Maths n Stats

[Ram]

1.The no.of ways selecting 10 balls out of an unlimited no.of white,red,green and blue balls is
a) 236    b)256   c)276   d)286

2. The no.of ways in which an examiner can assign 20 marks to 4 questions giving not less than    
two marks to a question is
a)280     b)365    c)455   d)545

3.Number of continuous functions characterized by the equation
x f (x) + 2 f (-x) = -1 , where x is any real number, is            (ISI 2011)
(a) 1, (b) 2, (c) 3, (d) None of these

4.Q.no. 24 of ISI 2011

5.In an analysis of bivariate data (X and Y) the following results were obtained.
Variance of X (s x2) = 9 , product of the regression coefficient of Y on X and X on Y is 0.36, and the regression coefficient from the regression of Y on X (b yx ) is 0.8.
The variance of Y is
(a) 16    (b) 4     (c) 1.69     d)3

Plz give explanations to your answers. Thanks in advance.....

[AJ]

2). 455

The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items(<=n)  is (n+r-1) C (r-1)

20 marks have to be divided in 4 questions.. have 2 minimum for each.... basically 12 marks have to b divided in 4 questions.. when each can have 0 to 12 marks... it becomes the abv formula..

15C3= 15!/12!3! = 455

(I dont know how we get that formula..trying to derive it)

[Manvendra]

Q1 :

A : 286.

Divide 10 things ( ~number of balls ) b/w 4 people ( ~ 4 different types of balls ):

13!/3! 10! = 13 . 12 . 11  / 6 = 22.13 = 286

[Manvendra]

Q5 :

In an analysis of bivariate data (X and Y) the following results were obtained.
Variance of X (s x2) = 9 , product of the regression coefficient of Y on X and X on Y is 0.36, and the regression coefficient from the regression of Y on X (b yx ) is 0.8.

A : 16

var ( X ) = 9

product of regression coefficient of Y on X and X on Y = 0.36

  {mean(XY) - mean(X) mean(Y)} / var(X) = regression coefficient of Y on X = 0.8
  {mean(XY) - mean(X) mean(Y)} / var(Y) = regression coefficient of X on Y.

Now,

0.36 = {mean(XY) - mean(X) mean(Y)} / var(X)  *  {mean(XY) - mean(X) mean(Y)} / var(Y)
0.36 = 0.8 * 0.8 * 9 / var(Y)

var(Y) = .8 * .8 * 9 / .36 = 16


 

[Ram]

@manvendra Thanks a lot got the concepts :)
@pinky hi i got the answer for the question follow this link http://www.pagalguy.com/forum/quantitative-questions-and-answers/55747-cat-2010-concepts-fundas-tips.html to see how the formula is derived...Thank you :)

[duck]

Hi Ram.. :)

Q3) The condition : x*f(x) + 2*f(-x) = -1 holds for all x (Given in the question)
Therefore, it will hold for x=-x also.
Subsitute this you'll get one more equation:  -x*f(-x) +2*f(x) = -1
Solve these two equations, you'll get one Continuous function f(x) = -(x+2)/(4+x^2)

[duck]

For Q24 ISI 2011

Given:|log x x_1| + |log x  x_2| + |log x / x_1| + |log x / x_2| = |log x_1 + log x_2|

LHS = |log x x_1| + |log x  x_2| + |log x / x_1| + |log x / x_2|
= |log x + log x_1| + |log x + log x_2| + |log x - log x_1| + |log x - log x_2|
= |log x + log x_1| + |log x - log x_1| + |log x + log x_2| + |log x - log x_2|
(Just changing the order in which the terms are written)
= |log x + log x_1| + |log x_1 - log x| + |log x + log x_2| + |log x_2 - log x| (since |a| = |-a|)
>=   |log x + log x_1 + log x_1 - log x| +  |log x + log x_2 + log x_2 - log x|
 (combining first two terms together and next two terms together using |a|+|b|>=|a+b|)
= |2log x_1| +  |2 log x_2|
= 2(|log x_1| +  |log x_2|)

RHS =  |log x_1 + log x_2| <=  |log x_1| + |log x_2|
since LHS = RHS, this implies that
|log x_1| + |log x_2| > = 2(|log x_1| +  |log x_2|)
this implies |log x_1| + |log x_2|=0
Thus, |log x_1| = |log x_2| = 0
Hence x_1 =  x_2 = 1 substituting in equation  |log x  x_1| + |log x  x_2| + |log x / x_1| + |log x / x_2| = |log x_1 + log x_2|
we get 4|log x| = 0 which gives us x = 1.
Thus there is a unique solution.

[anon_econ]

Duck plzz tell me how to go about question 17 of isi 2011 as well!!

[duck]

Hey Vasudha.. :)
i tried solving it.. but couldnt get it..

[Chocolate Frog]

For ISI 2011 question 17, I got f(x) = 3(x + x^2)/7, hence (d). However, my solution is way too lengthy (and way too inelegant) to be published. :P

Try integrating by parts multiple times.

[anon_econ]

I think it's better to leave it
Thanks anyway

[ritu]

hi vasudha....have u done me 2 part of 20011 nd 2012?????

[duck]

Thanks Chocolate Frog.. :)

Hey Vasudha.. :)
I tried doing like this>
f(x) = px+qx^2.
So, f(y) = py+qy^2.
Substituted this expression in the integral. And then proceeded. But i got none of the options as correct!
Some weird values came!! :O

[anon_econ]

Oh ok. Chalo thank u duck :)
And ritu i've done me2 for 2012 and i've tried most questions of 2011..