If 2 cot θ = 3, then find the value of \(\frac{\sqrt{13} \sin \theta-3 \tan \theta}{3 \tan \theta+\sqrt{13} \cos \theta}\).

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SSC CGL 2023 Tier-I Official Paper (Held On: 24 Jul 2023 Shift 1)
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  1. \(\frac{1}{\sqrt{13}}\)
  2. \(\frac{2}{\sqrt{13}}\)
  3. 0
  4. \(\frac{2}{3}\)

Answer (Detailed Solution Below)

Option 3 : 0
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Detailed Solution

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Given: 

2cotθ = 3 

Formula used: 

Sinθ = P/H 

Cosθ = B/H 

Tanθ = P/B 

Cotθ = B/P 

H2 = P2 + B2

where P = Perpendicular 

B = Base 

H = Height 

Calculation: 

2cotθ = 3 

⇒ cotθ = 3/2 

on comparing with (Cotθ = B/P), we get 

⇒ B = 3

⇒ P = 2 

According to Pythagorean theorem

⇒ H2 = P2 + B2

⇒ H2 = 22 + 32

⇒ H = √13 

⇒ Sinθ = 2/√13  

⇒ Cosθ = 3/√13  

⇒ Tanθ = 2/3 

Putting all the values in the given equation:

\(\frac{\sqrt{13} \sin \theta-3 \tan \theta}{3 \tan \theta+\sqrt{13} \cos \theta}\)= (√13 * 2/√13  - 3 *  2/3 )/ (3 *  2/3 + √13 * 3/√13)

⇒ (2 - 2)/(2 + 3)

⇒ 0

The value of the expression is 0.

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