126. If x = 999, y = 1000, z = 1001, then the value of
     

     x³ + y³ + z³ - 3xyz	is :
     x - y + z

1. 1. 1000

   
   
   

   2. 9000

   
   
   

   3. 1

   
   
   

   4. 9

   
   
   

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1. View Hint View Answer Discuss in Forum

   x³ + y³ + z³ – 3xyz = 1/2 (x + y + z) {(x – y)² + (y – z)² + (z – x)²}
   

   ∴	x³ + y³ + z³ – 3xyz
   	x - y + z

   
   

   =	1/2(x + y + z){(x-y)² + (y-z)² + (y-z)² + (z-x)²}
   	x - y + z

   
   

   =	1	(999 + 1000 + 100) =	{999 + 1000)² + (1000 - 10001)² + (1001 + 999)²}
   	2		(999 - 1000 + 1001)

   
   

   =	3000	× (1 + 1 + 4)
   	2 × 1000

   
   

   =	18	= 9
   	2

   Correct Option: D

   x³ + y³ + z³ – 3xyz = 1/2 (x + y + z) {(x – y)² + (y – z)² + (z – x)²}
   

   ∴	x³ + y³ + z³ – 3xyz
   	x - y + z

   
   

   =	1/2(x + y + z){(x-y)² + (y-z)² + (y-z)² + (z-x)²}
   	x - y + z

   
   

   =	1	(999 + 1000 + 100) =	{999 + 1000)² + (1000 - 10001)² + (1001 + 999)²}
   	2		(999 - 1000 + 1001)

   
   

   =	3000	× (1 + 1 + 4)
   	2 × 1000

   
   

   =	18	= 9
   	2

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127. If a + b + c = 0, then the value of (a³ + b³ + c³) is

1. 1. abc

   
   
   

   2. 2abc

   
   
   

   3. 3abc

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   If a + b + c = 0 then, a³ + b³ + c³ = 3abc
   ∵ a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)

   Correct Option: C

   If a + b + c = 0 then, a³ + b³ + c³ = 3abc
   ∵ a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)

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128. If x² – 4x + 1 = 0, then the value of		x6 + 1		is :
     		x³

1. 1. 48

   
   
   

   2. 52

   
   
   

   3. 55

   
   
   

   4. 58

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² – 4x + 1 = 0
   ⇒ x² + 1 = 4x
   

   ⇒	x² + 1	=	4x
   	x		x

   
   

   ⇒ x +	1	= 4
   	x

   
   On cubing both sides,
   

   	x +	1		³	= 64
   		x

   
   

   x³ +	1	+ 3		x +	1		³	= 64
   	x³				x

   
   

   ⇒ x³ +	1	+ 3 × 4 = 64
   	x³

   
   

   ⇒ x³ +	1	= 64 - 12 = 52
   	x³

   
   

   ⇒	x6 + 1	= 52
   	x³

   Correct Option: B

   x² – 4x + 1 = 0
   ⇒ x² + 1 = 4x
   

   ⇒	x² + 1	=	4x
   	x		x

   
   

   ⇒ x +	1	= 4
   	x

   
   On cubing both sides,
   

   	x +	1		³	= 64
   		x

   
   

   x³ +	1	+ 3		x +	1		³	= 64
   	x³				x

   
   

   ⇒ x³ +	1	+ 3 × 4 = 64
   	x³

   
   

   ⇒ x³ +	1	= 64 - 12 = 52
   	x³

   
   

   ⇒	x6 + 1	= 52
   	x³

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129. If x = 93, y = 93, z = 94 then the value of (x² – y² + 10xz + 10yz) is

1. 1. 104784

   
   
   

   2. 147840

   
   
   

   3. 174840

   
   
   

   4. 184740

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² – y² + 10xz + 10yz
   ⇒ (x + y) (x – y) + 10z (x + y)
   = (x + y) (x – y + 10z)
   = (93 + 93) (93 – 93 + 10 × 94)
   = 186 × 940 = 174840

   Correct Option: C

   x² – y² + 10xz + 10yz
   ⇒ (x + y) (x – y) + 10z (x + y)
   = (x + y) (x – y + 10z)
   = (93 + 93) (93 – 93 + 10 × 94)
   = 186 × 940 = 174840

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130. If x = 222, y = 223, z = 225 then the value of (x³ + y³ + z³ + 3xyz) is :

1. 1. 4590

   
   
   

   2. 4690

   
   
   

   3. 4950

   
   
   

   4. 4960

   
   
   

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1. View Hint View Answer Discuss in Forum

   x³ + y³ + z³ – 3xyz
   

   =	1	(x + y + z){(x-y)² + (y-z)² + (z-x)²}
   	2

   
   

   =	1	(222 + 223 + 225){(222-223)² + (223-225)² + (225-222)²}
   	2

   
   

   =	1	× 670(1 + 4 + 9)
   	2

   
   

   =	670 × 14	= 4690
   	2

   
   

   Correct Option: B

   x³ + y³ + z³ – 3xyz
   

   =	1	(x + y + z){(x-y)² + (y-z)² + (z-x)²}
   	2

   
   

   =	1	(222 + 223 + 225){(222-223)² + (223-225)² + (225-222)²}
   	2

   
   

   =	1	× 670(1 + 4 + 9)
   	2

   
   

   =	670 × 14	= 4690
   	2

   
   

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