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Question

# The number of real values of λ for which the system of linear equations. 2x+4y−λz=0, 4x+λy+2z=0,λx+2y+2z=0 has infinitely many solutions, is

A
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B
0
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C
2
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D
3
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Solution

## The correct option is A 1For infinitely many solutions, D=0,Dx=0,Dy=0 and Dz=0D=⎡⎢⎣24−λ4λ2λ22⎤⎥⎦=2(2λ−4)−4(8−2λ)−λ(8−λ2)=λ3+4λ−40If D=0, thenλ3+4λ−40=0Again, Dx=0Dx=⎡⎢⎣04−λ0λ2022⎤⎥⎦=0Similarly,Dy=0 and Dz=0The equation, λ3+4λ−40 has only one solutionSince λ(−∞)=−∞ and λ(∞)=∞Here α is only one solution, so λ(a)=0Using intermediate value propertyNow, differentiating λ3+4λ−40 w.r.t λ we get3λ2+4>0The equation can not have y.m, soλ(m)=0 and λ(y)=0Thus, the number of real value of λ is 1.

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