476. If a, b are rational numbers and ( a − 1) √2 + 3 = b√2 + a , the value of (a + b) is

1. 1. –5

   
   
   

   2. 3

   
   
   

   3. –3

   
   
   

   4. 5

   
   
   

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   (a − 1)√2 + 3 = b√2 + a
   ⇒  a = 3 ; a – 1 = b
   ⇒  3 – 1 = b ⇒ b = 2
   ∴  a + b = 3 + 2 = 5

   Correct Option: D

   (a − 1)√2 + 3 = b√2 + a
   ⇒  a = 3 ; a – 1 = b
   ⇒  3 – 1 = b ⇒ b = 2
   ∴  a + b = 3 + 2 = 5

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477. If   a =	√5 + 1	and  b =	√5 − 1	,   then the value of
     	√5 − 1		√5 − 1

     a2 + ab + b2	is
     a2 − ab + b2

1. 1. 3

      4

   
   
   

   2. 4

      3

   
   
   

   3. 3

      5

   
   
   

   4. 5

      3

   
   
   

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   a =	√5 + 1	=	√5 + 1	×	√5 + 1
   	√5 − 1		√5 − 1		√5 + 1

   
   

   =	(√5 + 1)2	=	5 + 1 + 2√5
   	5 − 1		4

   
   

   =	3 + √5
   	2

   
   

   ∴  b =	√5 − 1	=	3 − √5
   	2		2

   
   

   ∴  a + b =	3 + √5	+	3 − √5	= 3
   	2		2

   
   

   and   ab =	√5 + 1	×	√5 − 1	= 1
   	√5 − 1		√5 + 1

   
   

   ∴  Expression =	a2 + ab + b2	=	(a + b)2 − ab
   	a2 − ab + b2		(a + b)2 − 3ab

   
   

   =	9 − 1	=	8	=	4
   	9 − 3		6		3

   Correct Option: B

   a =	√5 + 1	=	√5 + 1	×	√5 + 1
   	√5 − 1		√5 − 1		√5 + 1

   
   

   =	(√5 + 1)2	=	5 + 1 + 2√5
   	5 − 1		4

   
   

   =	3 + √5
   	2

   
   

   ∴  b =	√5 − 1	=	3 − √5
   	2		2

   
   

   ∴  a + b =	3 + √5	+	3 − √5	= 3
   	2		2

   
   

   and   ab =	√5 + 1	×	√5 − 1	= 1
   	√5 − 1		√5 + 1

   
   

   ∴  Expression =	a2 + ab + b2	=	(a + b)2 − ab
   	a2 − ab + b2		(a + b)2 − 3ab

   
   

   =	9 − 1	=	8	=	4
   	9 − 3		6		3

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478. If   64x + 1 =	64	,   then the value of x is
     	4x

1. 1. 1

   
   
   

   2. 0

   
   
   

   3. 1

      2

   
   
   

   4. 2

   
   
   

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   (64)x + 1 =	64
   	4x

   
   ⇒  (4³)^{x + 1} × 4^x = 64
   ⇒  4^{3x + 3 + x} = 4³
   ⇒  4^{4x + 3} = 4³
   ⇒  4x + 3 = 3
   ⇒  x = 0

   Correct Option: B

   (64)x + 1 =	64
   	4x

   
   ⇒  (4³)^{x + 1} × 4^x = 64
   ⇒  4^{3x + 3 + x} = 4³
   ⇒  4^{4x + 3} = 4³
   ⇒  4x + 3 = 3
   ⇒  x = 0

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479. If ax² + bx + c = a (x – p)² , then the relation among a, b, c would be

1. 1. abc = 1
      

   
   
   

   2. b² = ac

   
   
   

   3. b² = 4ac

   
   
   

   4. 2b = a + c

   
   
   

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   ax² + bx + c = a (x – p)²
   ⇒  ax² + bx + c = a (x² – 2px + p²)
   ⇒  ax² + bx + c = ax² – 2apx + ap²
   Comparing the corresponding coefficients,
   b = – 2ap and c = ap²
   

   ⇒  b2 = 4a2p2 and p2 =	c
   	a

   
   

   ⇒  p2 =	b2	;
   	4a2

   
   

   ∴	b2 =	=	c	⇒  b2 = 4ac
   	4a2		a

   Correct Option: C

   ax² + bx + c = a (x – p)²
   ⇒  ax² + bx + c = a (x² – 2px + p²)
   ⇒  ax² + bx + c = ax² – 2apx + ap²
   Comparing the corresponding coefficients,
   b = – 2ap and c = ap²
   

   ⇒  b2 = 4a2p2 and p2 =	c
   	a

   
   

   ⇒  p2 =	b2	;
   	4a2

   
   

   ∴	b2 =	=	c	⇒  b2 = 4ac
   	4a2		a

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480. If a + b + c + d = 1, then the maximum value of (1 + a) (1 + b) (1 + c) (1 + d) is

1. 1. 1

   
   
   

   2. 	1		3
      	2

   
   
   

   3. 	3		3
      	4

   
   
   

   4. 	5		4
      	4

   
   
   

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   For maximum value,
   

   a = b = c = d =	1
   	4

   
   ∴  (1 + a) (1 + b) (1 + c) (1 + d)
   

   	5		4
   	4

   Correct Option: D

   For maximum value,
   

   a = b = c = d =	1
   	4

   
   ∴  (1 + a) (1 + b) (1 + c) (1 + d)
   

   	5		4
   	4

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