যদি a cot θ + b cosec θ = p and b cot θ + a cosec θ = q হয়, তাহলে p² - q² এর মান ______হবে।

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This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 17 Jul 2023 Shift 3)

Answer 1

প্রদত্ত:

a cot θ + b cosec θ = p

b cot θ + a cosec θ = q

অনুসৃত সূত্র:

Cosec² θ - cot² θ = 1

গণনা:

a cot θ + b cosec θ = p

উভয় পক্ষকে বর্গ করে,

(a cot θ + b cosec θ)² = (p)²

a² cot² θ + b2 cosec2 θ + 2 × ab cot θ × cosec θ = p² ----- (1)

b cot θ + a cosec θ = q

উভয় পক্ষকে বর্গ করে,

(b cot θ + a cosec θ)2 = (q)2

b2 cot2 θ + a2 cosec2 θ + 2 × ab cot θ × cosec θ = q2 ----- (2)

সমীকরণ (1) এবং (2) বিয়োগ করে

⇒ (p² - q²) = a2 cot2 θ + b2 cosec2 θ + 2 × ab × cot θ × cosec θ - (b2 cot2 θ + a2 cosec2 θ + 2 × ab × cot θ × cosec θ)

⇒ a2 cot2 θ - a2 cosec2 θ + b2 cosec2 θ - b2 cot2 θ

⇒ a2 (cot2 θ - cosec2 θ) + b2 (cosec2 θ - cot2 θ)

⇒ b2 - a2

∴ সঠিক বিকল্প হল 3

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