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28-10-2005, 06:11 PM

Hi there, i got some problems with this q:

11. (b) A large quantity of shoes from brand B is inspected before been distributed to slaes outlets. It is found that 1% of the left side shoes are defective while 2% of the right side shoes are defective. By assuming that the defects occur independently and using suitable approximation, calculate the probability that in a random sample of 50 pairs of shoes,

(i) there are two pairs of shoes with defects on their left side.
(ii) there is a pair of shoes with defects on both sides.

[stpm 1999]

(i)For left side shoes : n=50,p(defective)=0.01 => np=0.5...then we can use poisson distribution as approximation...

P(X=2)=[e^(-0.5)]*[(0.5)^2]*/2 = 0.075816332
= 0.076
(#) Here,n means number of left side shoe,X means number of left side shoe which is(are) defective.

(ii)Lets consider each pair of shoes...
p(both defective)=0.01*0.02=0.0002
p(not both defective)= 1 - 0.0002 = 0.9998 (or u can add up 0.01*0.98,0.99*0.02,0.98*0.99)

m=50,p=0.0002 =>mp=0.01
Poisson again......
P(Y=1)=[e^(-0.01)]*[0.01] = 0.0099

(#) Here,m means number of pairs,Y means number of pairs which is(are) both defective.
There r maybe other ways to solve (ii)......maybe there r better n more accurate solution....but this is wat i can get
since i din study that for a long time...Btw,The answer given is definitely strange....dun u feel that?
________
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kimsiang

29-10-2005, 03:30 PM

another q:

Given two parallel lines l1 and l2 passing through (5, 0) and (-5, 0) respectively, and meet the line 4x + 3y = 25 respectively at P and Q. If PQ equals to 5 units, find the possible slopes of l1 and l2.

[stpm 2001 p1 q5]

sorry for flooding so many q =.= but thanks in advance.
The answer/working uses the angle n tangent thing....but is there any other easier to understand way?

I think using the method using angle wil be easier compare to others...
For the questions regarding statistic above,Z is obviously not equal to X + Y.So,juz ignore it.
When talking about approximation,we r approximating binomial to others...means from discrete to continous.......but Why?? for example.when n = 500 ....i dun think our calculator can give us the value of 500 C 25 in binomial.......but by using poisson,when the p is 0.01,we can calculate the probability easily...isn`t?
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confused-freaker

29-10-2005, 07:37 PM

aah i see, thanks for the explaination :) ...

for the coordinate geometry q above, i find their working seems a little weird....i shall paste in the solution for you to spot if there is any probs ~

confused-freaker

30-10-2005, 07:44 AM

books solution :
files/chickens-maths2001q5soln.jpg

worked out solution:
files/chickens-maths2001q5mysoln.jpg

need judgement, thx in advance kimsiang

kimsiang

01-11-2005, 01:01 PM

oups...sorry for late reply......
yr solution is ok...but for the sample solution,1 of the mistakes is that it juz simply state that since theta can be positive n negative,n then come out with 2 solutions.Instead of doing that,1 shd draw another line where the slope is negative,n then start to evaluate the theta,beta,etc...shd come out with the same answer.
The way u used is a normal way,which is *secure*.But sometimes,do do it in other alternative ways.It helps to improve yr math... :wink:
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yama

04-11-2005, 12:43 PM

these are the questions tat i face problems , hope someone can help me to find the solution soon . thank you very muz !!

1. Find the points of intersection of the curve y^2 = 3ax and x^2 = 3ay. Calculate the are of the region bounded by the curves .

2. Find the set of the values of k such that the roots of the equation x^2 + (k+1)x +k + 4 = 0 are positive.

3. Expand (2+x)^-2 in ascending powers of x^-1 until and including the term in x^-4. state the range of values of x for which this expansion is valid. Find the coefficient of x^-18 in the expansion of (3x+x^-2)^12.

4. If ( x+iy )^3 = x + iy, find the real values of x and y .

5. Prove that x^2 + y^2 > 2xy if x, y E R and x is not equal to y. if x^2 + y^2 = 1 and w^2 + z^2 = 1 , show that xw + yz <1.

6. Show that (cosA + cos B)^2 + (sinA + sinB)^2 = 4cos^2 ? (A-B) . Hence, show that cos 15 = ? (route 2 + route6)

7. In a chemical process, substance A is continuously changing into the substance B at a rate which is proportional to the mass of A and inversely proportional to the mass of B , at any time t. the total mass of A and B remains constant and equals to M. the mass pf the B at time is m.
Obtain a differential equation relating m and t.
Given that M = 120 grams, 20 grams of substance B remains after 2 minutes, find the time taken for which the masses of both substances are equal.

8. If a and b are real roots of the equation x^2 + (k+2)x + (k+5) = 0 and k ER, find the ranges of k such that
(i) both a and b are positive,
(ii) both a and b are positive and less than 7

9. The parametric equations of a curve is given by x= t(t-2) and y= 2(t-1). If P is a point on the curve and O is the origin, show that the equation of the locus of Q which divides OP internally in the ratio 1:2 is given by 9y^2 = 4(3x+1). If the normal at P passes through the point A ( 1,8 ) , find the coordinates of P.

reign226

07-11-2005, 09:26 PM

yama

07-11-2005, 10:04 PM

I have been thinking over this problem for a VERY long time and the answers keep contradicting. It refers to linear combination of two independant normal random variables.

X=N(10, 0.1)

Let's say X is distributed normally with mean = 10 and SD = 0.1

Now let's say random variable Y = 10X.

That means:
E(Y) = 10E(X)

No problems so far. But:
Var(Y) = Var(10X)
= 100Var(X)

Right? But in some books, I the way they get the variance for Y, Var(Y) is to simply multiply Var(X) by 10, as there are ten X in Y. Which do you think is correct? Personally I go with 100Var(X) as it's mathematically proven. But I still have my doubt. Somebody help me!!

in this type of question

if the Q states that ten of x,
Y = X1+X2+X3+X4+....+X10
that mean Var(Y)= 10Var(X)

but

if the Q states that x is repeated ten times ,
Y = 10X
that means Var(Y)= Var(10X) = 100Var(X)

hope that u can understand that i trying to say

reign226

08-11-2005, 08:49 AM

The Q was: "Random variable X = such and such, which represents the weight of a chocolate bar. Random variable Y is the weight of 4 chocolate bars."

So what is the variance for Y? Isn't Y = 10x?

Hmm, then again i think i understand what you're trying to say. I just have to get into my brain that

Y = X1 + X2 + X3 + X4

is not equal to

Y = 4X.

*sigh* Thanks for the help!

reign226

08-11-2005, 02:03 PM

Here's another little devil:

if y = x + (1/x), show that x^4 - 2x^3 - 6x^2 - 2x + 1 = 0 can be transformed into a quadratic in terms of y

i got until this far only:
y^2(x^2 - 2x) + y(2-8x) + 8 = 0

(by long division with X + (1/X)) And now i'm stuck. Help!

kimsiang

09-11-2005, 01:13 AM

Here's another little devil:

if y = x + (1/x), show that x^4 - 2x^3 - 6x^2 - 2x + 1 = 0 can be transformed into a quadratic in terms of y

i got until this far only:
y^2(x^2 - 2x) + y(2-8x) + 8 = 0

Since y = x + (1/x) ------(1)
we can get y^2 = x^2 + (1/x^2) + 2
==>y^2 - 2 = x^2 + (1/x^2) ------(2)
then,x^4-2x^3-6x^2-2x+1 = x^2[x^2 - 2x - 6 - (2/x) + (1/x^2)]=0
Since y = x + (1/x),we can deduce that x not = 0.
then we can get [x^2 - 2x - 6 - (2/x) + (1/x^2)]=0
<=> [x^2 + (1/x^2) - 2x - (2/x) - 6]=0
<=> [x^2 + (1/x^2) - 2(x+(1/x)) - 6]=0
Substitute (1)and (2) into it,
we can get <=> [y^2 - 2 - 2y - 6]=0
<=> [y^2 - 2y - 8]=0
<=>(y-4)(y+2)=0
<=> y=4 or y = -2
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reign226

09-11-2005, 06:06 AM

Brilliant! Thanks a lot kimsang. Unfortunately I could never have thought of this way to do it...ever. *Sigh* Thanks a lot dude!

kimsiang

09-11-2005, 05:24 PM

You are welcome.......... :D
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Nelson

10-11-2005, 01:11 AM

2. Find the set of the values of k such that the roots of the equation x^2 + (k+1)x +k + 4 = 0 are positive.

I think the solution is as follows:

Given that the equation x^2 + (k+1)x +k + 4 = 0 has roots, this means that b^2-4ac≥0.
By taking a = 1, b = (k+1) and c = k+4
we have
(k+1)^2-4(1)(k+4)≥0
k^2+2k+1-4k-16≥0
k^2-2k-15≥0
(k+3)(k-5)≥0
Hence k≤-3 or k≥5

Let α and β be the roots of this equation
Given that those roots must be positive
Therefore α>0 and β>0
α+β = -k-1>0
k<-1
αβ=k+4>0
k>-4
The solution is -4<k<-1

By representing these solutions on a real number line,
we have the set of values of k is {}.

DecentMerson

10-11-2005, 01:47 AM

I have been thinking over this problem for a VERY long time and the answers keep contradicting. It refers to linear combination of two independant normal random variables.

X=N(10, 0.1)

Let's say X is distributed normally with mean = 10 and SD = 0.1

Now let's say random variable Y = 10X.

That means:
E(Y) = 10E(X)

No problems so far. But:
Var(Y) = Var(10X)
= 100Var(X)

Right? But in some books, I the way they get the variance for Y, Var(Y) is to simply multiply Var(X) by 10, as there are ten X in Y. Which do you think is correct? Personally I go with 100Var(X) as it's mathematically proven. But I still have my doubt. Somebody help me!!

about the Var(Y) = Var (10X) question... it is definitely
Var(Y) = Var(10X)
Var(Y) = 100Var(X)

but if Var(Y) = Var (X1 + X2 + X3 + ... + X10), assuming that Xi's are independent and identically distributed(i.i.d) random variable... then, the variance of the sum = sum of the variance because covariance = 0 when the random variables are independent...

so, Var (Y) = Var (X1) + Var (X2) + ... + Var (X10)
= 10 Var (X) (since that they are iid)

DecentMerson

10-11-2005, 02:46 AM

6. Show that (cosA + cos B)^2 + (sinA + sinB)^2 = 4cos^2 ? (A-B) . Hence, show that cos 15 = ? (route 2 + route6)

by using the identity
cos a + cos b = 2cos(a+b/2)cos(a-b/2)
sin a + sin b = 2sin(a+b/2)cos(a-b/2)

u get (cosA + cos B)^2 + (sinA + sinB)^2 = 4 [ cos^2 (A+B/2) cos^2(A-B/2) + sin^2(A+B/2)cos^2(A-B/2)]

as sin^2 X = 1 - cos^2 X... by applying this identity... and expanding it a little, u will come to 4cos^2 ? (A-B)

hence, using the equation u just proved... A-B/2 = 15... A-B = 30...
so, u can let A= 90 B=60, or A=30, B=0... and substitute the value into the equation... and u shld be able to get the answer...

reign226

10-11-2005, 10:10 AM

About the variance question again:

So what does this statement mean: "X = random variable representing mean number of hours studying by Males"

Y= random variable representing mean number of studies by 10 males".

Is Y = 10X or Y = X1 + X2...+ X10?

I saw this question in our textbook and they say Y= 10X. But the thing is, in another book, it's practically the same question (replace mean number of hours studying with weight of chocolate) and Y = random variable representing 10 chocolate bars and they use Y = X1 + X2 etc.

@<hidden>@<hidden>

It would be best if the question just state Y = what. But I'm not counting on it. Arrgh!

DecentMerson

10-11-2005, 11:02 AM

About the variance question again:

So what does this statement mean: "X = random variable representing mean number of hours studying by Males"
sMaths Worksheet Discrete Math Mathlearndirect W W And Learn Szh %C3%83%C2%A0dministrator Components Com Joomla Flash Uploader Install Joomla Flash Uploader Php Math Learn Direct [STPM] Mx T [Archive] - ReCom.orgi Math Learn Direct %D1%82%D0%B5%D0%BB%D0%BE+%D0%B1%D1%80%D0%BE%D1%88%D0%B5%D0%BD%D0%BE+%D0%B2%D0%B5%D1%80%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE+%D0%B2%D0%B2%D0%B5%D1%80%D1%85+%D1%81%D0%BE+%D1%81%D0%BA%D0%BE%D1%80%D0%BE%D1%81%D1%82%D1%8C%D1%8E+%D0%BA%D0%BE%D0%B3%D0%B4%D0%B0+%D1%82%D0%B5%D0%BB%D0%BE+%D0%B4%D0%BE%D1%81%D1%82%D0%B8%D0%B3%D0%BB%D0%BE+%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%B5%D0%B9+%D1%82%D1%80%D0%B0%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%98 bMaths Worksheet Discrete Math Mathlearndirect W W And Learn Szh %C3%83%C2%A0dministrator Components Com Joomla Flash Uploader Install Joomla Flash Uploader Php Math Learn Direct [STPM] Mx T [Archive] - ReCom.orgz g Direct w Xp%20luna%20royale%20perfection%20v2%20sp3%20descargar Learn Math Learn Direct

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