Vector Algebra · JEE Advanced previous year questions

2025 2 questions

1. Q162025 · 18 May 2025 · Paper 1

   Let $\vec{w}=\hat{ı}+\hat{ȷ}-2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u}\times \vec{v}=\vec{w}$ and $\vec{v}\times \vec{w}=\vec{u}$. Let $\alpha ,\beta ,\gamma$, and $t$ be real numbers such that $$\vec{u}=\alpha \hat{ı}+\beta \hat{ȷ}+\gamma \hat{k},-t\alpha +\beta +\gamma =0,\alpha -t\beta +\gamma =0,\text{and }\alpha +\beta -t\gamma =0.$$ Match each entry in List-I to the correct entry in List-II and choose the correct option. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) }|\vec{v}{|}^2\text{ is equal to}& \text{(1) }0\\ \text{(Q) If }\alpha =\sqrt{3}\text{, then }{\gamma }^2\text{ is equal to}& \text{(2) }1\\ \text{(R) If }\alpha =\sqrt{3}\text{, then }(\beta +\gamma {)}^2\text{ is equal to}& \text{(3) }2\\ \text{(S) If }\alpha =\sqrt{2}\text{, then }t+3\text{ is equal to}& \text{(4) }3\\ & \text{(5) }5\end{array}$$

   1. A.(P) → (2) (Q) → (1) (R) → (4) (S) → (5)
   2. B.(P) → (2) (Q) → (4) (R) → (3) (S) → (5)
   3. C.(P) → (2) (Q) → (1) (R) → (4) (S) → (3)
   4. D.(P) → (5) (Q) → (4) (R) → (1) (S) → (3)
   Show answer

   Answer: A

2. Q122025 · 18 May 2025 · Paper 2· Numerical answer

   Consider the vectors $$\vec{x}=\hat{ı}+2\hat{ȷ}+3\hat{k},\vec{y}=2\hat{ı}+3\hat{ȷ}+\hat{k},\text{and}\vec{z}=3\hat{ı}+\hat{ȷ}+2\hat{k}.$$ For two distinct positive real numbers $\alpha$ and $\beta$, define $$\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\text{and}\vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}.$$ If the vectors $\vec{X},\vec{Y}$, and $\vec{Z}$ lie in a plane, then the value of $\alpha +\beta -3$ is ____.

   Show answer

   Answer: -2