Q977: Assertion (A): Let $f(x) = e^{x}$ and $g(x) = \log x$. Then ...

Class CBSE Class 12 Mathematics Relations and Functions Q #977

KNOWLEDGE BASED

REMEMBER

1 Marks 2025 AISSCE(Board Exam) ASSERTION REASON

Assertion: Assertion (A): Let $f(x) = e^{x}$ and $g(x) = \log x$. Then $(f + g)x = e^{x} + \log x$ where domain of $(f + g)$ is $\mathbb{R}$.

Reason: Reason (R): $\text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is NOT the correct explanation of A.

(D) A is false but R is true.

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Correct Answer: D

AI Tutor Explanation

Detailed Solution

Step 1: Analyze the Assertion

The assertion states that $(f+g)(x) = e^x + \log x$ and that the domain of $(f+g)$ is $\mathbb{R}$. We know that $f(x) = e^x$ has a domain of $\mathbb{R}$, and $g(x) = \log x$ has a domain of $(0, \infty)$. Therefore, the domain of $(f+g)(x)$ is not $\mathbb{R}$.

Step 2: Analyze the Reason

The reason states that $\text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g)$. This is a correct statement.

Step 3: Determine the Correctness of Assertion and Reason

The assertion is false because the domain of $(f+g)(x)$ is not $\mathbb{R}$. The reason is true.

Final Answer: Assertion is false, but the Reason is true.

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the definitions of the domain of exponential and logarithmic functions and the rule for the domain of the sum of two functions.

Knowledge Dimension: CONCEPTUAL

Justification: The question tests the understanding of the concept of the domain of a function and how it is affected by operations on functions, specifically the sum of two functions.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the concepts of functions and their domains, which are fundamental topics in the syllabus. The question tests the student's understanding of these basic concepts.

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