Miscellaneous
Misc 2 Important
Misc 3 Important
Misc 4 Important
Misc 5 Important
Question 1 Important Deleted for CBSE Board 2025 Exams
Question 2 Deleted for CBSE Board 2025 Exams
Question 3 Important Deleted for CBSE Board 2025 Exams
Question 4 Deleted for CBSE Board 2025 Exams
Question 5 Important Deleted for CBSE Board 2025 Exams
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Question 7 Important Deleted for CBSE Board 2025 Exams
Question 8 Important Deleted for CBSE Board 2025 Exams
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Question 11 Important Deleted for CBSE Board 2025 Exams
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Question 13 Important Deleted for CBSE Board 2025 Exams
Question 14 Important Deleted for CBSE Board 2025 Exams
Question 15 Deleted for CBSE Board 2025 Exams
Question 16 Important Deleted for CBSE Board 2025 Exams
Question 17 (MCQ) Important Deleted for CBSE Board 2025 Exams
Question 18 (MCQ) Important Deleted for CBSE Board 2025 Exams
Miscellaneous
Last updated at April 16, 2024 by Teachoo
Misc 1 Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b. Angle between the lines with direction ratios a1, b1, c1 and a2, b2, c2 is given by cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 + 𝒄_𝟏 𝒄_𝟐)/(√(𝒂_𝟏^𝟐 + 𝒃_𝟏^𝟐 + 𝒄_𝟏^𝟐 ) √(𝒂_𝟏^𝟐 + 𝒃_𝟏^𝟐 + 𝒄_𝟏^𝟐 ))| Given, 𝑎1 = 𝑎, 𝑏1 = 𝑏, c1 = c and 𝑎2 = 𝑏 − 𝑐, 𝑏2 = 𝑐 − 𝑎, c2 = a – b So, cos θ = |(𝑎(𝑏 − 𝑐) + 𝑏(𝑐 − 𝑎) + 𝑐(𝑎 − 𝑏))/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √((𝑏 − 𝑐)2 + (𝐶 − 𝑎)2 + (𝑎 − 𝑏)2))| = |(𝒂𝒃 − 𝒂𝒄 + 𝒃𝒄 − 𝒂𝒃 + 𝒄𝒂 − 𝒃𝒄)/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √(𝑏^2 + 𝑐2 − 2𝑏𝑐 + 𝑐^2 + 𝑎^2 − 2𝑐𝑎 + 𝑎^2 + 𝑏^2 − 2𝑎𝑏 ))| = |𝟎/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √(𝑏^2 + 𝑐2 − 2𝑏𝑐 + 𝑐^2 + 𝑎^2 − 2𝑐𝑎 + 𝑎^2 + 𝑏^2 − 2𝑎𝑏 ))| = 0 Since cos θ = 0 So, θ = 90° Therefore, angle between the given pair of lines is 90°