Mistakes

Question 1

When implicit differntiwjton, and don’t have numbers, what to do

Answer 1

Sub back into equation to find them

Question 2

Don’t assume the thing they drew is the x and y axis, what to do

Answer 2

Don’t assume, so if you want max height find the difference between LOWEST point and highest, which could be neagtive…

Question 3

How to find min max range from domain

Answer 3

Find normal max range see if it lies

Find the smallest part by taking the bounds of the domain, draw it

Question 4

How to show change of sign exact wording

Answer 4

FUNCTION IS CONITNOUS between,
Change of sign

Question 5

Don’t lack, when integrating after so,ging something, must read q

Answer 5

Might have changed it, take some cknstsnt or divide andwer as a result

Question 6

Finding Cartesian equation of the plane of trajectory

Answer 6

Must take it from the GROUND

So whatever height it was , say 5, the trajector would be 5 - this , where x is the height ABOVE THE GROUND

model height abive as x, and find the difference in height, and work out from there

DONT LACK

Question 7

Proof it’s a sqaure

Answer 7

Angles, must do ALL OF THE ANGLES , so use gradients to prove perpendicualr

Then sides, two adjacent sides must be the same, say tangent, or radius

Question 8

How to explain why binomial expansion valid again

Answer 8

You find the values of x for which it’s valid, set the WHOLE x expression modded to be less than x

Work out the values of x

And show bith limits in there

Question 9

Explain why the value of the accerlwtion ( they said a so need to find a) immdiestly before velocity becomes constant is likely to mean that model B is a better model than model A?

Answer 9

Find the accerlwtion using differntiation

Find accerlwtion at that time, should be 0
Which means the accerlwtion was decreasing, and so the car reaches maximum speed WITHOUT sudden change in speed that was seen before

Question 10

How to show two linewr liens never intersect

Answer 10

If the grsdient is the same this means they’re parallel so jever intersect!

Question 11

How to show how SPEED increases

Answer 11

Find accerlwtion
If negative and already newgtive, then velocity more negative so speed increases etc

Question 12

Remember about series arithmetic geometric discrete CONTINOUS

Answer 12

They are DISCRETE

Hence if you tryna find the last value for which n is positive, set Un to be >0
Then go to the NEXT TERM

Then find the sum

Question 13

Don’t lack for inverse transformations

Answer 13

Write it out first, then perform transformation, might have to do 1 / instead

Just write it out fully

Question 14

Cartesian equation

Answer 14

I’m terms of y

Btw you can express any point not in a parametric curve in terms font too

Question 15

How to find the angle easily suvat

Answer 15

Just set new variables again to max range equation to form a suvat ewuation and can solve for theta hella easilly

Question 16

Remember check for degrees rwdians , draw out the graph hi cal to find range

Answer 16

Convert back to TIME

Question 17

How to set things in I and j to 0

Answer 17

Remember equilibrium make i equal rn j equal

And equilbroum not matter directions resuktsnt forces are 0

Question 18

When a sequence repeats what it called

Answer 18

Periodic, with period etc

Question 19

Remember CINVERGENT means converges at a point , but divergent?

Answer 19

Divergent means DOESNT converge, so do all the values out of the mod r is less than 1

Question 20

When does a series /sequence convergent when r =1

Answer 20

The sequence is convergent at r =1, but the SERIES is divergent , hende have to ude less than 1

So be CAREFUL !

Question 21

Why did they not find the roots

Why is statement invalid if you just say change of sign

Answer 21

Change of signs were missed as intervals were too far away from them

Did not mention it’s CONTINOUS

Question 22

Why does Newton raphsom fail to give correct root?

How to get rhe proper root

Answer 22

Because rue start value has grsdient TOO CLOSE TO A STATIONARY point, and so tangent meets x axis really far away from the root

It does converge to a root but not the correct one

Start closer to the root, hence in between rhide values

Question 23

Remember ball in ther air v time graoh

Answer 23

V t represents grsdient which si CONSTSNT so CONSTSNT until it reaches the ground at - initsl velocity again!

Question 24

Why would a balloon keep increasing volume be a problem

Answer 24

It would burst lmao

Might need to use differential equation if need a new expression dint lack

Question 25

Always give answers as numbers

Answer 25

Don’t leave exact unless they tell you

Question 26

BRUH displace rn time graph, DONT COUNT AREA FOR DISTANCE

Answer 26

DOMT BE STUPIDDD

Question 27

Remember POINTS OF INTERSECTION are quire what

Answer 27

Not only x but Y too!

Question 28

Partial fraction technique for binomial

Answer 28

Is not invalid, just rememebr 1 /something can be expanded too

This means you are adding two things indtead of multiplying which could get lengthy

Question 29

What does modelled as a particle mean !

What does uniform Planck mean

IMPORTANT

Answer 29

Have negligible size so contact force acts at a point

Size and shape does not karrer

2; this means the centre of mass lies in the centre due to uniform mass DISTRIbution !

Question 30

Bruh remember thst things like bricks are discrete, so if you have 4.48 bricks at distance x we know that it must be 4 bricks so take moment sand find new distance

Answer 30

Don’t mess up silly mistakes here olease

Question 31

READ q , about magentiude, or if it JUST SAYS TIME

Answer 31

Then need ONLY ONE TIME , don’t lack there

When particle at origin means 0, remember velocity or displacement,emf or whatever is not positve or negative at 0, similarly it wouldn’t be north east at 0, has to be somewhere else

0 is 0

Question 32

FINAL RATE ANSWER, MAKE SURE TI DO WHAT

Answer 32

ADD UNITS

Question 33

Proof by contradiction of prime number other than 3 which is 1 less than a sqaure number

Answer 33

Set up contradiction, assume there is a prime number p
Let p = n2 -1
SEE DOTS, always factorise
And now show the case for when a factor could be one, as that’s the only way to be prime, remember 1 and itself
This only occurs when n is 2 and p is 3p
But any time above n it won’t create a prime, hence contradicts, and thus n is only like this at 2

Question 34

Remember every time agaisnt slope

Answer 34

There is component of the weight, which reduces accerlwtion, so max distance is less 😭

Question 35

How to prove by contradiction irrational number root 2 ,3

Answer 35

Assume root 2 rational hence can be in the form a/b where a and b are whole and a /b is in its simplest form

Therefore a and b can not be multiples of each other

Let 2 = A2/ B2
So a2 = 2b2 , hence a2 is even
Hence a is even and written as 2p

Sub back in, show what b is
B2 is even, b is even, 2Q

Now multiples of Esch other hence contradiction

Question 36

Prove infinitely many primes contradiction

Answer 36

Assume finite list prime p1 p2 to pn

Let P be a number multiply all + 1

P is either prime or not confirmed

If prime, we found nee number not in list = contradiction

Id mit Prime, P Must be able to be divided by a prime fsctor ( by prime division theory).

However Any factor must also be a factor of 1, which is not possible

Hence contradiction

Question 37

Remember transformations

Answer 37

Replace the x all totgeht

Or the whole function y

So need to do different things if you first