Q1674: The domain of $f(x)=\cos^{-1}(2x-5)$ is:...

Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #1674

COMPETENCY BASED

APPLY

1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE

The domain of $f(x)=\cos^{-1}(2x-5)$ is:

(A) [-1,1]

(B) [4, 6]

(D) [2, 3]

Prev Next

AI Tutor Explanation

Detailed Solution

Step 1: Identify the condition for the domain of the inverse cosine function

The function $f(x) = \cos^{-1}(u)$ is defined only when the argument $u$ lies in the closed interval $[-1, 1]$. Therefore, for $f(x) = \cos^{-1}(2x-5)$, we must satisfy the inequality: $$-1 \leq 2x - 5 \leq 1$$

Step 2: Solve the inequality for $x$

Add 5 to all parts of the inequality: $$-1 + 5 \leq 2x - 5 + 5 \leq 1 + 5$$ $$4 \leq 2x \leq 6$$

Step 3: Isolate $x$

Divide all parts of the inequality by 2: $$\frac{4}{2} \leq x \leq \frac{6}{2}$$ $$2 \leq x \leq 3$$

Step 4: Conclusion

The domain of the function is the interval $[2, 3]$. Comparing this with the given options, it matches option (D).

Final Answer: [2, 3]

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student must apply the known domain constraints of inverse trigonometric functions to a specific algebraic expression.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the execution of a step-by-step algebraic procedure (solving a compound inequality) to arrive at the solution.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This tests the student's ability to manipulate inverse trigonometric function definitions beyond standard textbook examples.