bac-s-maths 2013 Q2

bac-s-maths · France · pondichery 4 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation
For each question, four answer options are given, of which only one is correct. For each question, indicate, without justification, the correct answer on your paper. A correct answer is worth 1 point. An incorrect answer or the absence of an answer gives neither points nor deducts any points.
Space is referred to an orthonormal coordinate system. $t$ and $t ^ { \prime }$ denote real parameters. The plane (P) has equation $x - 2 y + 3 z + 5 = 0$. The plane (S) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t + 2 t ^ { \prime } \\ y & = - t - 2 t ^ { \prime } \\ z & = - 1 - t + 3 t ^ { \prime } \end{aligned} \right.$ The line (D) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t \\ y & = - t \\ z & = - 1 - t \end{aligned} \right.$ We are given the points in space $\mathrm { M } ( - 1 ; 2 ; 3 )$ and $\mathrm { N } ( 1 ; - 2 ; 9 )$.
  1. A parametric representation of the plane (P) is: a. $\left\{ \begin{array} { r l r } x & = & t \\ y & = & 1 - 2 t \\ z & = & - 1 + 3 t \end{array} \right.$ b. $\left\{ \begin{array} { r l r } x & = t + 2 t ^ { \prime } \\ y & = 1 - t + t ^ { \prime } \\ z & = - 1 - t \end{array} \right.$ c. $\left\{ \begin{aligned} x & = t + t ^ { \prime } \\ y & = 1 - t - 2 t ^ { \prime } \\ z & = 1 - t - 3 t ^ { \prime } \end{aligned} \right.$ d. $\left\{ \begin{array} { l } x = 1 + 2 t + t ^ { \prime } \\ y = 1 - 2 t + 2 t ^ { \prime } \\ z = - 1 - t ^ { \prime } \end{array} \right.$
  2. a. The line (D) and the plane (P) are secant at point A(-8;3;2). b. The line (D) and the plane (P) are perpendicular. c. The line (D) is a line of the plane (P). d. The line (D) and the plane (P) are strictly parallel.
  3. a. The line (MN) and the line (D) are orthogonal. b. The line (MN) and the line (D) are parallel. c. The line (MN) and the line (D) are secant. d. The line (MN) and the line (D) are coincident.
  4. a. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are parallel. b. The line $( \Delta )$ with parametric representation $\left\{ \begin{aligned} x & = t \\ y & = - 2 - t \\ z & = - 3 - t \end{aligned} \right.$ is the line of intersection of the planes (P) and (S). c. The point M belongs to the intersection of the planes (P) and (S). d. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are perpendicular.
For each question, four answer options are given, of which only one is correct. For each question, indicate, without justification, the correct answer on your paper. A correct answer is worth 1 point. An incorrect answer or the absence of an answer gives neither points nor deducts any points.

Space is referred to an orthonormal coordinate system. $t$ and $t ^ { \prime }$ denote real parameters.\\
The plane (P) has equation $x - 2 y + 3 z + 5 = 0$.\\
The plane (S) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t + 2 t ^ { \prime } \\ y & = - t - 2 t ^ { \prime } \\ z & = - 1 - t + 3 t ^ { \prime } \end{aligned} \right.$\\
The line (D) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t \\ y & = - t \\ z & = - 1 - t \end{aligned} \right.$\\
We are given the points in space $\mathrm { M } ( - 1 ; 2 ; 3 )$ and $\mathrm { N } ( 1 ; - 2 ; 9 )$.

\begin{enumerate}
  \item A parametric representation of the plane (P) is:\\
a. $\left\{ \begin{array} { r l r } x & = & t \\ y & = & 1 - 2 t \\ z & = & - 1 + 3 t \end{array} \right.$\\
b. $\left\{ \begin{array} { r l r } x & = t + 2 t ^ { \prime } \\ y & = 1 - t + t ^ { \prime } \\ z & = - 1 - t \end{array} \right.$\\
c. $\left\{ \begin{aligned} x & = t + t ^ { \prime } \\ y & = 1 - t - 2 t ^ { \prime } \\ z & = 1 - t - 3 t ^ { \prime } \end{aligned} \right.$\\
d. $\left\{ \begin{array} { l } x = 1 + 2 t + t ^ { \prime } \\ y = 1 - 2 t + 2 t ^ { \prime } \\ z = - 1 - t ^ { \prime } \end{array} \right.$
  \item a. The line (D) and the plane (P) are secant at point A(-8;3;2).\\
b. The line (D) and the plane (P) are perpendicular.\\
c. The line (D) is a line of the plane (P).\\
d. The line (D) and the plane (P) are strictly parallel.
  \item a. The line (MN) and the line (D) are orthogonal.\\
b. The line (MN) and the line (D) are parallel.\\
c. The line (MN) and the line (D) are secant.\\
d. The line (MN) and the line (D) are coincident.
  \item a. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are parallel.\\
b. The line $( \Delta )$ with parametric representation $\left\{ \begin{aligned} x & = t \\ y & = - 2 - t \\ z & = - 3 - t \end{aligned} \right.$ is the line of intersection of the planes (P) and (S).\\
c. The point M belongs to the intersection of the planes (P) and (S).\\
d. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are perpendicular.
\end{enumerate}