236. If   a +	1	= 1 and b +	1	= 1, then c +	1	is equal to
     	b		c		a

1. 1. 0

   
   
   

   2. 1

      2

   
   
   

   3. 1

   
   
   

   4. 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   a +	1	= 1 ⇒ 1 -	1	=	b - 1
   	b		b		b

   
   

   ⇒	1	=	b	and b +	1	= 1 ⇒	1	= 1 - b ⇒ c =	1
   	a		b - 1		c		c		1 - b

   
   

   ∴ c +	1	=	1	+	b
   	a		1 - b		b - 1

   
   

   =	1	-	b	=	1 - b	= 1
   	1 - b		1 - b		1 - b

   Correct Option: C

   a +	1	= 1 ⇒ 1 -	1	=	b - 1
   	b		b		b

   
   

   ⇒	1	=	b	and b +	1	= 1 ⇒	1	= 1 - b ⇒ c =	1
   	a		b - 1		c		c		1 - b

   
   

   ∴ c +	1	=	1	+	b
   	a		1 - b		b - 1

   
   

   =	1	-	b	=	1 - b	= 1
   	1 - b		1 - b		1 - b

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237. If the expression x2+ x + 1 is written in the form		x +	1		2	+ q2,
     			2

     then the possible values of q are

1. 1. ±	1
      	3

   
   
   

   2. ±	√3
      	2

   
   
   

   3. ±	2
      	√3

   
   
   

   4. ±	1
      	2

   
   
   

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1. View Hint View Answer Discuss in Forum

   x² + x + 1
   

   = x2 + 2.x.	1	+	1	+	3
   	2		4		4

   
   

   =		x +	1		2	+		±	√3		2
   			2						2

   
   

   ∴		x +	1		2	+		±	√3		2
   			2						2

   
   

   =		x +	1		2	+ q2
   			2

   
   

   ⇒  q = ±	√3
   	2

   Correct Option: B

   x² + x + 1
   

   = x2 + 2.x.	1	+	1	+	3
   	2		4		4

   
   

   =		x +	1		2	+		±	√3		2
   			2						2

   
   

   ∴		x +	1		2	+		±	√3		2
   			2						2

   
   

   =		x +	1		2	+ q2
   			2

   
   

   ⇒  q = ±	√3
   	2

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238. If x and y are positive real numbers and xy = 8, then the minimum value of 2x + y is

1. 1. 9

   
   
   

   2. 17

   
   
   

   3. 10

   
   
   

   4. 8

   
   
   

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1. View Hint View Answer Discuss in Forum

   xy = 8 = 1 × 8 = 2 × 4
   

   =	1	× 16 =	1	× 24
   	2		3

   
   ∴  Minimum value of 2x + y
   = 2 × 2 + 4 = 8

   Correct Option: D

   xy = 8 = 1 × 8 = 2 × 4
   

   =	1	× 16 =	1	× 24
   	2		3

   
   ∴  Minimum value of 2x + y
   = 2 × 2 + 4 = 8

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239. If   x = 3 + √8, then   x2 +	1	is equal to
     	x2

1. 1. 38

   
   
   

   2. 36

   
   
   

   3. 34

   
   
   

   4. 30

   
   
   

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1. View Hint View Answer Discuss in Forum

   x = 3 + √8
   

   ∴	1	=	1	=	3 − √8
   	x		3 + √8		(3 + √8)(3 − √8)

   
   

   =	3 − √8	= 3 − √8
   	9 − 8

   
   

   Now,   x2 +	1	=		x +	1		2	− 2
   	x2				x

   
   = (3 + √8 + 3 − √8)² – 2
   = 36 – 2 = 34

   Correct Option: C

   x = 3 + √8
   

   ∴	1	=	1	=	3 − √8
   	x		3 + √8		(3 + √8)(3 − √8)

   
   

   =	3 − √8	= 3 − √8
   	9 − 8

   
   

   Now,   x2 +	1	=		x +	1		2	− 2
   	x2				x

   
   = (3 + √8 + 3 − √8)² – 2
   = 36 – 2 = 34

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240. If	xy	= a,	xz	= b and	yz	= c,
     	x + y		x + z		y + z

     where a, b, c are all non-zero numbers, then x equals to

1. 1. 2abc

      ab + bc − ac

   
   
   

   2. 2abc

      ab + ac − bc

   
   
   

   3. 2abc

      ac + bc − ab

   
   
   

   4. 2abc

      ab + bc + ac

   
   
   

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1. View Hint View Answer Discuss in Forum

   xy	= a ⇒	x + y	=	1
   x + y		xy		a

   
   

   ⇒	1	+	1	=	1	...(i)
   	y		x		a

   
   

   xz	= b     ⇒	x + z	=	1
   x + z		xz		b

   
   

   ⇒	1	+	1	=	1	...(ii)
   	z		x		b

   
   Similarly,
   

   1	+	1	=	1
   z		y		c

   
   

   ⇒	1	=	1	−	1	...(iii)
   	y		c		z

   
   By substitution method
   From equations (i) and (iii),
   

   1	−	1	=	1	−	1
   a		x		c		z

   
   

   ⇒	1	−	1	=	1	−	1	+	1
   	a		x		c		b		x

   
   [From equation (ii)]
   

   ⇒	1	+	1	=	1	−	1	+	1
   	x		x		a		c		b

   
   

   ⇒	2	=	bc − ab + ac
   	x		abc

   
   

   ⇒  x =	2abc
   	bc + ac − ab

   Correct Option: C

   xy	= a ⇒	x + y	=	1
   x + y		xy		a

   
   

   ⇒	1	+	1	=	1	...(i)
   	y		x		a

   
   

   xz	= b     ⇒	x + z	=	1
   x + z		xz		b

   
   

   ⇒	1	+	1	=	1	...(ii)
   	z		x		b

   
   Similarly,
   

   1	+	1	=	1
   z		y		c

   
   

   ⇒	1	=	1	−	1	...(iii)
   	y		c		z

   
   By substitution method
   From equations (i) and (iii),
   

   1	−	1	=	1	−	1
   a		x		c		z

   
   

   ⇒	1	−	1	=	1	−	1	+	1
   	a		x		c		b		x

   
   [From equation (ii)]
   

   ⇒	1	+	1	=	1	−	1	+	1
   	x		x		a		c		b

   
   

   ⇒	2	=	bc − ab + ac
   	x		abc

   
   

   ⇒  x =	2abc
   	bc + ac − ab

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