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Conics and applications of differentiation

Posted by White Group Mathematics on February 13, 2016 at 9:05 PM Comments comments (0)



I am having considerable difficulty grappling with this OCR problem. Could you kindly assist?









Student X















Worked it out for you, hope it helps. Peace.







Best Regards,

Mr Koh


Integrating cosec ^4 x wrt x

Posted by White Group Mathematics on February 2, 2016 at 6:00 AM Comments comments (0)


I was given some basic calculus revision questions to do for an undergrad module, but since there's nothing new in them, so I decided to give these questions an A-level tag. I've been working on them for a while now, but I'm stuck at this. Any help would be appreciated; I'm not particularly bothered if you want to post the answers as it's not an assessed homework, and I'm starting to get tired of these. :/


Student X







Please find the integral worked out as follows. Hope it helps. Peace.



 





Best Regards,

Mr Koh

Poisson distribution approximated to normal distribution: Supply and demand problem

Posted by White Group Mathematics on January 24, 2016 at 11:50 AM Comments comments (0)

 


Sales of cooking oil bought in a shop during a week follow a Poisson distribution with mean 100. How many units should be kept in stock to be at least 99% certain that supply will be able to meet demand? Use normal approximation to Poisson distribution, if appropriate.


 

 

 

I have already approximated the distribution to Y ~ N (100, 100). Then, I have tried to use P(Y>y) = 0.99 . That didn't work so I'm trying to use P( Y = y) = 0.99. However, with this, I'm stuck with P [ (y-100.5)/10 < Z < (y - 99.5)/10] = 0.99


 

 

 

The correct answer is 123. Please help.



 

 

Student X





 

 

 

 

 

 

You kinda got your understanding of stuff mixed up. Firstly, recognize that it is the demand that varies, not the supply. It is the amount demanded that follows a Poisson distribution, which is subsequently approximated to that of a normal distribution.



 


The part about defining the original underlying distribution modelling demand as Y ~ N (100, 100) is correct.


 


 

If you let k be the number of units you wish to store as stock, then the required inequality satisfying the requirements of the question should be P(Y ≤ k) ≥ 0.99, which after adjusting for continuity correct would yield P(Y ≤ k+0.5) ≥ 0.99





 

As such, k+0.5 ≥ InvNorm of a normal distribution curve with mean and variance both =100 *

  ie after standardization, solve (k+0.5-100)/ 10 ≥ InvNorm ( 0.99, 0, 1)

 

  

 

Hope this clarifies. Peace.


 

 

* The graphic calculator should help you compute the inequality in k here; unless you are using basic tables, then standardization is needed.





 

 

Best Regards,

 

Mr Koh

 

 

 

Locus of points equidistant from x-y and x-z plane

Posted by White Group Mathematics on November 4, 2015 at 10:10 AM Comments comments (0)


Hi Mr Koh,


                  I am unsure how about how the solution for part (a) is formulated. Could you please help?


                   

                  





Student X

   










I have drafted a detailed set of explanations-please see as attached below. Hope it helps. Peace.






Best Regards,

Mr Koh












Discovering vertices of a rectangle in an Argand diagram

Posted by White Group Mathematics on August 4, 2015 at 9:40 PM Comments comments (0)


Hi Mr Koh,




                  How do I do this question- In an Argand diagram, the points A, B, C and D represent the complex numbers a,  -2+5i,  c   and  3/2 - 1/2 i   respectively. Given that ABCD is a rectangle described in the clockwise sense with BC=2AB, find a and c.





                  I tried drawing B and D on the same diagram but they don't even have a right angle between them, so how can it form a rectangle? And I can't place points A and C on the diagram because they said ABCD is read clockwise but it is also mentioned BC=2AB....so I am lost. Haha. Thanks Mr Koh!








Student X











                     Based on the construct described, the rectangle's length is twice its breadth. I would recommend drawing a generic upright rectangle instead of trying to place the points specifically in Argand space. Do note that vectors are involved in the solving process. I have drafted a solution template for you:




                     


                 



                         Hope this helps. Peace.







Best Regards,

Mr Koh

Discerning winning probabilities in national lottery

Posted by White Group Mathematics on July 19, 2015 at 10:40 PM Comments comments (0)


Hi,

 



Here in my country, when you buy lottery tickets, you try to guess the six (6) combination of numbers out of 42 numbers. Unlike in Sweepstakes, the order of numbers does not really matter.


 




http://www.philippinepcsolotto.com/6-42-lotto-result-summary






 

Some say that the probability of winning the jackpot is simply 6/42 = 14.29%. That is high!






 

Since I was in high school, that formula seemed to make sense for me. After all, what is the probability of guessing one of the six combinations? 1/42. That is the probability of guessing each number in the combination of six. That is the result when you add the probability for each number.






 

(1/42) X 6 = 6/42

 

1/42 + 1/42 + 1/42 + 1/42 + 1/42 + 1/42 = 6/42





 

But I've come across another possibility today. Without repetitions of numbers, you can create 5,245,786 combinations of six digits from 1 to 42. What is the probability that you will guess the right combination out of more than 5 million combinations? 1 out of 5,245,786. That is 1.906 X 10 raised to negative 7.




 

I'm confused now because both solutions seem to make sense, but this second idea makes more sense. Each one is a computation based on a different point of view.





 

I hope you can enlighten me on this one.





 

Thanks,



Student X















The first manner of calculating the required probability which you cited is flawed. If say, each of these 6 combinations can be recycled, the probability would be computed as (1/42)^6 = 1/5489031744.






 

However, I am assuming they cannot be recycled, as such there will be 42 possible combinations for the first number, 41 for the second, 40 for the third, so on and so forth. That said, the probability would be computed as (1/42)*(1/41)*(1/40)*(1/39)*(1/38)*(1/37)= 1/3776965920.







You may therefore question: why isn't this equivalent to 1/5245786, which is considerably larger in fact? The reason is (1/42)*(1/41)*(1/40)*(1/39)*(1/38)*(1/37) actually examines the instance when sequencing of numbers matters, as opposed to 1/( 42 C 6) = 1/5245786 which doesn't pay any regard to the ordering of numbers. Clearly, the former would involve much greater stakes and thus even lower likelihood of winning.





 

Hope this clarifies. Peace.

 




 






Best Regards,

Mr Koh

 


Tiny abstract vectors problem

Posted by White Group Mathematics on July 14, 2015 at 10:25 PM Comments comments (0)


Hi Mr Koh,



                   I encountered this question. It says: if  a = (a • b) b, where a and b are vectors, what are the possible angles between a and b?  Also, what is the magnitude of vector b?





Student X

 








a = (a • b) b  actually implies a is parallel to b.  Note that a • b is in fact a scalar quantity,  such that a=kb, where we can set a • b = k . In view of this, the only possible angles between a and b would be 0 and 180 degrees.




To discover |b|, we shall proceed to take modulus of both sides of the original equation a = (a • b) b ; this gives us |a| = |a • b| |b| = [ |a| |b| ] |b| = |a| |b|^2 . Cancelling |a| on both sides leads to |b|^2 =1, and thus |b|=1 (shown)




Hope this helps. Peace.





Best Regards,

Mr Koh







How to evaluate the integral of 1/(1+x^4) ?

Posted by White Group Mathematics on June 20, 2015 at 10:00 PM Comments comments (0)


I am having problems trying to figure out a way to solve for the integral of 1/(1+x^4) ; attempted some form of substitution but it didn't work out. Could you advise on how to get started properly?




Student X








You can consider factorization of the quartic polynomial, followed by the employment of partial fractions to further break things down. Thereafter , it shouldn't be all that hard to continue the solving process.




Here's what I mean:










Hope this helps. Peace.







Best Regards,

Mr Koh





How to generate aČ-bČ in this complex numbers proof?

Posted by White Group Mathematics on May 17, 2015 at 11:05 PM Comments comments (0)



I am having problems trying to achieve the expression a²-b² for part (ii) of this problem. Please help.







Student X











Please find the detailed solution below, hope it helps. Peace.









Best Regards,


Mr Koh

For two given complex numbers a and b, is it true that |a-b| = |a|-|b| ?

Posted by White Group Mathematics on April 24, 2015 at 12:40 PM Comments comments (0)



For two given complex numbers a and b, is it true that  |a-b| = |a|-|b| ?




Student X









Generally, it isn't true. However, in specific instances, |a-b|  can be equal to  |a|-|b|.  This happens when arg(a)= arg(b). Bearing in mind |a-b| always represents geometrically the physical distance between the two complex numbers a and b, here is a visual representation for this unique case:









Another instance worth noting is when arg(b)= arg(a) - π, then |a-b| actually becomes |a|+|b|:




Hope the above clarifies. Peace.





Best Regards,


Mr Koh


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