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QJEEM23Ma2766: JEE Main 2023 Mathematics - Class 12 JEEM23Ma2766 Solution: Let S denote the set of all real values of λ such that the…

Let S denote the set of all real values of λ such that the system of equations

λx+y+z=1

x+λy+z=1

x+y+λz=1

is inconsistent, then ∑λ∈S(|λ|2+|λ|) is equal to

Multiple Choice Options

  • A 12
  • B 2
  • C 4
  • D 6

Step-by-Step Verified Solution

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λ(λ2−1)−1(λ−1)+1(1−λ)=0⇒λ3−λ−λ+1+1−λ=0⇒λ3−3λ+2=0⇒(λ−1)(λ2+λ−2)=0

⇒\Rightarrow λ=1,−2\lambda=1,-2

For λ=1⇒∞\lambda=1 \Rightarrow \infty solution

λ=−2⇒\lambda=-2 \Rightarrow no solution

∑λ∈S∣λ∣2+∣λ∣=6\sum\limits_{\lambda \in S}|\lambda|^{2}+|\lambda|=6

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