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3D Geometry question

2024 · Shift 1 · Q33
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  1. PYQ Library
  2. /JEE Advanced
  3. /Mathematics
  4. /3D Geometry
  5. /2024 · Shift 1 · Q33

3D Geometry question

2024 · Shift 1 · Q33

JEE AdvancedMathematics3D GeometryMCQ+3 / −1

Let γ∈R\gamma \in \mathbb{R}γ∈R be such that the lines L1:x+111=y+212=z+293L_1: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}L1​:1x+11​=2y+21​=3z+29​ and L2:x+163=y+112=z+4γL_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}L2​:3x+16​=2y+11​=γz+4​ intersect. Let R1R_1R1​ be the point of intersection of L1L_1L1​ and L2L_2L2​. Let O=(0,0,0)O=(0,0,0)O=(0,0,0), and n^\hat{n}n^ denote a unit normal vector to the plane containing both the lines L1L_1L1​ and L2L_2L2​.

Match each entry in List-I to the correct entry in List-II.

List-I List-II
(P) γ\gammaγ equals (1) −i^−j^+k^-\hat{i} - \hat{j} + \hat{k}−i^−j^​+k^
(Q) A possible choice for n^\hat{n}n^ is (2) 32\sqrt{\frac{3}{2}}23​​
(R) OR1→\overrightarrow{OR_1}OR1​​ equals (3) 111
(S) A possible value of OR1→⋅n^\overrightarrow{OR_1} \cdot \hat{n}OR1​​⋅n^ is (4) 16i^−26j^+16k^\frac{1}{\sqrt{6}} \hat{i} - \frac{2}{\sqrt{6}} \hat{j} + \frac{1}{\sqrt{6}} \hat{k}6​1​i^−6​2​j^​+6​1​k^
(5) 23\sqrt{\frac{2}{3}}32​​

The correct option is :
  1. A
    (P)→(3)(Q)→(4)(R)→(1)(S)→(2)(\mathrm{P}) \rightarrow(3) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(2)(P)→(3)(Q)→(4)(R)→(1)(S)→(2)
  2. B
    (P)→(5)(Q)→(4)(R)→(1)(S)→(2)(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(2)(P)→(5)(Q)→(4)(R)→(1)(S)→(2)
  3. C
    (P)→(3)(\mathrm{P}) \rightarrow(3) \quad(P)→(3)(Q) →(4)(R)→(1)\rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad→(4)(R)→(1)(S) →(5)\rightarrow(5)→(5)
  4. D
    (P)→(3)(Q)→(1)(R)→(4)(\mathrm{P}) \rightarrow(3) \quad(\mathrm{Q}) \rightarrow(1) \quad(\mathrm{R}) \rightarrow(4) \quad(P)→(3)(Q)→(1)(R)→(4)(S) →(5)\rightarrow(5)→(5)
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