MHT CET · Maths · Mathematical Reasoning

\(\mathrm{S}_1:\) If -7 is an integer, the \(\sqrt{-7}\) is a complex number
\(\mathrm{S}_2:-7\) is not an integer or \(\sqrt{-7}\) is a complex number

A \(\mathrm{S}_1\) and \(\mathrm{S}_2\) are converse statements of each other
B \(S_1\) and \(S_2\) are negations of each other
C \(\mathrm{S}_1\) and \(\mathrm{S}_2\) are equivalent statements
D \(S_1\) and \(S_2\) are contrapositive of each other

Verified Solution

Answer & Solution

Correct Answer

(C) \(\mathrm{S}_1\) and \(\mathrm{S}_2\) are equivalent statements

Step-by-step Solution

Detailed explanation

Let \(p:-7\) is an integer.
\(\mathrm{q}: \sqrt{-7}\) is a complex number.
Logical form of \(\mathrm{S}_1 \equiv \mathrm{p} \rightarrow \mathrm{q}\)
Logical form of \(\mathrm{S}_2 \equiv \sim \mathrm{p} \vee \mathrm{q}\)
We know that \(\mathrm{p} \rightarrow \mathrm{q} \equiv \sim \mathrm{p} \vee \mathrm{q}\)
\(\therefore \mathrm{S}_1\) and \(\mathrm{S}_2\) are equivalent statements.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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\(\mathrm{S}_1:\) If -7 is an integer, the \(\sqrt{-7}\) is a complex number\(\mathrm{S}_2:-7\) is not an integer or \(\sqrt{-7}\) is a complex number

Step-by-step Solution

\(\mathrm{S}_1:\) If -7 is an integer, the \(\sqrt{-7}\) is a complex number
\(\mathrm{S}_2:-7\) is not an integer or \(\sqrt{-7}\) is a complex number