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)

(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

Solution

(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

Solution

Simplify: 

(20)  (2a + 3b + 4c)(4a2 + 9b2 + 16c2 - 6ab - 12bc - 8ca)

(21)  (x −2y + 3z)(x2 + 4y2 + 9z2 + 2xy + 6yz −3xz)

Solution

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

Solution

(25)  If 2x + (2/x)  =  3, what is the value of x2 + 1/x2 ?

Solution

(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 ?

Solution

(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

Solution

(36) 4a2 - 20ab + 25b2

Solution

(37)  2x2 + 12xy + 18y2

Solution 

(38)  2ab2 - 16ab + 32a

Solution

(39)  x2 - 25y2

Solution

(40)  9m2 - 16n2

Solution

(41) x4 - y4

Solution

42)  8p4 - 18q4

Solution

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)  x- 8y+ 27z+ 18xyz

22)  -630

23)  9/4

24)  48xyz

25)  x+ 1/x2 = 1/4

26)   16

27)   1304

28)  2(x+ y2) = 26

29)  ab = 54

30)  25

31)  8ab(a+ 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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