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Hard Direction ratios and direction cosines and the angle between two intersecting lines MCQs for JEE

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Hard Direction ratios and direction cosines and the angle between two intersecting lines MCQs train multi-concept problem solving — attempt after mastering easy and medium sets (8+ on Goodmarks).

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Part of Three Dimensional Geometry in JEE Main Mathematics.

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Q1MathsUnit 11: Three Dimensional Geometry
The plane x−2y+z−6=0\boldsymbol{x}-\mathbf{2} \boldsymbol{y}+\boldsymbol{z}-\boldsymbol{6}=\mathbf{0} and the line x1=y2=z3\frac{x}{1}=\frac{y}{2}=\frac{z}{3} are related as
Q2MathsUnit 11: Three Dimensional Geometry
Let⁡A(2i^+3j^+5k^)B(−i^+3j^+2k^)\operatorname{Let} \boldsymbol{A}(\mathbf{2} \hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+\mathbf{5} \hat{\boldsymbol{k}}) \boldsymbol{B}(-\hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+2 \hat{\boldsymbol{k}}) and C(λi^+5j^+μk^)C(\lambda \hat{i}+5 \hat{j}+\mu \hat{k}) are vertices of aa triangle and its median through AA is equally inclined to the positive directions of the axes. The value of λ+\lambda+ μ\mu is equal to
Q3MathsUnit 11: Three Dimensional Geometry
The direction ratios of the joining A(1,2,1)A(1,2,1) and (2,1,2) are
Q4MathsUnit 11: Three Dimensional Geometry
The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is
Q5MathsUnit 11: Three Dimensional Geometry
The planes 2x−y+4z=52 x-y+4 z=5 and 5x−5 x- 2.5y+10z=62.5 y+10 z=6 are
Q6MathsUnit 11: Three Dimensional Geometry
The number of straight lines that are equally inclined to the threedimensional coordinate axes, is
Q7MathsUnit 11: Three Dimensional Geometry
If a ray makes angles α,β,γ\alpha, \beta, \gamma and δ\delta with the four diagonals of a cube and A:cos⁡2α+cos⁡2β+cos⁡2γ+cos⁡2δ\mathbf{A}: \cos ^{2} \boldsymbol{\alpha}+\cos ^{2} \boldsymbol{\beta}+\cos ^{2} \boldsymbol{\gamma}+\cos ^{2} \boldsymbol{\delta} B:sin⁡2α+sin⁡2β+sin⁡2γ+sin⁡2δ\mathbf{B}: \sin ^{2} \boldsymbol{\alpha}+\sin ^{2} \boldsymbol{\beta}+\sin ^{2} \boldsymbol{\gamma}+\sin ^{2} \boldsymbol{\delta} C:cos⁡2α+cos⁡2β+cos⁡2γ+cos⁡2δ\mathbf{C}: \cos 2 \boldsymbol{\alpha}+\cos 2 \boldsymbol{\beta}+\cos 2 \gamma+\cos 2 \boldsymbol{\delta} Arrange A,B,CA, B, C in descending order
Q8MathsUnit 11: Three Dimensional Geometry
If θ\theta is the angle between the lines AB,ACA B, A C where A,B,CA, B, C are the three points with coordinates (1,2,-1),(2,0,3),(3,-1,2) respectively, then 462cos⁡θ\sqrt{462} \cos \theta is equal to

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