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16 Jul 2026, 10:06
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IMO A
I. If x=-1, (-2)(y+3)(-y+3) =0
y=-3 or y=3, or |y| = 3 is a Prime, I. Must be true
II. If |y| not prime, then x=1 or y=-3/x... here x may or may not be prime, II. may or may not be true - out
III. If x is prime , y=-3 or y=-3/x, x could be = 1,-1,3,-3, so y/x could be -3, 3,1,-1...all integers, so this is alos out
Hence Ans = A. I. only
I. If x=-1, (-2)(y+3)(-y+3) =0
y=-3 or y=3, or |y| = 3 is a Prime, I. Must be true
II. If |y| not prime, then x=1 or y=-3/x... here x may or may not be prime, II. may or may not be true - out
III. If x is prime , y=-3 or y=-3/x, x could be = 1,-1,3,-3, so y/x could be -3, 3,1,-1...all integers, so this is alos out
Hence Ans = A. I. only
Bunuel
16 Jul 2026, 10:10
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if , (x-1) ( y+3) ( xy+3) =0 , either , x-1 or y+3 or xy-3 must be equal to zero.
so , x=1
y=-3
xy=-3
from statement one ,
( -1-1).(y+3) .(y+3) =0
or , (y+3) (Y-3) =0
either , y = -3 or y=3 , in both case , | y| = |-3| =3 =|3|
so , statement one is true .
now , statement two , |y| and x are not prime number . ( xy+3) should be zero . so , xy=-3
either x is 3 , y is -1 or x is -1 ,t is 3
if x is 3 y is |-1| =1 which is not prime number nd vice versa.
so , statement 2 is also false .
and from statement three, y+3 =0 or , y=-3
suppose , x is 3 , a prime number . now the equation will be = ( -3-1).(-3+3).(-9+3)= 0
so , the value of x = 3 and y = -3 , supports the equation .
but , y/x =- 3/3 =-1 which is an integer
so statement three is also false .
only statement 1 is right one
so , x=1
y=-3
xy=-3
from statement one ,
( -1-1).(y+3) .(y+3) =0
or , (y+3) (Y-3) =0
either , y = -3 or y=3 , in both case , | y| = |-3| =3 =|3|
so , statement one is true .
now , statement two , |y| and x are not prime number . ( xy+3) should be zero . so , xy=-3
either x is 3 , y is -1 or x is -1 ,t is 3
if x is 3 y is |-1| =1 which is not prime number nd vice versa.
so , statement 2 is also false .
and from statement three, y+3 =0 or , y=-3
suppose , x is 3 , a prime number . now the equation will be = ( -3-1).(-3+3).(-9+3)= 0
so , the value of x = 3 and y = -3 , supports the equation .
but , y/x =- 3/3 =-1 which is an integer
so statement three is also false .
only statement 1 is right one
16 Jul 2026, 10:17
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Let's analyze the given equation: \((x - 1)(y + 3)(xy + 3) = 0\)
For the product of these three factors to be zero, at least one of the factors must be zero. This gives us three possible scenarios (keeping in mind \(x\) and \(y\) must be integers):
Now, let's evaluate each statement to see which must be true:
I. If x = -1, |y| is a prime number
If \(x = -1\), then Scenario 1 (\(x = 1\)) is impossible. Therefore, we must satisfy either Scenario 2 or Scenario 3:
If Scenario 2: \(y = -3 \implies |y| = |-3| = 3\)
If Scenario 3: \(xy = -3 \implies (-1)y = -3 \implies y = 3 \implies |y| = |3| = 3\)
In both valid scenarios, \(|y| = 3\). Since 3 is a prime number, this statement MUST be true.
Statement I is TRUE.
II. If |y| is not a prime number, x is not a prime number
To prove this false, we just need to find one counterexample where \(|y|\) is not prime, but \(x\) is prime.
Looking at the pairs from Scenario 3 above, we have the pair \((3, -1)\).
Here, \(y = -1 \implies |y| = 1\) (1 is not a prime number).
However, \(x = 3\) (which is a prime number).
Since we found a case that directly contradicts the statement, it does not have to be true.
Statement II is FALSE.
III. If x is a prime number, y/x is not an integer
To prove this false, we just need a counterexample where \(x\) is prime and \(y/x\) is an integer.
From Scenario 2, we know \(y = -3\) is a valid solution for any integer \(x\).
Let's choose \(x = 3\) (a prime number).
If \(x = 3\) and \(y = -3\), then \(y/x = -3/3 = -1\).
Since -1 is an integer, this statement does not have to be true.
Statement III is FALSE.
Since only Statement I must be true, the correct option is A.
For the product of these three factors to be zero, at least one of the factors must be zero. This gives us three possible scenarios (keeping in mind \(x\) and \(y\) must be integers):
- \(x - 1 = 0 \implies x = 1\) (and \(y\) can be any integer)
- \(y + 3 = 0 \implies y = -3\) (and \(x\) can be any integer)
- \(xy + 3 = 0 \implies xy = -3\). The possible integer pairs for \((x, y)\) are \((1, -3)\), \((-1, 3)\), \((3, -1)\), and \((-3, 1)\).
Now, let's evaluate each statement to see which must be true:
I. If x = -1, |y| is a prime number
If \(x = -1\), then Scenario 1 (\(x = 1\)) is impossible. Therefore, we must satisfy either Scenario 2 or Scenario 3:
If Scenario 2: \(y = -3 \implies |y| = |-3| = 3\)
If Scenario 3: \(xy = -3 \implies (-1)y = -3 \implies y = 3 \implies |y| = |3| = 3\)
In both valid scenarios, \(|y| = 3\). Since 3 is a prime number, this statement MUST be true.
Statement I is TRUE.
II. If |y| is not a prime number, x is not a prime number
To prove this false, we just need to find one counterexample where \(|y|\) is not prime, but \(x\) is prime.
Looking at the pairs from Scenario 3 above, we have the pair \((3, -1)\).
Here, \(y = -1 \implies |y| = 1\) (1 is not a prime number).
However, \(x = 3\) (which is a prime number).
Since we found a case that directly contradicts the statement, it does not have to be true.
Statement II is FALSE.
III. If x is a prime number, y/x is not an integer
To prove this false, we just need a counterexample where \(x\) is prime and \(y/x\) is an integer.
From Scenario 2, we know \(y = -3\) is a valid solution for any integer \(x\).
Let's choose \(x = 3\) (a prime number).
If \(x = 3\) and \(y = -3\), then \(y/x = -3/3 = -1\).
Since -1 is an integer, this statement does not have to be true.
Statement III is FALSE.
Since only Statement I must be true, the correct option is A.
