UPSC CSE Prelims
Number Types Previous Year Questions (PYQs)
Practice solved questions for Number Types with detailed step-by-step solutions, key insights, and trend analysis for UPSC CSE PRELIMS.
Solved Previous Year Questions
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If $x$ and $y$ are integers, then is $x$ even? Statement I : $x^2 y^2$ is even. Statement II : $1 + x^2 + y^2$ is odd.
Detailed Explanation:
We need to determine if x is even.
- Statement I (x²y² is even): At least one of x or y is even, but x could be odd (y even) or even (y odd/even) — insufficient alone.
- Statement II (1 + x² + y² is odd): x² + y² must be even, so x and y have the same parity (both even or both odd) — insufficient alone.
- Combining both: At least one is even (Statement I) and both have same parity (Statement II) → both x and y must be even.
- Therefore, both statements together uniquely determine that x is even, making Option 3 the correct answer.
Three variables $x$, $y$ and $z$ take values $2, 3, 4$ or $5$ such that their values are always distinct. If $M$ and $N$ denote the largest possible value and the smallest possible value, respectively, for the expression ${(x \times y) + z}$; then $M - N$ is
Detailed Explanation:
To maximize $(x \times y) + z$, choose the two largest values for the product: $x = 5, y = 4$ gives $5 \times 4 = 20$, and the largest remaining value $z = 3$ gives $M = 20 + 3 = 23$.
To minimize $(x \times y) + z$, choose the two smallest values for the product: $x = 2, y = 3$ gives $2 \times 3 = 6$, and the smallest remaining value $z = 4$ gives $N = 6 + 4 = 10$.
Therefore, $M - N = 23 - 10 = 13$, making Option 3 the correct answer.
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