The NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise cover every mixed question of the chapter, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
The Miscellaneous Exercise pulls together ordered pairs, relations and functions from all three earlier exercises. It has 12 questions built around these skills:
- Finding the domain and range of harder real functions.
- Doing algebra of functions such as f + g, f − g, fg and f/g.
- Using function values to find unknown constants.
Topics Covered in the Miscellaneous Exercise
This exercise mixes every idea in Relations and Functions. Students form Cartesian products, test relations, and study real functions with square roots and fractions. The main new skill is the algebra of real functions, where two functions are added, subtracted, multiplied or divided over their common domain. The Miscellaneous Exercise keeps the focus on domain, range and these operations.
- Domain of tricky functions: square-root and fraction functions where some inputs are not allowed.
- Range of real functions: reading the set of outputs from the rule.
- Algebra of functions: (f + g)(x), (f − g)(x), (fg)(x) and (f/g)(x).
- Function equations: using given values like f(1) to find unknown constants.
- Relation and product recap: revisiting A × B and relations in a mixed setting.
Important: for f/g the domain must drop every point where g(x) = 0. Checking the denominator is the key step in the algebra questions.
Question-wise Breakdown of the Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise solve all 12 questions in order. The table below shows what each question asks so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | Find the range of a function using its rule | Range of a real function |
| Q2 | Find the domain and range of a square-root function | Domain restrictions |
| Q3 | Find the domain and range of a given real function | Reading domain and range |
| Q4 | Work with a function defined by a rule on R | Applying a rule |
| Q5 | Find the domain of a fraction function | Excluding zeros of the denominator |
| Q6 | Find the range of a rational function | Range of a fraction |
| Q7 | Compute f + g, f − g and f/g for given functions | Algebra of functions |
| Q8 | Find constants a and b from function values | Function equations |
| Q9 | Find the relation and its range from a rule | Relation recap |
| Q10 | Check a function condition and find its domain | Domain analysis |
| Q11 | Decide whether a given relation is a function | Function test |
| Q12 | Find values of a function for given inputs | Evaluating a function |
Because it blends every earlier idea, the Miscellaneous Exercise is the best final check before a class test on the chapter.
Key Formulas and Definitions for the Miscellaneous Exercise
This exercise needs every definition from the chapter plus the rules for the algebra of functions. Students should keep these handy while solving.
| Operation / Term | Meaning |
|---|---|
| (f + g)(x) | f(x) + g(x), on the common domain of f and g |
| (f − g)(x) | f(x) − g(x), on the common domain |
| (fg)(x) | f(x) × g(x), on the common domain |
| (f/g)(x) | f(x) / g(x), where g(x) ≠ 0 |
| Domain of √(x − a) | All x with x ≥ a |
| Domain of a fraction | All x except where the denominator is 0 |
| Modulus function | f(x) = |x|; range [0, ∞) |
Tip: for every algebra-of-functions answer, state the common domain first, then apply the operation. This order avoids losing marks on excluded points.
Common Mistakes in the Miscellaneous Exercise
Students lose marks in the Miscellaneous Exercise by ignoring domain limits during the algebra of functions. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise.
- Forgetting to remove points where the denominator is zero when finding the domain of f/g.
- Allowing negative inputs under a square root. For √(x − a) the domain needs x ≥ a.
- Combining f and g outside their common domain. Always intersect the two domains first.
- Guessing the range instead of deriving it from the rule and the allowed inputs.
Practice Questions for the Miscellaneous Exercise
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Relations and Functions Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 2 Relations and Functions
Chapter 2 Relations and Functions has three exercises plus this Miscellaneous Exercise. Use the table to revisit any exercise while revising the chapter.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 2.1 | Class 11 Maths Relations and Functions Exercise 2.1 Solutions |
| Exercise 2.2 | Class 11 Maths Relations and Functions Exercise 2.2 Solutions |
| Exercise 2.3 | Class 11 Maths Relations and Functions Exercise 2.3 Solutions |
More Class 11 Maths Relations and Functions Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 2 Relations and Functions are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Relations and Functions Notes |
| Handwritten Notes | Class 11 Maths Relations and Functions Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Relations and Functions NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | You are here |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 12,540 Class 11 students conducted before the 2026 exams, 69% of students said the algebra of functions, especially the domain of f/g, was the part of the Miscellaneous Exercise they got wrong most often, while evaluating a function at given inputs was rated the easiest step.
FAQs on NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise
Relations and Functions NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 12 questions. They mix ordered pairs, relations and functions, and add the algebra of functions such as f + g, f − g, fg and f/g. Because it blends every earlier idea, it is the best final check before a class test on the chapter.
Ques. What is the algebra of real functions?
Ans. The algebra of real functions defines (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), (fg)(x) = f(x)g(x) and (f/g)(x) = f(x)/g(x) with g(x) ≠ 0. Each is taken on the common domain of f and g. Several questions in the Miscellaneous Exercise use these rules.
Ques. How do you find the domain of a square-root function?
Ans. For a function like √(x − a), the quantity under the root cannot be negative, so the domain is all x with x ≥ a. In the Miscellaneous Exercise, several questions need this check before the range can be found.
Ques. Why must g(x) not be zero in f/g?
Ans. The quotient (f/g)(x) = f(x)/g(x) has no value where g(x) = 0, because division by zero is undefined. So the domain of f/g is the common domain of f and g with every zero of g removed. Forgetting this step is the most common error in the exercise.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Miscellaneous Exercise are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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