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Example 31  - If a, b, c, are in AP, find value of |2y+4 5y+7 - Whole row/column zero

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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise
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Example 31 (Method 1) If a, b, c, are in A.P, find value of 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 2y+4﷮5y+7﷮8y+a﷮3y+5﷮6y+8﷮9y+b﷮4y+6﷮7y+9﷮10y+c﷯﷯ Multiply & Divide by 2 = 2﷮2﷯ 2y+4﷮5y+7﷮8y+a﷮3y+5﷮6y+8﷮9y+b﷮4y+6﷮7y+9﷮10y+c﷯﷯ Multiplying 2 to R2 = 1﷮2﷯ 2y+4﷮5y+7﷮8y+a﷮𝟐(3y+5)﷮𝟐(6y+8)﷮𝟐(9y+b)﷮4y+6﷮7y+9﷮10y+c﷯﷯ = 1﷮2﷯ 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﷯ 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﷯ 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﷯ 2𝑦+4﷮5𝑦+7﷮8𝑦+𝑎﷮0﷮0﷮𝟐𝒃−𝒂−𝒄﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ = 1﷮2﷯ 2𝑦+4﷮5𝑦+7﷮8𝑦+𝑎﷮0﷮0﷮𝟎﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ If any row as column of determinant are zero, then value of determinant is also zero. = 1﷮2﷯ × 0 = 0 Thus, the value of determinant is 0 Example 31(Method 2) If a, b, c, are in A.P, find value of 2𝑦+4﷮5𝑦+7﷮8𝑦+𝑎﷮3𝑦+5﷮6𝑦+8﷮9𝑦+𝑏﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ We need to find value of 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 Consider 2𝑦+4﷮5𝑦+7﷮8𝑦+𝑎﷮3𝑦+5﷮6𝑦+8﷮9𝑦+𝑏﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ Applying R1 →R1 + R3 – 2R2 = 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﷯﷯ = 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﷯﷯ = 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﷯﷯ = 0﷮0﷮𝒂+𝒄−2𝑏﷮3𝑦+5﷮6𝑦+8﷮9𝑦+𝑏﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ = 0﷮0﷮𝟐𝒃−2𝑏﷮3𝑦+5﷮6𝑦+8﷮9𝑦+𝑏﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ = 0﷮0﷮0﷮3𝑦+5﷮6𝑦+8﷮9𝑦+𝑏﷮4𝑦+6﷮7𝑦+9﷮10𝑦+𝑐﷯﷯ If any row as column of determinant are zero, then value of determinant is also zero. = 0 Hence, value of determinant is 0

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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