576. If a, b, c be all positive integers, then the least positive value of a³ + b³ + c³ – 3abc is
     

1. 1. 1
      

   
   
   

   2. 2
      

   
   
   

   3. 4
      

   
   
   

   4. 3

   
   
   

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1. View Hint View Answer Discuss in Forum

   a³ + b³ + c³ – 3abc will be minimum if a = b = 1, c = 2
   ∴ Least value = 1 + 1 + 8 – 3 × 1
   × 1 × 2 = 10 – 6 = 4

   Correct Option: C

   a³ + b³ + c³ – 3abc will be minimum if a = b = 1, c = 2
   ∴ Least value = 1 + 1 + 8 – 3 × 1
   × 1 × 2 = 10 – 6 = 4

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577. When f(x) = 12x³ – 13x² – 5x + 7 is divided by (3x + 2), then the remainder is
     

1. 1. 2
      

   
   
   

   2. 0
      

   
   
   

   3. –1
      

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   By remainder theorem,
   

   Remainder = f		-2
   		3

   
   ∵ f(x) = 12x³ – 13x² – 5x + 7
   

   ∴ f		-2		= 12		-2		3	- 13		-2		2	- 5		-2		+ 7
   		3				3					3					3

   
   

   Remainder = -	12 × 8	-	13 × 4	+	10	+ 7
   	27		9		3

   
   

   Remainder = -	32	-	52	+	10	+ 7
   	9		9		3

   
   

   Remainder =	-32 - 52 + 30 + 63	=	9	= 1
   	9		9

   
   Second Method :
   

   Correct Option: D

   By remainder theorem,
   

   Remainder = f		-2
   		3

   
   ∵ f(x) = 12x³ – 13x² – 5x + 7
   

   ∴ f		-2		= 12		-2		3	- 13		-2		2	- 5		-2		+ 7
   		3				3					3					3

   
   

   Remainder = -	12 × 8	-	13 × 4	+	10	+ 7
   	27		9		3

   
   

   Remainder = -	32	-	52	+	10	+ 7
   	9		9		3

   
   

   Remainder =	-32 - 52 + 30 + 63	=	9	= 1
   	9		9

   
   Second Method :
   

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578. If ab + bc + ca = 0, then the value of	1	+	1	+	1	is
     	( a² - bc )		( b² - ac )		( c² - ab )

     
     

1. 1. 2
      

   
   
   

   2. –1
      

   
   
   

   3. 0
      

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   ab + bc + ca = 0
   ⇒ ab + ca = – bc
   ∴ a² – bc = a² + ab + ca
   ⇒ a² – bc = a (a + b + c)
   Similarly,
   b² – ac = b (a + b + c)
   c² – ab = c (a + b + c)
   

   ∴ Expression =	1	+	1	+	1
   	( a² - bc )		( b² - ac )		( c² - ab )

   
   

   Expression =	1	+	1	+	1
   	a (a + b + c)		b (a + b + c)		c (a + b + c)

   
   

   Expression =	bc + ac + ab	= 0    { ∴ ab + bc + ca = 0 }
   	abc (a + b + c)

   
   

   Correct Option: C

   ab + bc + ca = 0
   ⇒ ab + ca = – bc
   ∴ a² – bc = a² + ab + ca
   ⇒ a² – bc = a (a + b + c)
   Similarly,
   b² – ac = b (a + b + c)
   c² – ab = c (a + b + c)
   

   ∴ Expression =	1	+	1	+	1
   	( a² - bc )		( b² - ac )		( c² - ab )

   
   

   Expression =	1	+	1	+	1
   	a (a + b + c)		b (a + b + c)		c (a + b + c)

   
   

   Expression =	bc + ac + ab	= 0    { ∴ ab + bc + ca = 0 }
   	abc (a + b + c)

   
   

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579. If the equation 2x2 – 7x + 12 = 0 has two roots α and β, then the value of	α	+	β	is :
     	β		α

     
     

1. 1. 	7
      	2

      
      

   
   
   

   2. 	1
      	24

      
      

   
   
   

   3. 	7
      	24

      
      

   
   
   

   4. 	97
      	24

   
   
   

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1. View Hint View Answer Discuss in Forum

   2x² – 7x + 12 = 0
   

   ∴ α + β =	7
   	2

   
   

   αβ =	12	= 6
   	2

   
   [In equation ax² + bx + c = 0 ]
   

   ∴ α + β =	-b	and	αβ =	c
   	a			a

   
   

   ∴	α	+	β	=	α² + β²
   	β		α		αβ

   
   

   ⇒	α	+	β	=	( α + β )² - 2αβ
   	β		α		αβ

   
   

   ⇒	α	+	β	=	( 7 / 2)2 - 2 × 6
   	β		α		6

   
   

   ⇒	α	+	β	=	( 49 / 4) - 12
   	β		α		6

   
   

   ⇒	α	+	β	=	49 - 48	=	1
   	β		α		24		24

   Correct Option: B

   2x² – 7x + 12 = 0
   

   ∴ α + β =	7
   	2

   
   

   αβ =	12	= 6
   	2

   
   [In equation ax² + bx + c = 0 ]
   

   ∴ α + β =	-b	and	αβ =	c
   	a			a

   
   

   ∴	α	+	β	=	α² + β²
   	β		α		αβ

   
   

   ⇒	α	+	β	=	( α + β )² - 2αβ
   	β		α		αβ

   
   

   ⇒	α	+	β	=	( 7 / 2)2 - 2 × 6
   	β		α		6

   
   

   ⇒	α	+	β	=	( 49 / 4) - 12
   	β		α		6

   
   

   ⇒	α	+	β	=	49 - 48	=	1
   	β		α		24		24

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