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1.  

In each question two equations I and II are given. Solve both and choose: A) x > y, B) x < y, C) x ≥ y, D) x ≤ y, E) x = y or the relation cannot be established.

I. x² = 625 II. y³ = 15625

Correct Answer: D [x ≤ y]

Explanation:

x = ±25; y = 25 (a cube has one real root).

2.  
I. x² − 17x + 72 = 0 II. y = √49

Correct Answer: A [x > y]

Explanation:

x = 8, 9; y = 7 (√ means the positive root only) → x > y.

3.  
I. 3x + 4y = 25 II. 4x + 3y = 24

Correct Answer: B [x < y]

Explanation:

II − I: x − y = −1 → x < y (x = 3, y = 4).

4.  
I. 10x² − 7x + 1 = 0 II. 15y² − 8y + 1 = 0

Correct Answer: E [x = y or the relation cannot be established]

Explanation:

x = 1/5, 1/2; y = 1/5, 1/3.

5.  
I. x³ = 512 II. y² = 64

Correct Answer: C [x ≥ y]

Explanation:

x = 8 (one real cube root); y = ±8.

6.  
I. 4x² = 9 II. 2y² − 9y + 9 = 0

Correct Answer: D [x ≤ y]

Explanation:

x = ±3/2; y = 3/2, 3. Pairs: −3/2 is below both y-roots, 3/2 = 3/2, 3/2 < 3 → x ≤ y.

7.  
I. 3x² − 21x + 36 = 0 II. 5y² − 5y − 30 = 0

Correct Answer: C [x ≥ y]

Explanation:

Divide I by 3: x² − 7x + 12 = 0 → x = 3, 4.

8.  
I. 12x² − 7x + 1 = 0 II. 30y² − 11y + 1 = 0

Correct Answer: A [x > y]

Explanation:

x = 1/4, 1/3; y = 1/6, 1/5.

9.  
I. 2x + y = 15 II. x + 2y = 15

Correct Answer: E [x = y or the relation cannot be established]

Explanation:

I − II: x − y = 0 → x = y = 5.

10.  
I. 2x² + 11x + 15 = 0 II. y² + 5y + 6 = 0

Correct Answer: E [x = y or the relation cannot be established]

Explanation:

x = −3, −5/2; y = −3, −2. −3 = −3, −3 < −2, but −5/2 > −3 → mixed → relation cannot be established.

11.  
I. x² + x − 56 = 0 II. y² − 17y + 72 = 0

Correct Answer: B [x < y]

Explanation:

x = −8, 7; y = 8, 9. Largest x (7) < smallest y (8) → x < y, even though x has a negative root.

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