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Class 9 Maths Part 1 Chapter 4 – Exploring Algebraic Identities | Ganita Manjari Worksheet

Practise Class 9 Maths Part 1 Chapter 4 – Exploring Algebraic Identities with this Ganita Manjari Chapter 4 Worksheet. This worksheet includes important questions and revision exercises to strengthen understanding of algebraic identities, improve mathematical practice and support effective preparation for class tests and examinations.

Class 9 Maths – Chapter 4 Worksheet

📘 Class 9 Mathematics

Chapter 4 – Exploring Algebraic Identities

Worksheet

Name: __________________________    Date: ______________

100%

📝 Instructions

  • Read every question carefully.
  • Use suitable algebraic identities wherever possible.
  • Show all important steps.
  • Do not use a calculator unless your teacher permits it.

Section A – Multiple Choice Questions

1 mark
1.
The expansion of (a + b)² is:

a² + b²
a² + ab + b²
a² + 2ab + b²
a² − 2ab + b²

1 mark
2.
Which identity represents the difference of two squares?

a² + b² = (a+b)²
a² − b² = (a+b)(a−b)
a² − b² = (a−b)²
a² + b² = (a+b)(a−b)

1 mark
3.
Which of the following is equal to (x − y)²?

x² + 2xy + y²
x² − 2xy + y²
x² − y²
x² + y²

1 mark
4.
The factorisation of x² + 7x + 12 is:

(x+2)(x+6)
(x+3)(x+4)
(x−3)(x−4)
(x+1)(x+12)

1 mark
5.
Which identity gives x³ − y³?

(x−y)(x²+xy+y²)
(x+y)(x²+xy+y²)
(x−y)(x²−xy+y²)
(x+y)³

Section B – Fill in the Blanks

6. Complete the identities.

(a + b)² = a² + ______ + b²

(a − b)² = a² − ______ + b²

(a + b)(a − b) = ______

x³ − y³ = (x − y)(________________)

x³ + y³ = (x + y)(________________)

7. Complete:
(a + b + c)² = _______________________________

Section C – Expand

8. Expand:
(7x + 4y)²
9. Expand:
(3x − 2y + 4z)²
10. Expand:
(−3m + 4k − l)²
11. Expand:
(p + 3q + 7r)²

Section D – Use Identities to Calculate

12. Find 117² using a suitable identity.
13. Find 78² using a suitable identity.
14. Find 198² using a suitable identity.
15. Find 214² using a suitable identity.
16. Find 1104² using a suitable identity.
17. Find 1120² using a suitable identity.

Section E – Products Using Identities

18. Find:
17 × 21
19. Find:
104 × 96
20. Find:
24 × 16
21. Find:
147³
22. Find:
199³
23. Find:
127³
24. Find:
(−107)³
25. Find:
(−299)³

Section F – Factorisation

26. Factorise:
16y² + 4y + 1/16
27. Factorise:
9m²/25 − n²
28. Factorise:
27b³/64 − 1/216
29. Factorise:
x² + 5x/6 + 1/6
30. Factorise:
64y³ + 125z³
31. Factorise:
p³ + q³ + r³ − 3pqr

Section G – Simplification

32. Simplify the following rational algebraic expression:
(x² − 7x + 12) / (5x² + 5x − 100)

Show all factorisation steps.

33. Simplify:
(9a³ − 3b³) / (9a² − 36)

Section H – Application Questions

34. A rectangular pool is such that its breadth is 4 metres less than its length and its area is 96 square metres. Find the length and breadth of the pool.
35. Find possible expressions for the length and breadth of a rectangle whose area is:
25a² − 30ab + 9b²
36. Find possible expressions for the length and breadth of a rectangle whose area is:
36s² − 49t²
37. A 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.

Section I – Challenge Questions ⭐

38. If a + b + c = 5 and ab + bc + ca = 10, prove that
a³ + b³ + c³ − 3abc = −25
39. By factoring the expression, check that n³ − n is always divisible by 6 for all natural numbers n. Give reasons.
40. Find the value of
x³ + y³ − 12xy + 64
when x + y = −4.
41. Find the value of
x³ − 8y³ − 36xy − 216
when x = 2y + 6.

✅ Answer Key

Section A

  1. a² + 2ab + b²
  2. a² − b² = (a+b)(a−b)
  3. x² − 2xy + y²
  4. (x+3)(x+4)
  5. (x−y)(x²+xy+y²)

Section B

  1. 2ab; 2ab; a²−b²; x²+xy+y²; x²−xy+y²
  2. a²+b²+c²+2ab+2bc+2ca

Section C

  1. 49x² + 56xy + 16y²
  2. 9x² + 4y² + 16z² − 12xy + 24xz − 16yz
  3. 9m² + 16k² + l² − 24mk + 6ml − 8kl
  4. p² + 9q² + 49r² + 6pq + 42qr + 14pr

Section D

  1. 13689
  2. 6084
  3. 39204
  4. 45796
  5. 1218816
  6. 1254400

Section E

  1. 357
  2. 9984
  3. 384
  4. 147³ = 3176523
  5. 199³ = 7880599
  6. 127³ = 2048383
  7. (−107)³ = −1225043
  8. (−299)³ = −26730899

Section F

  1. (4y + 1/4)²
  2. (3m/5 − n)(3m/5 + n)
  3. (3b/4 − 1/6)(9b²/16 + b/8 + 1/36)
  4. (x + 1/2)(x + 1/3)
  5. (4y + 5z)(16y² − 20yz + 25z²)
  6. (p+q+r)(p²+q²+r²−pq−qr−rp)

Section G

For rational expressions, factor the numerator and denominator first, then cancel only common non-zero factors.

Section H

34. Length = 12 m, Breadth = 8 m.

35. 25a² − 30ab + 9b² = (5a−3b)²

36. 36s² − 49t² = (6s−7t)(6s+7t)

37. Area of path = (40+2s)² − 40² = 160s + 4s²

Section I

38. Using a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca), the value is −25.

39. n³−n = n(n−1)(n+1). These are three consecutive integers, so their product is divisible by both 2 and 3, and therefore by 6.

40. Since x+y=−4, the expression simplifies to −64 + 12xy − 12xy + 64 = 0.

⭐ Student Reflection

Which identity do you find easiest?

Which identity do you need more practice with?

One thing I learnt from this chapter:

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