131. If p = 99 then, the value of p(p² + 3p + 3) is :

1. 1. 989898

   
   
   

   2. 988899

   
   
   

   3. 999999

   
   
   

   4. 998889

   
   
   

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1. View Hint View Answer Discuss in Forum

   p = 99 (Given)
   Expression = p(p² + 3p + 3)
   = p³ + 3p² + 3p
   = p³ + 3p² + 3p + 1 – 1
   = (p + 1)³ – 1
   = (99 + 1)³ – 1 = (100)³ – 1
   = 1000000 – 1 = 999999

   Correct Option: C

   p = 99 (Given)
   Expression = p(p² + 3p + 3)
   = p³ + 3p² + 3p
   = p³ + 3p² + 3p + 1 – 1
   = (p + 1)³ – 1
   = (99 + 1)³ – 1 = (100)³ – 1
   = 1000000 – 1 = 999999

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132. If   x +	1	= 1 then the value of	x2 + 3x + 1	is
     	x		x2 + 7x + 1

1. 1. 1

   
   
   

   2. 3

      7

   
   
   

   3. 1

      2

   
   
   

   4. 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x +	1	= 1   (Given)
   	x

   
   

   Expression =	x2 + 3x + 1
   	x2 + 7x + 1

   
   

   =	x +	1	+ 3
   		x
   	x +	1	+ 7
   		x

   
   (Dividing numerator and denominator by x)
   

   =	1 + 3	=	4	=	1
   	1 + 7		8		2

   Correct Option: C

   x +	1	= 1   (Given)
   	x

   
   

   Expression =	x2 + 3x + 1
   	x2 + 7x + 1

   
   

   =	x +	1	+ 3
   		x
   	x +	1	+ 7
   		x

   
   (Dividing numerator and denominator by x)
   

   =	1 + 3	=	4	=	1
   	1 + 7		8		2

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133. If	m − a2	+	m − b2	+	m − c2	= 3 , then the value of m is
     	b2 + c2		c2 + a2		a2 + b2

1. 1. a² + b² − c²

   
   
   

   2. a² + b²

   
   
   

   3. a² + b² + c²

   
   
   

   4. a² − b² − c²

   
   
   

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1. View Hint View Answer Discuss in Forum

   m − a2	+	m − b2	+	m − c2	− 3 = 0
   b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − a2	− 1 +	m − b2	− 1 +	m − c2	− 1 = 0
   	b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − a2 − b2 − c2	+	m − b2 − c2 − a2	+	m − c2 − a2 − b2	= 0
   	b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − (a2 + b2 + c2)	+	m − (a2 + b2 + c2)	+	m −(a2 + b2 + c2)	= 0
   	b2 + c2		c2 + a2		a2 + b2

   
   ∴  Each term = 0
   

   ∴	m − (a2 + b2 + c2)	= 0
   	b2 + c2

   
   ⇒  m − (a² + b² + c²) = 0
   ⇒  m = (a² + b² + c²)

   Correct Option: C

   m − a2	+	m − b2	+	m − c2	− 3 = 0
   b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − a2	− 1 +	m − b2	− 1 +	m − c2	− 1 = 0
   	b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − a2 − b2 − c2	+	m − b2 − c2 − a2	+	m − c2 − a2 − b2	= 0
   	b2 + c2		c2 + a2		a2 + b2

   
   

   ⇒	m − (a2 + b2 + c2)	+	m − (a2 + b2 + c2)	+	m −(a2 + b2 + c2)	= 0
   	b2 + c2		c2 + a2		a2 + b2

   
   ∴  Each term = 0
   

   ∴	m − (a2 + b2 + c2)	= 0
   	b2 + c2

   
   ⇒  m − (a² + b² + c²) = 0
   ⇒  m = (a² + b² + c²)

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134. If x =	1	, y =	1	, then the value of 8xy (x2 + y2) is
     	2 + √3		2 − √3

1. 1. 196

   
   
   

   2. 290

   
   
   

   3. 112

   
   
   

   4. 194

   
   
   

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1. View Hint View Answer Discuss in Forum

   x =	1
   	2 + √3

   
   

   =	1	×	1	=	2 − √3
   	2 + √3		2 − √ 3		4 − 3

   
   = 2 − √ 3
   

   ∴  y =	1	= 2 + √ 3
   	2 − √ 3

   
   ∴  x + y = 2 – √ 3 + 2 + √ 3 = 4
   xy = (2 – √ 3 ) (2 + √ 3 )
   = 4 – 3 = 1
   ∴  8xy (x² + y²)
   = 8xy [(x + y)² – 2 xy]
   = 8 × 1 (4² – 2 × 1)
   = 8 (16 – 2) = 8 × 14 = 112

   Correct Option: C

   x =	1
   	2 + √3

   
   

   =	1	×	1	=	2 − √3
   	2 + √3		2 − √ 3		4 − 3

   
   = 2 − √ 3
   

   ∴  y =	1	= 2 + √ 3
   	2 − √ 3

   
   ∴  x + y = 2 – √ 3 + 2 + √ 3 = 4
   xy = (2 – √ 3 ) (2 + √ 3 )
   = 4 – 3 = 1
   ∴  8xy (x² + y²)
   = 8xy [(x + y)² – 2 xy]
   = 8 × 1 (4² – 2 × 1)
   = 8 (16 – 2) = 8 × 14 = 112

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135. If a =	√x + 2 + √x − 2	, then the value of (a2 – ax) is
     	√x + 2 − √x − 2

1. 1. 1

   
   
   

   2. 2

   
   
   

   3. – 1

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   a =	√x + 2 + √x − 2
   	√x + 2 − √x − 2

   
   By componendo and dividendo,
   

   a + 1

   a − 1

   
   

   =	√x + 2 + √x − 2 + √x + 2 − √x − 2
   	√x + 2 + √x − 2 − √x + 2 + √x − 2

   
   

   ⇒	a + 1
   	a − 1

   
   

   =	2√x + 2	=	√x + 2
   	2 √x − 2		√x − 2

   
   On squaring both sides,
   

   =	a2 + 2a + 1	=	x + 2
   	a2 − 2a + 1		x − 2

   
   

   ⇒	a2 + 1 + 2a	=	x + 2
   	a2 + 1 − 2a		x − 2

   
   By componendo and dividendo,
   

   2(a2 + 1)	=	2x
   4a		4

   
   

   ⇒	a2 + 1	= x
   	a

   
   ⇒  a² + 1 = ax
   ⇒  a² – ax = – 1

   Correct Option: C

   a =	√x + 2 + √x − 2
   	√x + 2 − √x − 2

   
   By componendo and dividendo,
   

   a + 1

   a − 1

   
   

   =	√x + 2 + √x − 2 + √x + 2 − √x − 2
   	√x + 2 + √x − 2 − √x + 2 + √x − 2

   
   

   ⇒	a + 1
   	a − 1

   
   

   =	2√x + 2	=	√x + 2
   	2 √x − 2		√x − 2

   
   On squaring both sides,
   

   =	a2 + 2a + 1	=	x + 2
   	a2 − 2a + 1		x − 2

   
   

   ⇒	a2 + 1 + 2a	=	x + 2
   	a2 + 1 − 2a		x − 2

   
   By componendo and dividendo,
   

   2(a2 + 1)	=	2x
   4a		4

   
   

   ⇒	a2 + 1	= x
   	a

   
   ⇒  a² + 1 = ax
   ⇒  a² – ax = – 1

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