NCERT Solutions for Class 11 Maths Chapter 4: Principle of Mathematical Induction

NCERT Class 11 Maths Chapter 4: Principle of Mathematical Induction: This chapter explores fundamental concepts of mathematical reasoning, contrasting inductive and deductive approaches. Inductive reasoning involves analyzing individual cases to form generalized conclusions through systematic observation, while deduction applies universal principles to specific scenarios. Algebraic formulations frequently express results in terms of positive integers (n), requiring mathematical induction for rigorous validation - a core technique detailed in these NCERT solutions.

The solutions emphasize mastery through structured practice, particularly focusing on two critical components:

• Base case verification (establishing initial validity)
• Inductive step execution (proving implications for successive cases)

These resources help students develop competency in constructing formal proofs and understanding real-world applications of induction principles. The chapter's exercises are carefully designed to reinforce logical reasoning skills through progressively challenging problems.

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction - Question with answers

Q. Define relation.
A relation from set A to set B is a subset of the Cartesian product A × B.

Q. What is a function?
A function is a relation where each element of the domain has a unique image in the codomain.

Q. What is the domain of a relation?
The set of all first elements (inputs) of the ordered pairs in a relation.

Q. What is the range of a relation?
The set of all second elements (outputs) of the ordered pairs in a relation.

Q. Differentiate between codomain and range.
Codomain is the set where output values come from, while range is the actual set of output values.

Q. Define Cartesian product.
The set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Q. How many elements will be in A × B if A has 3 elements and B has 4?
3 × 4 = 12 elements.

Q. If A = {1, 2} and B = {x, y}, find A × B.
A × B = {(1, x), (1, y), (2, x), (2, y)}

Q. Find the number of relations from A = {1, 2} to B = {3, 4, 5}.
Number of relations = 2^(n(A) × n(B)) = 2^(2×3) = 64

Q. If f(x) = x², what is f(3)?
f(3) = 3² = 9

Q. If f(x) = 2x + 5, find f(−2).
f(−2) = 2×(−2) + 5 = −4 + 5 = 1

Q. Is the relation R = {(1,2), (2,3), (3,4)} a function?
Yes, since each first element is mapped to exactly one second element.

Q. Is the relation R = {(1,2), (1,3)} a function?
No, because 1 maps to both 2 and 3.

Q. Find domain and range of f(x) = √(x − 2).
Domain: [2, ∞), Range: [0, ∞)

Q. Let A = {1, 2}, B = {3, 4}, define a function f: A → B by f(x) = x + 2. Find f.
f(1) = 3, f(2) = 4 → f = {(1, 3), (2, 4)}

Q. Every relation is a function.
❌ False. A function is a special type of relation.

Q. A function can map two different inputs to the same output.
✅ True. e.g., f(x) = x² ⇒ f(2) = f(−2) = 4

Q. If A has 3 elements and B has 2, there can be 8 relations from A to B.
❌ False. Number of relations = 2^(3×2) = 64

Q. If f(x) = 3x – 4, find x such that f(x) = 11.
3x − 4 = 11 ⇒ x = 5

Q. Let f: R → R be defined as f(x) = x². Is this one-one?
No. Because f(2) = f(−2) = 4

Q. Check whether the function f(x) = 2x is onto when f: R → R.
Yes. Every real number has a pre-image: x = y/2

Q. Is the vertical line test used to verify a function?
✅ Yes. If a vertical line cuts the graph at more than one point, it’s not a function.

Q. Sketch the graph of f(x) = |x| and state its domain and range.
Domain: R, Range: [0, ∞)

Q. If a function f: A → B is such that every element of B has at least one pre-image in A, then f is?
Onto (Surjective)

Q. If f(x) = 1/x, find domain and range.
Domain = R − {0}, Range = R − {0}

Q. Give an example of a many-to-one function.
f(x) = x² (e.g., f(2) = f(−2) = 4)

Q. Define an injective function with example.
One-one function. e.g., f(x) = x + 5

Q. What is a constant function?
A function that maps every element of the domain to the same element in the codomain. Example: f(x) = 3

Q. If f(x) = sin x, find domain and range.
Domain: R, Range: [−1, 1]

Q. The total number of functions from A = {1, 2} to B = {x, y, z} is ___
Answer: 3² = 9

Q. A function is also called a ___ relation.
Answer: well-defined

Q. If each element of A is paired with a unique element of B, the relation is a ___
Answer: function

Q. If A has ‘m’ elements and B has ‘n’, total number of functions is ___
Answer: n^m

Q. For f(x) = |x|, f(−4) = ___
Answer: 4

Q. f(x) = x³ is an example of a ___ function.
Answer: one-one and onto (bijective)