## Sunday, February 04, 2007

### MATHS QUIZ CLASS XI SOLUTIONS

MATHS QUIZ CLASS XI

SOLUTIONS

1. First which almost all have attempted correctly had no logic except that you had to fill 1 and 8 in the center boxes and the rest follows. Note that in the solution given below order of the set of nos can interchange: (7182);[(64)(35)]

4 6
7 1 8 2
3 5

1. This was plain application of G.P The point to note is that the horizontal displacement is NOT being asked here. Also note that the ball travels 1m once but after that every other distance twice(while going up and while coming down)

D = 1 + 2 ( 1/3 + 1/9 + 1/27 + ..............∞)

= 1 + 2[(1/3) / (1 – 1/3)]

= 1 + 1

=2 meters

An interesting but non-mathematical problem.The catch is that as the water level rises, the ship also automatically rises and with it rises the ladder. Hence the max no of rungs submerged under water at any point is 3

4. The amount of wine in water is the same as the amount of water in wine.

There are two almost similar approaches to this question. I will be writing both: Let the volume of the cup is 1 cup, and the volume of each bucket is 10 cups. Clunking our way through this (and you don't really need to pay attention here), we see that after the first transfer the proportion of wine to total mixture in the water bucket is 1/11. Thus, on the return trip, the proportion of water to total mixture in the cup is 10/11. After dumping it in the wine bucket we get back to the original volume of 10 cups. The proportion of water to total mixture is

`     (amount of water in wine bucket)/(total volume in wine bucket) =(10/11)/10  = 1/11`
`Well, what do you know. This is the same proportion as wine to total mixture in thewater bucket. Since both buckets are back to the same, original volumes, the amountof wine in the water bucket is exactly the as the amount of water in the winebucket. `

Assume that each glass contained 100 units of liquid and that the spoon held 10 units. The spoon first removes 10 units of water, so the water glass contains 90 units of water, and the wine glass contains 100 units of wine and 10 units of water.

With 110 units in the wine glass, the spoon will remove 1/11 of each liquid in that glass. Thus it will transfer to the water glass 9 1/11 units of wine and 10/11 units of water. The water glass will then contain 90 10/11 units of water and 9 1/11 units of wine, and the wine glass will contain 90 10/11 units of wine and 9 1/11 units of water.

5. let xi (where I=1 ,2,…….,10) be the last digit of the 10 numbers not divisible by 10.

1<= xi <= 9

we have to now prove that x1+x2+x3+…..+x10 = 10k

By multinomial series method

LHS = (x^0 +x +x^2 +x^3 + ……+x^9) (x^0 +x +x^2 +x^3 + ……+x^9)…….. (x^0 +x +x^2 +x^3 + ……+x^9) (x^0 +x +x^2 +x^3 + ……+x^9) 10 times

==(x^0 +x +x^2 +x^3 + ……+x^9)^10

We require a coefficient of x^10 or x^10k in (x^0 +x +x^2 +x^3 + ……+x^9)^10

(x^0 +x +x^2 +x^3 + ……+x^9)^10

==[[1(1 – x^10)/(1-x)]]^10

=={[1- x^10]^10}{[1-x]^(-10)}

Now in the above series we can always find a coefficient of x^10 or x^10k

6. We first line them up and ask first (n-1) persons to divide the object into 1/n parts each.The last person cannot divide and we have n parts.Then the “choosing” of the part follows the reverse order i.e. the last person chooses first and then the second-last and so on. The first person virtually has no option but to take away the left piece.The first person does not choose and the last person does not cut!

7. A = 3(area of sector describing 30 degrees) – 2(area of equilateral triangle)

= 3(π/6) – 2(root3 / 4) since radius of circle = length of side of triangle = 1 unit

= π/2 - root3 / 2= 0 .705 sq.units

8 . f(32)=f(16+16)=f(16*16)=f(256)=37656

f(68)=f(64+4)=f(64*4)=f(256)=f(32)=37656