Sample Questions of ISI ME I (Mathematics) 2010 Discussion

[XIPP]

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]

[shreya lahiri]

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?

[shreya lahiri]

if u pls explain me the Sol of question no 2 given above that of f and g function..

[duck]

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.

[XIPP]

@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?

[don123]

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[neha 1]

Hi
please explain ques 30

[rongmon]

Could someone share solution workings for Q 21/ 28/ 29.


Grateful!

[Granpa Simpson]

[Granpa Simpson]

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

[rongmon]

Thanks.

The second step, you interchanged a1 and a2 - what is the reasoning behind it?

[Granpa Simpson]

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...!!!

[soumen08]

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

[soumen08]

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?

[Dreyfus]

@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)

[soumen08]

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?

[Dreyfus]

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.....

[soumen08]

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

[Jim_Moriarity]

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

[Jim_Moriarity]

SUm no 18. 2010 Maths,