3 questions from 2025
Let \( \vec{W}=\hat{i}+\hat{j}-2\hat{k} \), and \( \vec{u} \) and \( \vec{v} \) be two vector, 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 \( \vec{u}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k} \), \( -t\alpha+\beta+\gamma=0 \), \( \alpha-t\beta+\gamma=0 \), and \( \alpha+\beta-t\gamma=0 \).
Match each entry in List-I to the correct entry in List-II and choose the correct option.
Consider the vectors \( \vec{x}=\hat{i}+2\hat{j}+3\hat{k},\quad \vec{y}=2\hat{i}+3\hat{j}+\hat{k},\; \text{and}\; \vec{z}=3\hat{i}+\hat{j}+2\hat{k}. \)
For two distinct positive real numbers \( \alpha \) and \( \beta \), define \( \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\; \text{and}\; \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}. \)
If the vectors \( \vec{X}, \vec{Y} \) and \( \vec{Z} \) lie in a plane, then the value of \( \alpha+\beta-3 \) is ________.
For any two points \( M \) and \( N \) in the \( XY \)-plane, let \( \overrightarrow{MN} \) denote the vector from \( M \) to \( N \), and \( \vec{0} \) denote the zero vector. Let \( P. Q \) and \( R \) be three distinct points in the \( XY \)-plane. Let \( S \) be a point inside the triangle \( \Delta PQR \) such that \( \overrightarrow{SP}+5\overrightarrow{SQ}+6\overrightarrow{SR}=\vec{0} \).
Let \( E \) and \( F \) be the mid-points of the sides \( PR \) and \( QR \), respectively. Then the value of \(\displaystyle \frac{\text{length of the line segment } EF}{\text{length of the segment } ES}\) is ________.
