ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper

Analysis of document file formats:

• Incorrect: .msw is not a standard document extension.
• Incorrect: .wrd is not used by Microsoft Word.
• Incorrect: .wor is a MapInfo workspace file format.
• Correct: The standard file extension identifying a document created in Microsoft Word is .doc (or modern OpenXML .docx).

Analysis of algebraic properties of inequalities:
Given $0 \lt a \lt b$:

• Correct: The square root function $f(x) = \sqrt{x}$ is strictly monotonically increasing for $x \gt 0$. Therefore, $0 \lt a \lt b \implies \sqrt{a} \lt \sqrt{b}$.
• Incorrect: Reversed inequality contradicts monotonicity.
• Incorrect: Since $a \lt b$ and both are positive, squaring preserves order: $a^2 \lt b^2$, so $b^2 \lt a^2$ is false.
• Incorrect: $a^2 \lt b^2 \implies a^2 - b^2 \lt 0$, so $a^2 - b^2 \gt 0$ is false.

Analysis of theorems on sequence limits:
Let $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$.

• Correct: By the fundamental limit theorem for sequences, if $(a_n)$ and $(b_n)$ converge, their sum $(a_n + b_n)$ converges and $\lim_{n \to \infty} (a_n + b_n) = A + B$.
• Incorrect: If $B \ne 0$, $(a_n / b_n)$ converges to $A / B$.
• Incorrect: If $a_n \to 0$, $(b_n / a_n)$ may diverge (e.g. division by zero limit).
• Incorrect: The difference $(a_n - b_n)$ is always convergent to $A - B$.