Assertion Reason Questions For Class 9 Maths Chapter 4 Exploring Algebraic Identities

Assertion and Reason questions for Class 9 Maths Chapter 4, Exploring Algebraic Identities are available in this article. It is meant to make the chapter easier to understand and help students get better at solving identity-based questions. The questions cover important topics such as algebraic identities, expanding expressions, factorisation, and the correct use of standard identities. With detailed solutions prepared according to the CBSE syllabus and NCERT textbook, students can strengthen their concepts and practise the most important question types. A free downloadable PDF is also available for quick revision and practice.

Assertion and Reason Questions on Exploring Algebraic Identities  Chapter 4 for Class 9

Directions: In each question below, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:

(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.

Question 1.
Assertion (A): The expression (x + y)² = x² + 2xy + y² is an identity.
Reason (R): An algebraic identity is an equation that holds true for all values of the variables appearing in it.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: The reason gives the meaning of an algebraic identity correctly. Since (x + y)² = x² + 2xy + y² is true for every value of x and y, it is an identity. So, R clearly explains why A is true.

Question 2
Assertion (A): (5x + 2y)² = 25x² + 20xy + 4y²
Reason (R): (a + b)² = a² + 2ab + b², where a and b can represent any algebraic terms.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: If we take a = 5x and b = 2y, then the identity gives 25x² + 20xy + 4y². So the assertion is correct, and the reason explains it perfectly.

Question 3
Assertion (A): 43² can be calculated as (40 + 3)² = 1600 + 240 + 9 = 1849.
Reason (R): (a – b)² = a² – 2ab + b² is the identity used to find 43².
Answer: (c) A is true; R is false.
Explanation: The calculation in A is correct, but it uses the identity (a + b)² = a² + 2ab + b², not the one given in R. So the assertion is right, but the reason is not.

Question 4
Assertion (A): x² + 4x + 4 = (x + 2)²
Reason (R): The identity a² + 2ab + b² = (a + b)² can be used in reverse to factorise algebraic expressions.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: The expression matches the pattern a² + 2ab + b² with a = x and b = 2. So it can be written as (x + 2)². The reason explains exactly how this factorisation works.

Question 5
Assertion (A): a² – b² = (a + b)(a – b)
Reason (R): (a + b)² – (a – b)² = 2ab.
Answer: (c) A is true but R is false.
Explanation: The assertion is a standard identity. But the reason is incorrect, because
(a + b)² – (a – b)² = 4ab, not 2ab.

Question 6
Assertion (A): (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Reason (R): This identity is derived by treating (b + c) as a single term d, and applying (a + d)² = a² + 2ad + d², then expanding d² = (b + c)².
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: The reason shows a neat way to expand the expression step by step. So it explains the identity in a clear and correct way.

Question 7
Assertion (A): 50p² + 60pq + 18q² = 2(5p + 3q)²
Reason (R): Before applying an identity, it is sometimes necessary to first take out a common numerical factor from all terms of the expression.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: First, factor out 2: 50p² + 60pq + 18q² = 2(25p² + 30pq + 9q²). Then the bracket becomes (5p + 3q)². So the reason explains the method used in the assertion.

Question 8
Assertion (A): x² + 11x + 30 = (x + 5)(x + 6)
Reason (R): To factorise x² + bx + c into (x + p)(x + q), we need p + q = b and p × q = c.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Here, 5 + 6 = 11 and 5 × 6 = 30, so the factorisation is correct. The reason gives the exact rule used to find the factors.

Question 9
Assertion (A): (a + b)³ = a³ + 3a²b + 3ab² + b³
Reason (R): (a – b)³ = a³ – 3a²b + 3ab² – b³ is obtained simply by replacing b with –b in the expansion of (a + b)³.
Answer: (b) Both A and R are true, but R does not explain A.
Explanation: Both statements are correct. But the reason is only another related identity; it does not explain why the first expansion is true.

Question 10
Assertion (A): x² – 5x + 6 = (x – 2)(x – 3)
Reason (R): When the coefficient of x is negative in x² + bx + c, the values of p and q in (x + p)(x + q) must both be negative.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Since –2 + –3 = –5 and (–2)(–3) = 6, the factorisation is correct. The reason gives the right pattern for choosing the factors.

Question 11
Assertion (A): x³ – y³ = (x – y)(x² + xy + y²)
Reason (R): x³ + y³ = (x + y)(x² – xy + y²), and the factorisation of x³ – y³ is obtained by replacing y with –y in this identity.
Answer: (c) A is true; R is false.
Explanation: The assertion is correct. But the reason is not stated properly, because the factorisation of x³ – y³ comes from the identity itself or by substituting y with –y carefully. As written, the reasoning is not accurate.

Question 12
Assertion (A): If x + y + z = 0, then x³ + y³ + z³ = 3xyz
Reason (R): The identity x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) holds for all real values of x, y, and z.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: If x + y + z = 0, then the right-hand side becomes 0, which gives x³ + y³ + z³ – 3xyz = 0. So the assertion follows directly from the identity in the reason.

Question 13
Assertion (A): The expression p³ + 6p²q + 12pq² + 8q³ represents the volume of a cube whose side is (p + 2q) units.
Reason (R): The volume of a cube of side a is a³, so to find the side from a given volume expression, one must express the volume in the form (a + b)³ = a³ + 3a²b + 3ab² + b³.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: The given expression matches the expansion of (p + 2q)³ exactly. So the volume belongs to a cube of side (p + 2q), and the reason explains how to identify it.

Question 14
Assertion (A): (x² – 7x + 12) ÷ (5x² + 5x – 100) simplifies to (x – 3) ÷ [5(x + 5)], provided the denominator is not zero.
Reason (R): The common factor (x – 4) can be cancelled from the numerator and denominator only when it is confirmed to be non-zero, which follows from the condition 5x² + 5x – 100 ≠ 0.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: The numerator becomes (x – 3)(x – 4) and the denominator becomes 5(x – 4)(x + 5). After cancelling the common factor (x – 4), we get the simplified form. The reason correctly explains why this cancellation is valid.

Question 15
Assertion (A): For any three consecutive integers, if we add the smallest and largest squares and then subtract twice the middle square, the result is always 2.
Reason (R): If the three consecutive integers are (n – 1), n, and (n + 1), then (n – 1)² + (n + 1)² – 2n² = 2, which follows from expanding using the identities (a – b)² and (a + b)².
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Expanding the squares gives a neat cancellation, and the result is always 2. The reason shows the algebra clearly, so it explains the assertion properly.

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