316. The simple value of tan 1°. tan 2°. tan 3°....... tan 89° is

1. 1. 	1
      	2

   
   
   

   2. 0

   
   
   

   3. 1

   
   
   

   4. 	2
      	3

   
   
   

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   tan 1° . tan 2° . tan 3° ... tan 89°
   = (tan 1° . tan 89°) (tan 2°. tan88°) ... tan 45°
   = (tan 1°. tan(90° – 1°) ) (tan 2° tan (90° – 2°) .. tan 45°
   = (tan 1° . cot 1°) (tan 2°. cot 2°) .. tan 45°
   = 1.1..1. = 1
   [tan (90° – θ) = cot θ]

   Correct Option: C

   tan 1° . tan 2° . tan 3° ... tan 89°
   = (tan 1° . tan 89°) (tan 2°. tan88°) ... tan 45°
   = (tan 1°. tan(90° – 1°) ) (tan 2° tan (90° – 2°) .. tan 45°
   = (tan 1° . cot 1°) (tan 2°. cot 2°) .. tan 45°
   = 1.1..1. = 1
   [tan (90° – θ) = cot θ]

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317. If x sin² 60° – (3 / 2) sec 60° tan² 30° + (4 / 5) sin² 45° tan² 60° = 0 then x is

1. 1. -	1
      	15

   
   
   

   2. – 4

   
   
   

   3. 
      
      
      4. – 2

      
      
      

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      1. View Hint View Answer Discuss in Forum

      -	4
      	15

      x sin²60° –	3	sec 60° . tan²30°
      	2

      
      

      +	4	sin²45° . tan² 60° = 0
      	5

      
      
      

      ⇒		3x	-	3	× 2 ×	1	+		4	×	1	× 3 = 0
      		4		2		3			5		2

      
      

      ⇒		3x	- 1 +	6	= 0
      		4		5

      
      

      ⇒		3x	= 1 -	6	=	5 - 6	=		- 1
      		4		5		5			5

      
      

      ⇒ x = -		1	×	4	=	- 4
      		5		3		15

      
      

   Correct Option: C

   x sin²60° –	3	sec 60° . tan²30°
   	2

   
   

   +	4	sin²45° . tan² 60° = 0
   	5

   
   
   

   ⇒		3x	-	3	× 2 ×	1	+		4	×	1	× 3 = 0
   		4		2		3			5		2

   
   

   ⇒		3x	- 1 +	6	= 0
   		4		5

   
   

   ⇒		3x	= 1 -	6	=	5 - 6	=		- 1
   		4		5		5			5

   
   

   ⇒ x = -		1	×	4	=	- 4
   		5		3		15

   
   

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318. If 7 sinα = 24 cos α; 0 < α < (π / 2) , then the value of 14 tan α – 75 cos α – 7 sec α is equal to

1. 1. 3

   
   
   

   2. 4

   
   
   

   3. 1

   
   
   

   4. 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   7 sin α = 24 cos α
   

   ⇒		sin α	=	24	⇒ tanα =	24
   		cosα		7		7

   
   
   

   ∴ cosα =		1	=	7
   		sec α		7

   
   ∴ 14 tanα – 75 cosα – 7 secα
   

   = 14 ×		24	- 75 ×	7	- 7 ×	25
   		7		25		7

   
   = 48 – 21 – 25 = 2

   Correct Option: D

   7 sin α = 24 cos α
   

   ⇒		sin α	=	24	⇒ tanα =	24
   		cosα		7		7

   
   
   

   ∴ cosα =		1	=	7
   		sec α		7

   
   ∴ 14 tanα – 75 cosα – 7 secα
   

   = 14 ×		24	- 75 ×	7	- 7 ×	25
   		7		25		7

   
   = 48 – 21 – 25 = 2

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319. The value of x which satisfies the equation 2 cosec² 30° + x sin² 60° – (3 / 4) tan² 30° = 10 is

1. 1. 2

   
   
   

   2. 3

   
   
   

   3. 0

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   2 cosec²30° + x sin²60° -	3	tan²30° = 10
   	4

   
   
   

   ⇒ 8 +		3x	-	3	×	1	= 10
   		4		4		3

   
   

   ⇒		3x	=	9	⇒ 3x = 9
   		4		4

   
   

   ⇒ x =	9	= 3
   	3

   
   

   Correct Option: B

   2 cosec²30° + x sin²60° -	3	tan²30° = 10
   	4

   
   
   

   ⇒ 8 +		3x	-	3	×	1	= 10
   		4		4		3

   
   

   ⇒		3x	=	9	⇒ 3x = 9
   		4		4

   
   

   ⇒ x =	9	= 3
   	3

   
   

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320. If 2 sin θ + cos θ = (7 / 3) then the value of (tan² θ – sec² θ) is

1. 1. 0

   
   
   

   2. –1

   
   
   

   3. 	3
      	7

   
   
   

   4. 	7
      	3

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan²θ – sec²θ = – (sec²θ – tan²θ) = –1.
   

   Correct Option: B

   tan²θ – sec²θ = – (sec²θ – tan²θ) = –1.
   

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