CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 1, Spring , 2006

• Your exam shall consist of problems similar to the homework and quiz problems. All homework problems are posted on the course web site.

• Monday, February 13th will be a review day. Please ask any questions you may have from the sections 2.4-2.7(W), 12.1-12.5(S)

• The exam will be held on Wednesday February 15th in Porter Hall 100.

• Notes and texts will not be permitted during the exam.

• Calculators or other elextronic devices are not permitted.

• You will have 50 minutes for the exam, and must stop when told to stop.

• Please consult the course web page for exam policies - these will be enforced. What follows is a practice exam. This is designed for you to take in 50 minutes. The actual exam will consist of different questions. To fully prepare please practice many homework problems.

1. Multiply:
   
   $$\left[ \begin{array}{rrr} 1 & 3 & -1 \\ 2 & 1 & 4 \end{arra... ...begin{array}{rr} 2 & 3 \\ 0 & 1 \\ -1 & 4 \end{array}\right]$$
   
   

2. Solve the following system of equations by applying Gaussian elimination on an appropriate augmented matrix. Express your result in vector form.
   
   $$\begin{array}{rrrrrrrrrrr} x_1 & + & 2x_2 & - & x_3 & + & 2x... ...+ & 4x_2 & - & 4x_3 & - & 3x_4 & + & 12x_5 & = & -2 \end{array}$$
   
   

3. Find the inverse (if it exists) of the matrix:
   
   $$\left[ \begin{array}{rrr} 2 & 3 & -9 \\ -4 & -5 & 15 \\ 2 & 1 & -6 \end{array}\right]$$
   
   

4. Find the center and radius of the sphere given by the equation $x^2-4x+y^2+y+z^2+4=0$.

5. Find the vector projection ${\bf {proj} }_{ \vec{a}} \vec{b}$ of the vector $\vec{b}=\langle 2, -3, -1 \rangle$ onto the vector $\vec{a}=\langle 2,-2,1 \rangle$.

6. Find the volume of the parallelepiped that has adjacent edges $PQ$, $PR$, and $PS$, with $P(3,-1,2)$, $Q(5,3,1)$, and $R(-1, 0, 1)$, and $S(-2,-5,0)$.

7. Find the parametric equations of the line containing the points $P(2,4,2)$ and $Q(3,4,0)$.

8. Find the equation of the plane containing the points $P(2,3,1)$, $Q(5,0,-2)$, and $R(1,9,0)$.