Question 10 (MCQ) - Miscellaneous - Chapter 4 Class 12 Determinants
Last updated at August 8, 2026 by Teachoo
Miscellaneous
Misc 1
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Chapter 4 Class 12 Determinants
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Misc 17 (Method 1) Choose the correct answer. If a, b, c, are in A.P., then the determinant |ā 8(š„+2&š„+3&š„+2š@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| is A. 0 B. 1 C. x D. 2x Since a, b & c are in A.P Then, b ā a = c ā b b ā a ā c + b = 0 2b ā a ā c = 0 (Common difference is equal) ā¦(1) Solving |ā 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 |ā 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 |ā 8(š„+2&š„+3&š„+2š@š(š„+3)&š(š„+4)&š(š„+2š)@š„+4&š„+5&š„+2š)| = 1/2 |ā 8(š„+2&š„+3&š„+2š@2š„+6&2š„+8&2š„+4š@š„+4&š„+5&š„+2š)| Applying R2 ā R2 ā R1 ā R3 = 1/2 |ā 8(š„+2&š„+3&š„+2š@2š„+6ā(š„+2)ā(š„+4 )&2š„+8ā(š„+3)ā(š„+5)&2š„+4šā(š„+2š)ā(š„+2š)@š„+4&š„+5&š„+2š)| = 1/2 |ā 8(š„+2&š„+3&š„+2š@2š„+6āš„ā2āš„ā4&2š„+8āš„ā3āš„ā5&2š„+4šāš„ā2šāš„ā2š@š„+4&š„+5&š„+2š)| = 1/2 |ā 8(š„+2&š„+3&š„+2š@0&0&4šā2šā2š@š„+4&š„+5&š„+2š)| = 1/2 |ā 8(š„+2&š„+3&š„+2š@0&0&2(ššāšāš)@š„+4&š„+5&š„+2š)| = 1/2 |ā 8(š„+2&š„+3&š„+2š@0&0&2(š)@š„+4&š„+5&š„+2š)| (From (1): 2b ā b ā c = 0) = 1/2 |ā 8(š„+2&š„+3&š„+2š@0&0&0@š„+4&š„+5&š„+2š)| If any row or column of determinant are zero, then value of determinant is also zero. = 1/2 Ć 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 |ā 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 ā b = c ā b b + b = c + a 2b = a + c (Common difference is equal) ā¦(1) Consider |ā 8(š„+2&š„+3&š„+2š@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| Applying R1 āR1 + R3 ā 2R2 = |ā 8((š„+2)+(š„+4)ā2(š„+3)&(š„+3)+(š„+5)ā2(š„+4)&(š„+2š)+(š„+2š)ā2(š„+2š)@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| = |ā 8(š„+2+š„+4ā2š„ā6&š„+3+š„+5ā2š„ā8&š„+2š+š„+2šā2š„ā4š@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| = |ā 8(2š„ā2š„+6ā6&2š„ā2š„+8ā8&2š„ā2š„+2š+2šā4š@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| = |ā 8(0&0&0+2(š+šā2š)@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| = |ā 8(0&0&2(ššā2š)@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| = |ā 8(0&0&0@š„+3&š„+4&š„+2š@š„+4&š„+5&š„+2š)| 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)
