Short Introduction
Algebraic identities are powerful mathematical tools that help simplify calculations, factorise expressions, and solve complex algebraic problems quickly. The Chapter End Exercise of “Exploring Algebraic Identities” tests students’ understanding of identities, factorisation techniques, and algebraic manipulations.
In this article, we provide detailed step-by-step solutions for Questions 1, 2 and 3 of the Chapter End Exercise.
Quick Information Box
| Particular | Details |
|---|---|
| Class | 9 |
| Subject | Mathematics |
| Chapter | Exploring Algebraic Identities |
| Section | Chapter End Exercise |
| Part | 1 |
| Questions Covered | Q1, Q2, Q3 |
| Difficulty Level | Easy to Moderate |
Concepts Used (Topics Covered)
✓ Algebraic Identities
✓ Expansion of Expressions
✓ Factorisation
✓ Difference of Squares
✓ Perfect Square Trinomials
✓ Rational Algebraic Expressions
✓ Simplification Techniques
Important Formulas
Identity 1
(a + b)² = a² + 2ab + b²
Identity 2
(a − b)² = a² − 2ab + b²
Identity 3
(a + b)(a − b) = a² − b²
Identity 4
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Identity 5
a³ + b³ + c³ − 3abc
= (a+b+c)(a²+b²+c²−ab−bc−ca)
Question 1
Use suitable identities to find the following products:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Question 2
Find the values using suitable identities:
Question 3
Factor the following algebraic expressions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
Question 4
Simplify the following:
(i)
(ii)
(iii)
Question 5
Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.
(i) 25a2 – 30ab + 9b2
(ii) 36s2 – 49t2
Question 6
Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.
(i) 6a2 – 24b2
(ii) 3ps2 – 15ps + 12p
Question 7
The village playground is shaped as a square of side 40 metres. A path of width s metres is created around the playground for people to walk. Find an expression for the area of the path in terms of s.
Question 8
If a number plus its reciprocal equals 10/3 , find the number.
Question 9
A rectangular pool has area 2×2 + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length. Hasta was a unit used to measure length.
Question 10
If both x – 2 and x – (1/2) are factors of px2 + 5x + r, show that p = r.
Question 11
If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 –3abc = – 25.
Question 12
By factoring the expression, check that n3 – n is always divisible by 6 for all natural numbers n. Give reasons.
Question 13
Find the value of
(i) x3 + y3 – 12xy + 64, when x + y = – 4
(ii) x3 – 8y3 – 36xy – 216, when x = 2y + 6
Key Learning from Chapter 4
After completing this chapter, students should be able to:
✅ Expand algebraic expressions quickly.
✅ Factorise complex expressions.
✅ Apply identities in numerical calculations.
✅ Verify algebraic identities.
✅ Simplify rational expressions efficiently.
Common Mistakes
1. Forgetting the Middle Term
Students often write:
(a+b)² = a²+b²
Correct form:
(a+b)² = a²+2ab+b²
2. Sign Errors
Incorrect handling of negative signs while using:
(a−b)²
3. Incomplete Factorisation
Not factoring expressions completely before cancellation.
4. Cancelling Terms Instead of Factors
Only factors can be cancelled.
5. Incorrect Expansion of Three-Term Expressions
Missing terms:
2ab, 2bc and 2ca
in
(a+b+c)²
Exam Tips
Tip 1
Memorise all identities thoroughly.
Tip 2
Write the identity before solving.
Tip 3
Always factorise completely.
Tip 4
Check signs carefully.
Tip 5
Verify answers by substitution whenever possible.
Tip 6
Practice mental calculations using identities.
Tip 7
Attempt factorisation questions systematically.
Practice MCQs
1. Which identity is correct?
A. (a+b)²=a²+b²
B. (a+b)²=a²+2ab+b²
C. (a+b)²=a²−2ab+b²
D. None
✅ Answer: B
2. Factorise x²−25
A. (x−5)²
B. (x+5)²
C. (x+5)(x−5)
D. None
✅ Answer: C
3. Value of 99² is
A. 9801
B. 9901
C. 9701
D. 9601
✅ Answer: A
4. Which identity is used in
(a+b)(a−b)
A. Perfect Square
B. Difference of Squares
C. Cube Identity
D. None
✅ Answer: B
5. Number of mixed terms in
(a+b+c)²
A. 1
B. 2
C. 3
D. 4
✅ Answer: C
6. Factorise x²+7x+12
A. (x+3)(x+4)
B. (x+2)(x+6)
C. (x+1)(x+12)
D. None
✅ Answer: A
7. Which identity gives
a²−b² ?
A. (a+b)²
B. (a−b)²
C. (a+b)(a−b)
D. None
✅ Answer: C
8. 102² equals
A. 10404
B. 10204
C. 10004
D. 10604
✅ Answer: A
9. Factorise 4x²−12x+9
A. (2x−3)²
B. (2x+3)²
C. (4x−9)
D. None
✅ Answer: A
10. Which chapter concept is most important?
A. Identities
B. Graphs
C. Geometry
D. Statistics
✅ Answer: A
Frequently Asked Questions (FAQ)
Q1. Why are algebraic identities important?
They simplify calculations and help solve algebraic problems quickly.
Q2. Which identity is used most frequently?
(a+b)² = a²+2ab+b²
Q3. How can I improve factorisation skills?
Practice identifying patterns and applying identities regularly.
Q4. Are algebraic identities important for Class 10?
Yes. They are extensively used in Class 10 Mathematics.
Q5. Can identities be used for mental calculations?
Yes. They greatly reduce calculation time.
Q6. What is the most common exam mistake?
Ignoring the middle term (2ab).
Q7. Is Chapter 4 important for competitive exams?
Yes. The concepts form the foundation for higher algebra.
Chapter Summary
Chapter 4 introduces students to powerful algebraic identities that help simplify calculations, expand expressions, factorise polynomials and solve algebraic problems efficiently. Mastering these identities improves speed, accuracy and confidence in mathematics.
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