@mohitjain said:A 6-6 GRID is cut from a 8-8 chessboard.In how many ways can we put 2 identical coins one on d white square and d other on d black in d grid such dat they are not placed in d same row or column?
36c1*12c1/2! = 216??
@bodhi_vriksha said:Now I got to know that "Certified Pagals" are better (at least in experience) than "Hardcore Pagals" @iLoveTorres Gaurav...this one(s) is for you:How many integral solutions exist for 3x + 7y = 1000 such that x > 100 and y > 50?For how many positive integers 'n', the equation 12x + ny = 31 has no integral solutions for n Team BV
1)17
2)39?
2)39?
@The_Loser said:Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds
l1 + l2/(x + y)= 23
l1/x= 23
l2/y= 17
27x + 17y = 23(x + y)
4x = 6y
x/y = 3/2 ?
@iLoveTorres said:1)172)39?
First one is correct. Good going Gaurav (GgG 😃 )
Second one is not upto mark. May be I've not told yet what I asked/expected for..anyway keep trying.
Good Night. :)
Second one is not upto mark. May be I've not told yet what I asked/expected for..anyway keep trying.
Good Night. :)
@bodhi_vriksha said:Now I got to know that "Certified Pagals" are better (at least in experience) than "Hardcore Pagals" @iLoveTorres Gaurav...this one(s) is for you:How many integral solutions exist for 3x + 7y = 1000 such that x > 100 and y > 50?For how many positive integers 'n', the equation 12x + ny = 31 has no integral solutions for n Team BV
2)45?
what i did was n cant take values depending on what values "x" takes
If x=-1 then "n" cant be 43
If x=0 "n" cant be 31
If x=1 "n" cant be 19
if x=2 "n" cant be 5
Therefore 49-4 = 45
what i did was n cant take values depending on what values "x" takes
If x=-1 then "n" cant be 43
If x=0 "n" cant be 31
If x=1 "n" cant be 19
if x=2 "n" cant be 5
Therefore 49-4 = 45
@bodhi_vriksha said:Now I got to know that "Certified Pagals" are better (at least in experience) than "Hardcore Pagals" @iLoveTorres Gaurav...this one(s) is for you:How many integral solutions exist for 3x + 7y = 1000 such that x > 100 and y > 50?For how many positive integers 'n', the equation 12x + ny = 31 has no integral solutions for n Team BV
1. 16 solutions ?
3x + 7y = 1000
x = 1000 - 7y/3
for y = 52.....x = 212
.......
........
y = 97....x = 107
so no. of solutions = 97 - 52/3 + 1 = 15 + 1 = 16 ?
@bodhi_vriksha said:
1) How many integral solutions exist for 3x + 7y = 1000 such that x > 100 and y > 50?2) For how many positive integers 'n', the equation 12x + ny = 31 has no integral solutions for n Team BV
1) 3x + 7y = 1000
x > 100, x = 101 + k
y > 50, y = 51 + n
3k + 7n = 340
n = 3a + 1, where a can vary from 0 to 15
So, 16 solutions
2) n should not be coprime to 12
So, 48 - 48(1/2)(2/3) = 48 - 16 = 32 different possible values of n
@bodhi_vriksha said:First one is correct. Good going Gaurav (GgG )Second one is not upto mark. May be I've not told yet what I asked/expected for..anyway keep trying.Good Night. Team BV
I think it will be 16 for the first one
Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
@chillfactor said:1) 3x + 7y = 1000x > 100, x = 101 + ky > 50, y = 51 + n3k + 7n = 340n = 3a + 1, where a can vary from 0 to 15So, 16 solutions2) n should not be coprime to 12So, 48 - 48(1/2)(2/3) = 48 - 16 = 32 different possible values of nI think it will be 16 for the first one Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
Yes...you are correct.
For first one it is 16.
Explanation goes like this (almost similar to yours)
3x + 7y = 1000, x > 100, y > 50
One solution for (x, y) is = (331, 1)
So general cases will be
x = 331 - 7n > 100 i.e. n
y = 1 + 3n > 50 i.e. n >= 17
So n varies from 17 to 32 i.e. 16 values. :)
=============================================
For second one, it's correct that "n should not be coprime to 12"
=============================================
Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
As RHS is multiple of 4, so must be LHS. In LHS, except 3a, other two terms are multiple of 4 irrespective of variables b and c. So 'a' must be multiple of 4.
Say a = 4a'
Also RHS is multiple of 3, so must be LHS. In LHS, except 4b, other two terms are multiple of 3 irrespective of variables a and c. So 'b' must be multiple of 3.
Say b = 3b'
After these modifications, the new equation becomes
12a' + 12b' + 12c = 432
i.e. a' + b' + c = 36 where a', b', c are non negative integers.
And required number of solutions of above equation is simply C(38, 2) = 703. :)
Team BV
For first one it is 16.
Explanation goes like this (almost similar to yours)
3x + 7y = 1000, x > 100, y > 50
One solution for (x, y) is = (331, 1)
So general cases will be
x = 331 - 7n > 100 i.e. n
y = 1 + 3n > 50 i.e. n >= 17
So n varies from 17 to 32 i.e. 16 values. :)
=============================================
For second one, it's correct that "n should not be coprime to 12"
=============================================
Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
As RHS is multiple of 4, so must be LHS. In LHS, except 3a, other two terms are multiple of 4 irrespective of variables b and c. So 'a' must be multiple of 4.
Say a = 4a'
Also RHS is multiple of 3, so must be LHS. In LHS, except 4b, other two terms are multiple of 3 irrespective of variables a and c. So 'b' must be multiple of 3.
Say b = 3b'
After these modifications, the new equation becomes
12a' + 12b' + 12c = 432
i.e. a' + b' + c = 36 where a', b', c are non negative integers.
And required number of solutions of above equation is simply C(38, 2) = 703. :)
Team BV
@chillfactor said:1) 3x + 7y = 1000x > 100, x = 101 + ky > 50, y = 51 + n3k + 7n = 340n = 3a + 1, where a can vary from 0 to 15So, 16 solutions2) n should not be coprime to 12So, 48 - 48(1/2)(2/3) = 48 - 16 = 32 different possible values of nI think it will be 16 for the first one Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
odd+even+even = odd
hence 3a would be 6k form
=> 6k+4b+12c = 432
=> 3k+2b+6c = 216
again
6x+2b+6c = 216
3x+b+3c = 108
b = 3(36 - x - c)
b should also be 3k form... (odd+odd+odd)
3x+3b+3c=108
x+b+c=36
so 38c2 solutions??
@mohitjain said:A 6-6 GRID is cut from a 8-8 chessboard.In how many ways can we put 2 identical coins one on d white square and d other on d black in d grid such dat they are not placed in d same row or column?
6*6 => 36 squares ==> 36c1 ways ;
second one ==> same row same colum ==> ll hav 6 blacks out of total 18 blacks ==> so we can place in the available 12 grids ==> 12c1 ways
so is it 36c1*12c1 @bodhi_vriksha pls correct me if am wrong
second one ==> same row same colum ==> ll hav 6 blacks out of total 18 blacks ==> so we can place in the available 12 grids ==> 12c1 ways
so is it 36c1*12c1 @bodhi_vriksha pls correct me if am wrong
@The_Loser said:Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds
L1 / s1 = 27 ; L2/s2 = 17 ; L1+L2 / s1+s2 = 23 ; ==> L1+L2 = 23S1 + 23S2
==> 27s1 + 17S2 = 23S1 + 23S2 ==> 4s1 = 6s2 ==> s1/s2 = 3:2
==> 27s1 + 17S2 = 23S1 + 23S2 ==> 4s1 = 6s2 ==> s1/s2 = 3:2
@The_Loser said:A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is
A => 100 m ,( t -8 ) sec ; Sa = 5 kmph ;
B => 92 m , t sec ; Sb =??
D=s*t ==> 100 = 5*5/18 * (t-8 ) ===> t=80 sec .
92 = Sb*80 ==>Sb = 92/80 m/s = 92/80 * 18/5 = 4.14 kmph
B => 92 m , t sec ; Sb =??
D=s*t ==> 100 = 5*5/18 * (t-8 ) ===> t=80 sec .
92 = Sb*80 ==>Sb = 92/80 m/s = 92/80 * 18/5 = 4.14 kmph
@chillfactor said:
Find the number of non-negative integral solutions for the equation 3a + 4b + 12c = 432
3a + 4b = 0
0 , 0
3a + 4b = 12
0 , 3
4 0
3a + 4b = 24
0 , 6
4 , 3
8 , 0
.
.
.
3a + 4b = 432
=> 37 soln.
total = 37*38/2 = 703
let f be a function defined on the set of all integers, and assume that it satisfies the following properties:
A. f(0) (not equal) 0
B. f(1)=3
C. f(x)f(y)=f(x+y) + f(x-y) for all integers x and y.
A. f(0) (not equal) 0
B. f(1)=3
C. f(x)f(y)=f(x+y) + f(x-y) for all integers x and y.
determine f(7).
@ravi.theja said:36c1* 26c1 ??
Please check your solution again. Think...how many choices of white squares do we have in the 6*6 grid and how many choices of black squares do we have after removing the corresponding row and column of the chosen white square?
@The_Loser said:A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is?
Sc= s ; St = 1.5s ; 75 = S* t ; 75 =1.5* s* ( t - 5/24 )
==> s*t = 1.5*s*(t-5/24 ) ==> t = 1.5t - 5/16 ==> 0.5t = 5/16 ==> t = 5/8 ;
75 = s*5/8 ==> s=120kmph
==> s*t = 1.5*s*(t-5/24 ) ==> t = 1.5t - 5/16 ==> 0.5t = 5/16 ==> t = 5/8 ;
75 = s*5/8 ==> s=120kmph
@jain4444 said:let f be a function defined on the set of all integers, and assume that it satisfies the following properties:A. f(0) (not equal) 0B. f(1)=3C. f(x)f(y)=f(x+y) + f(x-y) for all integers x and y.determine f(7).
f(1)f(0)= 2f(1)=> f(1){f(0)-2}=0=> f(0)=2
Also, {f(x)}^2 =f(2x)+f(0)=f(2x)+2 => f(2)=3^3 - 2 =7 and f(4)=7^2 -2 =47
Also, f(2)f(1)=f(3)+f(1)=>f(3)=18
Hence, 47*18=f(7) + 3=> f(7)=843
@jain4444 said:3a + 4b = 0 0 , 0 3a + 4b = 12 0 , 3 4 0 3a + 4b = 24 0 , 6 4 , 3 8 , 0...3a + 4b = 432 => 37 soln. total = 37*38/2 = 703 let f be a function defined on the set of all integers, and assume that it satisfies the following properties:A. f(0) (not equal) 0B. f(1)=3C. f(x)f(y)=f(x+y) + f(x-y) for all integers x and y.determine f(7).
843??
Generally speaking linear diophantine equations ax+by=k can be solved for integer (x,y) iff k is a multiple of gcd (a,b).
Consider the example that we discussed earlier 12x+ny=31. Clearly, for this equation to yield integer solutions, we must have gcd(12,n)=31/m, where m is an integer.
Consider the example that we discussed earlier 12x+ny=31. Clearly, for this equation to yield integer solutions, we must have gcd(12,n)=31/m, where m is an integer.
=>gcd (12,n)=1, for the above equation to have integer solutions
Also, please note that if gcd(a,b)=1, we can always find two integers x and y such that ax+by=1
@jain4444 said:let f be a function defined on the set of all integers, and assume that it satisfies the following properties:A. f(0) (not equal) 0B. f(1)=3C. f(x)f(y)=f(x+y) + f(x-y) for all integers x and y.determine f(7).
843
In how many ways one can choose 2 white squares and one black square from a chessboard ?