cmi-entrance 2024 Q10

cmi-entrance · India · ugmath 2 marks Complex Numbers Arithmetic Geometric Interpretation and Triangle/Shape Properties

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points]
Options:

The empty set
A singleton set
A line
A pair of lines
A circle
A plane perpendicular to $\mathbf{u}$
A plane parallel to $\mathbf{u}$
An infinite cone
A finite cone
A sphere
None of the above

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.

Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.

$\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points]

Options:
\begin{enumerate}
  \item The empty set
  \item A singleton set
  \item A line
  \item A pair of lines
  \item A circle
  \item A plane perpendicular to $\mathbf{u}$
  \item A plane parallel to $\mathbf{u}$
  \item An infinite cone
  \item A finite cone
  \item A sphere
  \item None of the above
\end{enumerate}

Paper Questions

QB1 QB2 QB3 QB4 QB5 QB6 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22