Example 34 - Chapter 4 Class 12 Determinants - Prove that |a + bx

Example 34 - Chapter 4 Class 12 Determinants - Part 2
Example 34 - Chapter 4 Class 12 Determinants - Part 3

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Transcript

Question 15 Prove that Δ = |■8(a+bx&c+dx&p+qx@ax+b&cx+d&px+q@u&v&w)| = (1 – x2) |■8(a&c&p@b&d&q@u&v&w)| Solving L.H.S Δ = |■8(a+bx&c+dx&p+qx@ax+b&cx+d&px+q@u&v&w)| Applying R1 → R1 − xR2 = |■8(𝑎+𝑏𝑥−𝑥 (𝑎𝑥+𝑏)&𝑐+𝑑𝑥−𝑥&𝑝+𝑞𝑥−𝑥 (𝑝𝑥+𝑞)@𝑎𝑥+𝑏&𝑐𝑥+𝑑&𝑝𝑥+𝑞@𝑢&𝑣&𝑤)| = |■8(𝑎+𝑏𝑥−𝑎𝑥2 −𝑏𝑥&𝑐+𝑑𝑥−𝑐𝑥2−𝑑𝑥&𝑝+𝑞𝑥−𝑝𝑥2−𝑝𝑥@𝑎𝑥+𝑏&𝑐𝑥+𝑑&𝑝𝑥+𝑞@𝑢&𝑣&𝑤)| = |■8(a−𝑎𝑥2 &c−cx2&p−px2@ax+b&cx+d&px+q@u&v&w)| = |■8(a (𝟏−𝒙𝟐) &c(𝟏−𝐱𝟐)&p(𝟏−𝐱𝟐)@ax+b&cx+d&px+q@u&v&w)| Taking (1 – x2) common from R1 = (1 – x2) |■8(a&c&p@ax+b&cx+d&px+q@u&v&w)| Applying R2 → R2 – xR1 = |■8(𝑎+𝑏𝑥−𝑎𝑥2 −𝑏𝑥&𝑐+𝑑𝑥−𝑐𝑥2−𝑑𝑥&𝑝+𝑞𝑥−𝑝𝑥2−𝑝𝑥@𝑎𝑥+𝑏&𝑐𝑥+𝑑&𝑝𝑥+𝑞@𝑢&𝑣&𝑤)| = |■8(a−𝑎𝑥2 &c−cx2&p−px2@ax+b&cx+d&px+q@u&v&w)| = |■8(a (𝟏−𝒙𝟐) &c(𝟏−𝐱𝟐)&p(𝟏−𝐱𝟐)@ax+b&cx+d&px+q@u&v&w)| Taking (1 – x2) common from R1 = (1 – x2) |■8(a&c&p@ax+b&cx+d&px+q@u&v&w)| Applying R2 → R2 – xR1 = (1 – x2) |■8(𝑎&𝑐&𝑝@𝑎𝑥+𝑏−𝑥𝑎&𝑐𝑥+𝑑−𝑐𝑥&𝑝𝑥+𝑞−𝑝𝑥@𝑢&𝑣&𝑤)| = (1 – x2) |■8(𝑎&𝑐&𝑝@𝑏&𝑑&𝑞@𝑢&𝑣&𝑤)| = R.H.S Hence Proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.