401. 168. If a + b = 1, find the value of a³ – b³ - ab – (a² - b²)²
     

1. 1. –1
      

   
   
   

   2. 1
      

   
   
   

   3. 0
      

   
   
   

   4. 2

   
   
   

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   a + b = 1 (Given)
   Expression = a³ + b³ – ab - (a² – b²)²
   Expression = (a + b)(a² – ab + b²) – ab - (a² – b²)²
   Expression = 1.(a² – ab + b²) – ab - (a + b)²(a – b)²
   Expression = a² – ab + b² – ab - (a² – 2ab + b²)
   Expression = a² – 2ab + b² –a² + 2ab - b² = 0

   Correct Option: C

   a + b = 1 (Given)
   Expression = a³ + b³ – ab - (a² – b²)²
   Expression = (a + b)(a² – ab + b²) – ab - (a² – b²)²
   Expression = 1.(a² – ab + b²) – ab - (a + b)²(a – b)²
   Expression = a² – ab + b² – ab - (a² – 2ab + b²)
   Expression = a² – 2ab + b² –a² + 2ab - b² = 0

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402. 	p-1q2		1 / 3		÷		p6q-3		1 / 3		= pa qb , then the value of a + b, where p and q are different
     	p3q-2						p-2q3

     
     positive primes, is

1. 1. –1
      

   
   
   

   2. 2
      

   
   
   

   3. 1
      

   
   
   

   4. 0

   
   
   

   5. None of these

   
   
   

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   	p-1q2		1 / 3		÷		p6q-3		1 / 3		= pa qb
   	p3q-2						p-2q3

   
   ⇒ (p^{-1 - 3}q^{2 + 2})^{1 / 3} ÷ (p^{6 + 2}q^{-3 - 3})^{1 / 3} = p^a q^b
   ⇒ (p^-4q⁴)^{1 / 3} ÷ (p⁸q^-6)^{1 / 3} = p^a q^b
   

   ⇒	p-4 / 3q4 / 3	= pa qb
   	p8/ 3q-6 / 3

   
   ⇒ p^{{ (-4 / 3) - (8 / 3) }}q^{{ (4 / 3) + (6 / 3) }} = p^aq^b
   ⇒ p^-4 q^{(10 / 3)} = p^a q^b
   

   ⇒ a = -4 , b =	10
   	3

   
   

   ∴ a + b = -4 +	10	=	-2
   	3		3

   Correct Option: E

   	p-1q2		1 / 3		÷		p6q-3		1 / 3		= pa qb
   	p3q-2						p-2q3

   
   ⇒ (p^{-1 - 3}q^{2 + 2})^{1 / 3} ÷ (p^{6 + 2}q^{-3 - 3})^{1 / 3} = p^a q^b
   ⇒ (p^-4q⁴)^{1 / 3} ÷ (p⁸q^-6)^{1 / 3} = p^a q^b
   

   ⇒	p-4 / 3q4 / 3	= pa qb
   	p8/ 3q-6 / 3

   
   ⇒ p^{{ (-4 / 3) - (8 / 3) }}q^{{ (4 / 3) + (6 / 3) }} = p^aq^b
   ⇒ p^-4 q^{(10 / 3)} = p^a q^b
   

   ⇒ a = -4 , b =	10
   	3

   
   

   ∴ a + b = -4 +	10	=	-2
   	3		3

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403. If a -	1	= 5 , then the value of (a - 3)3 -	1	is
     	(a - 3)		(a - 3)3

     
     

1. 1. 5
      

   
   
   

   2. 7
      

   
   
   

   3. 2
      

   
   
   

   4. 14

   
   
   

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   a -	1	= 5
   	(a - 3)

   
   

   ⇒ (a - 3) -	1	= 5 - 3 = 2
   	(a - 3)

   
   On cubing both sides,
   

   ⇒		(a - 3) -	1		3	= 8
   			(a - 3)

   
   

   ⇒ (a - 3)3 -	1	- 3 × (a - 3) ×	1		(a - 3) -	1		= 8
   	(a - 3)3		(a - 3)			(a - 3)

   
   [∴ (a – b)³ = a³ – b³ – 3ab(a – b)]
   

   ⇒ (a - 3)3 -	1	- 3 × 2 = 8
   	(a - 3)3

   
   

   ⇒ (a - 3)3 -	1	= 8 + 6 = 14
   	(a - 3)3

   Correct Option: D

   a -	1	= 5
   	(a - 3)

   
   

   ⇒ (a - 3) -	1	= 5 - 3 = 2
   	(a - 3)

   
   On cubing both sides,
   

   ⇒		(a - 3) -	1		3	= 8
   			(a - 3)

   
   

   ⇒ (a - 3)3 -	1	- 3 × (a - 3) ×	1		(a - 3) -	1		= 8
   	(a - 3)3		(a - 3)			(a - 3)

   
   [∴ (a – b)³ = a³ – b³ – 3ab(a – b)]
   

   ⇒ (a - 3)3 -	1	- 3 × 2 = 8
   	(a - 3)3

   
   

   ⇒ (a - 3)3 -	1	= 8 + 6 = 14
   	(a - 3)3

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404. If (3x – 2y) : (2x + 3y) = 5 : 6, then one of the values of			³√x + ³√y		2	is
     			³√x - ³√y

     
     

1. 1. 	1
      	5

      
      

   
   
   

   2. 5
      

   
   
   

   3. 25
      

   
   
   

   4. 	1
      	25

      
      

   
   
   

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   ∴	3x - 2y	=	5
   	3x - 2y		6

   
   ⇒ 18x – 12y = 10x +15y
   ⇒ 18x – 10x = 12y + 15y
   ⇒ 8x = 27y
   

   ⇒	x	=	27
   	y		8

   
   On taking cube root of both sides,
   

   	³√x	=	³√27	=	3
   	³√y		³√8		2

   
   By componendo and dividendo,
   

   	³√x + ³√y	=	3 + 2	=	5
   	³√x - ³√y		3 - 2		1

   
   On squaring both sides,
   

   ⇒			³√x + ³√y		2	= 5 × 5 = 25
   			³√x - ³√y

   
   

   Correct Option: C

   ∴	3x - 2y	=	5
   	3x - 2y		6

   
   ⇒ 18x – 12y = 10x +15y
   ⇒ 18x – 10x = 12y + 15y
   ⇒ 8x = 27y
   

   ⇒	x	=	27
   	y		8

   
   On taking cube root of both sides,
   

   	³√x	=	³√27	=	3
   	³√y		³√8		2

   
   By componendo and dividendo,
   

   	³√x + ³√y	=	3 + 2	=	5
   	³√x - ³√y		3 - 2		1

   
   On squaring both sides,
   

   ⇒			³√x + ³√y		2	= 5 × 5 = 25
   			³√x - ³√y

   
   

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405. If	144	=	14.4	, then the value of x is
     	0.144		x

1. 1. 144

   
   
   

   2. 14.4

   
   
   

   3. 1.44

   
   
   

   4. 0.0144

   
   
   

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   144	=	14.4
   0.144		x

   
   ⇒  144 × x = 14.4 × 0.144
   

   ⇒  x =	14.4 × 0.144
   	144

   
   

   =	144 × 144	= 0.0144
   	144 × 10000

   Correct Option: D

   144	=	14.4
   0.144		x

   
   ⇒  144 × x = 14.4 × 0.144
   

   ⇒  x =	14.4 × 0.144
   	144

   
   

   =	144 × 144	= 0.0144
   	144 × 10000

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