Match the LIST-I with LIST-II

LIST-I		LIST-II
A. The number of integer solutions of 2x² − x − 1 = 0	I.	0
B. The number of integer solutions of x² − 5x + 6 = 0	II.	1
C. The number of positive integer solutions of |x − 4| = |x − 3| + 1	III.	2
D. The number of integer solutions of |x + 1| = x	IV.	3

Choose the correct answer from the options given below:

Correct option is D

Given:
A. 2x² − x − 1 = 0
B. x² − 5x + 6 = 0
C. |x − 4| = |x − 3| + 1
D. |x + 1| = x
Solution:
For A:
2x² − x − 1 = 0
Factoring:
(2x + 1)(x − 1) = 0
Therefore,
x = −1/2 or x = 1
Only x = 1 is an integer.
Hence, the number of integer solutions is 1.
Therefore, A → II.
For B:
x² − 5x + 6 = 0
Factoring:
(x − 2)(x − 3) = 0
Therefore,
x = 2 or x = 3
Both values are integers.
Hence, the number of integer solutions is 2.
Therefore, B → III.
For C:
|x − 4| = |x − 3| + 1
For x = 1:
3 = 2 + 1
Hence, x = 1 is a solution.
For x = 2:
2 = 1 + 1
Hence, x = 2 is a solution.
For x = 3:
1 = 0 + 1
Hence, x = 3 is a solution.
Thus, there are 3 positive integer solutions.
Therefore, C → IV.
For D:
|x + 1| = x
Since the left-hand side is non-negative, x must be greater than or equal to 0.
For x ≥ 0:
x + 1 = x
This is impossible.
For x < 0, the right-hand side is negative, whereas the left-hand side is non-negative. Hence, no solution exists.
Therefore, the number of integer solutions is 0.
Thus, D → I.
Hence, the correct matching is:
A-II, B-III, C-IV, D-I
Correct Answer: (d)