456. For what value(s) of a is   x +	1	√x + a2   a perfect square ?
     	4

1. 1. ±	1
      	18

   
   
   

   2. 1

      8

   
   
   

   3. −	1
      	5

   
   
   

   4. 1

      4

   
   
   

---

1. View Hint View Answer Discuss in Forum

   x +	1	√x + a2
   	4

   
   

   = (√x)2 + 2.√x.	1	+ (a)2
   	8

   
   

   Clearly a =	1
   	8

   
   

   Then, expression =		√x +	1		2
   			8

   Correct Option: B

   x +	1	√x + a2
   	4

   
   

   = (√x)2 + 2.√x.	1	+ (a)2
   	8

   
   

   Clearly a =	1
   	8

   
   

   Then, expression =		√x +	1		2
   			8

---

457. If a ≠ b, then which of the following statements is true ?

1. 1. a + b	= √ab
      2

   
   
   

   2. a + b	< √ab
      2

   
   
   

   3. a + b	> √ab
      2

   
   
   

   4. All of the above

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Arithmetic mean (AM) =	a + b
   	2

   
   Geometric mean (GM) = √ab
   As AM > GM
   

   a + b	> √ab
   2

   Correct Option: C

   Arithmetic mean (AM) =	a + b
   	2

   
   Geometric mean (GM) = √ab
   As AM > GM
   

   a + b	> √ab
   2

---

---

458. If

     a

     +

     b

     +

     c

     = 1, then the value of

     1

     +

     1

     +

     1

     is

     1 − a

     1 − b

     1 − c

     1 − a

     1 − b

     1 − c

1. 1. 1

   
   
   

   2. 2

   
   
   

   3. 3

   
   
   

   4. 4

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Tricky Approach
   

   a	+	b	+	c	= 1
   1 − a		1 − b		1 − c

   
   

   ⇒

   a

   + 1

   +

   b

   + 1

   +

   c

   + 1

   = 3 + 1 = 4

   1 − a

   1 − b

   1 − c

   
   

   ⇒	a + 1 − a	+	b + 1 − b	+	c + 1 − c	= 4
   	1 − a		1 − b		1 − c

   
   

   ⇒	1	+	1	+	1	= 4
   	1 − a		1 − b		1 − c

   Correct Option: D

   Tricky Approach
   

   a	+	b	+	c	= 1
   1 − a		1 − b		1 − c

   
   

   ⇒

   a

   + 1

   +

   b

   + 1

   +

   c

   + 1

   = 3 + 1 = 4

   1 − a

   1 − b

   1 − c

   
   

   ⇒	a + 1 − a	+	b + 1 − b	+	c + 1 − c	= 4
   	1 − a		1 − b		1 − c

   
   

   ⇒	1	+	1	+	1	= 4
   	1 − a		1 − b		1 − c

---

459. If x, y are two positive real numbers and x^1/3 = y^1/4 , then which of the following relations is true?

1. 1. x³ = y⁴

   
   
   

   2. x³ = y

   
   
   

   3. x = y⁴

   
   
   

   4. x²⁰ = y¹⁵

   
   
   

---

1. View Hint View Answer Discuss in Forum

   x^1/3 = y^1/4
   ⇒ (x^1/3)¹² = (y^1/4)¹² ⇒ x⁴ = y³
   ⇒ (x⁴)⁵ = (y³)⁵ ⇒ x²⁰ = y¹⁵

   Correct Option: D

   x^1/3 = y^1/4
   ⇒ (x^1/3)¹² = (y^1/4)¹² ⇒ x⁴ = y³
   ⇒ (x⁴)⁵ = (y³)⁵ ⇒ x²⁰ = y¹⁵

---

---

460. If x is real, then the minimum value of (x² – x + 1) is

1. 1. 3

      4

   
   
   

   2. 0

   
   
   

   3. 1

   
   
   

   4. 1

      4

   
   
   

---

1. View Hint View Answer Discuss in Forum

   For expression ax² + bx + c, a > 0, the minimum value is given by
   

   4ac − b2

   4a

   
   Here, for x² – x + 1
   a = 1, b = –1, c = 1
   

   ∴  Minimum value =	4 × 1 × 1 − 1	=	3
   	4 × 1		4

   Correct Option: A

   For expression ax² + bx + c, a > 0, the minimum value is given by
   

   4ac − b2

   4a

   
   Here, for x² – x + 1
   a = 1, b = –1, c = 1
   

   ∴  Minimum value =	4 × 1 × 1 − 1	=	3
   	4 × 1		4

---