# Sample Questions of ISI ME I (Mathematics) 2010 Discussion Classic List Threaded 128 messages 1 ... 34567
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Duck, just have a doubt. Why other posts call you Nidhi? How they got to know your name? Will they also know my real name without being mentioned myself?(Also I think you have completed your masters, right?) By the way, as Shreya's olders posts suggest, she wants to ask the prev. Amit Sir's questions : If a < b < c < d, then the equation (x − a)(x − b) + 2(x − c)(x − d) = 0 has (a) both the roots in the interval [a, b], (b) both the roots in the interval [c, d], (c) one root in the interval (a, b) and the other root in the interval (c, d), (d) one root in the interval [a, b] and the other root in the interval [c, d]. Let f and g be two differentiable functions on (0, 1) such that f(0) = 2, f(1) = 6, g(0) = 0 and g(1) = 2. Then there exists θ ∈ (0, 1) such that f'(θ) equals (a) (1/2)g'(θ), (b) 2g'(θ), (c) 6g'(θ), (d) (1/6)g'(θ). The minimum value of log(x)a + log(a)x, for 1 < a < x, is (a) less than 1, (b) greater than 2, (c) greater than 1 but less than 2. (d) None of these. The value of ∫(1/(2x(1+√x)))dx over [4, 9] equals (a) log(e)3 − log(e)2, (b) 2log(e)2 − log(e)3, (c) 2log(e)3 − 3log(e)2, (d) 3log(e)3 − 2log(e)2. Read log(x)a as log value of a when x is the base, And likewise others. Read ∫(1/(2x(1+√x)))dx over [4, 9] as the value of definite integral of the said function over the range [4, 9]
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by duck he duck,for the above question f(1)=1..real valued function,for f(x)=x²,if x=2,then why f(x) will b an irrational no.?? and is the ans of the min value of log(x)a+log(a)x is greater than 2?
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by XIPP if u pls explain me the Sol of question no 2 given above that of f and g function..
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by shreya lahiri HI.. :) 23) Dont consider that example. Its wrong. Let me give you a rough idea. Its given that the function can take only rational values , its continuous and f(1)=1. Now, if you take any function which has "x" term in it.. then, it can take irrational values. Therefore, it must be the case that f(x) is a constant function. And as f(1)=1⇒ f(x)=1. Therefore, f(2)=1. 14) Yes, the minimum value is greater than 2. 13) Take f(x)= 2+4x and g(x)=2x Now, solve the question by ruling out options. :)
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 @Duck, Really stuck, please help for 14) See here that the function (log a/log x) + (log x/log a) has no global minima at 1 < a < x, how did you derive that it is greater than 2?
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Amit Goyal CONTENTS DELETED The author has deleted this message.
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by duck Hi please explain ques 30
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Could someone share solution workings for Q 21/ 28/ 29. Grateful!
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion "I don't ride side-saddle. I'm as straight as a submarine"
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by rongmon for Q28) for all x, x-box(x)>=0, thus g(x)=1+x-box(x)>0, thus f(g(x))=1, since f(x)=1 for all x>0  "I don't ride side-saddle. I'm as straight as a submarine"
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Granpa Simpson Thanks. The second step, you interchanged a1 and a2 - what is the reasoning behind it?
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Actually i have substituted x=-x in the second one and then subtracted both the equations by multiplying a1 and a2..you can either eliminate f(x) or f(-x) whichever you want here i have eliminated f(x) to get f(-x) so i have put (-b1/b2) in the final step...however you can also eliminate f(-x) and find out f(x) in that case you will have to substitute x=(b1/b2) in the final step...!!!  "I don't ride side-saddle. I'm as straight as a submarine"
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Amit Goyal Hello Sir, What then is the answer to the question? Is it continuously differentiable at x=0? Since the function does not exist for 0- , isnt the answer (d)? Thanks, Soumen
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Amit Goyal Amit Goyal wrote Hi Sonal, a(n) will converge to 0 if the X(i)'s are independent. Here cov(X(i),X(j)) = p and Simple computation will give the following value of variance of a(n), a(n) = [n(sq(s)) + (sq(n) - n)p]/(sq(n)) = (sq(s)/n) + p - (p/n) Hence, lim(n→∞) a(n) = p Are you sure on this one? You seem to be thinking of X(i) as numbers. They are distributions (meaning that they are a sequence of numbers). Intuitively, when you add two distributions, the variance of the sum must be greater than their individual variances. The formula is actually - Var(A+B) = Var(A) + Var(B) + 2*Cov(A,B) Extending, to n terms, we will get Var(A1+A2+....An) = n*sigma^2 + n*rho But we have 1/n outside. So, we get Sigma^2+rho which is option (d) I'd like someone to confirm or deny this. Am I right? Am I missing something?
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 @soumen...actually u r not applying the var formula correctly Var(mean of Xs) =( 1/n²)( varx1 + varx2 + varx3+........varxn+ sum of all covariance terms) Sum of all covariance pair is equal to n(n-1)*p as all cov b/w diff pairs of x's are same. And sum of variances is equal to n*sigma² So Var (mean of Xs) = (1/n²)*(n*sigma² + n(n-1)*p)
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Dreyfus wrote @soumen...actually u r not applying the var formula correctly Var(mean of Xs) =( 1/n²)( varx1 + varx2 + varx3+........varxn+ sum of all covariance terms) Sum of all covariance pair is equal to n(n-1)*p as all cov b/w diff pairs of x's are same. And sum of variances is equal to n*sigma² So Var (mean of Xs) = (1/n²)*(n*sigma² + n(n-1)*p) Thanks Dreyfus! I understand that now. I was considering (wrongly) that the variance of the mean is the mean of the variance. The n^2 is what I had missed. Can you help me with this question? Consider the following 2-variable linear regression where the error e(i)’s are independently and identically distributed with mean 0 and variance 1; y(i) = a + b(x(i) − Mean(x)) + e(i), i = 1, 2, . . . , n. where Mean(x) = (x(1) + x(2) + ....x(n))/n Let a^ and b^ be ordinary least squares estimates of a and b respectively. Then the correlation coefficient between a^ and b^ is (a) 1, (b) 0, (c) −1, (d) 1/2. How is the answer zero?
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Use the formulas of slope estimate, intercept estimate, the cov formulae in expected form and use the fact that mean of (xi- mean is xi) is zero...,use these formulas ull get ur result.....
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 The formulas are - a = (x_bar*y_bar - (x*y)_bar)/(x_bar^2 - (x^2)_bar ) and b = y_bar - a*x_bar From here, I cant follow your logic. I understand that the mean of the mean is just the mean and thus the expression you wrote is zero, but how do I apply it here? Can you please explain? Thanks, Soumen
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Can anyone give the answer list of 2010. Like give for other years.