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CBSE · Class 9 · Mathematics

Exploring Algebraic Identities

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
(x + 6)3 expands to:
  1. x3 + 18x2 + 18x + 216
  2. x3 + 216
  3. x3 + 6x2 + 36x + 216
  4. x3 + 18x2 + 108x + 216
2.
(3x − 9)3 expands to:
  1. 27x3 − 243x2 + 729x − 729
  2. 27x3 − 729
  3. 27x3 + 243x2 − 729x − 729
  4. 27x3 − 243x2 − 729x − 729
3.
(a + b)3 equals:
  1. a3 + 3a2b + b3
  2. a3 + 3ab + b3
  3. a3 + 3a2b + 3ab2 + b3
  4. a3 + b3
4.
Using (a + b)3, 133 equals:
  1. 2206
  2. 1027
  3. 2197
  4. 1690
5.
Since 6 + 4 + (−10) = 0, the value of 63 + 43 + (−10)3 is:
  1. −720
  2. −240
  3. 0
  4. 720
6.
If x + y = 7 and xy = 7, then x2 + y2 is:
  1. 42
  2. 49
  3. 63
  4. 35
7.
Using an identity, 1022 equals:
  1. 10444
  2. 10204
  3. 10004
  4. 10404
8.
64x3 + 27 factorises as:
  1. (4x + 3)(16x2 − 12x + 9)
  2. (4x − 3)(16x2 + 12x + 9)
  3. (4x + 3)(16x2 + 12x + 9)
  4. (4x + 3)3
Section B — Short answer8 × 2 = 16 marks
9.
Expand (x + 6y + 3z)2.
10.
Expand (3x − 4)3.
11.
Find 10023 using an identity.
12.
If x + y = 11 and xy = 18, find x3 + y3.
13.
Factorise x2 + 10x + 16.
14.
Factorise 64x2 − 81.
15.
Factorise 64x3 − 125.
16.
Without cubing, find 103 + 63 + (−16)3.
Section C — Word problems4 × 4 = 16 marks
17.
A cube of side (x + 5) cm is built from small cubes of side 1 cm. Write its volume as an expanded polynomial, then find the volume when x = 17.
18.
A square plot of side 39 m has a square pond of side 32 m dug in it. Use an identity to find the area left, quickly.
19.
Kabir picks two numbers whose sum is 5 and whose product is 6, and says the sum of their squares is 13. Show this is right using an identity, without finding the numbers.
20.
A box measures (x + 4) by (x + 2) by (x + 3) cm. Expand its volume as a polynomial in x.
AmbifulWorksheets
Practice that never repeats
CBSE · Class 9 · Mathematics

Answer key

Exploring Algebraic Identities · AW-9M-IDT-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) x3 + 18x2 + 108x + 216a = x, b = 6: a3 + 3a2b + 3ab2 + b3
2(a) 27x3 − 243x2 + 729x − 729a = 3x, b = 9: a3 − 3a2b + 3ab2 − b3
3(c) a3 + 3a2b + 3ab2 + b3—
4(c) 2197(10 + 3)3
5(a) −7203abc = 3 × 6 × 4 × (−10)
6(d) 3572 − 2 × 7
7(d) 1040410000 + 400 + 4
8(a) (4x + 3)(16x2 − 12x + 9)a = 4x, b = 3: (a + b)(a2 − ab + b2)
9x2 + 36y2 + 9z2 + 12xy + 36yz + 6zxa2 + b2 + c2 + 2ab + 2bc + 2ca
1027x3 − 108x2 + 144x − 64(3x)3 − 3(3x)2(4) + 3(3x)(4)2 − 43
111006012008(1000 + 2)3 = 1000000000 + 6000000 + 12000 + 8
12737(x + y)3 − 3xy(x + y) = 1331 − 594
13(x + 2)(x + 8)8 + 2 = 10, 8 × 2 = 16
14(8x + 9)(8x − 9)a2 − b2 with a = 8x, b = 9
15(4x − 5)(16x2 + 20x + 25)a3 − b3 = (a − b)(a2 + ab + b2)
16−2880the numbers add to 0, so the sum of cubes is 3 × 10 × 6 × (−16)
17x3 + 15x2 + 75x + 125; 10648 cm3(x + 5)3; 223 = 10648
18497 m2392 − 322 = (39 + 32)(39 − 32) = 71 × 7
1913x2 + y2 = (x + y)2 − 2xy = 25 − 12
20x3 + 9x2 + 26x + 24x3 + (4 + 2 + 3)x2 + (8 + 6 + 12)x + 4 × 2 × 3