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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)

Miscellaneous

Misc 1

Misc. 2 Important

Misc 3

Misc 4

Misc 5

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11 Important

Misc 12 Important

Misc. 13

Misc 14

Misc. 15 Important

Misc. 16 Important

Misc 17 Important You are here

Misc 18

Misc 19 Important

Matrices and Determinants - Formula Sheet and Summary

Chapter 4 Class 12 Determinants

Serial order wise

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.