31. If x = √3 + √2 , then the value of x3 -	1		is :
    	x3

    
    

1. 1. 14√2
      

   
   
   

   2. 14√3
      

   
   
   

   3. 22√2
      

   
   
   

   4. 10√2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x = √3 + √2
   

   ∴	1	=	1
   	x		√3 + √2

   
   

   =	√3 - √2
   	( √3 + √2 )( √3 - √2 )

   
   = √3 - √2
   

   ∴ x +	1	= √3 + √2 + √3 + √2 = 2√2
   	x

   
   Cubing both sides,
   

   ⇒		x -	1		3	= 16√2
   			x

   
   

   ⇒ x3 -	1	- 3		x -	1		= 16√2
   	x3				x

   
   

   ⇒ x3 -	1	- 3 × 2√2 = 16√2
   	x3

   
   

   ⇒ x3 -	1	= 16√2 + 6√2 = 22√2
   	x3

   
   

   Correct Option: C

   x = √3 + √2
   

   ∴	1	=	1
   	x		√3 + √2

   
   

   =	√3 - √2
   	( √3 + √2 )( √3 - √2 )

   
   = √3 - √2
   

   ∴ x +	1	= √3 + √2 + √3 + √2 = 2√2
   	x

   
   Cubing both sides,
   

   ⇒		x -	1		3	= 16√2
   			x

   
   

   ⇒ x3 -	1	- 3		x -	1		= 16√2
   	x3				x

   
   

   ⇒ x3 -	1	- 3 × 2√2 = 16√2
   	x3

   
   

   ⇒ x3 -	1	= 16√2 + 6√2 = 22√2
   	x3

   
   

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32. If x2 + 1 = 2x , then the value of	[ x2 + ( 1 / x2 ) ]		is
    	x2 - 3x + 1

    
    

1. 1. 0
      

   
   
   

   2. 1
      

   
   
   

   3. 2
      

   
   
   

   4. –2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

   Correct Option: D

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

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33. If x2 + 1 = 2x , then the value of	[ x2 + ( 1 / x2 ) ]		is
    	x2 - 3x + 1

    
    

1. 1. 0
      

   
   
   

   2. 1
      

   
   
   

   3. 2
      

   
   
   

   4. –2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

   Correct Option: D

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

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34. If x2 + 1 = 2x , then the value of	[ x2 + ( 1 / x2 ) ]		is
    	x2 - 3x + 1

    
    

1. 1. 0
      

   
   
   

   2. 1
      

   
   
   

   3. 2
      

   
   
   

   4. –2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

   Correct Option: D

   x² + 1 = 2x (Given)
   

   ⇒ x +	1	= 2 ...(i)
   	x

   
   

   Expression =	x4 +	1
   		x2
   	x2 - 3x + 1

   
   

   =	x6 + 1
   	x2
   	x2 - 3x + 1

   
   

   Expression =	x6 + 1
   	( x2 + 1 - 3x ).x2

   
   

   Expression =	x6 + 1	=	x6 + 1
   	( 2x - 3x ).x2		-x3

   
   

   Expression = -		x6 + 1		= -		x6	+	1
   		x3				x3		x3

   
   

   = -		x3 +	1
   			x3

   
   

   Expression = -			x +	1			3	- 3		x +	1
   				x							x

   
   = – [2³ – 3 × 2]
   = – 2
   

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35. If x = 1 – √2 , the value of		x -	1		3	is :
    			x

    
    

1. 1. – 8
      

   
   
   

   2. 8
      

   
   
   

   3. 2√2
      

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   x = 1 – √2
   On multiplying and divided by ( 1 + √2 )
   

   ∴	1	=	1 × ( 1 + √2 )
   	x		( 1 – √2 )( 1 + √2 )

   
   = 1 – √2
   

   ∴		x -	1		3	= ( 1 – √2 + 1 + √2 )3
   			x

   
   = 2³ = 8

   Correct Option: B

   x = 1 – √2
   On multiplying and divided by ( 1 + √2 )
   

   ∴	1	=	1 × ( 1 + √2 )
   	x		( 1 – √2 )( 1 + √2 )

   
   = 1 – √2
   

   ∴		x -	1		3	= ( 1 – √2 + 1 + √2 )3
   			x

   
   = 2³ = 8

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