Exercise Set 4.1
Question 1: Using the identity (a + b)2 = a2 + 2ab + b2, expand the following:
(i) (7x + 4y)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = 7x, b = 4y
Step-by-step expansion:
(7x + 4y)2 = (7x)2 + 2(7x)(4y) + (4y)2
= 49x2 + 56xy + 16y2
Final Answer: 49x2 + 56xy + 16y2
(ii) (7/5x + 3/2y)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = 7/5x, b = 3/2y
Step-by-step expansion:
(7/5x + 3/2y)2 = (7/5x)2 + 2(7/5x)(3/2y) + (3/2y)2
= 49/25x2 + 21/5xy + 9/4y2
Final Answer: 49/25x2 + 21/5xy + 9/4y2
(iii) (2.5p + 1.5q)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = 2.5p, b = 1.5q
Step-by-step expansion:
(2.5p + 1.5q)2 = (2.5p)2 + 2(2.5p)(1.5q) + (1.5q)2
= 6.25p2 + 7.5pq + 2.25q2
Final Answer: 6.25p2 + 7.5pq + 2.25q2
(iv) (3/4s + 8t)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = 3/4s, b = 8t
Step-by-step expansion:
(3/4s + 8t)2 = (3/4s)2 + 2(3/4s)(8t) + (8t)2
= 9/16s2 + 12st + 64t2
Final Answer: 9/16s2 + 12st + 64t2
(v) (x + 1/2y)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = x, b = 1/2y
Step-by-step expansion:
(x + 1/2y)2 = x2 + 2(x)(1/2y) + (1/2y)2
= x2 + x/y + 1/4y2
Final Answer: x2 + x/y + 1/4y2
(vi) (1/x + 1/y)2
Formula: (a + b)2 = a2 + 2ab + b2
Identify terms: a = 1/x, b = 1/y
Step-by-step expansion:
(1/x + 1/y)2 = (1/x)2 + 2(1/x)(1/y) + (1/y)2
= 1/x2 + 2/xy + 1/y2
Final Answer: 1/x2 + 2/xy + 1/y2
Question 2: Using the same identity, find the values of the following:
(i) (64)2
Split term: 64 = 60 + 4
(60 + 4)2 = 602 + 2(60)(4) + 42 = 3600 + 480 + 16 = 4096
Final Answer: 4096
(ii) (105)2
Split term: 105 = 100 + 5
(100 + 5)2 = 1002 + 2(100)(5) + 52 = 10000 + 1000 + 25 = 11025
Final Answer: 11025
(iii) (205)2
Split term: 205 = 200 + 5
(200 + 5)2 = 2002 + 2(200)(5) + 52 = 40000 + 2000 + 25 = 42025
Final Answer: 42025
Exercise Set 4.2
Question 1: Factor completely:
(i) 9x2 + 24xy + 16y2
Pattern: (3x)2 + 2(3x)(4y) + (4y)2
Final Answer: (3x + 4y)2
(ii) 4s2 + 20st + 25t2
Pattern: (2s)2 + 2(2s)(5t) + (5t)2
Final Answer: (2s + 5t)2
(iii) 49x2 + 28xy + 4y2
Pattern: (7x)2 + 2(7x)(2y) + (2y)2
Final Answer: (7x + 2y)2
(iv) 64p2 + 32/3pq + 4/9q2
Pattern: (8p)2 + 2(8p)(2/3q) + (2/3q)2
Final Answer: (8p + 2/3q)2
(v) 3a2 + 4ab + 4/3b2
Factor out common fraction: 1/3(9a2 + 12ab + 4b2) = 1/3((3a)2 + 2(3a)(2b) + (2b)2)
Final Answer: 1/3(3a + 2b)2
(vi) 9/5s2 + 6sv + 5v2
Factor out common fraction: 1/5(9s2 + 30sv + 25v2) = 1/5((3s)2 + 2(3s)(5v) + (5v)2)
Final Answer: 1/5(3s + 5v)2
Question 2: Find the values of the following using the identity (a – b)2 = a2 – 2ab + b2:
(i) (79)2
(80 – 1)2 = 802 – 2(80)(1) + 12 = 6400 – 160 + 1 = 6241
Final Answer: 6241
(ii) (193)2
(200 – 7)2 = 2002 – 2(200)(7) + 72 = 40000 – 2800 + 49 = 37249
Final Answer: 37249
(iii) (299)2
(300 – 1)2 = 3002 – 2(300)(1) + 12 = 90000 – 600 + 1 = 89401
Final Answer: 89401
Exercise Set 4.3
Question 1: Find the following squares using one of the above identities:
(i) 1172 = (120 – 3)2 = 1202 – 2(120)(3) + 32 = 14400 – 720 + 9 = 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
Question 2: Factor using suitable identities:
(i) 16y2 – 24y + 9 = (4y – 3)2
(ii) 9/4s2 + 6st + 4t2 = (3/2s + 2t)2
(iii) m2/9 + mk/3 + k2/4 + 3nk + 2mn + 9n2 = (m/3 + k/2 + 3n)2
(iv) p2/16 – 2 + 16/p2 = (p/4 – 4/p)2
(v) 9a2 + 4b2 + c2 – 12ab + 6ac – 4bc = (3a – 2b + c)2
Question 3: Expand using (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca:
(i) (p + 3q + 7r)2 = p2 + 9q2 + 49r2 + 6pq + 42qr + 14rp
(ii) (3x – 2y + 4z)2 = 9x2 + 4y2 + 16z2 – 12xy – 16yz + 24zx
Question 4: Is this an identity? (a + b – c)2 + (a – b + c)2 + (a – b – c)2 = 2a2 + 2b2 + 2c2
Answer: No. Expanding LHS yields 3a2 + 3b2 + 3c2 – 2ab – 2bc + 2ca, which does not equal 2a2 + 2b2 + 2c2.
Exercise Set 4.4
Question 1: Fill in the blanks:
(i) s2 – 11s + 24 = (s – 3)(s – 8)
(ii) (3x – 7)(x + 1) = 3x2 – 4x – 7
(iii) 10x2 – 11x – 6 = (2x – 3)(5x + 2)
(iv) 6x2 + 7x + 2 = (2x + 1)(3x + 2)
Question 2: Select and use identity to find products without direct multiplication:
(i) (41)2 = (40 + 1)2 = 1681
(ii) (27)2 = (30 – 3)2 = 729
(iii) 23 × 17 = (20 + 3)(20 – 3) = 202 – 32 = 391
(iv) (135)2 = (130 + 5)2 = 18225
(v) (97)2 = (100 – 3)2 = 9409
(vi) 18 × 29 = (20 – 2)(20 + 9) = 522
(vii) 34 × 43 = (30 + 4)(30 + 13) = 1462
(viii) (205)2 = (200 + 5)2 = 42025
Question 3: Factor the following:
(i) 9a2 + b2 + 4c2 – 6ab + 12ac – 4bc = (-3a + b + 2c)2 or (3a – b – 2c)2
(ii) 16s2 + 25t2 – 40st = (4s – 5t)2
(iii) r2 – r – 42 = (r – 7)(r + 6)
(iv) 49g2 + 14gh + h2 = (7g + h)2
(v) 64u2 + 121v2 + 4w2 – 176uv – 32uw + 44vw = (8u – 11v – 2w)2
Exercise Set 4.5
Question 1: Simplify the following rational expressions:
(i) (3p2 – 3pq – 18q2)/(p2 + 3pq – 10q2) = 3(p – 3q)(p + 2q)/(p + 5q)(p – 2q)
(ii) (n3 – 3n2m + 3nm2 – m3)/(5m2 – 10mn + 5n2) = n – m/5
(iii) (w3 – v3 + x3 + 3wvx)/(w2 + v2 + x2 – 2wv – 2vx + 2wx) = (w2 + v2 + x2 + wv + vx – wx)/(w – v + x)
(iv) (4y2 – 20yz + 25z2)/(25z2 – 4y2) = 5z – 2y/5z + 2y
(v) (x2 + x – 6)(x2 – 7x + 12)/(x2 – 6x + 8)(x2 – 9) = 1
(vi) (p4 – 16)/(p2 – 4p + 4) = (p + 2)(p2 + 4)/p – 2
End-of-Chapter Exercises
Question 1: Use suitable identities to find the following products:
(i) (-3x + 4)2 = 16 – 24x + 9x2
(ii) (2s + 7)(2s – 7) = 4s2 – 49
(iii) (p2 + 1/2)(p2 – 1/2) = p4 – 1/4
(iv) (2n + 7)(2n – 7) = 4n2 – 49
(v) (s – 2t)(s2 + 2st + 4t2) = s3 – 8t3
(vi) (1/2r – 4r)2 = 1/4r2 – 4 + 16r2
(vii) (-3m + 4k – l)2 = 9m2 + 16k2 + l2 – 24mk – 8kl + 6lm
(viii) (x – 1/3y)3 = x3 – x2y + 1/3xy2 – 1/27y3
(ix) (7/2k – 2/3m)3 = 343/8k3 – 49/2k2m + 14/3km2 – 8/27m3
Question 2: Find values using suitable identities:
(i) 17 × 21 = 357
(ii) 104 × 96 = 9984
(iii) 24 × 16 = 384
(iv) 1472 = 21609
(v) 1993 = 7880599
(vi) 1272 = 16129
(vii) (-107)3 = -1225043
(viii) (-299)3 = -26730899
Question 3: Factor the following expressions:
(i) 4y2 + 1 + 1/16y2 = (2y + 1/4y)2
(ii) 9m2 – 1/25n2 = (3m – 1/5n)(3m + 1/5n)
(iii) 27b3 – 1/64b3 = (3b – 1/4b)(9b2 + 3/4 + 1/16b2)
(iv) x2 + 5x/6 + 1/6 = (x + 1/2)(x + 1/3)
(v) 27u3 – 1/125 – 27u2/5 + 9u/25 = (3u – 1/5)3
(vi) 64y3 + 1/125z3 = (4y + 1/5z)(16y2 – 4/5yz + 1/25z2)
(vii) p3 + 27q3 + r3 – 9pqr = (p + 3q + r)(p2 + 9q2 + r2 – 3pq – 3qr – rp)
(viii) 9m2 – 12m + 4 = (3m – 2)2
(x) 4x2 + 9y2 + 36z2 + 12xz + 36yz + 24xy = (2x + 3y + 6z)2
(xi) 27u3 – 1/216 – 9u2/2 + u/4 = (3u – 1/6)3
Question 4: Simplify:
(i) (4×2 + 4x + 1)/(4x2 – 1) = 2x + 1/2x – 1
(ii) 9(3a3 – 24b3)/(9a2 – 36b2) = 3(a2 + 2ab + 4b2)/a + 2b
(iii) (s3 + 125t3)/(s2 – 2st – 35t2) = s2 – 5st + 25t2/s – 7t
Question 5: Length and breadth of rectangles:
(i) Area: 25a2 – 30ab + 9b2
Dimensions: Length = 5a – 3b, Breadth = 5a – 3b
(ii) Area: 36s2 – 49t2
Dimensions: Length = 6s + 7t, Breadth = 6s – 7t
Question 6: Length, breadth, and height of cuboids:
(i) Volume: 6a3 – 24b2 = 6a(a2 – 4b2)
Dimensions: 6a, (a – 2b), (a + 2b)
(ii) Volume: 3ps2 – 15ps + 12p = 3p(s – 4)(s – 1)
Dimensions: 3p, (s – 4), (s – 1)
Question 7: Path area around square playground:
Outer side = 40 + 2s
Area of path = (40 + 2s)2 – 402 = 4s2 + 160s sq. metres
Question 8: Number plus reciprocal equals 10/3:
Equation: x + 1/x = 10/3
Final Answer: x = 3 or x = 1/3
Question 9: Rectangular pool length:
Length = (2×2 + 7x + 3)/(2x + 1) = x + 3 hastas
Question 10: Show p = r if x – 2 and x – 1/2 are factors of px2 + 5x + r:
f(2) = 4p + 10 + r = 0 ⇒ 4p + r = -10
f(1/2) = p/4 + 5/2 + r = 0 ⇒ p + 4r = -10
4p + r = p + 4r ⇒ 3p = 3r ⇒ p = r (Proved)
Question 11: If a + b + c = 5 and ab + bc + ca = 10, prove a3 + b3 + c3 – 3abc = -25:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) ⇒ 25 = a2 + b2 + c2 + 20 ⇒ a2 + b2 + c2 = 5
a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – (ab + bc + ca)) = 5(5 – 10) = -25 (Proved)
Question 12: Divisibility of n3 – n by 6:
n3 – n = (n – 1)n(n + 1)
Product of three consecutive natural numbers always contains at least one multiple of 2 and exactly one multiple of 3, so it is divisible by 2 × 3 = 6.
Question 13: Value of expressions:
(i) x3 + y3 – 12xy + 64 when x + y = -4:
Since x + y + 4 = 0, x3 + y3 + 43 – 3(x)(y)(4) = 0
(ii) x3 – 8y3 – 36xy – 216 when x = 2y + 6:
Since x – 2y – 6 = 0, x3 + (-2y)3 + (-6)3 – 3(x)(-2y)(-6) = 0
