266. If cos²θ – sin²θ = (1 / 3) , where 0 ≤ q ≤ (π / 2) , then the value of cos⁴θ – sin⁴θ is

1. 1. 	1
      	3

   
   
   

   2. 	2
      	3

   
   
   

   3. 	1
      	9

   
   
   

   4. 	2
      	9

   
   
   

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1. View Hint View Answer Discuss in Forum

   cos² θ - sin² θ =	1
   	3

   
   cos⁴ θ – sin⁴ θ
   = (cos² θ + sin² θ) (cos² θ – sin² θ)
   

   = 1 ×		1	=	1
   		3		3

   
   

   Correct Option: A

   cos² θ - sin² θ =	1
   	3

   
   cos⁴ θ – sin⁴ θ
   = (cos² θ + sin² θ) (cos² θ – sin² θ)
   

   = 1 ×		1	=	1
   		3		3

   
   

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267. If tan θ =		1	and 0 , < θ <	π	, then the value of	cosec² θ - sec² θ	is
     		√11		2		cosec² θ + sec² θ

1. 1. 	3
      	4

   
   
   

   2. 	4
      	5

   
   
   

   3. 	5
      	6

   
   
   

   4. 	6
      	7

      
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan θ =	1	; cotθ = √11
   	√11

   
   

   ∴	cosec²θ - sec²θ
   	cosec²θ + sec²θ

   
   

   =	1 + cot²θ - (1 - tan²θ)
   	1 + cot²θ + 1 + tan²θ

   
   

   =	cot²θ - tan²θ
   	cot²θ + tan²θ + 2

   
   
   

   =		120	=	5
   		144		6

   
   

   Correct Option: C

   tan θ =	1	; cotθ = √11
   	√11

   
   

   ∴	cosec²θ - sec²θ
   	cosec²θ + sec²θ

   
   

   =	1 + cot²θ - (1 - tan²θ)
   	1 + cot²θ + 1 + tan²θ

   
   

   =	cot²θ - tan²θ
   	cot²θ + tan²θ + 2

   
   
   

   =		120	=	5
   		144		6

   
   

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268. The value of
     

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. 2

   
   
   

   4. 	3
      	2

      
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   Expression =		1	sin	π	. cos
   		√2		6

   
   
   

   =

   1

   ×

   1

   ×

   1

   -

   1

   ×

   2

   +

   5 × 1

   √2

   2

   √2

   √3

   √3

   12 × 1

   
   

   =		1	-	2	+	5
   		4		3		12

   
   

   =	3 - 8 + 5	= 0
   	12

   
   

   Correct Option: A

   Expression =		1	sin	π	. cos
   		√2		6

   
   
   

   =

   1

   ×

   1

   ×

   1

   -

   1

   ×

   2

   +

   5 × 1

   √2

   2

   √2

   √3

   √3

   12 × 1

   
   

   =		1	-	2	+	5
   		4		3		12

   
   

   =	3 - 8 + 5	= 0
   	12

   
   

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269. If sin θ =		3	, then the value of	tan θ + cos θ	is equal to
     		5		cotθ + cosecθ

1. 1. 	29
      	60

   
   
   

   2. 	31
      	60

   
   
   

   3. 	34
      	60

   
   
   

   4. 	37
      	60

   
   
   

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1. View Hint View Answer Discuss in Forum

   sin θ =	3
   	5

   
   ∴ cosθ = √1 - sin²θ
   
   

   cot θ =		1	=	4
   		tan θ		3

   
   

   cosec θ =		1	=	5
   		sin θ		3

   
   
   

   =		31	×	3	=	31
   		20		9		60

   
   

   Correct Option: B

   sin θ =	3
   	5

   
   ∴ cosθ = √1 - sin²θ
   
   

   cot θ =		1	=	4
   		tan θ		3

   
   

   cosec θ =		1	=	5
   		sin θ		3

   
   
   

   =		31	×	3	=	31
   		20		9		60

   
   

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270. If a cos θ + b sin θ = p and a sin θ – b cos θ = q, then the relation between a, b, p and q is

1. 1. a² – b² = p² – q²

   
   
   

   2. a² + b² = p² + q²

   
   
   

   3. a + b = p + q

   
   
   

   4. a – b = p – q
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   a cos θ + b sin θ = p a sin θ – b cos θ = θ
   On squaring and adding, a² cos² θ + b² sin² θ + 2 a b sin θ . cos θ + a² sin² θ + b² cos² θ – 2 a b sin θ. cos θ
   = p² + q²
   ⇒ a² cos² θ + a² sin² θ + b² sin² θ + b² cos² θ = p² + q²
   ⇒ a² (cos²θ + sin²θ) + b² (sin²θ + cos²θ ) = p² + q²
   ⇒ a² + b² = p² + q²
   

   Correct Option: B

   a cos θ + b sin θ = p a sin θ – b cos θ = θ
   On squaring and adding, a² cos² θ + b² sin² θ + 2 a b sin θ . cos θ + a² sin² θ + b² cos² θ – 2 a b sin θ. cos θ
   = p² + q²
   ⇒ a² cos² θ + a² sin² θ + b² sin² θ + b² cos² θ = p² + q²
   ⇒ a² (cos²θ + sin²θ) + b² (sin²θ + cos²θ ) = p² + q²
   ⇒ a² + b² = p² + q²
   

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