3
Question
Absolute value of sum of roots of the equation ∣∣
∣∣x+22x+33x+42x+33x+44x+53x+55x+810x+17∣∣
∣∣=0 is ...........
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Solution
Applying R3→(R3−R2);R3→(R2−R1)
∣∣
∣
∣∣(x+2)(2x+3)(3x+4)(x+1)(x+1)(x+1)(x+2)2(x+2)6(x+2)∣∣
∣
∣∣=0
Applying C3→C3−C2;C2→C2−C1
⇒∣∣
∣
∣∣(x+2)(x+1)(x+1)(x+1)00(x+2)(x+2)6(x+2)∣∣
∣
∣∣=0⇒−(x+1){4(x+1)(x+2)−(x+2)(x+2)}=0⇒−3(x+1){(x+1)(x+2)}=0⇒x=−1,−1,−2
Hence sum of roots is −4 its absolute value is 4
∣∣ ∣ ∣∣(x+2)(2x+3)(3x+4)(x+1)(x+1)(x+1)(x+2)2(x+2)6(x+2)∣∣ ∣ ∣∣=0
Applying C3→C3−C2;C2→C2−C1
⇒∣∣ ∣ ∣∣(x+2)(x+1)(x+1)(x+1)00(x+2)(x+2)6(x+2)∣∣ ∣ ∣∣=0⇒−(x+1){4(x+1)(x+2)−(x+2)(x+2)}=0⇒−3(x+1){(x+1)(x+2)}=0⇒x=−1,−1,−2
Hence sum of roots is −4 its absolute value is 4
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