Check sibling questions

Example 31 - If a, b, c are in AP, find value of |2y + 4 5y + 7 8y+a|

Example 31 - Chapter 4 Class 12 Determinants - Part 2
Example 31 - Chapter 4 Class 12 Determinants - Part 3
Example 31 - Chapter 4 Class 12 Determinants - Part 4
Example 31 - Chapter 4 Class 12 Determinants - Part 5
Example 31 - Chapter 4 Class 12 Determinants - Part 6

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Transcript

Example 31 (Method 1) If a, b, c, are in A.P, find value of |■8(2y+4&5y+7&8y+a@3y+5&6y+8&9y+b@4y+6&7y+9&10y+c)| Given a, b & c are in A.P Then, b – a = c – b b – a – c + b = 0 2b – a – c = 0 Solving (Common difference is equal) |■8(2y+4&5y+7&8y+a@3y+5&6y+8&9y+b@4y+6&7y+9&10y+c)| Multiply & Divide by 2 = 2/2 |■8(2y+4&5y+7&8y+a@3y+5&6y+8&9y+b@4y+6&7y+9&10y+c)| Multiplying 2 to R2 = 1/2 |■8(2y+4&5y+7&8y+a@𝟐(3y+5)&𝟐(6y+8)&𝟐(9y+b)@4y+6&7y+9&10y+c)| = 1/2 |■8(2y+4&5y+7&8y+a@6y+10&12y+16&18y+2b@4y+6&7y+9&10y+c)| Applying R2 →R2 – R1 – R3 = 1/2 |■8(2y+4&5y+7&8y+a@6y+10−(2𝑦+4)−(4𝑦+6)&12y+16−(5𝑦+7)−(7𝑦+9)&18y+2b−(8y+a)−(10y+c)@4y+6&7y+9&10y+c)| = 1/2 |■8(2y+4&5y+7&8y+a@6y+10−2𝑦−4−4𝑦−6&12y+16−5𝑦−7−7𝑦−9&18y+2b−2𝑦−𝑎−10𝑦−𝑐@4y+6&7y+9&10y+c)| = 1/2 |■8(2𝑦+4&5𝑦+7&8𝑦+𝑎@0&0&𝟐𝒃−𝒂−𝒄@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = 1/2 |■8(2𝑦+4&5𝑦+7&8𝑦+𝑎@0&0&𝟎@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = 1/2 × 0 = 0 Thus, the value of determinant is 0 (From (1): 2b – a – c = 0) If any row or column of determinant are zero, then value of determinant is also zero. Example 31 (Method 2) If a, b, c, are in A.P, find value of |■8(2𝑦+4&5𝑦+7&8𝑦+𝑎@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| Given a, b & C are in A.P Then b – a = c – b b + b = a + c 2b = a + c (Common difference is equal) Consider |■8(2𝑦+4&5𝑦+7&8𝑦+𝑎@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| Applying R1 → R1 + R3 – 2R2 = |■8(2𝑦+4+(4𝑦+6)−2(3𝑦+5)&5𝑦+7+(7𝑦+9)−2(6𝑦+8)&8𝑦+𝑎+(10𝑦+𝑐)−2(9𝑦+𝑏)@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = |■8(2𝑦+4+4𝑦+6−6𝑦−10&5𝑦+7+7𝑦+9−12𝑦−16&8𝑦+𝑎+10𝑦+𝑐−18𝑦−2𝑏@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = |■8(6𝑦−6𝑦+10−10&12𝑦−12𝑦+16−16&18𝑦−18𝑦+𝑎+𝑐−2𝑏@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = |■8(0&0&𝒂+𝒄−2𝑏@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = |■8(0&0&𝟐𝒃−2𝑏@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| = |■8(0&0&0@3𝑦+5&6𝑦+8&9𝑦+𝑏@4𝑦+6&7𝑦+9&10𝑦+𝑐)| If any row or column of determinant are zero, then value of determinant is also zero. = 0 Hence, value of determinant is 0 (From (1): 2b = a + c)

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.