Vector Algebra · JEE Advanced previous year questions

2024 2 questions

1. Q162024 · 26 May 2024 · Paper 1

   Let $\gamma \in \mathbb{R}$ be such that the lines $L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma }$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$. Match each entry in List-I to the correct entry in List-II. List-I: (P) $\gamma$ equals (Q) A possible choice for $\hat{n}$ is (R) $\overline{O{R}_1}$ equals (S) A possible value of $\overline{O{R}_1}\cdot \hat{n}$ is List-II: (1) $-\hat{i}-\hat{j}+\hat{k}$ (2) $\sqrt{\frac{3}{2}}$ (3) 1 (4) $\frac{1}{\sqrt{6}}\hat{i}-\frac{2}{\sqrt{6}}\hat{j}+\frac{1}{\sqrt{6}}\hat{k}$ (5) $\sqrt{\frac{2}{3}}$ The correct option is

   1. A.(P) $\to$ (3) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (2)
   2. B.(P) $\to$ (5) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (2)
   3. C.(P) $\to$ (3) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (5)
   4. D.(P) $\to$ (3) \quad (Q) $\to$ (1) \quad (R) $\to$ (4) \quad (S) $\to$ (5)
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   Answer: C

2. Q112024 · 26 May 2024 · Paper 2· Numerical answer

   Let $\vec{p}=2\hat{i}+\hat{j}+3\hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha ,\beta$, and $\gamma$, we have $$15\hat{i}+10\hat{j}+6\hat{k}=\alpha (2\vec{p}+\vec{q})+\beta (\vec{p}-2\vec{q})+\gamma (\vec{p}\times \vec{q}),$$ then the value of $\gamma$ is ____.

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   Answer: 2