371. If x +	1	= √3 , then the value of x30 + x24 + x18 + x12 + x6 + 1 is
     	x

     
     

1. 1. √3
      

   
   
   

   2. - √3
      

   
   
   

   3. 1
      

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 8,
   

   x +	1	= √3
   	x

   
   On cubing both sides,
   

   ⇒ x3 +	1	+ 3		x +	1		= 3√3
   	x3				x

   
   

   ⇒ x3 +	1	+ 3√3 = 3√3
   	x3

   
   

   ⇒ x3 +	1	= 0
   	x3

   
   ∴ Expression = x³⁰ + x²⁴ + x¹⁸ + x¹² + x⁶ + 1
   Expression = x²⁴( x⁶ + 1 ) + x¹²( x⁶ + 1 ) + 1( x⁶ + 1 )
   Expression = ( x⁶ + 1 )( x²⁴ + x¹² + 1 )
   

   Expression = x3		x3 +	1		( x24 + x12 + 1 ) = 0
   			x3

   Correct Option: D

   Using Rule 8,
   

   x +	1	= √3
   	x

   
   On cubing both sides,
   

   ⇒ x3 +	1	+ 3		x +	1		= 3√3
   	x3				x

   
   

   ⇒ x3 +	1	+ 3√3 = 3√3
   	x3

   
   

   ⇒ x3 +	1	= 0
   	x3

   
   ∴ Expression = x³⁰ + x²⁴ + x¹⁸ + x¹² + x⁶ + 1
   Expression = x²⁴( x⁶ + 1 ) + x¹²( x⁶ + 1 ) + 1( x⁶ + 1 )
   Expression = ( x⁶ + 1 )( x²⁴ + x¹² + 1 )
   

   Expression = x3		x3 +	1		( x24 + x12 + 1 ) = 0
   			x3

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372. If		x -	1		2	= 3 , then the value of		x6 +	1		equals
     			x						x6

     
     

1. 1. 90
      

   
   
   

   2. 100
      

   
   
   

   3. 110
      

   
   
   

   4. 120

   
   
   

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1. View Hint View Answer Discuss in Forum

   		x -	1		2	= 3
   			x

   
   

   ⇒ x2 +	1	- 2 = 3
   	x2

   
   

   ⇒ x2 +	1	= 3 + 2 = 5
   	x2

   
   On cubing both sides,
   

   		x2 +	1		3	= (5)3
   			x2

   
   

   ⇒ x6 +	1	+ 3		x2 +	1		= 125
   	x6				x2

   
   

   ⇒ x6 +	1	+ 3 × 5 = 125
   	x6

   
   

   ⇒ x6 +	1	= 125 – 15 = 110
   	x6

   Correct Option: C

   		x -	1		2	= 3
   			x

   
   

   ⇒ x2 +	1	- 2 = 3
   	x2

   
   

   ⇒ x2 +	1	= 3 + 2 = 5
   	x2

   
   On cubing both sides,
   

   		x2 +	1		3	= (5)3
   			x2

   
   

   ⇒ x6 +	1	+ 3		x2 +	1		= 125
   	x6				x2

   
   

   ⇒ x6 +	1	+ 3 × 5 = 125
   	x6

   
   

   ⇒ x6 +	1	= 125 – 15 = 110
   	x6

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373. If x +	1	= 3 , then the value of x5 +	1	is :
     	x		x5

     
     

1. 1. 110
      

   
   
   

   2. 132
      

   
   
   

   3. 122
      

   
   
   

   4. 123

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 1 and 8,
   

   x +	1	= 3
   	x

   
   On squaring both sides,
   

   ⇒		x +	1		2	= 9
   			x

   
   

   ⇒ x2 +	1	+ 2 = 9
   	x2

   
   

   ⇒ x2 +	1	= 9 – 2 = 7 ... (i)
   	x2

   
   Again,
   

   ⇒		x +	1		3	= (3)3 = 27
   			x

   
   

   ⇒ x3 +	1	+ 3		x +	1		= 27
   	x3				x

   
   

   ⇒ x3 +	1	+ 3 × 3 = 27
   	x3

   
   

   ∴ x3 +	1	= 27 - 9 = 18 ...(ii)
   	x3

   
   

   ∴		x2 +	1			x3 +	1		= 18 × 7 = 126
   			x2				x3

   
   

   ⇒ x5 +	1	+ x +	1	= 126
   	x5		x

   
   

   ⇒ x5 +	1	= 126 – 3 = 123
   	x5

   Correct Option: D

   Using Rule 1 and 8,
   

   x +	1	= 3
   	x

   
   On squaring both sides,
   

   ⇒		x +	1		2	= 9
   			x

   
   

   ⇒ x2 +	1	+ 2 = 9
   	x2

   
   

   ⇒ x2 +	1	= 9 – 2 = 7 ... (i)
   	x2

   
   Again,
   

   ⇒		x +	1		3	= (3)3 = 27
   			x

   
   

   ⇒ x3 +	1	+ 3		x +	1		= 27
   	x3				x

   
   

   ⇒ x3 +	1	+ 3 × 3 = 27
   	x3

   
   

   ∴ x3 +	1	= 27 - 9 = 18 ...(ii)
   	x3

   
   

   ∴		x2 +	1			x3 +	1		= 18 × 7 = 126
   			x2				x3

   
   

   ⇒ x5 +	1	+ x +	1	= 126
   	x5		x

   
   

   ⇒ x5 +	1	= 126 – 3 = 123
   	x5

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374. When 2x +	2	= 3 , then value of		x3 +	1	+ 2		is
     	x				x3

     
     

1. 1. 	2
      	7

      
      

   
   
   

   2. 	7
      	8

      
      

   
   
   

   3. 	7
      	2

      
      

   
   
   

   4. 	8
      	7

   
   
   

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1. View Hint View Answer Discuss in Forum

   2x +	2	= 3
   	x

   
   On dividing by 2,
   

   x +	1	=	3
   	x		2

   
   On cubing both sides,
   

   ⇒		x +	1		3	=	27
   			x				8

   
   

   ⇒ x3 +	1	+ 3		x +	1		=	27
   	x3				x			8

   
   

   ⇒ x3 +	1	+	3 × 3	=	27
   	x3		2		8

   
   

   ⇒ x3 +	1	=	27	-	9
   	x3		8		2

   
   

   ⇒ x3 +	1	=	27 - 36
   	x3		8

   
   

   ⇒ x3 +	1	=	-9
   	x3		8

   
   

   ∴ x3 +	1	+ 2 = 2 -	9	=	16 - 9	=	7
   	x3		8		8		8

   Correct Option: B

   2x +	2	= 3
   	x

   
   On dividing by 2,
   

   x +	1	=	3
   	x		2

   
   On cubing both sides,
   

   ⇒		x +	1		3	=	27
   			x				8

   
   

   ⇒ x3 +	1	+ 3		x +	1		=	27
   	x3				x			8

   
   

   ⇒ x3 +	1	+	3 × 3	=	27
   	x3		2		8

   
   

   ⇒ x3 +	1	=	27	-	9
   	x3		8		2

   
   

   ⇒ x3 +	1	=	27 - 36
   	x3		8

   
   

   ⇒ x3 +	1	=	-9
   	x3		8

   
   

   ∴ x3 +	1	+ 2 = 2 -	9	=	16 - 9	=	7
   	x3		8		8		8

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375. If x = ³√x² + 11 - 2 , then the value of (x³ + 5x² + 12x) is
     

1. 1. 0
      

   
   
   

   2. 3
      

   
   
   

   3. 7
      

   
   
   

   4. 11

   
   
   

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1. View Hint View Answer Discuss in Forum

   x = ³√x² + 11 - 2
   ⇒ x + 2 = ³√x² + 11
   On cubing both sides,
   (x + 2)³ = x² + 11
   ⇒ x³ + 2³ + 3x² × 2 + 3x × 2² = x² + 11
   ⇒ x³ + 8 + 6x² + 12x = x² + 11
   ⇒ x³ + 5x² + 12x = 11 – 8 = 3
   

   Correct Option: B

   x = ³√x² + 11 - 2
   ⇒ x + 2 = ³√x² + 11
   On cubing both sides,
   (x + 2)³ = x² + 11
   ⇒ x³ + 2³ + 3x² × 2 + 3x × 2² = x² + 11
   ⇒ x³ + 8 + 6x² + 12x = x² + 11
   ⇒ x³ + 5x² + 12x = 11 – 8 = 3
   

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