6. If x +	1	=	3	, find the value of 8x3 +	1
   	4x		2		8x3

1. 1. 18
      

   
   
   

   2. 36
      

   
   
   

   3. 24
      

   
   
   

   4. 16

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 8,
   

   x +	1	=	3
   	4x		2

   
   Multiplying both sides by 2
   

   ⇒ 2x +	1	= 3
   	2x

   
   On cubing both sides,
   

   ∴ 8x3 +	1	+	3 × 2x ×	1	×		2x +	1		= 27
   	8x3			2x				2x

   
   

   ⇒ 8x3 +	1	+ 3 × 3 = 27
   	8x3

   
   

   ⇒ 8x3 +	1	= 27 – 9 = 18
   	8x3

   Correct Option: A

   Using Rule 8,
   

   x +	1	=	3
   	4x		2

   
   Multiplying both sides by 2
   

   ⇒ 2x +	1	= 3
   	2x

   
   On cubing both sides,
   

   ∴ 8x3 +	1	+	3 × 2x ×	1	×		2x +	1		= 27
   	8x3			2x				2x

   
   

   ⇒ 8x3 +	1	+ 3 × 3 = 27
   	8x3

   
   

   ⇒ 8x3 +	1	= 27 – 9 = 18
   	8x3

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7. If x4 +	1	= 119 ,and x > l, then the value of x3 -		1	is
   	x4			x3

1. 1. 54

   
   
   

   2. 18

   
   
   

   3. 72

   
   
   

   4. 36

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule1 and 8,
   

   x4 +	1	= 119
   	x4

   
   

   ⇒		x2 +	1		2	- 2 = 119
   			x2

   
   

   ⇒		x2 +	1		2	= 121
   			x2

   
   

   ⇒ x2 +	1	= 11
   	x2

   
   

   ⇒		x -	1		2	+ 2 = 11
   			x

   
   

   ⇒		x -	1		2	= 9
   			x

   
   

   ⇒ x -	1	= 3
   	x

   
   

   ∴ x3 -	1	- 3		x -	1		= 27
   	x3				x

   
   

   ⇒ x3 -	1	- 3 × 3 = 27
   	x3

   
   

   ⇒ x3 -	1	= 27 + 9 = 36
   	x3

   
   

   Correct Option: D

   Using Rule1 and 8,
   

   x4 +	1	= 119
   	x4

   
   

   ⇒		x2 +	1		2	- 2 = 119
   			x2

   
   

   ⇒		x2 +	1		2	= 121
   			x2

   
   

   ⇒ x2 +	1	= 11
   	x2

   
   

   ⇒		x -	1		2	+ 2 = 11
   			x

   
   

   ⇒		x -	1		2	= 9
   			x

   
   

   ⇒ x -	1	= 3
   	x

   
   

   ∴ x3 -	1	- 3		x -	1		= 27
   	x3				x

   
   

   ⇒ x3 -	1	- 3 × 3 = 27
   	x3

   
   

   ⇒ x3 -	1	= 27 + 9 = 36
   	x3

   
   

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8. If x + y = z, then the expression x³ + y³ - z³ + 3xyz will be equal to :

1. 1. 0
      

   
   
   

   2. 3xyz
      

   
   
   

   3. –3xyz
      

   
   
   

   4. z³

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 20,
   x + y = z ⇒ x + y + (–z) = 0
   ∴ x³ + y³ - z³ + 3xyz = x³ + y³ + ( - z )³ – 3x.y (–z) = 0
   

   Correct Option: A

   Using Rule 20,
   x + y = z ⇒ x + y + (–z) = 0
   ∴ x³ + y³ - z³ + 3xyz = x³ + y³ + ( - z )³ – 3x.y (–z) = 0
   

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9. If 3x +	1	= 5 , then the value of 8x3 +		1	is
   	2x			27x3

   
   

1. 1. 118	1
      	2

      
      

   
   
   

   2. 30	10
      	27

   
   
   

   3. 0

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   Using Rule 8,
   

   3x +	1	= 5
   	2x

   
   On multiplying both sides by ( 2 / 5 )
   

   ⇒ 2x +	1	=	10
   	3x		3

   
   Cubing both sides,
   

   ∴ 8x3 +	1	+	3 × 2x ×	1	×		2x +	1		=	1000
   	27x3			3x				3x			27

   
   

   ⇒ 8x3 +	1	+ 3 × 3 = 27
   	8x3

   
   

   ⇒ 8x3 +	1	+ 2 ×	10	=	1000
   	27x3		3		27

   
   

   ⇒ 8x3 +	1	=	1000	-	20
   	27x3		27		3

   
   

   ⇒ 8x3 +	1	=	1000 - 180	=	820	= 30	10
   	27x3		27		27		27

   Correct Option: B

   Using Rule 8,
   

   3x +	1	= 5
   	2x

   
   On multiplying both sides by ( 2 / 5 )
   

   ⇒ 2x +	1	=	10
   	3x		3

   
   Cubing both sides,
   

   ∴ 8x3 +	1	+	3 × 2x ×	1	×		2x +	1		=	1000
   	27x3			3x				3x			27

   
   

   ⇒ 8x3 +	1	+ 3 × 3 = 27
   	8x3

   
   

   ⇒ 8x3 +	1	+ 2 ×	10	=	1000
   	27x3		3		27

   
   

   ⇒ 8x3 +	1	=	1000	-	20
   	27x3		27		3

   
   

   ⇒ 8x3 +	1	=	1000 - 180	=	820	= 30	10
   	27x3		27		27		27

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10. If a +	1	+ 1 = 0 ( a ≠ 0 ) ,then the value of ( a4 - a ) is :
    	a

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 2

   
   
   

   4. -1

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given , a +	1	+ 1 = 0
   	a

   
   ⇒ a² + a + 1 = 0
   ⇒ a⁴ - a = a(a³ – 1)
   = a(a – 1)(a² + a + 1 ) = 0

   Correct Option: A

   Given , a +	1	+ 1 = 0
   	a

   
   ⇒ a² + a + 1 = 0
   ⇒ a⁴ - a = a(a³ – 1)
   = a(a – 1)(a² + a + 1 ) = 0

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