# Misc 17

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 17 Choose the correct answer. If a, b, c, are in A.P., then the determinant 𝑥+2𝑥+3𝑥+2𝑎𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 is A. 0 B. 1 C. x D. 2x Since a, b & c are in A.P Then b – a = c – b b – a – c + b = 0 2b – a – c = 0 Solving x+2x+3x+2ax+3x+4x+2bx+4x+5x+2c Multiplying and dividing 2 = 22 x+2x+3x+2ax+3x+4x+2bx+4x+5x+2c Multiplying R2 by 2 = 12 𝑥+2𝑥+3𝑥+2𝑎𝟐(𝑥+3)𝟐(𝑥+4)𝟐(𝑥+2𝑏)𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎2𝑥+62𝑥+82𝑥+4𝑏𝑥+4𝑥+5𝑥+2𝑐 Applying R2 → R2 – R1 – R3 = 12 𝑥+2𝑥+3𝑥+2𝑎2𝑥+6− 𝑥+2−(𝑥+4 )2𝑥+8− 𝑥+3−(𝑥+5)2𝑥+4𝑏− 𝑥+2𝑎−(𝑥+2𝑐)𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎2𝑥+6−𝑥−2−𝑥−42𝑥+8−𝑥−3−𝑥−52𝑥+4𝑏−𝑥−2𝑎−𝑥−2𝑐𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎004𝑏−2𝑎−2𝑐𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎002(𝟐𝒃−𝒂−𝒄)𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎002(𝟎)𝑥+4𝑥+5𝑥+2𝑐 = 12 𝑥+2𝑥+3𝑥+2𝑎000𝑥+4𝑥+5𝑥+2𝑐 If any row as column of determinant are zero, then value of determinant is also zero. = 12 × 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 x+2x+3x+2ax+3x+4x+2bx+4x+5x+2c is A. 0 B. 1 C. x D. 2x We need to find value of x+2x+3x+2ax+3x+4x+2bx+4x+5x+2c Since a, b & c are in A.P Then a – b = c – b b + b = c + a 2b = a + c Consider 𝑥+2𝑥+3𝑥+2𝑎𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 Applying R1 →R1 + R3 – 2R2 = 𝑥+2+ 𝑥+4−2(𝑥+3) 𝑥+3+ 𝑥+5−2(𝑥+4) 𝑥+2𝑎+ 𝑥+2𝑐−2(𝑥+2𝑏)𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 = 𝑥+2+𝑥+4−2𝑥−6𝑥+3+𝑥+5−2𝑥−8𝑥+2𝑎+𝑥+2𝑐−2𝑥−4𝑏𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 = 2𝑥−2𝑥+6−62𝑥−2𝑥+8−82𝑥−2𝑥+2𝑎+2𝑐−4𝑏𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 = 000+2(𝒂+𝒄−2𝑏)𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 = 002(𝟐𝒃−2𝑏)𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 = 000𝑥+3𝑥+4𝑥+2𝑏𝑥+4𝑥+5𝑥+2𝑐 If any row as column of determinant are zero, then value of determinant is also zero. = 0 Hence, value of determinant is 0 Correct Answer is A

Chapter 4 Class 12 Determinants

Serial order wise

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