Three Dimensional Geometry
Consider the lines and the planes Let ax + by + cz = d be the equation of the plane passing through the point of intersection of lines L_{1} and L_{2}, and perpendicular to planes P_{ 1} and P_{2}.
Match List − I with List − II and select the correct answer using the code given below the lists:

List I 

List II 
P. 
a = 
1. 
13 
Q. 
b = 
2. 
–3 
R. 
c = 
3. 
1 
S. 
d = 
4. 
–2 
(I)  ${l}_{1}\equiv x1=\frac{y}{2}\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x+1=y$  (i)  ${\mathrm{cos}}^{1}\left(\frac{1}{\sqrt{12}}\right)$  (P)  $\frac{1}{\sqrt{3}}$ units 
(II)  l_{1} ≡ x + 1 = z l_{2} ≡ y = z 
(ii)  ${\mathrm{cos}}^{1}\left(\frac{3}{\sqrt{10}}\right)$  (Q)  0 units 
(III)  ${l}_{1}\equiv x=y1\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv \left(x+2\right)=\frac{y}{2}=z$  (iii)  $\frac{\pi}{2}$  (R)  $\frac{2\sqrt{30}}{15}$ units 
(IV)  ${l}_{1}\equiv \frac{x+2}{2}=y=z\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x=\frac{y}{2}$  (iv)  $\frac{\pi}{3}$  (S)  $\frac{1}{\sqrt{11}}$ units 
For a line that is equally inclined to y and zaxes and belongs to one of the pair of equations (Column 1), which of the following combination is correct?
(I)  ${l}_{1}\equiv x1=\frac{y}{2}\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x+1=y$  (i)  ${\mathrm{cos}}^{1}\left(\frac{1}{\sqrt{12}}\right)$  (P)  $\frac{1}{\sqrt{3}}$ units 
(II)  l_{1} ≡ x + 1 = z l_{2} ≡ y = z 
(ii)  ${\mathrm{cos}}^{1}\left(\frac{3}{\sqrt{10}}\right)$  (Q)  0 units 
(III)  ${l}_{1}\equiv x=y1\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv \left(x+2\right)=\frac{y}{2}=z$  (iii)  $\frac{\pi}{2}$  (R)  $\frac{2\sqrt{30}}{15}$ units 
(IV)  ${l}_{1}\equiv \frac{x+2}{2}=y=z\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x=\frac{y}{2}$  (iv)  $\frac{\pi}{3}$  (S)  $\frac{1}{\sqrt{11}}$ units 
For a line on which all the points have the ordinate 1 more than the abscissa and that belongs to one of the pair of equations (Column 1), which of the following combination is correct?
Let $\frac{x1}{2}=\frac{y4}{3}=\frac{z3}{4}$ be a straight line. Match the expressions in column I with the corresponding points in column II.
Column I  Column II  
P. 
Point on the line at a distance of $\sqrt{29}$ unit from (3, 7, 7) is ... 
1.  (3, 7, 7) 
Q. 
Point on the line common to the plane $x+y+z17=0$ is ... 
2.  (1, 4, 3) 
R. 
Point on the line at a distance of $\sqrt{26}$ from the origin is ... 
3.  (5, 10, 11) 
S. 
Point on the line common to the plane $x+y+z35=0$ is ... 
4.  (7, 13, 15) 
(I)  ${l}_{1}\equiv x1=\frac{y}{2}\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x+1=y$  (i)  ${\mathrm{cos}}^{1}\left(\frac{1}{\sqrt{12}}\right)$  (P)  $\frac{1}{\sqrt{3}}$ units 
(II)  l_{1} ≡ x + 1 = z l_{2} ≡ y = z 
(ii)  ${\mathrm{cos}}^{1}\left(\frac{3}{\sqrt{10}}\right)$  (Q)  0 units 
(III)  ${l}_{1}\equiv x=y1\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv \left(x+2\right)=\frac{y}{2}=z$  (iii)  $\frac{\pi}{2}$  (R)  $\frac{2\sqrt{30}}{15}$ units 
(IV)  ${l}_{1}\equiv \frac{x+2}{2}=y=z\phantom{\rule{0ex}{0ex}}{l}_{2}\equiv x=\frac{y}{2}$  (iv)  $\frac{\pi}{3}$  (S)  $\frac{1}{\sqrt{11}}$ units 
For a line on which all the points have (−2) times the abscissa equal to 4 more than the ordinate and that belongs to one of the pair of equations (Column 1), which of the following combination is correct?