along those lines.. An very old IMO question when IMO wasn't that hard.
Show that infinite set of integers a exist such that the sequence Z_n=n^4+a ,n>=0 contains no prime numbers.

Reply 51

OP

Strong induction v/s weak induction

Okay, so we're taught this version of induction in school/at A-Level:

We have a proposition

P(n)

, that we want to show is true for all

n \in \mathbb{N}

, to do so:
Base case: Show that

P(1)

is true.
Induction: Assume that

P(k)

is true and show that this assumption implies that

P(k+1)

is true.

This then means that

P(1) \Rightarrow P(2) \Rightarrow P(3) \Rightarrow \cdots \Rightarrow P(n)

for all natural

n

and this makes sense to us.

However, sometimes, this isn't enough and we need to use a different kind of induction with a slight twist:

We have a proposition

P(n)

, that we want to show is true for all

n \in \mathbb{N}

, to do so:

Base case: Show that

P(1)

is true.
Induction: Assume that

P(1), P(2), \ldots, P(k)

is true and show that this assumption implies that

P(k+1)

is true.Here we're using that not only the first case is true, but that the second, third up to the

k

th term is true and these being true implies the next one is true, and the next - etc... it's not a very big leap to make from the normal induction we know.

When is it useful?

Strong induction is useful when the result for

n = k-1

depends on the result for some smaller value of

n

which isn't

k-2

.

Examples?

Let's go fundamental :cool:

Let

P(n)

be the proposition that every positive integer

n \geq 2

can be expressed as a product of one or more prime factors.

Base:

2 = 2

Induction: Suppose every integer between

2

and

k

can be written as a product of prime factors. We now have two cases:

Case 1:

k+1

is prime:

k+1 = k+1

and we are done.

Case 2:

k+1

is not prime: then

k+1 = ab

for some

1 < a,b < k+1

but by our induction hypothesis we can say that

a = p_1^{\alpha_1} p_2^{\alpha_2} p_3^{\alpha_3} \cdots p_a^{\alpha_a}

and

b = q_1^{\beta_1}q_2^{\beta_2} \cdots q_b^{\alpha_b}

so that

k+1 = p_1^{\alpha_1}q_1^{\beta_1} \cdots p_a^{\alpha_a}q_b^{\beta_b}

. In both cases we have what we want and so we are done.

Example of usage: STEP III, 2008, Q2

(edited 8 years ago)

Reply 52

DylanJ42

Original post by 16Characters....

That's the question I first met it on (also, I loved III 2002). It's come up a few times though.

yea i worked through a few of the questions (most partially) because i couldn't be bothered to do a mock yesterday :spank:

i really enjoy those types of manipulations ie "tan-ing" both sides or more commonly, taking logs of both sides. idk why it just feels good :facepalm:

Reply 53

Insight314

7

@Zacken, @Duke Glacia,

I was literally trying to find a thread/website with STEP Tips and Tricks yesterday. Thank you. I am really glad that you've decided to make one, and lucky that Duke Glacia accidentally tagged me in the OP (what is up with that?) since I do not regularly visit TSR to look for such thread.

Zacken, I think it is really helpful to give an "extension" problem like you did in "2. Trigonometrical symmetry and substitution for integration" since the main point of STEP questions is largely to learn a technique and then be able to apply it.

Recommendation: Do you have any Tricks and Hacks on questions with simultaneous equations, I often just skip them because I am not sure how to attempt them yet. Here is an example (Question 1 STEP III 2008) which I just tried doing before going to sleep and was easy at first (due to the hint) until you get all the equations and have no idea what to do with them. I know this is quite a general recommendation but I guess you could refer to different techniques of factorisation or whatever that is needed when attempting such questions.

I think it would be nice for people to recommend tricks on here, or maybe another thread linked to this.

If we get quite a few Tricks and Hacks on here, it is also possible to make it into a pdf through ShareLatex and type it out in LaTeX just like Stephen Siklos did; I wouldn't mind typing it out, if you want me to. It would be pretty useful to revise through on the day of the exam. :smile:

And, again, thank you for making this thread - you have no idea how useful this is going to be for me!

Reply 54

Abscissa

6

Original post by Insight314

@Zacken, @Duke Glacia,

I was literally trying to find a thread/website with STEP Tips and Tricks yesterday. Thank you. I am really glad that you've decided to make one, and lucky that Duke Glacia accidentally tagged me in the OP (what is up with that?) since I do not regularly visit TSR to look for such thread.

Zacken, I think it is really helpful to give an "extension" problem like you did in "2. Trigonometrical symmetry and substitution for integration" since the main point of STEP questions is largely to learn a technique and then be able to apply it.

Recommendation: Do you have any Tricks and Hacks on questions with simultaneous equations, I often just skip them because I am not sure how to attempt them yet. Here is an example (Question 1 STEP III 2008) which I just tried doing before going to sleep and was easy at first (due to the hint) until you get all the equations and have no idea what to do with them. I know this is quite a general recommendation but I guess you could refer to different techniques of factorisation or whatever that is needed when attempting such questions.

I think it would be nice for people to recommend tricks on here, or maybe another thread linked to this.

If we get quite a few Tricks and Hacks on here, it is also possible to make it into a pdf through ShareLatex and type it out in LaTeX just like Stephen Siklos did; I wouldn't mind typing it out, if you want me to. It would be pretty useful to revise through on the day of the exam. :smile:

And, again, thank you for making this thread - you have no idea how useful this is going to be for me!

I used the hint on the third equation and then found the sum and product of x and y to make an quadratic equation for which I could solve.

Reply 55

OP

Original post by Insight314

Zacken, I think it is really helpful to give an "extension" problem like you did in "2. Trigonometrical symmetry and substitution for integration" since the main point of STEP questions is largely to learn a technique and then be able to apply it.

Awesome, I'll keep that in mind - I wrote all my posts yesterday in a bit of a rush and didn't have time to come up with interesting questions for all of them, but I'll try and go back and add questions to my existing posts, so keep an eye on them.

Recommendation: Do you have any Tricks and Hacks on questions with simultaneous equations, I often just skip them because I am not sure how to attempt them yet. Here is an example (Question 1 STEP III 2008) which I just tried doing before going to sleep and was easy at first (due to the hint) until you get all the equations and have no idea what to do with them. I know this is quite a general recommendation but I guess you could refer to different techniques of factorisation or whatever that is needed when attempting such questions.

I have some stuff to say on simultaneous equations - I'll add that to my list of planned posts.

I think it would be nice for people to recommend tricks on here, or maybe another thread linked to this.

Definitely, that'd be helpful, so far I've got some stuff to say on differential equations, inequalities, simultaneous equations, modular arithmetic and some other stuff that I've forgotten now - a list of recommendations/requests would be welcome.

And, again, thank you for making this thread - you have no idea how useful this is going to be for me!

Glad it's helpful!

Reply 56

Imperion

17

Great work guys! Really like this.

Reply 57

Imperion

17

Original post by aymanzayedmannan

Not a STEP-er but I'm joining this thread to pick up on useful techniques. :biggrin:

Here's an easy one that might be helpful:

Factorising

x^{4} + 1

and polynomials of the like -

It's fairly easy to write out

\displaystyle x^{4} + 1 \equiv \left ( x^{2} + \alpha x +1 \right )\left( x^{2}+\beta x +1 \right )

and then compare coefficients using C3 techniques, but in case you're feeling adventurous,

This could come in handy in certain integrals. Try the classic

\displaystyle \int \sqrt{\tan x} \ \mathrm{d}x

using this. :h:

This is genius bruh :adore:

Reply 58

OP