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.
(C) A is true but R is false.
(D) A is false but R is true.
Prev Next
Correct Answer: D

AI Tutor Explanation

Powered by Gemini

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.

AI generated content. Review strictly for academic accuracy.

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.

More from this Chapter

SA
Let R be a relation defined over N, where N is set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, $n\in N$." Find whether R is reflexive, symmetric and transitive or not.
SA
Let R be a relation defined on a set N of natural numbers such that $R=\{(x,y): xy \text{ is a square of a natural number, } x, y\in N\}$. Determine if the relation R is an equivalence relation.
LA
34. (a) If N denotes the set of all natural numbers and R is the relation on $N \times N$ defined by $(a, b) R (c, d)$, if $ad(b+c)=bc(a+d)$. Show that R is an equivalence relation.
LA
Show that a function $f:R\rightarrow R$ defined by $f(x)=\frac{2x}{1+x^{2}}$ is neither one-one nor onto. Further, find set A so that the given function $f:R\rightarrow A$ becomes an onto function.
LA
Show that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as $f(x)=\frac{5x-3}{4}$ is both one-one and onto.
View All Questions