CBSE Class 12 Sample Paper for 2025 Boards
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CBSE Class 12 Sample Paper for 2025 Boards
Last updated at Oct. 3, 2024 by Teachoo
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Question 25 The two co-initial adjacent sides of a parallelogram are 2ı ˆ−4ȷ ˆ−5𝑘 ˆ and 2ı ˆ+2ȷ ˆ+3𝑘 ˆ. Find its diagonals and use them to find the area of the parallelogram.Let ABCD be the parallelogram Where 𝑎 ⃗ = 2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ 𝑏 ⃗ = 2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ We need to find diagonals (𝑑_1 ) ⃗ and (𝑑_2 ) ⃗ Finding diagonal (𝒅_𝟏 ) ⃗ Now, (𝒅_𝟏 ) ⃗=𝒂 ⃗ + 𝒃 ⃗ = (2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ) + (2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ) = 𝟒𝚤 ˆ−𝟐𝚥 ˆ−𝟐𝒌 ˆ Finding diagonal (𝒅_𝟐 ) ⃗ Now, (𝒅_𝟐 ) ⃗=𝒂 ⃗ − 𝒃 ⃗ = (2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ) − (2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ) = 2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ − 2𝚤 ˆ−2𝚥 ˆ−3𝑘 ˆ = −𝟔𝚥 ˆ−𝟖𝒌 ˆ Now, We need to find area of parallelogram using diagonals Area of parallelogram = 𝟏/𝟐 |(𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ | Finding (𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ (𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@4&−2&−2@0&−6&−8)| = 𝑖 ̂ ((−2) × (−8) − (−6) × (−2)) − 𝑗 ̂ (4 × (−8) − 0 × (−2)) + 𝑘 ̂ (4 × (−6) − 0 × (−2)) = 𝑖 ̂ (16 − 12) − 𝑗 ̂ (−32 − 0) + 𝑘 ̂ (−24 − 0) = 4𝑖 ̂ + 32𝑗 ̂ − 24𝑘 ̂ = 4(𝒊 ̂ + 8𝒋 ̂ − 6𝒌 ̂) Now, Area of Parallelogram ABCD = 𝟏/𝟐 |(𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ | = 1/2 × 4√(12+82+(−6)2) = 2√(1+64+36) = 𝟐√𝟏𝟎𝟏 square units
