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3e-spring2011-exam_1_sample1

3e-spring2011-exam_1_sample1 - Math 3E Exam#1 Sample Laney...

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Math 3E — Exam #1 Sample Laney College, Fall 2006 Fred Bourgoin 1. Consider the following system of equations. y + 2 kz = 0 x + 2 y + 6 z = 2 kx + 2 z = 1 For what values of k does the system have no solutions, exactly one solution, and infinitely many solutions? Explain your answers. 2. Circle the matrix (or matrices) in reduced row echelon form. A = 1 1 0 1 0 0 1 0 0 0 0 2 , B = 1 0 0 0 0 1 0 1 0 0 1 2 0 0 0 0 , C = 1 2 0 0 0 0 0 0 0 0 1 0 , D = 1 2 0 0 0 0 1 0 0 0 0 1 3. Compute the products. (a) 1 1 - 1 - 5 1 1 1 - 5 3 1 2 3 (b) 1 0 - 1 0 1 1 1 - 1 - 2 1 2 3 3 2 1 2 1 3 4. Determine whether each of the given types of geometric transformations in R 2 is invertible. Briefly justify your answers. (a) Rotation (b) Reflection (c) Projection 1
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5. Let ⃗x = [ 1 2 ] and y = [ - 1 2 ] be vectors in R 2 . For each matrix below, identify the type of geometric transformation it represents (be precise!), and determine the images of ⃗x and y . (a) [ 0 - 1 1 0 ] (b) [ 0 . 8 0 . 6 0 . 6 - 0 . 8 ] (c) [ 0 . 5 0 . 5 0 . 5 0 . 5 ] 6. What is the rank of the matrix A = 1 2 0 2 3 0 0 1 3 2 0 0 1 4 - 1 0 0 0 0 1 ? 7. Find a diagonal matrix D such that D 5 3 - 9 = 2 0 1 . 8. Find the inverse of A = 2 5 0 0 1 3 0 0 0 0 1 2 0 0 2 5 . 9. If A is an invertible matrix and c is a nonzero scalar, is the matrix cA invertible? If so, what is the relationship between A 1 and ( cA ) 1 ? 10. The momentum P of a system of 2 particles in space with masses m 1 and m 2 , and velocities ⃗v 1 and ⃗v 2 is defined as P = m 1 ⃗v 1 + m 2 ⃗v 2 . Consider two elementary particles with velocities ⃗v 1 = 1 1 1 a nd ⃗v 2 = 4 7 10 . The particles collide. After the collision, their respective velocities are observed to be ⃗w 1 = 4 7 4 a nd ⃗w 2 = 2 3 8 . 2
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Assume that the momentum of the system is conserved throughout the collision. What does this experiment tell you about the masses of the two particles?
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