DSE 2012 Doubts

[Ayushya Kaul]

Hey, couldn't wrap my head around these questions:

QUESTION 14.  Integral of x^n.sin(x) dx for limits 0 to 1.
(a) Does not exist
(b) Is necessarily greater than 1
(c) Is greater than 1/(n + 1)
(d) Is less than 1/(n + 1)

Can anyone show the steps for solving this?

QUESTION 23. Consider a homogenous goods market with the demand
function Q = 30  P, where Q and P denote quantity and price respectively.
There are two firms playing a price game in the following manner: rm 1
quotes a price and then rm 2 chooses a price. When they charge the same
price they share the market equally and otherwise the market demand goes
to the firm charging lower price. Firm 1 has a capacity constraint at the
output level 5 units such that upto five units the marginal cost of production
is Rs 3 per unit of output, however beyond 5 units it cannot produce any
output. Firm 2 does not have any capacity constraint, it can produce any
amount with the marginal cost Rs 6. What would be the equilibrium price
in the market?
(a) 3
(b) 6
(c) 6 -e , where e is very small positive number
(d) 3 +e , where e is very small positive number

QUESTION 46. What is the total number of local maxima and local min-
ima of the following function
f(x) =(2 + x)^3  if  -3 <x<= -1
        x^2/3        if  -1<x< 2
(a) 1
(b) 2
(c) 3
(d) 4

Also questions 36 and 39 part A of DSE 2012.
The questions have too many cumbersome signs for it to simply copy and paste so here's the link:
http://econdse.org/wp-content/uploads/2012/07/2012-Option-A.pdf

Once again, any help is greatly appreciated.

[MR]

The 23rd question is on bertrand model price competition duopoly. As you must have read that in Bertrand model by the end due to cut throat competition amongst firms the nash equilibrium is to charge P=MCi, where i denotes the number of firm like i=1,2,3.....n. But this is a situation with capacity constraint. So the nash equilibrium amongst these firms will be to charge the higher MC as their price. As the firm with the lower MC gets the entire market share as the products are homogenous. So though this firm has lower MC it should not cut the price of the rival firm because as it is, it cannot by itself satisfy the entire market. So the answer is 6.

[maahi]

QUESTION 46. i think 2 . for d first function we differentiate 3 times and find local minima at x = -2 and for d odr fn local minima
at x = 0

 btw if u know Q  27 then plese help with it .

QUESTION 14.  Integral of x^n.sin(x) dx for limits 0 to 1.
use ilate method , integ by parts
u = sin x  dv= x^n dx  --> v =  x^n+1/n+1     du = =cosx dx
int = sin x * x^n+1/n+1  - int x^n+1/n+1 (- cos x ) dx
 if u look at d exp it does implies d

http://discussion-forum.2150183.n2.nabble.com/DSE-2012-Ques-39-plzz-help-tp7581555.html;cid=1371550585574-761
 36 ) no where ginni coeff is mentioned , cumulative dist fn sums up d exp fr every individual so we don't know about any one person's exp .


i m nt sure .  

[Ayushya Kaul]

I believe question 45 and 27 have been discussed before.
But I didn't get 46.
My intuition told me 2 as the answer, but mathematically, I couldnt prove it.
I could only show -2 as the point of inflection.

[Ayushya Kaul]

Bump.

Could someone please take a look at question 46 and 36, and help me out?
Any help would be greatly appreciated :)