261. If x sin 60° tan 30° – tan²45° = cosec 60° cot 30° – sec²45°, then x =

1. 1. 2

   
   
   

   2. - 2

   
   
   

   3. 6

   
   
   

   4. - 4

   
   
   

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   x sin60° tan 30° – tan²45°
   = cosec 60°. cot30° – sec²45°
   

   ⇒ x .		√3	.	1	- 1
   		2		√3

   
   

   =	2	× √3- (√2)²
   	√3

   
   

   ⇒	x	- 1 = 2 - 2 = 0
   	2

   
   

   ⇒	x	= 1 ⇒ x = 2
   	2

   
   

   Correct Option: A

   x sin60° tan 30° – tan²45°
   = cosec 60°. cot30° – sec²45°
   

   ⇒ x .		√3	.	1	- 1
   		2		√3

   
   

   =	2	× √3- (√2)²
   	√3

   
   

   ⇒	x	- 1 = 2 - 2 = 0
   	2

   
   

   ⇒	x	= 1 ⇒ x = 2
   	2

   
   

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262. If x = a secα cosβ, y = b secα sinβ, z = c tan α, then the value of
     

     		x²	+	y²	-	z²	is
     		a²		b²		c²

1. 1. 2

   
   
   

   2. 0

   
   
   

   3. 1

   
   
   

   4. - 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   x = a secα . cosβ
   

   ⇒	x	= secα . cosβ
   	α

   
   Similarly,
   

   		y	= secα . sinβ,	z	= tanα
   		b		c

   
   

   ∴		x²	+	y²	-	z²
   		a²		b²		c²

   
   = sec²α. cos²β + sec²α . sin²β – tan²α
   = sec²α (cos²β + sin²β) – tan²α
   = sec²α – tan²α = 1

   Correct Option: C

   x = a secα . cosβ
   

   ⇒	x	= secα . cosβ
   	α

   
   Similarly,
   

   		y	= secα . sinβ,	z	= tanα
   		b		c

   
   

   ∴		x²	+	y²	-	z²
   		a²		b²		c²

   
   = sec²α. cos²β + sec²α . sin²β – tan²α
   = sec²α (cos²β + sin²β) – tan²α
   = sec²α – tan²α = 1

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263. If θ = 60°, then (1 / 2)√1 + sin θ + (1 / 2)√1 - sin θ is equal to

1. 1. cot	θ
      	2

   
   
   

   2. sec	θ
      	2

   
   
   

   3. sin	θ
      	2

   
   
   

   4. cos	θ
      	2

   
   
   

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   		1	√1 + sin θ	1	√1 1 sin θ
   		2		2

   
   

   =	1	(1 + sin 60°) + √1 - sin 60°
   	2

   
   
   

   =	1	(√3 + 1 + √3 - 1)
   	4

   
   

   =		2√3	=	√3	= cos 30°
   		4		2

   
   

   = cos	θ
   	2

   
   

   Correct Option: D

   		1	√1 + sin θ	1	√1 1 sin θ
   		2		2

   
   

   =	1	(1 + sin 60°) + √1 - sin 60°
   	2

   
   
   

   =	1	(√3 + 1 + √3 - 1)
   	4

   
   

   =		2√3	=	√3	= cos 30°
   		4		2

   
   

   = cos	θ
   	2

   
   

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264. The value of	cos² 60° + 4sec² 30° - tan² 45°	is
     	sin² 30° + cos² 30°

1. 1. 	64
      	√3

   
   
   

   2. 	55
      	12

   
   
   

   3. 	67
      	12

   
   
   

   4. 	67
      	10

   
   
   

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1. View Hint View Answer Discuss in Forum

   Expression
   

   =	cos² 60° + 4sec² 30° - tan² 45°
   	sin² 30° + cos² 30°

   
   
   [∵ sin²θ + cos²θ = 1]
   

   =		1	+	16	- 1
   		4		3

   
   

   =		3 + 64 - 12	=	55
   		12		3

   
   

   Correct Option: B

   Expression
   

   =	cos² 60° + 4sec² 30° - tan² 45°
   	sin² 30° + cos² 30°

   
   
   [∵ sin²θ + cos²θ = 1]
   

   =		1	+	16	- 1
   		4		3

   
   

   =		3 + 64 - 12	=	55
   		12		3

   
   

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265. The value of sin²30° cos²45° + 5tan²30° + (3 / 2) sin²90° – 3 cos²90° is

1. 1. 3	7
      	24

   
   
   

   2. 3	3
      	24

   
   
   

   3. 3	1
      	24

   
   
   

   4. 3	5
      	24

      
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin² 30° cos² 45° + 5 tan² 30° +	3	sin² 90° - 3cos² 90°
   	2

   
   
   

   	3	× 1 - 3 × 0
   	2

   
   

   =		1	×	1	+ 5 ×	1	+	3
   		4		2		3		2

   
   

   =		1	+	5	+	3	=	3 + 40 + 36
   		8		3		2		24

   
   

   =		79	= 3	7
   		24		24

   
   

   Correct Option: A

   sin² 30° cos² 45° + 5 tan² 30° +	3	sin² 90° - 3cos² 90°
   	2

   
   
   

   	3	× 1 - 3 × 0
   	2

   
   

   =		1	×	1	+ 5 ×	1	+	3
   		4		2		3		2

   
   

   =		1	+	5	+	3	=	3 + 40 + 36
   		8		3		2		24

   
   

   =		79	= 3	7
   		24		24

   
   

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