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ICSE-VIII-Mathematics

12: Algebraic Identities Class 8 Maths

with Solutions -
Qstn# B-1 Prvs-QstnNext-Qstn
  • #1
    Expand : (i) (2a + b)2 (ii) (a - 2b)2 (iii) (a + 1/2a)2 (iv) (2a - 1/a)2 (v) (a + b - c)2 (vi) (a - b + c)2 (vii) (3x + 1/3x)2 (viii) (2x - 1/2x)2
    Ans : (i) (2a + b)2 = (2a)2 +
    (b)2 + 2 × 2a × b [(a + b)2 = a2 + b2 + 2ab]
    = 4a2 + b2 + 4ab (ii) (a - 2b)2 =
    (a)2 + (2b)2 - 2×a×2b [(a - b)2 = a2 + b2 - 2ab]
    = a2 + 4b2 - 4ab (iii) (a + 1/2a)2
    =
    (a)2 + (1/2a)2 + 2×a× 1/2a
    = a2 + 1/4a2 + 2a/2a
    = a2 + 1/4a2 + 1 (iv) (2a - 1/a)2
    = (2a)2 + (1/a)2 - 2×2a× 1/a
    = 4a2 + 1/a2 - 4 (v) (a + b - c)2
    =
    (a)2 +
    (b)2 + (-c)2 + 2×a× b + 2×b×(-c) + 2 × (-c) ×
    (a)
    = a2 + b2 + c2 + 2ab - 2bc - 2ca
    Note: (a + b + c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca (vi) (a - b + c)2 =
    (a)2 + (-b)2 +
    (c)2 + 2×a×-b + 2(-b)
    (c) + 2×c×a
    = a2 + b2 + c2 - 2ab - 2bc + 2ca (vii) (3x + 1/3x)2
    = (3x)2 + (1/3x)2 + 2×3x× 1/3x
    = 9x2 + 1/9x2 + 2 (viii) (2x - 1/2x)2
    = (2x)2 + (1/2x)2 - 2×2x× 1/2x
    = 4x2 + 1/4x2 - 2
  • #1-i
    (2a + b)2
    Ans : (2a + b)2 = (2a)2 +
    (b)2 + 2 × 2a × b [(a + b)2 = a2 + b2 + 2ab]
    = 4a2 + b2 + 4ab
  • #1-ii
    (a - 2b)2
    Ans : (a - 2b)2 =
    (a)2 + (2b)2 - 2×a×2b [(a - b)2 = a2 + b2 - 2ab]
    = a2 + 4b2 - 4ab
  • #1-iii
    (a + 1/2a)2
    Ans : (a + 1/2a)2
    =
    (a)2 + (1/2a)2 + 2×a× 1/2a
    = a2 + 1/4a2 + 2a/2a
    = a2 + 1/4a2 + 1
  • #1-iv
    (2a - 1/a)2
    Ans : (2a - 1/a)2
    = (2a)2 + (1/a)2 - 2×2a× 1/a
    = 4a2 + 1/a2 - 4
  • #1-v
    (a + b - c)2
    Ans : (a + b - c)2
    =
    (a)2 +
    (b)2 + (-c)2 + 2×a× b + 2×b×(-c) + 2 × (-c) ×
    (a)
    = a2 + b2 + c2 + 2ab - 2bc - 2ca
    Note: (a + b + c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca
  • #1-vi
    (a - b + c)2
    Ans : (a - b + c)2 =
    (a)2 + (-b)2 +
    (c)2 + 2×a×-b + 2(-b)
    (c) + 2×c×a
    = a2 + b2 + c2 - 2ab - 2bc + 2ca
  • #1-vii
    (3x + 1/3x)2
    Ans : (3x + 1/3x)2
    = (3x)2 + (1/3x)2 + 2×3x× 1/3x
    = 9x2 + 1/9x2 + 2
  • #1-viii
    (2x - 1/2x)2
    Ans : (2x - 1/2x)2
    = (2x)2 + (1/2x)2 - 2×2x× 1/2x
    = 4x2 + 1/4x2 - 2