PYQ Vault
EbooksTutor

Parameter values for which a homogeneous system has a non-trivial solution

This type of question, asked 8 times: 2021 Paper 10, 2021 Paper 23, 27 Jun 2022, 8 Apr 2023, 4 Apr 2024, 6 Apr 2024, 5 Apr 2026 S1, 6 Apr 2026 S2. All 8 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 10

    63
    Let the system of linear equations
    4x+λy+2z=04x +\lambda y + 2z = 0
    2x−y+z=02x - y + z = 0
    μx+2y+3z=0,λ,μ∈R\mu x + 2y + 3z = 0,\lambda,\mu\in R. has a non-trivial solution. Then which of the following is true?

    Tap an option to check your answer.

  2. Question 2

    2021 Paper 23

    61
    If α+β+γ=2π\alpha + \beta + \gamma = 2\pi, then the system of equations
    x+(cos⁡γ)y+(cos⁡β)z=0x + (\cos\gamma)y + (\cos\beta)z = 0
    (cos⁡γ)x+y+(cos⁡α)z=0(\cos\gamma)x + y + (\cos\alpha)z = 0
    (cos⁡β)x+(cos⁡α)y+z=0(\cos\beta)x + (\cos\alpha)y + z = 0
    has:

    Tap an option to check your answer.

  3. Question 3

    27 Jun 2022

    153
    Let for some real numbers α\alpha and β,a=α−iβ\beta,a = \alpha - i\beta. If the system of equations 4ix+(1+i)y=04ix + (1 + i)y = 0 and 8(cos⁡2π3+isin⁡2π3)x+aˉy=08\left( \cos\frac{2\pi}{3}+ i\sin\frac{2\pi}{3} \right)x +\bar{a}y = 0 has more than one solution then αβ\frac{\alpha}{\beta} is equal to:

    Tap an option to check your answer.

  4. Question 4

    8 Apr 2023

    157
    Let SS be the set of all values of θ∈[−π,π]\theta \in \lbrack - \pi,\pi\rbrack for which the system of linear equations
    x+y+3z=0x + y +\sqrt{3}z = 0
    −x+(tan⁡θ)y+7z=0- x + (\tan\theta)y +\sqrt{7}z = 0
    x+y+(tan⁡θ)z=0x + y + (\tan\theta)z = 0
    has non-trivial solution. Then 120π∑θ∈Sθ\frac{120}{\pi}\sum_{\theta \in S} \theta is equal to

    Tap an option to check your answer.

  5. Question 5

    4 Apr 2024

    67
    If the system of equations
    x+(2sin⁡α)y+(2cos⁡α)z=0x + (\sqrt{2}\sin\alpha)y + (\sqrt{2}\cos\alpha)z = 0
    x+(cos⁡α)y+(sin⁡α)z=0x + (\cos\alpha)y + (\sin\alpha)z = 0
    x+(sin⁡α)y−(cos⁡α)z=0x + (\sin\alpha)y - (\cos\alpha)z = 0
    has a non-trivial solution, then α∈(0,π2)\alpha \in\left( 0,\frac{\pi}{2} \right) is equal to:

    Tap an option to check your answer.

  6. Question 6

    6 Apr 2024

    81
    Let αβγ=45;α,β,γ∈R\alpha\beta\gamma = 45;\alpha,\beta,\gamma \in R. If x(α,1,2)+y(1,β,2)x(\alpha,1,2) + y(1,\beta,2) +z(2,3,γ)=(0,0,0)+ z(2,3,\gamma) = (0,0,0) for some x,y,z∈R,xyz≠x,y,z \in R,xyz \neq 0 , then 6α+4β+γ6\alpha + 4\beta + \gamma is equal to

  7. Question 7

    5 Apr 2026 S1

    54
    Consider the system of linear equations in x,y,zx,y,z : x+2y+tz=0x + 2y + tz = 0, 6x+y+5tz=06x + y + 5tz = 0, 3x+t2y+f(t)z=03x +t^{2}y + f(t)z = 0, Where f:R→Rf:R \rightarrow R is a differentiable function. If this system has infinitely many solutions for all t∈Rt\in R, then ff

    Tap an option to check your answer.

  8. Question 8

    6 Apr 2026 S2

    54
    The sum of all possible values of θ∈[0,2π]\theta\in \lbrack 0,2\pi\rbrack, for which the system of equations :
    xcos3θ−8y−12z=0xcos3\theta- 8y- 12z= 0
    xcos2θ+3y+3z=0xcos2\theta+ 3y+ 3z= 0
    x+y+3z=0x+y+ 3z= 0
    has a non-trivial solution, is equal to:

    Tap an option to check your answer.