official quant thread for cat08

[The Last Romeo]

Originally Posted by linksuresh

New Problem !

f(x) = sqrt[x+2005*sqrt{x+2005*sqrt(x+2005*sqrt...............

where f(x)>0.

Find the remainder when

f(0)f(2006)f(4014)f(6024)f(8036) is divided by 1000.


I neither have the answer nor the solution.

Hi Suresh,
My solution is also on the expected lines of what Prakhar said, but just elaborating on what Prakhar said.
F(x) = sqrt[x + 2005*f(x)]

F^2(x) = x + 2005 * F(x)
now
F^2(0) = 0 + 2005* F(0)
F(0) = 2005
similarly
F^2(2006) = 2006 + 2005 *F(2006)
let F(2006) = z
z^2 = 2006 + 2005z
z^2 - 2005z -2006=0
z^2-2006z + z -2006=0
z(z-2006) + 1(z-2006)=0
(z-2006) * (z + 1) = 0
hence z=2006 or -1
similarly for
F(4014) z=2007 or -2
f(6024) z=2008 or -3
F(8036) z=2009 or -4

so we have the eqaution
2005 * 2006 * 2007 * 2008 * 2009/1000
using cyclicity theorm

=>5 * 6 * 7 * 8 * 9/1000
=>1680 * 9/1000
=> 680 * 9/1000
=> 6120/1000
=>120
hence the answer, hope i am clear with the solution.

[STALWART]

Originally Posted by gaggi_85

I found this ques to be very interesting...
the answer should be in the form of 2k+3 with k=9, the ansr should be 21.

Pls confirm if i am right so that i can share my approach...

@ varun .. thanx

@ gaggi ...

yah can refer solution given by varun

100! + 2 = ? mod (1111)

1111=101*11 ( both are coprimes to each other)

100! + 2 = 2 mod(11) ... (1)

100! + 2 = 1 mod (101) .... (2)

form equations ....

N = 101 K1 + 1
N = 11K2 + 2

Solving ...

11 K2 = 101 K1 - 1 .... (3)

K1 = 17 and K2 = whatever .. (find out yahself but we just need any one of 'em )

N = 1718

1718 = 607 mod (1111)

So answer = 607

Sayonara

[implex]

Here I present Problem set 3 (question 151-200). This time considering the demand of all, key and solutions have been provided, you are welcome to post your queries and doubts here, and if you have alternate solutions please come forward and send it to me

Cheers
Rahul

[Varun Khullar]

Originally Posted by rahul_kash

Hey all,

I am not able solve this by Fermat's theorem

Please tell me how to calculate the value of
2^21 mod 23 ,with steps

22 is euler

2^-1 mod 23
2x = 23y+1
12 .. answer is 12

[chandu_IIMA]

q)A car goes 20 miles on a gallon of gas driving at 60m/h. If the car is driven at 75m/h it goes as far as 70%. How many gallons of gas will be required to travel 210m/h.
a)6.17 b)22.5 c)12 d)15



[quote=convivial;919607]hi frnds
as cat 07 is over and there r puys who want to gear up for cat 08....i am starting this official quant thread for cat 08....i request the administrator and other puys to make it an official thread.
best of luck





************cogito_ergo_sum*********************** ************

[chandu_IIMA]

Q)A train approaches a tunnel PQ which is 16m long. Two rabbits A and B are standing at points which are 12m and 8m inside the tunnel with respect to the entrance P. When the train is xm away from P. a starts running towards P & B towards Q. differences between the ratio of the speed of A to that of the Train and the ratio of the speed of b to that of the train is 1/8. How much can the distance x be,if both of them get caught at the ends of the tunnel?
a)30m b)16m c)48m d)none of these





Q)Rakesh and Siddharth row on a river simultaneously from the same point. Rakesh ows downstream and Siddharth rows upstream. In 15 minutes,they re 2.25 km apart. rakesh then turns to follow Siddhath and after 30 min from the beginning,the boats rowed together 3.5 km. If the speed of Rakesh and Siddharth and the stream are constant at how many km/hr does the stream flow?
a)3 b)2.5 c)2 d)5

[quote=chandu_IIMA;1169069]
q)A car goes 20 miles on a gallon of gas driving at 60m/h. If the car is driven at 75m/h it goes as far as 70%. How many gallons of gas will be required to travel 210m/h.
a)6.17 b)22.5 c)12 d)15

Originally Posted by convivial

hi frnds
as cat 07 is over and there r puys who want to gear up for cat 08....i am starting this official quant thread for cat 08....i request the administrator and other puys to make it an official thread.
best of luck





************cogito_ergo_sum*********************** ************

[aniruddha_recd]

Originally Posted by chandu_IIMA

Q)A train approaches a tunnel PQ which is 16m long. Two rabbits A and B are standing at points which are 12m and 8m inside the tunnel with respect to the entrance P. When the train is xm away from P. a starts running towards P & B towards Q. differences between the ratio of the speed of A to that of the Train and the ratio of the speed of b to that of the train is 1/8. How much can the distance x be,if both of them get caught at the ends of the tunnel?
a)30m b)16m c)48m d)none of these

When train covers X m ,rabbit A covers 12 m.

Speed of Train/Speed of A = X/12

Similarly when train covers X+16 m ,B covers 8m

Speed of Train/Speed of B = (X+16)/8

According to problem

12/X -8/(X+16 ) = 1/8

From options putting X = 48.satisfies above equation

[prince_chittu]

Q)Rakesh and Siddharth row on a river simultaneously from the same point. Rakesh ows downstream and Siddharth rows upstream. In 15 minutes,they re 2.25 km apart. rakesh then turns to follow Siddhath and after 30 min from the beginning,the boats rowed together 3.5 km. If the speed of Rakesh and Siddharth and the stream are constant at how many km/hr does the stream flow?
a)3 b)2.5 c)2 d)5

let speed of rakesh=r ,siddharth=s and river =u
in first 15 mins
15/60(s-u+r+u)=2.25
s+u=9

in 30 mins from begining
15/60(s-u+r+u)+15/60(s-u+r-u)=3.5
2u=4
u=2 km/h

[STALWART]

Another Piece of Cake ....

There are infinite number of Red , Green , Yellow and Black colored balls . In how many ways one can choose 20 balls ?

Sayonara

[STALWART]

Prime numbers are infinite . Prove it .... use any method i.e. induction , binomial ....... But just don say they are infinite caz natural numbers are infinite . The probability of finding 'em out decreases as we travel towards right side of the number line. But does it ever become 0 ? If not then prove it that they are infinite in numbers in a theortical manner. This was asked in IIM-Interview. Lets try it.