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Angle between a line and a plane is given: find the unknown

This type of question, asked 6 times: 11 May 2024 S2, 19 Apr 2025 S1, 22 Apr 2025 S2, 20 Apr 2026 S1, 14 May 2026 S1, 15 May 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    11 May 2024 S2

    112
    The angle between the line x+12=y−21=z−3−2\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2} and plane x−2y−λz=3x-2y-\lambda z=3 is cos⁡−1 ⁣(223)\cos^{-1}\!\left(\frac{2\sqrt{2}}{3}\right), then value of λ\lambda is:

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  2. Question 2

    19 Apr 2025 S1

    131
    If the angle θ\theta between the line x+11=y−12=z−22\frac{x+ 1}{1}=\frac{y- 1}{2}=\frac{z- 2}{2} and the plane 2x−y+λz+4=02x-y+\sqrt{\lambda}z+ 4 = 0 is such that sin⁡θ=13\sin\theta=\frac{1}{3}, then λ+1=\lambda+ 1 =

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  3. Question 3

    22 Apr 2025 S2

    104
    If the angle between the line x=y−12=z−3λx =\frac{y - 1}{2}=\frac{z - 3}{\lambda} and the plane x+2y+3z=4x + 2y + 3z = 4 is cos⁡−1514\cos^{- 1}\sqrt{\frac{5}{14}}, then the value of λ\lambda is

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  4. Question 4

    20 Apr 2026 S1

    93
    The acute angle between the line r⃗=(i^+2j^+k^)+λ(i^+j^+k^)\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k}) and the plane r⃗⋅(2i^+pj^+k^)=8\vec{r}\cdot(2\hat{i} + p\hat{j} + \hat{k}) = 8 is sin⁡−1(23)\sin^{-1}\left(\dfrac{\sqrt{2}}{3}\right), then the value of pp are...

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  5. Question 5

    14 May 2026 S1

    93
    If the angle θ\theta between the line x+11=y−12=z−22\dfrac{x + 1}{1} = \dfrac{y - 1}{2} = \dfrac{z - 2}{2} and the plane 2x−y+kz+4=02x - y + \sqrt{k}z + 4 = 0 is such that cos⁡θ=83\cos\theta = \dfrac{\sqrt{8}}{3}. Then k=k =

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  6. Question 6

    15 May 2026 S1

    92
    If the angle between the line x=y−52=z+8kx = \dfrac{y - 5}{2} = \dfrac{z + 8}{k} and the plane 2x−3y+3z=92x - 3y + 3z = 9 is cos⁡−1(1322)\cos^{-1}\left(\sqrt{\dfrac{13}{22}}\right), then k=…k = \ldots

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