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Math 240 Final Exam
Spring 2012
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for notes while you take this exam. No calculators, no course notes, no books, no help from your neighbors.
Show all work on multiple choice questions: one point will be given for clearly circling the correct answer,
and up to four points will be giving for the supporting work. Please clearly mark a multiple choice option
for each problem. Remember to put your name at the top of this page. Good luck.
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Question Points Your
Number Possible Score
Total 65
- Solve the equations for x. 2 x + 3 y + 2 z = 1 x + 0 y + 3 z = 2 2 x + 2 y + 3 z = 3 Hint: If A =
(^) then A−^1 =
a) x = − 1 b) x = 0 c) x = 2 d) x = 5 e) x = 7 f) x = 11 g) none of the above
- Calculate the outward flux of F~ across S if F~ (x, y, z) = 3xy^2 ~i + xez~j + z^3 ~k and S is the surface of the solid bounded by the cylinder y^2 + z^2 = 1 and the planes x = −1 and x = 2. a) 0 b) − π 4 c) 11 π 8 d) 3π e) 9 π 5 f) 9 π 2 g) none of the above
- Compute the outward flux of ∇ × F~ through the surface of the ellipsoid 2x^2 + 2y^2 + z^2 = 8 lying above the plane z = 0, where F~ = (3x − y)~i + (x + 3y)~j + (1 + x^2 + y^2 + z^2 )~k.
a) 0 b) 2π c) 3π d) 8π e) 12π f) 16π g) none of the above
- Identify all possible eigenvalues of an n × n matrix A if A which satisfies the following matrix equation:
A − 2 I = −A^2.
Must A be invertible? Record your answer in the box below and provide justification for your answer. a) λ = 0, 1 b) λ = 0, 2 c) λ = 0, 1 , − 2 d) λ = 1, − 2 e) λ = 0, 1 , − 3 f) λ = 1, − 3 g) none of the above
Is A invertible?
- Solve the differential equation 9 x^2 y′′^ + 2y = 0 on the interval (0, ∞) subject to the initial conditions y(1) = 1 and y′(1) = 43. a) y = 2x (^23) − 3 x (^13) b) y = 3x (^23) − 2 x (^13) c) y = 3x (^32) − 3 x^3 d) y = 3x (^32) − 2 x^3 e) y = 2x^2 − 3 x f) y = 3x^2 − 2 x g) none of the above
- Let ~ω := 〈 1 , 2 , 3 〉, and let ~r(t) = 〈x(t), y(t), z(t)〉. Now consider the differential equation
d dt ~r = ~ω × ~r.
Select the answer which correctly expresses this system of equations in matrix notation when
X^ ~(t) =
x(t) y(t) z(t)
Do not solve the system.
a) d dt
X~ =
(^) X~ b) d dt
X~ =
(^) X~ c) d dt
X~ =
X~
d) d dt
X~ =
(^) X~ e) d dt
X~ =
(^) X~ f) d dt
X~ =
X~
g) none of the above
- Let y be a function satisfying y(0) = y′(0) = y′′(0) = 0 which is a solution of the ODE
y′′′^ − 4 y′′^ + 4y′^ = 4.
Compute y(1). a) y(1) = − 5 b) y(1) = 4 c) y(1) = − 3 d) y(1) = 2 e) y(1) = − 1 f) y(1) = 0 g) none of the above
- Solve the following system of differential equations subject to the initial conditions y 1 (0) = 1 and y 2 (0) = 3. Clearly state your solution. What is y 1 (1)? dy 1 dx = 3y 1 − y 2 dy 2 dx =^ y^1 +^ y^2
a) y 1 (1) = 2e b) y 1 (1) = 2e − 1 c) y 1 (1) = 3 d) y 1 (1) = 5e^2 e) y 1 (1) = 7e f) y 1 (1) = −e^2 g) none of the above
- Circle “T” for true or “F” for false in the space provided to the left of the following statements. You DO NOT need to justify your answer for full credit.
( T F ) Every 2 × 2 diagonalizable matrix with repeated eigenvalue is a diagonal matrix.
( T F ) There is a vector field F~ such that ∇ × F~ = 〈x, y, z〉.
( T F ) If det(A) = 0, then the system A X~ = 0 has infinitely many solutions.
( T F ) If y 1 and y 2 are solutions to a non-homogeneous linear differential equation, then y 1 + y 2 is also a solution.
( T F ) If A and B are square matrixes such that AB^2 = I, then B is invertible.
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