For a natural no. b, let N(b) denote the no. Of natural numbers a for which the equation x²+ax+b =0 has integer roots.what is the smallest value of b for which N(b)=20?
let A and B together have n marbles Where n is greater than zero A says to be if I give you some Marbles then you will have twice as many Marbles as I will have then B says to A if I give you some Marbles then you will have thrice as many Marbles as I will have what is the minimum possible value of n from the above statements if they are true - himanshu
Assume, there are n no. of marbles. Initially, A having x marbles . So, B get n-x marbles. Now , A gives a1 marbles to B therefore, 2(x-a1)=(n-x+a1).........(1) Now, B gives a2 marbles to A therefore, x+a2= 3(n-x-a2)........(2) From 1 and 2 after eleminating x ,we get n=12(a1+a2)/5 For n to be smallest natrural no. a1+a2=5 therefore, n=12
there are n - 1 Redballs n green balls and n + 1 blue balls in a bag the number of ways of choosing 2 balls from the bag that have different colour is 299 what is the value of n _himanshu
For a natural no. b, let N(b) denote the no. Of natural numbers a for which the equation x²+ax+b =0 has integer roots.what is the smallest value of b for which N(b)=20?
ReplyDeleteHow to solve above question.
Himanshu
Pre-RMO 2014, q. no 17
Deletelet A and B together have n marbles Where n is greater than zero A says to be if I give you some Marbles then you will have twice as many Marbles as I will have then B says to A if I give you some Marbles then you will have thrice as many Marbles as I will have what is the minimum possible value of n from the above statements if they are true
ReplyDelete- himanshu
For 1st. Condition
Deletex + 2x = m
=}3x=m ie 3 is a factor of m
For 2nd condition
x + 3x = m
=}4x = m ie 4 is also a factor of m
Least no- lcm of 3 & 4 = 12
1-(8,4)
2-(3,9)
Assume, there are n no. of marbles. Initially, A having x marbles . So, B get
Deleten-x marbles.
Now , A gives a1 marbles to B
therefore, 2(x-a1)=(n-x+a1).........(1)
Now, B gives a2 marbles to A
therefore, x+a2= 3(n-x-a2)........(2)
From 1 and 2 after eleminating x ,we get
n=12(a1+a2)/5
For n to be smallest natrural no.
a1+a2=5
therefore, n=12
there are n - 1 Redballs n green balls and n + 1 blue balls in a bag the number of ways of choosing 2 balls from the bag that have different colour is 299 what is the value of n
ReplyDelete_himanshu
(n+1)(n)+(n)(n-1)+(n-1)(n+1)=299
ReplyDeletetherefore n=10
This comment has been removed by the author.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteAnswer is 1.
ReplyDeleteSee the solution p-rmo 2015 Q.17 which I have published on my blog.
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