Question 13 - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 13 A bird flies through a distance in a straight line given by the vector ı ˆ+2ȷ ˆ+𝑘 ˆ.A man standing beside a straight metro rail track given by 𝑟 ⃗=(3+𝜆)ı ˆ+(2𝜆− 1) ȷ ˆ+3𝜆𝑘 ˆ is observing the bird. The projected length of its flight on the metro track is (A) 6/√1 units (B) 14/√6 units (C) 8/√14 units (D) 5/√6 unitsGiven Bird vector = 𝚤 ˆ+𝟐𝚥 ˆ+𝒌 ˆ And, Man standing on line 𝑟 ⃗=(3+𝜆)ı ˆ+(2𝜆− 1) ȷ ˆ+3𝜆𝑘 ˆ 𝒓 ⃗=(𝟑𝚤 ˆ−𝚥 ˆ) +𝝀(𝚤 ˆ+𝟐𝚥 ˆ+𝟑𝒌 ˆ) We need to find Projection of bird vector on line Thus, we need to find projection of Bird vector (𝚤 ˆ+𝟐𝚥 ˆ+𝒌 ˆ ) on parallel vector of line (𝚤 ˆ+𝟐𝚥 ˆ+𝟑𝒌 ˆ) Now, Projection of 𝒂 ⃗ along 𝒃 ⃗ = (𝒂 ⃗ . 𝒃 ⃗)/|𝒃 ⃗ | Here, 𝒂 ⃗ = 𝚤 ˆ+𝟐𝚥 ˆ+𝒌 ˆ 𝒃 ⃗ = 𝚤 ˆ+𝟐𝚥 ˆ+𝟑𝒌 ˆ Projection = |𝒂 ⃗ | 𝒄𝒐𝒔 θ =|𝒂 ⃗ | 𝒄𝒐𝒔 θ ×|𝒃 ⃗ |/|𝒃 ⃗ | =(𝒂 ⃗ . 𝒃 ⃗)/|𝒃 ⃗ | Now, Projection of 𝑎 ⃗ along 𝑏 ⃗ = (𝑎 ⃗ . 𝑏 ⃗)/|𝑏 ⃗ | = ((𝚤 ˆ + 2𝚥 ˆ + 𝑘 ˆ ).(𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ ))/|𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ | = ((𝚤 ˆ + 2𝚥 ˆ + 𝑘 ˆ ).(𝚤 ˆ + 2𝚥 ˆ + 3𝑘 ˆ ))/√(1^2 + 2^2 + 3^2 ) = (𝟏 × 𝟏 + 𝟐 × 𝟐 + 𝟏 × 𝟑)/√(𝟏^𝟐 + 𝟐^𝟐 + 𝟑^𝟐 ) = (1 + 4 + 3)/√(1 + 4 + 9) = 𝟖/√𝟏𝟒 units So, the correct answer is (C)