1. you didn't mention the number of bonds issued or the par value of the bonds issued. as such the question cannot be answered.

2. if the co-operation to form a business is for a specific project only it is joint venture. if they expect to continue to do the same business in future too(i.e. after completion of this venture) then it is a partnership. In this case in the abscence of any other information you can assume it is joint venture

3.U-shaped. short run tfc curve always goes down. but short run tvc curve goes to a minimum and then rises.

tc = tfc + tvc so u shaped\

4. np = 3 npq = 2 implies q = 2/3 implies p = 1/3 and n =9 so P(X<=2) = P(X=0)+P(X=1)+P(X=2) = (2/3)^9 + 9*(2^8/3^9) + 36*(2^7/3^9) = 0.3771

5. P(A) = 6/(6+9) = 6/15 = 2/5 P(B) = 1/(1+12) = 1/13 A and B are independent events ( the prabablity of B solving the problem doesn't depend on whether A solved it or not) the problem getting solved is P(AUB) = 6/15 + 1/13 - ((6/15)*(1/13)) = 29/65

6. (r^8-1) = 5*(r^4-1) implies r^4+1 = 5 implies r^4 = 4 imples r = root(2)

8. if the guy who is a doc and officer is included then the other 2 members can be choosen from anyone i.e. in 8c2 whays i.e. 28 ways. now suppose the guy is excluded, now either it is 2docs and 1 officer or 2 officers and 1 doc so 2*4c2*4c1 = 2*6*4 = 48 ways total = 28 + 48 = 76

9.the limit is of the form 0/0 so we have to use l'Hospitals rule, i.e. limf(x)/g(x) = limf'(x)/g'(x)
f(x) = e^(5x)-e^(3x)-e^(2x)+1 g(x) = x^2
f'(x) = 5e^(5x)-3e^(3x)-2e^(2x) and g'(x) = 2x but f'(0)/g'(0) is also of the form 0/0 so we have to apply the l'Hospitals rule once again. f''(x) = 25e^(5x)-9e^(3x)-4e^(2x) g''(x) = 2 ; f''(0)/g''(0) = (25-9-4)/2 = 6

10. This is similar to the 9th problem again applying l'Hospital's rule twice we get, ((ln(10))^2 - (ln(5))^2 - (ln(2))^2)/2 =   1.1155

11.A monopolist can control either the price at which he sells his product or the amount of quantity he produces.
If he controls the price of the good, then he cannot control the quantity and vice-versa.This is because the demand curve is decided not by the monopolist but by many general mocroeconmic factors and he can control one variable but the other will be dependent on the first.
 

7. First we try to find out the lpossible distribution of the number of balls in the boxes. there should be atleast 1 ball in each box. so we keep 3 balls in 3 boxes. now, the remaining two balls can be arranged in 2 ways. either they can be put in seperate boxes or in the same box. the first will result in a distribution 2-2-1 and the second choice will result in a distribution 3-1-1. let us seperate the problem into the above two cases and try to solve it.
Case 1: the distribution is 2-2-1. Now we can select the two boxes in which we keep two balls in 3c2 ways. now we can select 2 balls to keep in a box in 5c2 ways and two balls in other box in 3c2 ways. the remaining ball is kept in the third box. so number of ways is 3c2*5c2*3c2 = 3*3*10 = 90
Case 2: the distribution is 3-1-1. Now we can select the one box in which we keep the three balls in 3c1 i.e. 3 ways. now we can select 3 balls from 5 balls in 5c3 ways i.e. 10 ways. from the remaining 2 balls we can select one in 2 ways. the remaining ball will be placed in the 3rd box. so number of ways is 3*10*2 = 60
so total is 90 + 60 = 150 ways in which we can arrange 5 different coloured balls in 3 boxes so that every box has atleast one ball.

12. the marginal revenue is defined as p1*q1 - p*q so the answeris concave/convex when the demand curve is concave/convex so answer is a

exponential x^y means x raised to the power y

loss on redemption if the number of debentures is 20,000 is 20,000*2 = 40,000 so loss to be written off every year is 40,000/5 = 8000