51. If   2x +	1	=1, then the value of x2 +	1	is
    	4x		64x2

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 1

      4

   
   
   

   4. 2

   
   
   

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   2x +	1	= 1
   	4x

   
   On dividing by 2, we get
   

   x +	1	=	1
   	8x		2

   
   On squaring both sides, we get
   

   	x +	1		2	=	1
   		8x				4

   
   

   ⇒  x2 +	1	+ 2 × x ×	1	=	1
   	64x2		8x		4

   
   

   ⇒  x2 +	1	+	1	=	1
   	64x2		4		4

   
   

   ⇒  x2 +	1	=	1	−	1	= 0
   	64x2		4		4

   Correct Option: A

   2x +	1	= 1
   	4x

   
   On dividing by 2, we get
   

   x +	1	=	1
   	8x		2

   
   On squaring both sides, we get
   

   	x +	1		2	=	1
   		8x				4

   
   

   ⇒  x2 +	1	+ 2 × x ×	1	=	1
   	64x2		8x		4

   
   

   ⇒  x2 +	1	+	1	=	1
   	64x2		4		4

   
   

   ⇒  x2 +	1	=	1	−	1	= 0
   	64x2		4		4

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52. If   a + b + c = 2c, find	a	+	c	.
    	a − c		b − c

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 2

   
   
   

   4. –1

   
   
   

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1. View Hint View Answer Discuss in Forum

   a + b = 2c
   ⇒  a – c = c – b
   

   ∴	a	+	c
   	a − c		b − c

   
   

   =	a	+	c
   	c – b		b − c

   
   

   =	a	−	c
   	c – b		c – b

   
   

   =	a − c	=	c – b	= 1
   	c – b		c – b

   Correct Option: B

   a + b = 2c
   ⇒  a – c = c – b
   

   ∴	a	+	c
   	a − c		b − c

   
   

   =	a	+	c
   	c – b		b − c

   
   

   =	a	−	c
   	c – b		c – b

   
   

   =	a − c	=	c – b	= 1
   	c – b		c – b

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53. If   a + b + c = 0 then the value of	a2 + b2 + c2	is
    	ab + bc + ca

1. 1. 2

   
   
   

   2. –2

   
   
   

   3. 0

   
   
   

   4. 4

   
   
   

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1. View Hint View Answer Discuss in Forum

   a + b + c = 0
   ∴  (a + b + c)² = 0
   ⇒  a² + b² + c² + 2ab + 2bc + 2ca = 0
   ⇒  a² + b² + c² = – 2ab – 2bc – 2ca
   

   ∴	a2 + b2 + c2
   	ab + bc + ca

   
   

   =	−2(ab + bc + ca)	= −2
   	ab + bc + ca

   Correct Option: B

   a + b + c = 0
   ∴  (a + b + c)² = 0
   ⇒  a² + b² + c² + 2ab + 2bc + 2ca = 0
   ⇒  a² + b² + c² = – 2ab – 2bc – 2ca
   

   ∴	a2 + b2 + c2
   	ab + bc + ca

   
   

   =	−2(ab + bc + ca)	= −2
   	ab + bc + ca

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54. The value of (2a + b)² – (2a – b)² is :

1. 1. 8ab

   
   
   

   2. –8ab

   
   
   

   3. 8a² + 2b²

   
   
   

   4. 8a² – 2b²

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² – y² = (x + y) (x – y)
   ∴  (2a + b)² – (2a – b)² = (2a + b + 2a – b) (2a + b – 2a + b)
   = 4a × 2b = 8ab

   Correct Option: A

   x² – y² = (x + y) (x – y)
   ∴  (2a + b)² – (2a – b)² = (2a + b + 2a – b) (2a + b – 2a + b)
   = 4a × 2b = 8ab

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55. If   a + b + c = m and	1	+	1	+	1	= 0, then average of a2,b2 and c2 is
    	a		b		c

1. 1. m²

   
   
   

   2. m2

      3

   
   
   

   3. m2

      9

   
   
   

   4. m2

      27

   
   
   

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1. View Hint View Answer Discuss in Forum

   a + b + c = m
   

   and ,	1	+	1	+	1	= 0
   	a		b		c

   
   

   ⇒	bc + ac + ab	= 0
   	abc

   
   ⇒  bc + ac + ab = 0
   ∴  (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
   ⇒  m² = a² + b² + c² + 2 × 0
   ⇒  a² + b² + c² = m²
   

   ∴  Required average =	a2 + b2 + c2	=	m2
   	3		3

   Correct Option: B

   a + b + c = m
   

   and ,	1	+	1	+	1	= 0
   	a		b		c

   
   

   ⇒	bc + ac + ab	= 0
   	abc

   
   ⇒  bc + ac + ab = 0
   ∴  (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
   ⇒  m² = a² + b² + c² + 2 × 0
   ⇒  a² + b² + c² = m²
   

   ∴  Required average =	a2 + b2 + c2	=	m2
   	3		3

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