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Last updated at Jan. 23, 2020 by Teachoo

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Misc 17 (Method 1) Choose the correct answer. If a, b, c, are in A.P., then the determinant |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| is A. 0 B. 1 C. x D. 2x Since a, b & c are in A.P Then, b โ a = c โ b b โ a โ c + b = 0 2b โ a โ c = 0 (Common difference is equal) โฆ(1) Solving |โ 8(x+2&x+3&x+2a@x+3&x+4&x+2b@x+4&x+5&x+2c)| Multiplying and dividing 2 = 2/2 |โ 8(x+2&x+3&x+2a@x+3&x+4&x+2b@x+4&x+5&x+2c)| Multiplying R2 by 2 = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@๐(๐ฅ+3)&๐(๐ฅ+4)&๐(๐ฅ+2๐)@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@2๐ฅ+6&2๐ฅ+8&2๐ฅ+4๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| Applying R2 โ R2 โ R1 โ R3 = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@2๐ฅ+6โ(๐ฅ+2)โ(๐ฅ+4 )&2๐ฅ+8โ(๐ฅ+3)โ(๐ฅ+5)&2๐ฅ+4๐โ(๐ฅ+2๐)โ(๐ฅ+2๐)@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@2๐ฅ+6โ๐ฅโ2โ๐ฅโ4&2๐ฅ+8โ๐ฅโ3โ๐ฅโ5&2๐ฅ+4๐โ๐ฅโ2๐โ๐ฅโ2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@0&0&4๐โ2๐โ2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@0&0&2(๐๐โ๐โ๐)@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@0&0&2(๐)@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| (From (1): 2b โ b โ c = 0) = 1/2 |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@0&0&0@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| If any row or column of determinant are zero, then value of determinant is also zero. = 1/2 ร 0 = 0 Thus, the value of determinant is 0 Correct Answer is A Misc 17 (Method 2) Choose the correct answer. If a, b, c, are in A.P., then the determinant |โ 8(x+2&x+3&x+2a@x+3&x+4&x+2b@x+4&x+5&x+2c)| is A. 0 B. 1 C. x D. 2x Since a, b & c are in A.P Then a โ b = c โ b b + b = c + a 2b = a + c (Common difference is equal) โฆ(1) Consider |โ 8(๐ฅ+2&๐ฅ+3&๐ฅ+2๐@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| Applying R1 โR1 + R3 โ 2R2 = |โ 8((๐ฅ+2)+(๐ฅ+4)โ2(๐ฅ+3)&(๐ฅ+3)+(๐ฅ+5)โ2(๐ฅ+4)&(๐ฅ+2๐)+(๐ฅ+2๐)โ2(๐ฅ+2๐)@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = |โ 8(๐ฅ+2+๐ฅ+4โ2๐ฅโ6&๐ฅ+3+๐ฅ+5โ2๐ฅโ8&๐ฅ+2๐+๐ฅ+2๐โ2๐ฅโ4๐@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = |โ 8(2๐ฅโ2๐ฅ+6โ6&2๐ฅโ2๐ฅ+8โ8&2๐ฅโ2๐ฅ+2๐+2๐โ4๐@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = |โ 8(0&0&0+2(๐+๐โ2๐)@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = |โ 8(0&0&2(๐๐โ2๐)@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| = |โ 8(0&0&0@๐ฅ+3&๐ฅ+4&๐ฅ+2๐@๐ฅ+4&๐ฅ+5&๐ฅ+2๐)| If any row or column of determinant are zero, then value of determinant is also zero. = 0 Hence, value of determinant is 0 Correct Answer is A (From (1): 2b = a + c)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.