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Equation of a line; Skew lines, the shortest distance between them and its equation Mock Test for JEE

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Part of Three Dimensional Geometry in JEE Main Mathematics.

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Q1MathsUnit 11: Three Dimensional Geometry
The equation of the plane passing through the intersection of the planes x+y+z=6\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}=\boldsymbol{6} and 2x+3y+4z+5=\boldsymbol{2} \boldsymbol{x}+\boldsymbol{3} \boldsymbol{y}+\boldsymbol{4} \boldsymbol{z}+\boldsymbol{5}= 0,0, and the point (1,1,1) is
Q2MathsUnit 11: Three Dimensional Geometry
Find the equation of the plane through the points A(2,2−1),B(3,4,2)\boldsymbol{A}(\mathbf{2}, \mathbf{2}-\mathbf{1}), \boldsymbol{B}(\mathbf{3}, \mathbf{4}, \mathbf{2}) and C(7,0,6)\boldsymbol{C}(\boldsymbol{7}, \boldsymbol{0}, \boldsymbol{6})
Q3MathsUnit 11: Three Dimensional Geometry
x−21=y−31=z−4−1&x−1k=\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-1} \& \frac{x-1}{k}= y−42=z−52\frac{\boldsymbol{y}-\boldsymbol{4}}{\boldsymbol{2}}=\frac{\boldsymbol{z}-\boldsymbol{5}}{\boldsymbol{2}} are coplanar then k=?\mathbf{k}=?
Q4MathsUnit 11: Three Dimensional Geometry
The shortest distance between the lines x−54=y−7−5=z+3−5\frac{\boldsymbol{x}-\mathbf{5}}{\mathbf{4}}=\frac{\boldsymbol{y}-\mathbf{7}}{-\mathbf{5}}=\frac{\boldsymbol{z}+\mathbf{3}}{-\mathbf{5}} and x−84=\frac{\boldsymbol{x}-\mathbf{8}}{\mathbf{4}}= y−7−5=z−5−5\frac{y-7}{-5}=\frac{z-5}{-5} is
Q5MathsUnit 11: Three Dimensional Geometry
If the points (x,y,−3),(2,0,−1)(x, y,-3),(2,0,-1) and (4,2,3) lies on a straight line, then what are the values of xx and yy respectively?
Q6MathsUnit 11: Three Dimensional Geometry
The equation of a plane passing through the point A(2,−3,7)A(2,-3,7) and making equal intercepts on the axes, is?
Q7MathsUnit 11: Three Dimensional Geometry
If the distance between a point PP and the point (1,1,1) on the line x−13=\frac{x-1}{3}= y−14=z−112\frac{y-1}{4}=\frac{z-1}{12} is 13,13, then the coordinates of PP are
Q8MathsUnit 11: Three Dimensional Geometry
A plane which passes through the point (3,2,0) and the line x−31=y−75=\frac{x-3}{1}=\frac{y-7}{5}= z−44\frac{z-4}{4} is?
Q9MathsUnit 11: Three Dimensional Geometry
Consider three vectors p→=i+j+\overrightarrow{\boldsymbol{p}}=\boldsymbol{i}+\boldsymbol{j}+ k,q→=2i+4j−k\boldsymbol{k}, \overrightarrow{\boldsymbol{q}}=2 \boldsymbol{i}+\boldsymbol{4} \boldsymbol{j}-\boldsymbol{k} and r→=i+j+3k.\overrightarrow{\boldsymbol{r}}=\boldsymbol{i}+\boldsymbol{j}+\boldsymbol{3} \boldsymbol{k} . p,qp, q and rr denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
Q10MathsUnit 11: Three Dimensional Geometry
The following lines are r^=(i^+j^)+\hat{\boldsymbol{r}}=(\hat{\boldsymbol{i}}+\hat{\boldsymbol{j}})+ λ(i^+2j^−k^)+μ(−i^+j^−2k^)\lambda(\hat{i}+2 \hat{j}-\hat{k})+\mu(-\hat{i}+\hat{j}-2 \hat{k})
Q11MathsUnit 11: Three Dimensional Geometry
The foot of the perpendicular from the point A(7,14,5)\boldsymbol{A}(\mathbf{7}, \mathbf{1 4}, \mathbf{5}) to the plane 2x+4y−\mathbf{2} \boldsymbol{x}+\mathbf{4} \boldsymbol{y}- z=2z=2 is?
Q12MathsUnit 11: Three Dimensional Geometry
The line x−23=y+12=z−1−1\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1} intersects the curve xy=c2,z=0,\boldsymbol{x} \boldsymbol{y}=\boldsymbol{c}^{2}, \boldsymbol{z}=\mathbf{0}, if c=\boldsymbol{c}=
Q13MathsUnit 11: Three Dimensional Geometry
A line d.c's proportional to (2,1,2) meets each of the lines x=y+a=z\boldsymbol{x}=\boldsymbol{y}+\boldsymbol{a}=\boldsymbol{z} and x+a=2y=2z.x+a=2 y=2 z . Then the coordinates of each of the points of intersection are given by

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