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Values of two parameters (lambda, mu) for which a system of three equations has infinitely many solutions

This type of question, asked 26 times: 2021 Paper 20, 26 Jun 2022, 28 Jun 2022, 29 Jun 2022, 26 Jul 2022, 29 Jul 2022, 24 Jan 2023, 30 Jan 2023, 6 Apr 2023, 11 Apr 2023, 13 Apr 2023, 29 Jan 2024, 31 Jan 2024, 1 Feb 2024, 5 Apr 2024, 6 Apr 2024, 8 Apr 2024, 9 Apr 2024, 22 Jan 2025, 24 Jan 2025, 2 Apr 2025, 7 Apr 2025, 23 Jan 2026 S2, 2 Apr 2026 S2, 4 Apr 2026 S2, 8 Apr 2026 S2. All 31 questions below, oldest first.

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  1. Question 1

    2021 Paper 20

    86
    If the system of linear equations
    2x+y−z=32x + y - z = 3
    x−y−z=αx - y - z =\alpha
    3x+3y+βz=33x+ 3y+\beta z= 3
    has infinitely many solution, then α+β−αβ\alpha+\beta-\alpha\beta is equal to

  2. Question 2

    26 Jun 2022

    152
    If the system of equations αx+y+z=5,x+2y+3z=4,x+3y+5z=β\alpha x + y + z = 5,x + 2y + 3z = 4,x + 3y + 5z = \beta, has infinitely many solutions, then the ordered pair (α,β)(\alpha,\beta) is equal to:

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  3. Question 3

    28 Jun 2022

    176
    If the system of linear equations 2x−3y=γ+52x - 3y = \gamma + 5, αx+5y=β+1\alpha x + 5y = \beta + 1, where α,β,γ∈R\alpha,\beta,\gamma \in R has infinitely many solutions, then the value of ∣9α+3β+5γ∣|9\alpha + 3\beta + 5\gamma| is equal to

  4. Question 4

    29 Jun 2022

    67
    If the system of linear equations
    2x+y−z=72x + y - z = 7
    x−3y+2z=1x - 3y + 2z = 1
    x+4y+δz=kx + 4y + \delta z = k, where δ,k∈R\delta,k \in R has infinitely many solutions, then δ+k\delta + k is equal to:

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  5. Question 5

    26 Jul 2022

    63
    If the system of linear equations.
    8x+y+4z=−28x + y + 4z = - 2
    x+y+z=0x + y + z = 0
    λx−3y=μ\lambda x - 3y = \mu
    has infinitely many solutions, then the distance of the point (λ,μ,−12)\left( \lambda,\mu, -\frac{1}{2} \right) from the plane 8x+y+4z+2=08x + y + 4z + 2 = 0 is :

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  6. Question 6

    29 Jul 2022

    153
    If the system of equations
    x+y+z=6x + y + z = 6
    2x+5y+αz=β2x + 5y + \alpha z = \beta
    x+2y+3z=14x + 2y + 3z = 14
    has infinitely many solutions, then α+β\alpha + \beta is equal to:

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  7. Question 7

    24 Jan 2023

    152
    If the system of equations
    x+2y+3z=34x+3y−4z=48x+4y−λz=9+μ\begin{matrix} & x + 2y + 3z = 3 \\ & 4x + 3y - 4z = 4 \\ & 8x + 4y - \lambda z = 9 + \mu \end{matrix}
    has infinitely many solutions, then the ordered pair (λ,μ)(\lambda,\mu) is equal to :

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  8. Question 8

    30 Jan 2023

    158
    For α,β∈R\alpha,\beta \in R, suppose the system of linear equations
    x−y+z=5x - y + z = 5
    2x+2y+αz=82x + 2y + \alpha z = 8
    3x−y+4z=β3x - y + 4z = \beta
    has infinitely many solutions. Then α\alpha and β\beta are the roots of

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  9. Question 9

    6 Apr 2023

    76
    If the system of equations
    x+y+az=bx + y + az = b
    2x+5y+2z=62x + 5y + 2z = 6
    x+2y+3z=3x + 2y + 3z = 3
    has infinitely many solutions, then 2a+3b2a + 3b is equal to

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  10. Question 10

    11 Apr 2023

    158
    If the system of linear equations
    7x+11y+αz=135x+4y+7z=β175x+194y+57z=361\begin{matrix} & 7x + 11y + \alpha z = 13 \\ & 5x + 4y + 7z = \beta \\ & 175x + 194y + 57z = 361 \end{matrix}
    has infinitely many solutions, then α+β+2\alpha + \beta + 2 is equal to

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  11. Question 11

    13 Apr 2023

    151
    If the system of equations
    2x+y−z=52x−5y+λz=μx+2y−5z=7\begin{matrix} & 2x + y - z = 5 \\ & 2x - 5y + \lambda z = \mu \\ & x + 2y - 5z = 7 \end{matrix}
    has infinitely many solutions, then (λ+μ)2+(λ−μ)2(\lambda + \mu)^{2}+ (\lambda - \mu)^{2} is equal to

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  12. Question 12

    29 Jan 2024

    172
    Let for any three distinct consecutive terms a,b,ca,b,c of an A.P, the lines ax+by+c=0ax + by + c = 0 be concurrent at the point PP and Q(α,β)Q(\alpha,\beta) be a point such that the system of equations x+y+z=6x + y + z = 6, 2x+5y+αz=β2x + 5y + \alpha z = \beta and x+2y+3z=4x + 2y + 3z = 4, has infinitely many solutions. Then (PQ)2(PQ)^{2} is equal to

  13. Question 13

    31 Jan 2024

    68
    If the system of linear equations
    x−2y+z=−4x- 2y+z= - 4
    2x+αy+3z=52x+\alpha y+ 3z= 5
    3x−y+βz=33x-y+\beta z= 3
    has infinitely many solutions, then 12α+13β12\alpha+ 13\beta is equal to

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  14. Question 14

    1 Feb 2024

    167
    Let the system of equations x+2y+3z=5,2x+3y+z=9,4x+3y+λz=μx + 2y + 3z = 5,2x + 3y + z = 9,4x + 3y + \lambda z = \mu have infinite number of solutions. Then λ+2μ\lambda + 2\mu is equal to:

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  15. Question 15

    1 Feb 2024

    72
    If the system of equations
    2x+3y−z=52x + 3y - z = 5
    x+αy+3z=−4x + \alpha y + 3z = - 4
    3x−y+βz=73x - y + \beta z = 7
    has infinitely many solutions, then 13αβ13\alpha\beta is equal to

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  16. Question 16

    5 Apr 2024

    164
    The values of m,nm,n, for which the system of equations x+y+z=4x + y + z = 4, 2x+5y+5z=172x + 5y + 5z = 17,
    x+2y+mz=nx + 2y + mz = n
    has infinitely many solutions, satisfy the equation:

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  17. Question 17

    5 Apr 2024

    66
    If the system of equations
    11x+y+λz=−511x + y + \lambda z = - 5
    2x+3y+5z=32x + 3y + 5z = 3
    8x−19y−39z=μ8x - 19y - 39z = \mu
    has infinitely many solutions, then λ4−μ\lambda^{4}- \mu is equal to:

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  18. Question 18

    6 Apr 2024

    179
    If the system of equations
    2x+7y+λz=32x + 7y + \lambda z = 3
    3x+2y+5z=43x + 2y + 5z = 4
    x+μy+32z=−1x + \mu y + 32z = - 1
    has infinitely many solutions, then (λ−μ)(\lambda - \mu) is equal to

  19. Question 19

    8 Apr 2024

    157
    If the system of equations x+4y−z=λx+ 4y-z=\lambda, 7x+9y+μz=−3,5x+y+2z=−17x+ 9y+\mu z= - 3,5x+y+ 2z= - 1 has infinitely many solutions, then (2μ+3λ)(2\mu + 3\lambda) is equal to :

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  20. Question 20

    9 Apr 2024

    65
    Let λ,μ∈R\lambda,\mu \in R. If the system of equations
    3x+5y+λz=33x + 5y + \lambda z = 3
    7x+11y−9z=27x + 11y - 9z = 2
    97x+155y−189z=μ97x + 155y - 189z = \mu
    has infinitely many solutions, then μ+2λ\mu + 2\lambda is equal to:

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  21. Question 21

    22 Jan 2025

    139
    If the system of linear equations :
    x+y+2z=62x+3y+az=a+1 −x−3y+bz=2b\begin{matrix} & x + y + 2z = 6 \\ & 2x + 3y + az = a + 1 \\ & \ - x - 3y + bz = 2b \end{matrix}
    where a,b∈Ra,b \in R, has infinitely many solutions, then 7a+3b7a + 3b is equal to :

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  22. Question 22

    24 Jan 2025

    70
    If the system of equations
    2x−y+z=45x+λy+3z=12100x−47y+μz=212\begin{matrix} & 2x - y + z = 4 \\ & 5x + \lambda y + 3z = 12 \\ & 100x - 47y + \mu z = 212 \end{matrix}
    has infinitely many solutions, then μ−2λ\mu - 2\lambda is equal to

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  23. Question 23

    24 Jan 2025

    136
    If the system of equations
    x+2y−3z=22x+λy+5z=514x+3y+μz=33\begin{matrix} & x+ 2y- 3z= 2 \\ & 2x+\lambda y+ 5z= 5 \\ & 14x+ 3y+\mu z= 33 \end{matrix}
    has infinitely many solutions, then λ+μ\lambda+\mu is equal to:

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  24. Question 24

    2 Apr 2025

    133
    If the system of equation
    2x+λy+3z=52x+\lambda y+ 3z= 5
    3x+2y−z=73x+ 2y-z= 7
    4x+5y+μz=94x+ 5y+\mu z= 9
    has infinitely many solutions, then (λ2+μ2)\left( \lambda^{2}+\mu^{2} \right) is equal to :

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  25. Question 25

    2 Apr 2025

    64
    If the system of linear equations
    3x+y+βz=32x+αy−z=−3x+2y+z=4\begin{matrix} & 3x+y+\beta z= 3 \\ & 2x+\alpha y-z= - 3 \\ & x+ 2y+z= 4 \end{matrix}
    has infinitely many solutions, then the value of 22β−9α22\beta- 9\alpha is :

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  26. Question 26

    7 Apr 2025

    144
    Let the system of equations
    x+5y−z=14x+3y−3z=724x+y+λz=μ\begin{matrix} & x+ 5y-z= 1 \\ & 4x+ 3y- 3z= 7 \\ & 24x+y+\lambda z=\mu \end{matrix}
    λ,μ∈R\lambda,\mu\in R, have infinitely many solutions. Then the number of the solutions of this system, If x,y,zx,y,z are integers and satisfy 7≤x+y+z≤777 \leq x + y + z \leq 77, is

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  27. Question 27

    7 Apr 2025

    67
    Let the system of equations: 2x+3y+5z=97x+3y−2z=812x+3y−(4+λ)z=16−μ\begin{matrix} & 2x+ 3y+ 5z= 9 \\ & 7x+ 3y- 2z= 8 \\ & 12x+ 3y- (4 +\lambda)z= 16 -\mu \end{matrix} have infinitely many solutions. Then the radius of the circle centred at ( λ,μ\lambda,\mu ) and touching the line 4x=3y4x = 3y is.

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  28. Question 28

    23 Jan 2026 S2

    56
    The system of linear equations x+y+z=6x + y + z = 6, 2x+5y+az=362x + 5y + az = 36, x+2y+3z=bx + 2y + 3z = b has

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  29. Question 29

    2 Apr 2026 S2

    53
    If the system of equations
    x+5y+6z=4x+ 5y+ 6z= 4
    2x+3y+4z=72x+ 3y+ 4z= 7
    x+6y+az=bx+ 6y+az=b
    has infinitely many solutions, then the point (a,b)(a,b) lies on the line

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  30. Question 30

    4 Apr 2026 S2

    53
    If the system of equations : x+y+z=5x+y+z= 5, x+2y+3z=9x+ 2y+ 3z= 9, x+3y+λz=μx+ 3y+\lambda z=\mu has infinitely many solutions, then the value of λ+μ\lambda+\mu is :

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  31. Question 31

    8 Apr 2026 S2

    53
    If the system of linear equations : x+y+z=6x+y+z= 6, x+2y+5z=10x+ 2y+ 5z= 10, 2x+3y+λz=μ2x + 3y +\lambda z =\mu. has infinitely many solutions, then the value of λ+μ\lambda+\mu equals.

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