Oxford PAT 2016

OP

Original post by NatoHeadshot

Hey I saw you posted some snail question last night it looked interesting could you post it here or the link to the book?

Sure.
Link to the post : http://www.thestudentroom.co.uk/showthread.php?t=4198061

Link to the book : I am not sure if I am allowed post the link to the book this way. I found it in the google search. Anyway : https://docs.google.com/file/d/0B4IIN-Yl_suNRHNpOVByeVRCTHM/edit?pref=2&pli=1

And the problem :

Reply 162

OP

Is the following answer presented by physicsandmathstutor not fully correct or is there a flaw in my reasoning?

Reply 163

hellomynameisr

12

Original post by tangotangopapa2

Is the following answer presented by physicsandmathstutor not fully correct or is there a flaw in my reasoning?

I see what you mean but i don't understand why he multiplied the LHS of the inequality by (x-1)^2 instead of just (x-1) ?

Reply 164

OP

We don't know weather or not is (x-1) positive. If x-1 is positive then we don't flip inequality sign if it is negative then we flip. But if we multiply both sides by (x-1)^2 then we are multiplying by positive number so we don't have to flip the inequality sign.

Original post by hellomynameisr

I see what you mean but i don't understand why he multiplied the LHS of the inequality by (x-1)^2 instead of just (x-1) ?

Reply 165

Original post by hellomynameisr

I see what you mean but i don't understand why he multiplied the LHS of the inequality by (x-1)^2 instead of just (x-1) ?

common technique with inequalities, need to square denominators before multiplying to ensure that its positive in this case as otherwise multiplying through flips the inequality (in this case, if the denominator is always positive then thats fine)

Reply 166

φM23

7

Original post by tangotangopapa2

We don't know weather or not is (x-1) positive. If x-1 is positive then we don't flip inequality sign if it is negative then we flip. But if we multiply both sides by (x-1)^2 then we are multiplying by positive number so we don't have to flip the inequality sign.

When you test between 0 and 1 in the factorised cubic you get a positive answer which doesn't agree with the inequality.

(edited 7 years ago)

Reply 167

OP

Original post by PhyM23

When you test between 0 and 1 in the factorised cubic you get a positive answer which doesn't agree with the inequality.

The formula a/(1-r) is used in the solution which is true only if -1 < r < 1 i.e if x > 1. So, we can not conclude weather or not the given inequality is satisfied when x < 1 based on calculation using this formula.

If you take x = 0.5 (to the inequality of the question), L.H.S x + 2 is 2.5 but RHS is infinitely large. Clearly x + 2 is less than R.H.S.

(edited 7 years ago)

Reply 168

Original post by tangotangopapa2

The formula a/(1-r) is used in the solution which is true only if -1 < r < 1 i.e if x > 1. So, we can not conclude weather or not the given inequality is satisfied when x < 1 based on calculation using this formula.

If you take x = 0.5 (to the inequality of the question), L.H.S x + 2 is 2.5 but RHS is infinitely large. Clearly x + 2 is less than R.H.S.

correct, for x<1 sequence diverges so sum is infinitely large

Reply 169

OP

Original post by samb1234

correct, for x<1 sequence diverges so sum is infinitely large

Thanks
So, the ans is 0 < x < root 2, isn't it?

Reply 170

Original post by tangotangopapa2

Thanks
So, the ans is 0 < x < root 2, isn't it?

should be

Reply 171

hellomynameisr

12

Original post by samb1234

common technique with inequalities, need to square denominators before multiplying to ensure that its positive in this case as otherwise multiplying through flips the inequality (in this case, if the denominator is always positive then thats fine)

Original post by tangotangopapa2

We don't know weather or not is (x-1) positive. If x-1 is positive then we don't flip inequality sign if it is negative then we flip. But if we multiply both sides by (x-1)^2 then we are multiplying by positive number so we don't have to flip the inequality sign.

Ah okay I see. Thanks guys :biggrin:

Reply 172

OP

You're very welcome.

Original post by hellomynameisr

Ah okay I see. Thanks guys :biggrin:

Reply 173

φM23

7

Original post by tangotangopapa2

Thanks
So, the ans is 0 < x < root 2, isn't it?

I thought the sign was the other way round for some reason so I edited my reply. I feel there is a crucial property of infinite series we're failing to see. I will quote someone who knows more about this than me.

@Zacken

Please could you take a look at this. Have we been taught an over-simplified version of this topic in school which doesn't explain series like this? It is post 163. Thanks

(edited 7 years ago)

Reply 174