isi maths

[ritu]

q1..number of continuous functions characterized by equation---
x.f(x)+2.f(-x)=-1
where x is any real number is
a.1
b.2
c.3
d.none of these


q2..if x and y are real numbers such that x^2+y^2=1,then the maximum value of:IxI +IyI is-
a.1/2
b.root 2
c.1/root 2
d.2


q3..let f(x)=Ax^2+Bx+C,where A,B,C are real nos..if f(x) is an integer whenever x is an integer..then
a.2A & A+B  are integers,but C is not an integer.
b.A+B and C are integers,but 2A is not an integer
c.2A,A+B and C are all integers
d.none of these

q4..consider 2 consumers with identical income,M,and utility function U=xy where x is amout of restaurant good consumed and y is any other good consumed.the unit prices are given for both goods.consumers have two alternative plans--
plan A...eat together at restraunt and each pays his own bill
plan B...they eat together at restaurant but each pays one -half of the bill..find equilibrium consumption under each plan and exp[lain the difference in 2 schemes if any.

[anon_econ]

I'll give you some hints.
Q3: f(0), f(1) and f(2) are integers.
Q4: Let the price of y be 1. In the 2nd scenario, the 1st person's b.c would be p(x1+x2)/2 + y= M. His choice variables are x1 and y. He takes x2 to be given. Similarly u get the choice problem for the 2nd person. Find out the nash equlibrium.
Q3: I suppose we have to maximize |x| + |y|.
x^2+y^2=1 gives u all the points on the circle with radius 1 and centre 0. Now |x|+|y| takes on the same set of values in all 4 quadrants. The max value will also be attained once in every quadrant. So just maximize x+y where y=positive square root of (1-x^2).
I don't know about Q1.

[duck]

Q1> Since, the equation holds for every x, it must hold for -x as well.
So, put "-x" in place of x .. solve the two equations..you'll get f(x) which is continuous at all points.
So, there is only one continuous function.

[ritu]

oh thank u so much vasudha and duck....i dont know why i am not able to think like you guys...thanx a lot...:)))

[anon_econ]

It's not that. Just keep fiddling around with whatever info u r given in the question n u'll get it. Keep practicing the way u r right now :)

[s-2]

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[s-2]

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[anon_econ]

Hi shreya! RD sharma's books have MCQs at the end of each chapter..they are good. Then we can look at the questions asked in the b.stat n b.math entrances at isi (u can find them at http://www.isical.ac.in/~deanweb/SAMPLEQUESTIONS.HTML). I guess we can also practice from IIT prep books & IIT/AIEEE's previous year papers. And for functions i think even a CAT book would be good..