Algebra | ETE-889022705

Algebra
1. If a3 + b3 = 1344 and a + b = 28, then (a + b)2 – 9. If 40√5x 3 − 3√3y 3 = (2√5x − √3y) (Ax 2 + Bxy +
3ab is equal to: Cy 2 ), then what is the value of √B 2 + C 2 − A ?
(A) 24 (B) 16 (A) 11 (B) 7
(C) 32 (D) 48 (C) 8 (D) 9

2. If x+y+z=19, XYZ =216 and xy + yz + zx =114 10. If x + 1/x = 7, then x³ + 1/x³ is equal to:
then the value of √x 3 + y 3 + z 3 + xyz is: (A) 300 (B) 322
(A) 32 (B) 30 (C) 364 (D) 343
(C) 28 (D) 35
11. If 250√2x 3 − 5√5y 3 − [5√2x − √5y][Ax 2 + Bxy +
3. If 8x + y² – 12x – 4xy + 9 = 0 then the value of
2
Cy 2 ], then the value of (A + C√10B) is:
(14x – 5y) is: (A) 10 (B) 5
(A) 9 (B) 6 (C) √52 (D) √25
(C) 5 (D) 3
12. If x ≠ – 1, 2 and 5, then the simplified value of
4. If a + b + c = 4 and ab + bc + ca = 1, then find (x2 −8) x2 +2x+1 x2 +2x+4
{2 × ÷ } is equal to :
the value of a³ + b³ + c³ – 3abc is: x2 −x−2 x2 −4x−5 3x−15

(A) 50 (B) 60 (A) 1/6 (B) 6

(C) 3/2 (D) 2/3
(C) 52 (D) 47

13. If a³ + b³ = 110 and a + b = 5, then (a + b)² – 3ab

5. If 24√3x 3 + 2√2y 3 = (2 √3x + √2y)
is equal to:
(Ax 2 + Bxy + Cy 2 ), to (2A + √6B − C) is equal to:
(A) 52 (B) 32
(A) 10 (B) 14
(C) 42 (D) 22
(C) 6 (D) 8

14. If x² + 1 = 3x, then the value of (x4+x–2)8/(x²+5x

6. If A³ + B³ = 110 and A + B = 5, then (A+ B)² – 3AB
+1) is:
is equal to:
(A) 2 1/3 (B) 214
(A) 52 (B) 32
(C) 4 /12 (D) 3 /12
(C) 42 (D) 22

15. If 40√5x 3 − 3√3y 3 = (2√5x − √3y) × (Ax 2 + Bxy +

7. If A + B + C = 5 and A2 + B2 + C2 = 33, then what
is the value of A3 + B3 + C3 – 3ABC cy 2 ) , then what is the value of √B 2 + C 2 − A ?
(A) 11 (B) 7
(A) 195 (B) 180
(C) 8 (D) 9
(C) 192 (D) 185

8. If x² + 1 = 3x, then the value of (x4+x-2)/(x²+5x +1) 16. If a + b + c = 5 and a2 + b2 + c2 = 33, then what is
is: the value of a3 + b3 + c3 – 3abc?
(A) 2 1/3 (B) 2 1/4 (A) 195 (B) 180
(C) 192 (D) 185
(C) 4 1/2 (D) 3 1/2

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 Algebra | ETE-889022705
17. If a² + b² = 88 and ab = 6, (a > 0, b > 0) then what 29. If x = √3 − √2 , then the value of x³ – x– 3 is:
is the value of (a³ + b³) ? (A) 22√3 (B) -22√2
(A) 980 (B) 1180 (C) 22√2 (D) -22√3
(C) 820 (D) 1000
30. If (x + 7)³ + (2x + 8)³ + (2x + 3)³ = 3(x + 7) (2x + 8)
18. If x – 1/x = 10, then x³ – 1/x³ is equal to: (2x + 3), then what is the value of x?
(A) 970 (B) 1000 (A) –3.6 (B) 3.6
(C) 1030 (D) 1100 (C) 2.4 (D) –2.4

19. If (x – 7)³ + (x – 8)³ +(x + 6)³ = 3 (x – 7) (x – 8) (x + 31. If x = 2 − √3 then the value of x3 – x–3 is
6), then what is the value of x? (A) -30√3 (B) 30√3
(A) 6 (B) 8 (C) -30 √2 (D) 30√2
(C) 10 (D) 3
32. If (2x – 7)³ + ( 2x – 8)³ + (2x –3)³ = 3 (2x –7) (2x – 8)
20. If x4 + x–4 = 2207, (x > 0) then the value of x + x– (2x –3), then what is the value of x?
1
is: (A) 4 (B) 2
(A) 19 (B) 7 (C) 1 (D) 3
(C) 11 (D) 9
33. If x4 + x–4 = 1442, (x > 0) then the value of x + x –1
21. If x + x = 1442, (x > 0) then the value of x - x is:
4 –4 –1
is:
(A) 7 (B) 8 (A) 2√10 (B) 3√10
(C) 6 (D) 15 (C) 4√10 (D) 15

22. If (3x –7)³ + (3x – 8)³ + (3x + 6)³ = 3(3x –7)(3x – 34. If a² + b² = 135 and ab = 7, (a > 0, b > 0) then the
8)(3x + 6), then what is the value of x ? value of (a³ – b³) is :
(A) 3 (B) 1 (A) 1562 (B) 1600
(C) 4 (D) 2 (C) 1680 (D) 1350

23. If x + 1/x = 10, then x³ + 1/x³ is equal to : 35. If x = 2 + √3, then the value of x3 – x–3 is:
(A) 970 (B) 1030 (A) –52 (B) -30√3
(C) 1000 (D) 1100 (C) 30√3 (D) 52

24. If a² + b² = 99 and ab = 11, (a > 0, b > 0) then the 36. If (x – 7)3 + (2x + 8)3 +(2x – 3)3 = 3(x – 7) (2x + 8)
value of (a³ + b³) is: (2x – 3), then what is the value x?
(A) 1250 (B) 968 (A) 1.6 (B) 2.4
(C) 1100 (D) 1080 (C) 1.2 (D) 0.4

25. If 8 (a + b)3 + (a – b)3 = (3a + b) (Aa2 + Bab + Cb2), 37. If a3 + b3 = 1344 and a + b = 28, then (a + b)2 –
then what is the value of (A + B – C) ? 3ab is equal to:
(A) 2 (B) 4 (A) 24 (B) 16
(C) 10 (D) 11 (C) 32 (D) 48

26. If x2 – 6x + 1 = 0, then the value of (x4+1/x2) ÷ 38. If x4 + x–4 = 47, (x > 0), then the value of (2x – 3)2 is:
(x2+1) is: (A) 2 (B) 3
(A) 39 (B) 33 (C) 5 (D) 4
(C) 35 (D) 36
27. If x + y + z = 3 and xy + yz + zx = – 18, then what 39. If x=2+√5 then the value of (x3–x–3) is:
is the value of x3 + y3 + z3 – 3xyz = ? (A) –52 (B) 52
(A) 187 (B) 217 (C) 76 (D) –76
(C) 191 (D) 189
40. If (x – 8)³ + (2x + 16)³ + (2x –13)³ = 3(x – 8)(2x +
28. If (2x + 7)³ + (2x + 8)³ + (2x + 3)³ = 3(2x + 7) (2x + 16)(2x –13), then what is the value of x?
8) (2x + 3), then what is the value of x ? (A) 0.7 (B) –1
(A) –2 (B) 3 (C) 1 (D) 0
(C) 2 (D) –3

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 Algebra | ETE-889022705
41. If x=2+√3 then the value of x³ + x–3 is: 46. The value of the expression
(A) 52 (B) –52 1 1 2 1 2
{(a + ) − (a − ) }| is:
(C) -52√3 (D) 52√3 4 a a
(A) 1/2 (B) 1/4
42. If a3 – b3 = 899 and a – b = 29, then (a – b)2 + 3ab (C) 1 (D) 4
is equal to:
(A) 35 (B) 29 47. If a3 – b3 = 899 and a – b = 31, then (a – b)2 + 3ab
is equal to:
(C) 16 (D) 31
(A) 35 (B) 31
(C) 16 (D) 29
43. If x4 + x –4
= 1154, (x > 0), then the value of 2(x –
3)2 is:
(A) 16 (B) 12 48. If x4 + x –4 = 194, (x > 0), then the value of (2x – 4)2
(C) 20 (D) 15 is:
(A) 15 (B) 20
1 1 (C) 12 (D) 16
44. If x − = 7, then x 3 − is equal to
x x3
(A) 480 (B) 364
49. If x = 2+√5 then the value of x3+x–3 is:
(C) 376 (D) 500
(A) 40√5 (B) 34√5
(C) 46√5 (D) 36√5
45. If (3x + 1)3 + (x – 3)3 +(2x – 4)3 = 6(3x + 1) (x – 3) (x
– 2), then x is equal to:
50. What is the value of a - b when a2 + b2 - 6a - 6b +
(A) 3 (B) 1
18 = 0?
(C) 2 (D) –1/3
(A) 0 (B) 3
(C) 6 (D) 9

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 Algebra | ETE-889022705
Algebra (Solution)

1. Answer:(D) 110 = 5 [a² + b² – ab]

a3 + b3 = [a + b] [a2 + b2 – ab] a² + b² – ab = 22
(a + b)2 – 3ab = a2 + b2 + 2ab – 3ab [a + b]² – 3ab = 22.
= a2 + b2 – ab 7. Answer: (D)
A.T.Q a+b+c=5
1344 = 28[a2 + b2 – ab] Squaring both sides,
a2 + b2 – ab = 48 a² + b² + c² + 2(ab + bc + ca) = 25
2. Answer: (D) 33 + 2(ab + bc + ca) = 25
x + y + z =19 ab + bc + ca = –4
x2 +y2 +z2 +2 (xy+y2+zx)=361 a³ + b³ + c³ – 3abc = [a + b + c] [a² + b² + c² – ab
x2 +y2 +z2 =361-2×114=133 – bc – ca]
x3 +y3 +z3-3xyz =[x + y + z][x2 + y2 + z2 − xy − a³ + b³ + c³ – 3abc = 5 [33 + 4]
yz − zx] a³ + b³ + c³ – 3abc = 185
Add 4xyz both sides 8. Answer: (B)
X3+y3+z3+xyz =[19][133 − 114] + 4 × 216 x2 +1=3x
X3+y3+z3+xyz =19×19+864 x+1/x=3
X3+y3+z3+xyz=1225 Cubing both sides, we get
√x3 + y3 + z3 + xyz = 35 x3 +
1
= 27 − 9
x3
3. Answer: (B) 1
8x2 + y² –12x – 4xy + 9 = 0 x3 + = 18
x3
[2x – y]² + [2x – 3]² = 0 Now,
x = 3/2 and y = 2x (X4 +X2 )
X2 +5X+1
y=3 1
X3 + 3
14 x – 5y = 1
X

= 14×3/2–5×3 = 6 X+ +5
X
18 9 1
4. Answer: (C) = = =2
3+5 4 4
a+b+c=4 9. Answer: (B)
Squaring both sides, 40√5x3 − 3√3 y3 =[2√5x − √3y] × [Ax2 + Bxy +
a² + b² + c² + 2(ab + bc + ca) = 16
Cy2 ]
a² + b² + c² = 16 – 2 ×1 3 3
a² + b² + c² = 14 _____ (1) [(2√5 𝑥) − (√3 𝑦) ]
A.T.Q, = [2√5 x − √3y] × [Ax 2 + Bxy + Cy 2 ]
a³ + b³ + c³ – 3abc = [a + b + c] [a² + b² + c² – ab [2√5x − √3 𝑦] [20x2 + 2√15 𝑥𝑦 + 3y2 ]
– bc – ca] = [2√5 x − √3y] × [Ax 2 + Bxy + Cy 2 ]
a³ + b³ + c³ – 3abc = 4 [14 – 1] [20x2 + 3y2 + 2√15xy] = [Ax 2 + Bxy + Cy 2 ]
a³ + b³ + c³ – 3abc = 52 By comparing the coefficients
5. Answer: (A)
A = 20 , B = 2√15 , C = 3
24√3x3 + 2√2y3
∴ √B 2 + C 2 − A)
= (2√3x + √2y)(Ax2 + Bxy + Cy2 )
= √60 + 9 − 20
[(2√3x)3 + (√2y)3 ] =7
= [(2√3x) + (√2y)][Ax2 + Bxy + Cy3 ] 10. Answer: (B)
We know that a2 + b2 = [a + b][a2 + b2 − ab] 1
X+ =7
∴ [2√3 x + √2 𝑦][12x 2 + 2y 2 − 2√6xy] x

= [2√3 x + √2y][Ax 2 + Bxy + Cy 2 ] Cube both side

1 1 1
By comparing the coefficients, we get x3 + + 3x × x [x + ] = 343
x3 x x
A = 12, B = -2√6, C = 2 x3 +
1
= 343- 21
x3
A. T. Q 1
x +
3
= 322
2A + √6 B − C x3
2 × 12 − √6,× 2√6 − 2 11. Answer: (B)
=10 250√2x3 − 5√5y3 − [5√2x − √5y][Ax2 + Bxy +
6. Answer: (D) Cy2 ]
a² + b³ = [a + b] [a² + b² – ab]
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 Algebra | ETE-889022705
[(5√2 𝑥) − (5√5 𝑦) ]
3 3
16. Answer: (D)
a+b+c=5
= [5√2x − √5y][Ax 2 + Bxy + Cy 2 ]
Squaring both sides,
[a3 − b3 ] = [a − b][a2 + b2 + ab]
a² + b² + c² + 2(ab + bc + ca) = 25
[5√2x − √5 𝑦][50x2 + 5y2 + 5√10 xy] 33 + 2(ab + bc + ca) = 25
= [5√2x − √5y][Ax 2 + Bxy + Cy 2 ] ab + bc + ca = –4
[50x2 + 5y2 + 5√10xy] a³ + b³ + c³ – 3abc = [a + b + c] [a² + b² + c² – ab
= [Ax 2 + Bxy + Cy 2 ] – bc – ca]
A = 50 , B = 5√10 , C=5 a³ + b³ + c³ – 3abc = 5 [33 + 4]
A.T.Q a³ + b³ + c³ – 3abc = 185
= A + C − √10B 17. Answer: (C)
= 50 + 5 - 5 √10 × √10 a3 + b3 = [a + b] [a2 + b2 – ab]
=5 [a + b]2 = a2 + b2 + 2ab
12. Answer: (B) [a + b]2 = 88 + 2 × 6
2(x2 − 8) x2 + 2x + 1 x2 + 2x + 4 [a + b]2 = 100
( 2 )× 2 ÷ a + b = 10
x −x−2 x − 4x − 5 3x − 15
2(x−2)(x2 +4+2x)(x2 +2x+1) x2 +2x+4 a3 + b3 = [a + b] [a2 + b2 – ab]
=( ÷ ) = 10[88 –6]
(x2 −x−2)×(x2 −4x−5) 3(x−5)
2(x−2)(x2 +4+2x)(x2 +2x+1) x2 +2x+4 = 10 × 82
= ( ( )( )( )( ) ÷ ( ) )
x+1 x−2 x+1 x−5 3 x−5 = 820
2(x−2)(x2 +4+2x)(x2 +2x+1) ×3(x−5)
= ( (x+1)(x−2)(x+1)(x−5)(x2 +2x+4) ) 18. Answer: (C)
x – 1/x = 10
= 2× 3
Cube both side
=6
x³ – 1/x³ – 3(x – 1/x) = 1000
13. Answer: (D)
x³ – 1/x³ = 1030
a3 + b³ = [a + b] [a² + b² – ab]
19. Answer: (D)
110 = 5 [a² + b² – ab]
(x – 7)3 + (x – 8)3 +(x – 6)3 = 3 (x – 7) (x – 8) (x + 6)
a² + b² – ab = 22
In these type of question go through options.
[a + b]² – 3ab = 22.
x=3
14. Answer: (B)
(3 – 7)3 + (3 – 8)3 +(3 + 6)3 = 3 (3 – 7) (3 – 8) (3 +
x2 +1=3x
6)
x+1/x=3
– 64 –125 + 729 = 3 (–4)(–5)(9)
Cubing both sides, we get
1 540 = 540
x3 + = 27 − 9
x3 Value of x = 3 satisfied.
1
x3 + = 18 20. Answer: (B)
x3
Now, x4+1/x4=2207
(X4 +X2 ) Add 2 both side
X2 +5X+1 x4+1/x4+2=2209,x2+1/x2=47
1
X3 + 3 Add 2 again in both side
= X
x2+1/x2 + 2 = 49
1
X+ +5
X

=
18 9
= =2
1
x+1/x=7
3+5 4 4
15. Answer: (B) 21. Answer: (C)
x4+1/x4=1442
40√5x3 − 3√3y3
Add 2 from both side
=[2√5 x − √3y] × [Ax2 + Bxy + Cy2 ]
3 3 x4+1/x4+2=1444
[(2√5 𝑥) − (√3 𝑦) ] x2+1/x2 =38
= [2√5 x − √3y] × [Ax 2 + Bxy + Cy 2 ] Subtract 2 from both side
[2√5 x − √3 𝑦] [20x2 + 2√15 𝑥𝑦] x2+1/x2–2=36
= [2√5 x − √3y] × [Ax 2 + Bxy + Cy 2 ] x–1/x=6
[20x2 + 3y2 + 2√15xy] = [Ax 2 + Bxy + Cy 2 ] 22. Answer: (B)
By comparing the coefficients If (3x –7)3 + (3x – 8)3 + (3x + 6)3 = 3(3x –7) (3x –
8)(3x + 6)
A = 20 , B = 2√15 , C = 3
In these type of question go through option
∴ √B 2 + C 2 − A)
x=1
= √60 + 9 − 20 (3 –7)3 + (3 – 8)3 + (3 + 6)3 = 3(3 –7) (3 –8)(3 + 6)
=7 540 = 540
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 Algebra | ETE-889022705
Value for x = 1 satisfied. Let x = –3
23. Answer: (A) [2x – 3 + 7]3 + [2x – 3 + 8]3 + [2x – 3 + 3]3 +3[2x –
x + 1/x = 10 3 + 7] [2x – 3 + 8] [2x – 3 + 3]
Cube both side [1]3 + [2]3 + [–3]3 = 3 × 1 × 2 × (–3)
x³ – 1/x³ + 3(x – 1/x) = 1000 1 + 8 – 27 = –18
x³ – 1/x³ = 970 –18 = –18
24. Answer: (B) the value of x = –3 is satisfied
a3+b3=[a +b][a2+b2–ab] ________ (1) 29. Answer: (B)
[a + b]2 = a2 + b2 + 2ab x = √3 − √2 − − − − − (1)
[a + b]2 = 99 +2 ×11 1
= √3 + √2-----------(2)
x
a + b = 11 1
a3+b3=[a +b][a2+b2–ab] x − = √3 − √2 − √3 − √2
x
a3 + b3 = [11] [99 – 11] 1
x − = −2√2
a3 + b3 = 968 x
25. Answer: (C) Cube both side
1 1 1
8[a + b]3 + [a – b]3 = [3a + b] [Aa2 + Bab + Cb2] x3 − 3 − 3 × x × [x − ] = −16√2
[2(a + b)]3 + [a – b]3 = [3a + b] [Aa2 + Bab + Cb2] x x x
1
= [3a + b] [Aa2 + Bab + Cb2] 3
x 3 = −22√2
x
[2a + 2b + a – b] [2(a + b)]2+ [a – b] + (a + b) – (a
30. Answer: (A)
– b)]
We know that
= [3a + b] [Aa2 + Bab + Cb2]
If a3 + b3 + c3 = 3abc
[3a + b] [4a² + 4b² + 8ab + a² + b² – 2ab + 2a² –
⟹a+b+c=0
2b²] = [3a + b] [Aa2 + Bab + Cb2]
So here, (x + 7)³ + (2x + 8)³ + (2x + 3)³ = 3(x + 7)
[7a² + 3b² + 6ab] = [Aa2 + Bab + Cb2]
(2x + 8) (2x + 3)
By comparing
⟹ (x + 7) + (2x + 8) + (2x + 3) = 0
A = 7, B = 6 C = 3
⟹ 5x + 18 = 0
=A+B–C
⟹ x = –18/ 5 = –3.6
=7+6–3
31. Answer: (A)
= 10
x = 2 − √3
26. Answer: (B) 1
Now, x − = −2√3
x² - 6x + 1 = 0 x

Take x common Also, x+1/x=4 => x2+1/x2=16-2=14

1 1 1
x+1/x=6 ______ (1) So, x3 –x–3 = x3 − = (x − ) (x2 + + 1)
x3 x x2
A.T.Q = (–2√3) (14+1)
= (x4+1/x²)/(x²+1) =–30√3
Multiply by x and divide is domination 32. Answer: (D)
(x3+1/x3)/(x+1/x) (2x – 7)3 + (2x – 8)3 + (2x –3)3 = 3 (2x –7) (2x – 8)
From (1) (2x –3)
x3+1/x3 = 198 _____ (2) In these type of question it is better to go through
From (2) option.
= 198/6 = 33 Let x = 3
27. Answer: (D) (2×3 – 7)3 + (2×3 – 8)3 + (2×3 –3)3 = 3 (2×3 –7)
x+y+z=3 (2×3 – 8) (2×3 –3)
S.B.S (–1)3 + (–2)3 + (3)3 = 3 (–1) (–2) (3)
x2 + y2 + z2 + 2 (xy + yz + zx) = 9 18 = 18
x2 + y2 + z2 – 36 = 9 So, the value of x = 3
x2 + y2 + z2 = 45 _______ (1) 33. Answer: (A)
x3 + y3 + z3 – 3xyz = (x + y + z)(x² + y² + z² –xy - x4+ 1/x4 =1442
yz - zx ) add 2 both side
x3 + y3 + z3 – 3xyz = 3(45 + 18) x4 +1/x4+2=1442
x3 + y3 + z3 – 3xyz = 189 x2 +1/x2=38
add 2 both side
28. Answer: (D) x2 +1/x2+2=40
[2x + 7]3 + [2x + 8]3 + [2x + 3]3 = 3[2x + 7] (2x + 1 2
8] [2x + 3] [x + ] = 40
x
In these types of questions go through option
X+1/x=2√10
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 Algebra | ETE-889022705
34. Answer: (A) multiply by 4 both side
a3 – b3 = [a – b] [a2 + b2 + ab] ________ (1) 4x2 + 4 = 12x _______ (1)
A.T.Q A.T.Q
a2 + b2 = 135 = [2x – 3]2
subtract 2ab from both side. = 4x2 + 9 – 12x
a2 + b2 –2ab= 135 – 2 × 7 Put the value of 12x from (1)
[a – b]2 = 121 = 4x2 + 9 – 4x2 – 4
a – b = 11 =5
from (1) 39. Answer: (C)
a3 – b3 = [a – b] [a2 + b2 + ab] x = 2 + √5 − − − − − −(1)
= 11 [135 + 7] 1
= √5 − 2 − − − − − −(2)
x
= 11 × 142
From (1) and (2)
= 1562 1
x − = 2 + √5 − √5 + 2
35. Answer: (C) x
1
x = 2 + √3 x− =4
x
1/x=2-√3 Cube both side
1 1 1
Then x3 − − 3 × x × [x − ] = 64
x3 x x
1 3 1
x − = 2√3 x − = 64 + 3 × 4 = 76
x x3
Cube both side 40. Answer: (C)
3
x −
1 1 1
− 3 × x × [x − ] = 24√3 (x – 8)³ + (2x + 16)³ + (2x –13)³ = 3(x – 8)(2x +
x3
16)(2x –13)
x x
3 1
x − = 24√3 + 3 × 2√3
x3 a3 + b3 + c3 – 3abc = 0
1
3
x − = 30√3 a+b+c=0
x3
36. Answer: (D) [x – 8]³ + [2x + 16]³ + [2x –13]³ – 3(x – 8)(2x +
(x – 7)3 + (2x + 8)3 +(2x – 3)3 = 3(x – 7) (2x + 16)(2x –13) = 0
8) (2x – 3) A.T.Q
(x – 7)3 + (2x + 8)3 +(2x – 3)3 – 3(x – 7) (2x + x – 8 + 2x + 16 + 2x – 13 = 0
8) (2x – 3) = 0 5x – 5 = 0
a+b+c=0 5x = 5
a3 + b3 + c3 – 3abc = 0 x=1
then 41. Answer: (A)
x – 7 + 2x + 8 + 2x – 3 = 0 x=2+√3
5x = 2 1
= 2 − √3
x = 0.4 x
1
37. Answer: (D) x+ =4
x
a3 + b3 = [a + b] [a2 + b2 – ab] Cube both side
(a + b)2 – 3ab = a2 + b2 + 2ab – 3ab 1
= a2 + b2 – ab x3 + 3 = 64 − 3 × 4
x
A.T.Q 3
1
x + 3 = 52
1344 = 28[a2 + b2 – ab] x
a2 + b2 – ab = 48 42. Answer: (D)
38. Answer: (C) a3 – b3 = [a – b] [a2 + b2 + ab]
1 [a – b]2 + 3ab = a2 + b2 – 2ab + 3ab
x4 + 4 = 47
x [a – b]2 + 3ab = a2 + b2 + ab
add 2 both side then
1 A.T.Q
x4 + 4 + 2 = 49
x a3 + b3 = 899
Then from (i)
1
[x2 + 2 ] = 7 [a – b][a2 + b2 + ab] = 899
x 29[a2 + b2 + ab] = 899
Add 2 both side
1 [a2 + b2 + ab] = 31
x2 + 2 + 2 = 9 [a – b]2 + 3ab = 31
x
1 43. Answer: (A)
x+ =3 1
x x4 + =1154
x4
x2 +1=3x
add 2 both side
Best Defence Institute for AFCAT , CDS , NDA , CAPF , ACC , TA Mob no :- 87-09-49-64-74 Page | 7
 Algebra | ETE-889022705
x4 +
1
+2=1156 Add 2 both side
x4 1
1
x2 + =34 x2 + 2 + 2 = 16
x2 x
again add 2 both side 1
1 x + = 4 − − − − − (1)
x2 + +2=36 x
x2
1
x2 +1=4x
x + =6 multiply by 4 both side
x
x + 1 = 6x ______ (1)
2
4x2 + 4 = 16x _______ (2)
A.T.Q A.T.Q
= 2[x – 3]2 = (2x – 4)2
= 2[x2 + 9 –6x] = 4x2 + 16 – 16x
Put the value of 6x from (1) = 4x2 + 16 – 4x2 – 4
= 2[x2 + 9 –x2 – 1] = 12
= 16 49. Answer: (B)
44. Answer: (B) x=2+√5 --------- (1)
1 1
x− =7 = √5 − 2--------- (2)
x x

Cube both side Then

1 1 1
x3 − 3 − 3 × x × = 343 x + = 2√5
x x x
1 Cube both side
3
x − 3 − 3 × 7 = 343 1
x x3 + = 40√5-6√5
x3
3
1 1
x − 3 = 343 x3 + = 34√5
x x3
45. Answer: (B) 50. Answer:(A)
(3x + 1)3 + (x – 3)3 +(2x – 4)3 = 6(3x + 1) (x – 3) (x – a2 + b2 - 6a - 6b + 18 = 0
2) ⇒ a2 - 2 × a × 3 + 32 + b2 - 2 × b × 3 + 32 =0
⇒ (3x + 1)3 + (x – 3)3 +(2x – 4)3 = 3(3x + 1) (x – ⇒ (a - 3)2 + (b - 3)2 = 0
3) (2x – 4) ⇒ (a - 3)2 = 0 & (b - 3)2 = 0
Clearly the above equation is in the form of ⇒a-3=0&b-3=0
a3 + b3 + c3 = 3abc ⇒a=3&b=3
⇒ a+b+c=0 ∴a-b=0
So, (3x + 1) + (x – 3) + (2x – 4) = 0
⇒ 6x – 6 = 0
⇒ x=1
46. Answer: (C)
We know that, (a + b)2 – (a – b)2 = 4ab
So,
1 1 2 1 2
4
[(a + a) − (a − a) ]
1 1 1
= × [4 × a × ] = × 4 = 1
4 a 4
47. Answer: (D)
a3 – b3 = [a – b] [a2 + b2 + ab]
[a – b]2 + 3ab =a2 + b2 + 2ab + 3ab
= a2 + b2 + ab
A.T.Q
a3 – b3 = [a – b][a2 + b2 + ab]
899 = 31 [a2 + b2 + ab]
a2 + b2 + ab = 29
[a - b]2 + 3ab = 29
48. Answer: (C)
1
x4 + 4 = 194
x
add 2 both side
1
x4 + 4 + 2 = 196
x
2
1
[x + 2 ] = 14
x
Best Defence Institute for AFCAT , CDS , NDA , CAPF , ACC , TA Mob no :- 87-09-49-64-74 Page | 8