UPSC Prelims 2001 Q139: Science & Technology – Quantitative Aptitude
Solve the given equations:
x² + y² = 34
x⁴ − y⁴ = 544
The values of x and y are
- (a) ±4, ±3
- (b) ±5, ±3
- (c) ±3, ±5
- (d) ±3, ±4
Answer: (b) ±5, ±3
Explanation
Solving x² + y² = 34 and x⁴ − y⁴ = 544
| Step | Working |
|---|---|
| Given | x² + y² = 34 and x⁴ − y⁴ = 544 |
| Factorise | x⁴ − y⁴ = (x² + y²)(x² − y²) → 544 = 34 × (x² − y²) → x² − y² = 16 |
| Add and subtract | 2x² = 50 → x² = 25; 2y² = 18 → y² = 9 |
| Values | x = ±5, y = ±3 |
Given: x² + y² = 34 and x⁴ − y⁴ = 544. Writing x⁴ − y⁴ as (x² + y²)(x² − y²) gives x² − y² = 16. Solving the two equations in x² and y² gives x = ±5 and y = ±3.
Answer: (b) ±5, ±3.
Why each option is right or wrong
- (a) Incorrect. 4² + 3² = 25, not 34.
- (b) CORRECT. 25 + 9 = 34 and 625 − 81 = 544.
- (c) Incorrect. x = ±3, y = ±5 gives x⁴ − y⁴ = −544.
- (d) Incorrect. 3² + 4² = 25, not 34.
Key point
Identity used: a⁴ − b⁴ = (a² + b²)(a + b)(a − b).
Remember
x² = 25 and y² = 9, so x = ±5 and y = ±3.
Common mistakes
- Swapping x and y and getting a negative difference.
- Trying values without factorising first.
- Forgetting the ± signs on both x and y.
Exam tip
Check options by putting them into both equations.
Sources
- NCERT Class IX — Mathematics, chapter 'Polynomials' (algebraic identities)
- R. S. Aggarwal — Quantitative Aptitude, chapter 'Simplification'