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Please explain/discuss the following

1. If two of the four following expressions -- x+y; x+5y; x-y; 5x-y -- are chosen at random, what is the probability that the product will be of the form x^2 - (by)^2, where b is an integer?

a 1/2
b 1/3
c 1/4
d 1/5
e 1/6

I would say 1/4 but answer is e....

2. Store S sold a total of 90 copies during seven days of last week and it sold different numbers of copies on any two of the days. If for the seven days store S sold greatest number of copies on Saturday and second greatest number of copies on Friday, did store S sell more than 11 copies on Friday?

(1) Last week store S sold 8 copies of the book on Thursday
(2) Last week store S sold 38 copies

Answer is b however i think is c...

3. In the rectangular coordinate system, are the points (r,s) and (u,v) equidistant from the origin?

1. Answer e = 1/6. Only product of the expressions in the form mentioned in the question is x^2-y^2, where b = 1, is the integer. Now total number of ways of selecting two expression out of 4 are 4c2 = 6, so the probability is 1/6

2. Answer B, suppose store sell X copies on Saturday and Y copies on Firday, while for rest of the five days it sells Z copies each. we have X + Y + 5Z = 90, where X>Y>Z.

Case 1. We know that Z = 8 so X + Y = 50 in this case X can have any value, so we cant say whether Y is > or <11>Z

Maximum value of Z in this condition can by 8, making Y =12. If Z = 9 then Y = 7, which is not possible.

So Case 2. is sufficient.

Answer 3.

If (r,s) and (u,v) are equidistance from the center (0,0), then r^2 + s^2 = U^2 + V^2 ( equalising the distances from the origin)

Case 1. r+s = 1, doesnt tell anything about (u,v).. in sufficient

Case 2. though relation between different coordinate points can be formed but it doent give any clue on the distance from the origin.. insufficient.

Using both the conditions. U^2 = 1+r^2-2r, and V^2 = 1+s^-2s

Adding both these equations we have U^2 +v^2 = r^2 + s^2 +2 - 2(r+s)

From case 1, we have r+s = 1, so the equation will become U^2 + V^2 = r^2 + s^2 .... sufficient.

2. Store S sold a total of 90 copies during seven days of last week and it sold different numbers of copies on any two of the days. If for the seven days store S sold greatest number of copies on Saturday and second greatest number of copies on Friday, did store S sell more than 11 copies on Friday?

(1) Last week store S sold 8 copies of the book on Thursday (2) Last week store S sold 38 copies