Q1. Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier. (i) 1172 (ii) 782 (iii) 1982 (iv) 2142 (v) 11042 (vi) 11202
Answer:
(i) 1172 = (100 + 17)2 = 1002 + 2(100)(17) + 172 = 10000 + 3400 + 289 = 13689
(ii) 782 = (80 - 2)2 = 802 - 2(80)(2) + 22 = 6400 - 320 + 4 = 6084
(iii) 1982 = (200 - 2)2 = 2002 - 2(200)(2) + 22 = 40000 - 800 + 4 = 39204
(iv) 2142 = (200 + 14)2 = 2002 + 2(200)(14) + 142 = 40000 + 5600 + 196 = 45796
(v) 11042 = (1100 + 4)2 = 11002 + 2(1100)(4) + 42 = 1210000 + 8800 + 16 = 1218816
(vi) 11202 = (1100 + 20)2 = 11002 + 2(1100)(20) + 202 = 1210000 + 44000 + 400 = 1254400
Alternatively: 11202 = (1000 + 120)2 = 1000000 + 240000 + 14400 = 1254400
Q2. Factor using suitable identities: (i) 16y2 - 24y + 9 (iii) m2/9 + mk/3 + k2/4 + 3nk + 2mn + 9n2 (v) 9a2 + 4b2 + c2 - 12ab + 6ac - 4bc
Answer:
(i) 16y2 - 24y + 9 = (4y)2 - 2(4y)(3) + 32 = (4y - 3)2
(iii) m2/9 + mk/3 + k2/4 + 3nk + 2mn + 9n2 = (m/3)2 + (k/2)2 + (3n)2 + 2(m/3)(k/2) + 2(k/2)(3n) + 2(3n)(m/3) = (m/3 + k/2 + 3n)2
(v) 9a2 + 4b2 + c2 - 12ab + 6ac - 4bc = (3a)2 + (-2b)2 + c2 + 2(3a)(-2b) + 2(-2b)(c) + 2(c)(3a) = (3a - 2b + c)2
Q3. Expand the following using the identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (i) (p + 3q + 7r)2 (ii) (3x - 2y + 4z)2
Answer:
(i) (p + 3q + 7r)2 = p2 + (3q)2 + (7r)2 + 2(p)(3q) + 2(3q)(7r) + 2(7r)(p) = p2 + 9q2 + 49r2 + 6pq + 42qr + 14rp
(ii) (3x - 2y + 4z)2 = (3x)2 + (-2y)2 + (4z)2 + 2(3x)(-2y) + 2(-2y)(4z) + 2(4z)(3x) = 9x2 + 4y2 + 16z2 - 12xy - 16yz + 24zx
Q4. Is this an identity? (a + b - c)2 + (a - b + c)2 + (a - b - c)2 = 2a2 + 2b2 + 2c2.
Answer:
Let's expand the left-hand side using the identity (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx.
(a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca
(a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ca
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc + 2ca
Adding these three expressions:
LHS = (a2 + a2 + a2) + (b2 + b2 + b2) + (c2 + c2 + c2) + (2ab - 2ab - 2ab) + (-2bc - 2bc + 2bc) + (-2ca + 2ca + 2ca)
LHS = 3a2 + 3b2 + 3c2 - 2ab - 2bc + 2ca
RHS = 2a2 + 2b2 + 2c2
LHS ≠ RHS in general. Therefore, it is not an identity.