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Line lies in a plane: find the unknowns in the line or the plane

This type of question, asked 10 times: 2 May 2023 S1, 3 May 2023 S1, 9 May 2023 S1, 9 May 2023 S2, 10 May 2023 S1, 10 May 2024 S1, 22 Apr 2025 S1, 25 Apr 2025 S2, 12 May 2026 S2, 13 May 2026 S1. All 11 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2 May 2023 S1

    144
    If the line x−32=y+2−1=z+43\dfrac{x-3}{2} = \dfrac{y+2}{-1} = \dfrac{z+4}{3} lies in the plane lx+my−z=9lx + my - z = 9, then l2+m2l^2 + m^2 is

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  2. Question 2

    3 May 2023 S1

    137
    The value of m, such that x−41=y−21=2z−m3\frac{x-4}{1}=\frac{y-2}{1}=\frac{2z-m}{3} lies in the plane 2x−5y+2z=72x-5y+2z=7, is

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  3. Question 3

    9 May 2023 S1

    129
    The value of m, such that x−41=y−21=z−m2\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-m}{2} lies in the plane 2x−4y+z=72x-4y+z=7, is

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  4. Question 4

    9 May 2023 S2

    121
    The value of m such that x−41=y−21=z+m2\frac{x-4}{1} = \frac{y-2}{1} = \frac{z+m}{2} lies in the plane 2x−4y+z=72x-4y+z=7 is

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  5. Question 5

    10 May 2023 S1

    147
    If the line x−23=y−1−5=z+22\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2} lies in the plane x+3y−αz+β=0x+3y-\alpha z+\beta=0, then (α,β)=(\alpha,\beta)=

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  6. Question 6

    10 May 2023 S1

    130
    If the line x−32=y+21=z+43\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3} lies in the plane lx+my−z=9lx+my-z=9, then l2+m2l^2+m^2 is equal to

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  7. Question 7

    10 May 2024 S1

    140
    Line x−23=y−1−5=z+22\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2} lies in plane x+3y−αz+β=0x+3y-\alpha z+\beta=0. Value of α2+αβ+β2\alpha^2+\alpha\beta+\beta^2 is

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  8. Question 8

    22 Apr 2025 S1

    150
    Let the line x−23=y−1−5=z+22\frac{x- 2}{3}=\frac{y- 1}{- 5}=\frac{z+ 2}{2} lie in the plane x+3y−αz+β=0x+ 3y-\alpha z+\beta= 0, then the value of (β−α)(\beta-\alpha) is equal to

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  9. Question 9

    25 Apr 2025 S2

    125
    If the line x−32=y+5−1=z+22\frac{x- 3}{2}=\frac{y+ 5}{- 1}=\frac{z+ 2}{2} lies in the plane αx+3y−z+β=0\alpha x+ 3y-z+\beta= 0, then values of α\alpha and β\beta respectively are ....

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  10. Question 10

    12 May 2026 S2

    95
    If the line x−23=y−15=z+22\dfrac{x - 2}{3} = \dfrac{y - 1}{5} = \dfrac{z + 2}{2} lies in the plane x+3y−pz+q=0x + 3y - pz + q = 0, then the value of p−q=p - q =........

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  11. Question 11

    13 May 2026 S1

    93
    Let the line x−23=y−1−5=z+22\dfrac{x - 2}{3} = \dfrac{y - 1}{-5} = \dfrac{z + 2}{2} lie in the plane x+3y−αz+β=0x + 3y - \alpha z + \beta = 0. Then (α,β)(\alpha, \beta) equals

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