PRACTICE TEST USING ALGEBRAIC IDENTITIES
(1) Expand following
(i) (2x + 3y + 4z)2
(ii) (−p +2q + 3r)2
(iii) (2p + 3)(2p −4)(2p −5)
(iv) (3a +1)(3a −2)(3a + 4) Solution
(2) Using algebraic identity, find the coefficients of x2 , x and constant term without actual expansion.
(i) (x + 5)(x + 6)(x + 7)
(ii) (2x + 3)(2x −5)(2x −6) Solution
(2) If (x + a)(x + b)(x + c) = x3 + 14x2 + 59x + 70 , find the value of
(i) a + b + c
(ii) (1/a) + (1/b) + (1/c)
(iii) a2 + b2 + c2
(iv) (a/bc) + (b/ac) + (c/ab) Solution
(4) Expand (3x + 2y)3
(A) 27x³ + 9 x² y + 4 x y² + 8 y³
(B) 27x³ + 54 x² y + 16 xy² + 8 y³
(C) 9x2 + 24 xy + 9y²
(5) Expand (2a + b)³
(A) 8a³ + 12 a² b + 6a b² + b ³
(B) 27a³ + 27 a² b + 16 ab² + b³
(C) 25a² + 24 a²b + 12 ab² + 9b²
(6) Expand (S + T) ³
(A) S³ - 3 S² T + 3 ST ² + T ³
(B) S³ - 3 S² T - 3 S T ² + T ³
(C) S³ + 3 S² T + 3 S T ² + T ³
(7) Expand (3V + Q) ³
(A) 27 V ³ + 2 V ² Q + 3 V Q² + Q³
(B) 27V³ - 2 V ² Q - 3 V Q² + Q³
(C) 27 V³ - 2 V²Q - 3V Q² + Q³
(8) Expand (4T + 3Q) ³
(A) 64T³ - 96 T² Q - 108 T Q² + Q³
(B) 64T³ - 96 T² Q + 108 T Q² - Q³
(C) 64T³ + 96 T²Q + 108 T Q² + 27Q³
(9) Expand (p + 5q)3
(A) p³ - 15 p² q + 75 p q² - 125 q³
(B) p³ - 15 p² q + 75 p q² + 125 q³
(C) p³ + 15 p² q + 75 p q² + 125 q³
(10) (3a - 4b)3
(11) (x + (1/y))3
Evaluate the following by using identities:
(12) 983
(13) 10013
(14) If (x + y + z) = 9 and (xy + yz + zx) = 26, then find the value of x2 + y2 + z2
(15) Find 27a3 + 64b3 , if 3a + 4b = 10 and ab = 2
(16) Find x3 - y3, if x - y = 5 and xy = 14 Solution
(17) If a + (1/a) = 6, then find the value of a3 + 1/a3
(18) If x2 + 1/x2 = 23, then find the value of x + (1/x) and x3 + (1/x3) Solution
(19) If (y - (1/y))3 = 27, then find the value of y3 - (1/y)3
Simplify:
(20) (2a + 3b + 4c)(4a2 + 9b2 + 16c2 - 6ab - 12bc - 8ca)
(21) (x −2y + 3z)(x2 + 4y2 + 9z2 + 2xy + 6yz −3xz)
By using identity evaluate the following:
(22) 73 - 103 + 33
(23) 1 + (1/8) - (27/8) Solution
(24) If 2x −3y −4z = 0 , then find 8x3 - 27y3 - 64z3
(25) If 2x + (2/x) = 3, what is the value of x2 + 1/x2 ?
(26) If a+b+c = 6 and a2+b2+c2 = 14, what is the value of (a-b)2 + (b-c)2 + (c-a)2 ? Solution
(27) If x-y = 8 and xy = 5, what is the value of
x3-y3+8(x+y)2 Solution
(28) If x + y = 5 and xy = 6 and x > y, then find 2(x2+y2) Solution
(29) If a3-b3 = 513 and a-b = 3, what is the value of ab ? Solution
(30) If a2+b2+c2 = 9 and ab+bc+ca = 8, what is the vale of (a+b+c)2 Solution
(31) a+b = √7 and a-b = √5, then find the value of 8ab(a2+b2) Solution
(32) If a-b = 4 and ab = 60, what is the value of a+b ?
(33) If a+b = 9m and ab = 18m2, what is the value of a-b ? Solution
(34) Simplify :
(2345 x 2345 - 759x759)/(2345-759)
(35) x2 + 6xy + 9y2
(36) 4a2 - 20ab + 25b2
(37) 2x2 + 12xy + 18y2
(38) 2ab2 - 16ab + 32a
(39) x2 - 25y2
(40) 9m2 - 16n2
(41) x4 - y4
42) 8p4 - 18q4
Answer Key
1)
i) 4x2 + 9y2 + 16z2 + 12xy + 24yz + 16zx
ii) p2 + 4q2 + 9r2 - 4pq + 12qr - 6rp
iii) 8p3 -24p2 -14p + 60
iv) 27a3 + 27a2 -18a - 8
2)
i)
Coefficient of x2 = 18
Coefficient of x = 107
Constant term = 210
ii)
Coefficient of x2 = -32
Coefficient of x = -6
Constant term = 90
3)
i) abc = 70
(ii) (1/a) + (1/b) + (1/c) = 59/70
(iii) a2 + b2 + c2 = 78
(iv) (a/bc) + (b/ac) + (c/ab) = 39/35
4) 27 x3 + 54 x 2y + 36 y2 + 8y3
5) 8a3 + 12 a b + 6 a b2 + b3
6) (S + T)3 = S3 + 3 S2 T + 3 ST2 + T3
7) 27V3 + 27 V 2Q + 9 VQ2 + Q3
8) 64 T3 + 144 T 2 Q + 108 2 + 27 Q3
9) p3 + 15 p 2 q + 75 p q2) + 125q3
10) 27a3 - 108a2b + 144ab2 - 64b3
11) x3 - (3x2/y) + (3x/y2) - (1/y3)
12) 941192
13) 1003003001
14) 29
15) 280
16) 335
17) 198
18) 105
19) 18
20) 8a3 + 27b3 + 64c3 - 72abc
21) x3 - 8y3 + 27z3 + 18xyz
22) -630
23) 9/4
24) 48xyz
25) x2 + 1/x2 = 1/4
26) 16
27) 1304
28) 2(x2 + y2) = 26
29) ab = 54
30) 25
31) 8ab(a2 + b2) = 24
32) a + b = ±16
33) a - b = ±9m
34) 3104
35) (x + 3y)(x + 3y)
36) (2a - 5b)(2a - 5b)
37) 2(x + 3y)(x + 3y)
38) 2a(b - 4)2
39) (x + 5y)(x - 5y)
40) (3m + 4n)(3m - 4n)
41) (x2 + y2)(x + y)(x - y)
42) 2(4p2 + 9q2)(2p + 3q)(2p - 3q)
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