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Value of one parameter for which a system has infinitely many solutions

This type of question, asked 5 times: 2021 Paper 3, 2021 Paper 12, 30 Jan 2023, 23 Jan 2025, 29 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    87
    If the system of equations
    kx+y+2z=13x−y−2z=2 −2x−2y−4z=3\begin{matrix} & kx+y+ 2z= 1 \\ & 3x-y- 2z= 2 \\ & \ - 2x- 2y- 4z= 3 \end{matrix}
    has infinitely many solutions, then kk is equal to

  2. Question 2

    2021 Paper 12

    56
    The value of k∈Rk\in R, for which the following system of linear equations 3x−y+4z=33x - y + 4z = 3, x+2y−3z=−2x+ 2y- 3z= - 2, 6x+5y+kz=−36x+ 5y+kz= - 3, has infinitely many solutions, is:

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  3. Question 3

    30 Jan 2023

    72
    Let the system of linear equations
    x+y+kz=2x + y + kz = 2
    2x+3y−z=12x + 3y - z = 1
    3x+4y+2z=k3x + 4y + 2z = k
    have infinitely many solutions. Then the system
    (k+1)x+(2k−1)y=7(k + 1)x + (2k - 1)y = 7
    (2k+1)x+(k+5)y=10(2k + 1)x + (k + 5)y = 10
    has:

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  4. Question 4

    23 Jan 2025

    69
    If the system of equations
    (λ−1)x+(λ−4)y+λz=5λx+(λ−1)y+(λ−4)z=7 (λ+1)x+(λ+2)y−(λ+2)z=9\begin{matrix} & (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \\ & \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \\ & \ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \end{matrix}
    has infinitely many solutions, then λ2+λ\lambda^{2}+ \lambda is equal to

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  5. Question 5

    29 Jan 2025

    136
    Let α,β(α≠β)\alpha,\beta(\alpha \neq \beta) be the values of m , for which the equations x+y+z=1;x+2y+4z=mx + y + z = 1;x + 2y + 4z = m and x+4y+10z=m2x + 4y + 10z =m^{2} have infinitely many solutions. Then the value of ∑n=110(nα+nβ)\sum_{n = 1}^{10} \left( n^{\alpha}+n^{\beta} \right) is equal to :

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