Chapter 3: Expansions - Practice worksheet with complete answer key
According to your class and board
- Board: ICSE
- Class/Year: 9
- Subject: Mathematics
- Book + Publisher: Concise Mathematics (Selina)
- Chapter: Expansions
- Difficulty: Standard
- Goal: Practice
A1. Write the expansion of (a + b)².
A2. Expand (x − 3)² using the standard identity.
A3. What is the value of (a + b)(a − b)?
A4. Find the square of 105 using a suitable identity.
A5. Write the identity for (a + b + c)².
B1. Expand: (3x + 2y)² and simplify.
B2. Expand: (5a − 4b)² and identify each term.
B3. If x + 1/x = 5, find the value of x² + 1/x².
B4. Find the product (2x + 3)(2x − 3) using an identity.
B5. Expand (x + 2y + 3z)² and list all terms.
B6. Simplify: (a + b)² − (a − b)².
B7. If a − b = 3 and ab = 10, find a² + b².
B8. Evaluate 99² using the identity (a − b)².
B9. Expand (2x + 1)³ using the cube identity.
B10. Find the value of x² + y² if x + y = 7 and xy = 10.
C1. The side of a square is increased by 3 cm. Express the new area using a suitable identity and find the increase in area when the original side is 10 cm.
C2. If (x + y) = 8 and (x − y) = 2, find x² − y² and xy.
C3. Without actual multiplication, evaluate 203 × 197 using a suitable identity.
C4. If 2a + 3b = 12 and ab = 6, find 4a² + 9b².
C5. Prove that (a + b)² + (a − b)² = 2(a² + b²).
C6. A rectangular field has dimensions (x + 5) and (x + 3). Express the area as a polynomial and find it when x = 7.
C7. If a² + b² = 25 and ab = 12, find (a + b)² and (a − b)².
C8. Factorise using identities: 4x² − 12xy + 9y².
D1. If x − 1/x = 3, find the values of: (a) x² + 1/x² (b) x⁴ + 1/x⁴. Show all steps. (3 marks)
D2. Expand (2a + b − 3c)² and simplify completely. (3 marks)
D3. If a + b + c = 9 and a² + b² + c² = 35, find the value of ab + bc + ca. (3 marks)
D4. Without actual division, prove that (x³ + y³) is divisible by (x + y). Write the quotient. (2 marks)
D5. Simplify: (3x + 4y)² − (3x − 4y)² using identities. (2 marks)
E1. A student writes (a + b)² = a² + b². What error has the student made?
E2. Is (x − y)² the same as (y − x)²? Justify.
A1. (a + b)² = a² + 2ab + b².
A2. (x − 3)² = x² − 6x + 9.
A3. (a + b)(a − b) = a² − b².
A4. 105² = (100 + 5)² = 10000 + 1000 + 25 = 11,025.
A5. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
B1. (3x + 2y)² = 9x² + 12xy + 4y².
B2. (5a − 4b)² = 25a² − 40ab + 16b². Terms: 25a² (square of first), −40ab (twice the product), 16b² (square of second).
B3. (x + 1/x)² = x² + 2 + 1/x² → 25 = x² + 1/x² + 2 → x² + 1/x² = 23.
B4. (2x + 3)(2x − 3) = (2x)² − 3² = 4x² − 9.
B5. (x + 2y + 3z)² = x² + 4y² + 9z² + 4xy + 12yz + 6xz.
B6. (a + b)² − (a − b)² = (a² + 2ab + b²) − (a² − 2ab + b²) = 4ab.
B7. (a − b)² = a² − 2ab + b² → 9 = a² + b² − 20 → a² + b² = 29.
B8. 99² = (100 − 1)² = 10000 − 200 + 1 = 9,801.
B9. (2x + 1)³ = 8x³ + 12x² + 6x + 1.
B10. (x + y)² = x² + 2xy + y² → 49 = x² + y² + 20 → x² + y² = 29.
C1. New side = (a + 3), new area = (a + 3)² = a² + 6a + 9. Original area = a² = 100. New area = 169. Increase = 69 cm².
C2. x² − y² = (x + y)(x − y) = 8 × 2 = 16. From (x + y) and (x − y): x = 5, y = 3. xy = 15.
C3. 203 × 197 = (200 + 3)(200 − 3) = 200² − 3² = 40000 − 9 = 39,991.
C4. (2a + 3b)² = 4a² + 12ab + 9b² → 144 = 4a² + 9b² + 72 → 4a² + 9b² = 72.
C5. LHS: (a² + 2ab + b²) + (a² − 2ab + b²) = 2a² + 2b² = 2(a² + b²) = RHS. Proved.
C6. Area = (x + 5)(x + 3) = x² + 8x + 15. When x = 7: 49 + 56 + 15 = 120 sq. units.
C7. (a + b)² = a² + 2ab + b² = 25 + 24 = 49. (a − b)² = a² − 2ab + b² = 25 − 24 = 1.
C8. 4x² − 12xy + 9y² = (2x)² − 2(2x)(3y) + (3y)² = (2x − 3y)².
D1. (a) (x − 1/x)² = x² − 2 + 1/x² → 9 = x² + 1/x² − 2 → x² + 1/x² = 11. (b) (x² + 1/x²)² = x⁴ + 2 + 1/x⁴ → 121 = x⁴ + 1/x⁴ + 2 → x⁴ + 1/x⁴ = 119.
D2. (2a + b − 3c)² = (2a)² + (b)² + (−3c)² + 2(2a)(b) + 2(b)(−3c) + 2(2a)(−3c) = 4a² + b² + 9c² + 4ab − 6bc − 12ac.
D3. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) → 81 = 35 + 2(ab + bc + ca) → ab + bc + ca = 23.
D4. x³ + y³ = (x + y)(x² − xy + y²). Since (x + y) is a factor, x³ + y³ is divisible by (x + y). Quotient = x² − xy + y².
D5. Using (A + B)² − (A − B)² = 4AB: A = 3x, B = 4y → 4(3x)(4y) = 48xy.
E1. The student forgot the middle term. Correct: (a + b)² = a² + 2ab + b². The term 2ab is missing. — Why students get this wrong: They incorrectly distribute the square to each term separately, as if (a + b)² = a² + b², which only works for multiplication, not for squaring a sum.
E2. Yes, (x − y)² = (y − x)². Because (x − y) = −(y − x), and squaring removes the negative sign: [−(y − x)]² = (y − x)². — Why students get this wrong: They think the negative sign inside should make the result different, forgetting that (−k)² = k².