qpaper
Generate

Class 9 Maths Chapter 4: Exploring Algebraic Identities — Important Questions & Sample Paper

CBSE· Ganita Manjari· 44+ original questions readyहिन्दी में देखें

Practise important & sample questions with answers, see the CBSE marks distribution & blueprint, or generate a full sample paper — free, for 2026-27.

Reviewed by qpaper's CBSE curriculum team · Edited by Mohit · Updated June 2026

Quick answer

Yes — this page has 44+ original Class 9 Mathematics Chapter 4 (“Exploring Algebraic Identities”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 4 'Exploring Algebraic Identities' in the NCERT Ganita Manjari textbook for Class 9 introduces students to fundamental algebraic identities that simplify algebraic manipulation. These identities are powerful tools for expanding products of binomials and trinomials, factoring polynomials, and performing quick calculations. The chapter focuses on key identities such as (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², a² – b² = (a + b)(a – b), (x + a)(x + b) = x² + (a + b)x + ab, and the square of a trinomial (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. Through a variety of examples and exercises, students learn to apply these identities both forward (expansion) and backward (factorization). Exam questions typically ask students to expand expressions like (2x + 5y)² or (3p – 4q + 5r)², factorize completely expressions such as 9x² – 12x + 4 or 25p² + 30pq + 9q² – 16r², and use identities cleverly to evaluate numerical expressions like 106² – 94² without direct squaring. Mastery of these identities builds a strong foundation for higher-level algebra and problem-solving. Regular practice with varied problems, as provided on qpaper.in, helps students gain confidence and speed in tackling CBSE exam questions.

Generate a full sample paper for this chapter

Pick chapters, set your blueprint and marks distribution, and export a print-ready PDF or editable Word — with an answer key. Free to build.

Generate full paperSee a sample paper

Preview: a real paper we generate

This is the actual CBSE board-style layout you export — built from this chapter's own questions, with an answer key.

Mathematics — Exploring Algebraic Identities

Class 9Time: 3 hrsMax Marks: 80

SECTION A

  1. 1.

    Without directly squaring the numbers, evaluate 106^2 - 94^2 using a suitable identity.

    (a) 1200(b) 2400(c) 3600(d) 4800
    1
  2. 2.

    If (x+5)^2 = x^2 + kx + 25, what is the value of k?

    (a) 5(b) 10(c) 25(d) 20
    1
  3. 3.

    Factor 50a^2 + 60ab + 18b^2 completely.

    (a) 2(5a + 3b)^2(b) (5a + 3b)^2(c) 2(25a + 9b)^2(d) 2(5a - 3b)^2
    1

+ 41 more questions in the full paper

Generate full paper

Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question typeMarks eachIn our bank
Multiple Choice (MCQ)1 mark13
Assertion–Reason1 mark6
Short Answer2 marks8
Short Answer3 marks6
Long Answer5 marks5
Case Study4 marks6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

Real, exam-style questions to practise and revise — each with its answer. Generate a full paper for unlimited more.

  1. Q1. Without directly squaring the numbers, evaluate 106^2 - 94^2 using a suitable identity.

    1 mark
    Multiple Choice (MCQ)
    (A) 1200(B) 2400(C) 3600(D) 4800
    ▸ Answer

    2400

  2. Q2. If (x+5)^2 = x^2 + kx + 25, what is the value of k?

    1 mark
    Multiple Choice (MCQ)
    (A) 5(B) 10(C) 25(D) 20
    ▸ Answer

    10

  3. Q3. Factor 50a^2 + 60ab + 18b^2 completely.

    1 mark
    Multiple Choice (MCQ)
    (A) 2(5a + 3b)^2(B) (5a + 3b)^2(C) 2(25a + 9b)^2(D) 2(5a - 3b)^2
    ▸ Answer

    2(5a + 3b)^2

  4. Q4. Which of the following equations is an identity?

    1 mark
    Multiple Choice (MCQ)
    (A) (x+1)^2 = x^2 + 1 for all x(B) (a - b)^2 = a^2 - 2ab + b^2(C) x^2 - 1 = 0 has solution x = 1(D) 2x + 3 = 7 for x = 2
    ▸ Answer

    (a - b)^2 = a^2 - 2ab + b^2

  5. Q5. Assertion (A): For any integer n, (n+1)² – (n–1)² is divisible by 4. Reason (R): (n+1)² – (n–1)² = (n+1+n–1)(n+1–(n–1)) = 2n · 2 = 4n, which is a multiple of 4.

    1 mark
    Assertion–Reason
    (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.
    ▸ Answer

    Both A and R are true and R is the correct explanation of A.

  6. Q6. Factorise the algebraic expression x^2 + 10x + 25 completely.

    2 marks
    Short Answer
    ▸ Answer

    (x + 5)^2

  7. Q7. If x + y = 8 and xy = 15, find the value of x^2 - y^2 without solving for x and y individually.

    2 marks
    Short Answer
    ▸ Answer

    ±16

  8. Q8. Factor completely: 49m^2 + 42m + 9.

    3 marks
    Short Answer
    ▸ Answer

    (7m + 3)^2

  9. Q9. Factor completely: 12x^2 + 36xy + 27y^2.

    3 marks
    Short Answer
    ▸ Answer

    3(2x + 3y)^2

  10. Q10. Expand (3a + 2b - 4c)² using the identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Hence, find the value of the expression when a = 1/3, b = -1/2 and c = 0.

    5 marks
    Long Answer
    ▸ Answer

    Expanded form: 9a² + 4b² + 16c² + 12ab - 16bc - 24ca Value for given a, b, c: 0

  11. Q11. Prove the identity (a + b + c)² - (a - b - c)² = 4a(b + c). Use this result to evaluate (23)² - (17)² without direct calculation.

    5 marks
    Long Answer
    ▸ Answer

    Proof as shown in solution; value = 240

  12. Q12. In a craft project, students created a pattern by joining squares and rectangles. The total area of the pattern is given by the expression 18x^2 + 60xy + 50y^2. The teacher asks them to factorise it completely to find possible dimensions.

    4 marks
    Case Study
    1. (i) First take out the greatest common factor from the expression.1 mark
    2. (ii) Now factorise the remaining trinomial using a suitable identity.2 marks
    3. (iii) Hence, write the final factorised form.1 mark
    ▸ Answer

    Q1: 2(9x^2 + 30xy + 25y^2); Q2: (3x + 5y)^2; Q3: 2(3x + 5y)^2

Frequently asked questions

What are the most important algebraic identities from Chapter 4 that I must memorize for the Class 9 CBSE exam?

The essential identities are: (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², a² – b² = (a + b)(a – b), (x + a)(x + b) = x² + (a + b)x + ab, and (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. These are directly used in expansion, factorization, and simplification problems.

How do I factorize an expression like 9x² – 12x + 4 using identities?

Recognize it as a perfect square trinomial. Compare with (a – b)² = a² – 2ab + b². Here, a² = 9x² ⇒ a = 3x, b² = 4 ⇒ b = 2, and the middle term –12x should equal –2ab = –2·3x·2 = –12x. Since it matches, 9x² – 12x + 4 = (3x – 2)².

In the CBSE exam, are there questions where we need to use identities to evaluate numeric expressions without direct calculation?

Yes, such questions are common. For example, evaluate 106² – 94² using the identity a² – b² = (a + b)(a – b). Here, 106² – 94² = (106 + 94)(106 – 94) = 200 × 12 = 2400. You avoid squaring large numbers.

What is the identity for the square of a trinomial, and when is it asked in exams?

The identity is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. It is often asked in expansion problems, like expanding (3p – 4q + 5r)², which you handle by treating it as (3p + (–4q) + 5r)² and applying the identity carefully with sign handling.

More chapters

Class 9 Maths Ch4 — Important Questions & Sample Paper with Answers | Free PDF