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


Transcript

Example 31 (Method 1) If a, b, c, are in A.P, find value of |■8(2y+4&5y+7&[email protected]+5&6y+8&[email protected]+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&[email protected]+5&6y+8&[email protected]+6&7y+9&10y+c)| Multiply & Divide by 2 = 2/2 |■8(2y+4&5y+7&[email protected]+5&6y+8&[email protected]+6&7y+9&10y+c)| Multiplying 2 to R2 = 1/2 |■8(2y+4&5y+7&[email protected]𝟐(3y+5)&𝟐(6y+8)&𝟐(9y+b)@4y+6&7y+9&10y+c)| = 1/2 |■8(2y+4&5y+7&[email protected]+10&12y+16&[email protected]+6&7y+9&10y+c)| Applying R2 →R2 – R1 – R3 = 1/2 |■8(2y+4&5y+7&[email protected]+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&[email protected]+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&[email protected]𝑦+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.