276. If 3sinθ + 5cosθ = 5, then 5sinθ – 3cosθis equal to

1. 1. ± 3

   
   
   

   2. ± 5

   
   
   

   3. 1

   
   
   

   4. ± 2

   
   
   

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1. View Hint View Answer Discuss in Forum

   3 sinθ + 5 cosθ = 5 ...(i)
   5 sinθ – 3 cosθ = x ...(ii)
   On squaring and adding, 9 sin²θ + 25 cos²θ + 25 sin²θ + 9 cos²θ = 25 + x²
   ⇒ 9 (sin²θ + cos²θ ) + 25 (cos²θ + sin²θ ) = 25 + x²
   ⇒ 9 + 25 = 25 + x²
   ⇒ x² = 9
   ⇒ x = ±3

   Correct Option: A

   3 sinθ + 5 cosθ = 5 ...(i)
   5 sinθ – 3 cosθ = x ...(ii)
   On squaring and adding, 9 sin²θ + 25 cos²θ + 25 sin²θ + 9 cos²θ = 25 + x²
   ⇒ 9 (sin²θ + cos²θ ) + 25 (cos²θ + sin²θ ) = 25 + x²
   ⇒ 9 + 25 = 25 + x²
   ⇒ x² = 9
   ⇒ x = ±3

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277. If sinθ + sin²θ = 1, then the value of cos²θ + cos⁴ θ is

1. 1. 2

   
   
   

   2. 4

   
   
   

   3. 0

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   sinθ + sin²θ = 1
   ⇒ sinθ = 1 – sin²θ = cos²θ
   ∴ cos²θ + cos⁴θ
   = cos²θ + (cos²θ) 2
   = cos²θ + sin²θ = 1
   

   Correct Option: D

   sinθ + sin²θ = 1
   ⇒ sinθ = 1 – sin²θ = cos²θ
   ∴ cos²θ + cos⁴θ
   = cos²θ + (cos²θ) 2
   = cos²θ + sin²θ = 1
   

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278. If tan θ + cot θ = 2 then the value of θ is

1. 1. 45°

   
   
   

   2. 60°

   
   
   

   3. 90°

   
   
   

   4. 30°

   
   
   

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1. View Hint View Answer Discuss in Forum

   tan θ + cot θ = 2
   

   ⇒ tan θ +	1	= 2
   	tan θ

   
   

   ⇒	tan² θ + 1	- 2
   	tan θ

   
   ⇒ tan²θ + 1 = 2tanθ
   ⇒ tan²θ – 2 tan θ + 1 = 0
   ⇒ (tanθ – 1)² = 0
   ⇒ tan θ – 1 = 0
   ⇒ tan θ = 1 = tan 45°
   ⇒ θ = 45

   Correct Option: A

   tan θ + cot θ = 2
   

   ⇒ tan θ +	1	= 2
   	tan θ

   
   

   ⇒	tan² θ + 1	- 2
   	tan θ

   
   ⇒ tan²θ + 1 = 2tanθ
   ⇒ tan²θ – 2 tan θ + 1 = 0
   ⇒ (tanθ – 1)² = 0
   ⇒ tan θ – 1 = 0
   ⇒ tan θ = 1 = tan 45°
   ⇒ θ = 45

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279. If cos πx = x² – x + (5 / 4) , the value of x will be

1. 1. 0

   
   
   

   2. 1

   
   
   

   3. - 1

   
   
   

   4. None of the above

   
   
   

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1. View Hint View Answer Discuss in Forum

   cosπx = x² - x +	5
   	4

   
   

   = x² - 2.x.	1	+	1	-	1	+	5
   	2		4		4		4

   
   

   =		x -	1		²	+ 1 > 1
   			2

   
   - 1 ≤ cosx ≤ 1
   

   Correct Option: D

   cosπx = x² - x +	5
   	4

   
   

   = x² - 2.x.	1	+	1	-	1	+	5
   	2		4		4		4

   
   

   =		x -	1		²	+ 1 > 1
   			2

   
   - 1 ≤ cosx ≤ 1
   

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280. The numerical value of 1 +		1	- sec² 27° +	1	- cosec² 27° is
     		cot² 63°		sin² 63°

1. 1. 1

   
   
   

   2. 2

   
   
   

   3. - 1

   
   
   

   4. 0

   
   
   

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1. View Hint View Answer Discuss in Forum

   1 +	1	- sec²27° +	1	- cosec²27° = 1 + tan²63° - sec²27° + cosec²63° - cosec²27°
   	cot²63°		sin²63°

   
   = 1 + tan²(90° – 27°) – sec²27° + cosec2 (90° – 27°) – cosec²27°
   = 1 + cot²27° – sec²27° + sec²27° – cose²27°
   = 1 + cot²27° – cosec²27°
   = 1 – 1 = 0
   [∵ cosec²θ - cot²θ = 1]

   Correct Option: D

   1 +	1	- sec²27° +	1	- cosec²27° = 1 + tan²63° - sec²27° + cosec²63° - cosec²27°
   	cot²63°		sin²63°

   
   = 1 + tan²(90° – 27°) – sec²27° + cosec2 (90° – 27°) – cosec²27°
   = 1 + cot²27° – sec²27° + sec²27° – cose²27°
   = 1 + cot²27° – cosec²27°
   = 1 – 1 = 0
   [∵ cosec²θ - cot²θ = 1]

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