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What is the principal argument of −1−i?

• $\frac{-3\pi}{4}$
• $\frac{\pi}{4}$
• $\frac{-\pi}{4}$
• $\frac{3\pi}{4}$

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For $z^{4}$=−1, which coefficients of π list every argument once in [0,2π)?

• $\frac{1}{4}$, $\frac{3}{4}$, $\frac{5}{4}$, $\frac{7}{4}$
• $\frac{1}{4}$, $\frac{3}{4}$, $\frac{5}{4}$, $\frac{7}{4}$, $\frac{9}{4}$
• 0, $\frac{1}{2}$, 1, $\frac{3}{2}$
• $\frac{1}{4}$ only

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For A=$\begin{pmatrix}2&1\\0&2\end{pmatrix}$, which statement is justified?

• The eigenspace has dimension 1
• Two eigenvalues counted with multiplicity guarantee two independent vectors
• Every vector is an eigenvector because both diagonal entries are 2
• A has no eigenvectors because the determinant is nonzero

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For A=$\begin{pmatrix}2&1\\0&2\end{pmatrix}$, what is the upper-right entry of $A^{3}$?

• 12
• 1
• 6
• 8

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Where does the binomial series for [1−2(x−1)]⁻² converge?

• $\frac{1}{2}$<x<$\frac{3}{2}$
• −1<x<1
• $\frac{-1}{2}$<x<$\frac{1}{2}$
• $\frac{1}{2}$≤x≤$\frac{3}{2}$

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For y″−4y′+4y=$e^{2x}$, which trial can supply the forcing?

• Kx²$e^{2x}$
• K$e^{2x}$
• Kx$e^{2x}$
• Kx²

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One petal of r=2sin(3θ) is traced for 0≤θ≤$\frac{\pi}{3}$. What is its area?

• $\frac{\pi}{3}$
• $\frac{2\pi}{3}$
• π
• 2π

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Why does Newton’s method for $x^{2}$−5 fail at $x_{0}$=0?

• The derivative is zero, so the tangent has no x-intercept
• The function value is negative, so no real root exists
• Zero is already a root
• Newton’s method requires every starting value to be positive

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There are 3 outcomes in A∩B, 2 in A∩not B and 5 in not A∩B. What is P(A|B)?

• $\frac{3}{8}$
• $\frac{3}{5}$
• $\frac{3}{10}$
• $\frac{5}{8}$

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A discrete rejection rule has null rejection probability 0.0625 at nominal level 0.1. What does 0.0625 describe?

• The actual probability of rejecting when $H_{0}$ is true
• The probability that $H_{0}$ is true
• The probability that $H_{1}$ is false after rejection
• An error because every 10% test must reject exactly 10% under $H_{0}$

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A 2 kg particle has velocity −3 m/s. What is its kinetic energy?

• 9 J
• −9 J
• −6 J
• 3 J

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A minimum spanning tree has total weight 20. Which conclusion follows?

• It connects every vertex with minimum total edge weight
• Every pairwise route in it is shortest
• Its weight is the minimum travelling-salesperson tour
• It must contain a cycle through all vertices

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In an induction step proving P(n) for all n≥1, what may be assumed?

• P(k) for an arbitrary integer k≥1
• P(k+1), because that is the target
• P(1) alone, then the conclusion for every n
• P(n) for all n≥1

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What is the $x^{4}$ coefficient of ln(1+$x^{2}$)?

• $\frac{-1}{2}$
• $\frac{-1}{4}$
• $\frac{1}{2}$
• 0

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A plane transformation has determinant −2. What happens to area?

• It doubles; orientation reverses
• Area becomes negative
• It halves; orientation reverses
• It doubles; orientation stays the same

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The closest point on the infinite line P(t)=(4t,0,0) to Q=(6,1,0) has t=$\frac{3}{2}$. For 0≤t≤1, where is the closest point?

• t=1
• t=$\frac{3}{2}$
• t=0
• t=$\frac{1}{2}$

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B=P⁻¹AP, where P is invertible. Which statement must hold?

• A and B have the same eigenvalues
• A and B have the same entries
• Every eigenvector is unchanged
• B must be diagonal

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A=2I is a 2×2 matrix. What is the dimension of its eigenspace for eigenvalue 2?

• 2
• 1
• 0
• It depends on whether 2 is positive

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For invertible A and B, which expression equals (AB)⁻¹?

• B⁻¹A⁻¹
• A⁻¹B⁻¹
• AB
• A⁻¹+B⁻¹

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A 3D transformation has matrix diag(2,3,4). What is its volume scale factor?

• 24
• 9
• 12
• $\frac{1}{24}$

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For |z−2|<3 and Im(z)≥0, which boundary points belong to the region?

• Interior points of the real-axis diameter, but not the circular arc
• The whole arc and diameter
• The arc but not the diameter
• No point on the real axis

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How many complex numbers satisfy |z−1|=2 and Im(z)=2?

• 1
• 0
• 2
• Infinitely many

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Both z and w have principal argument $\frac{3\pi}{4}$. What is the principal argument of zw?

• $\frac{-\pi}{2}$
• $\frac{3\pi}{2}$
• $\frac{\pi}{2}$
• 9π²/16

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What can be lost by dividing cos x(2sin x−1)=0 by cos x on 0≤x<2π?

• $\frac{\pi}{2}$ and $\frac{3\pi}{2}$
• $\frac{\pi}{6}$ and $\frac{5\pi}{6}$
• Only x=0
• No solutions

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How many real solutions does cosh x=2 have before any domain restriction?

• 2
• 1
• 0
• Infinitely many

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At x=1, what happens to the Maclaurin series x−$x^{2}$/2+$x^{3}$/3−… for ln(1+x)?

• It converges to ln 2
• It diverges because every endpoint is excluded
• It converges to 1
• Every term after the first is zero

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What is the coefficient of $x^{2}$ in (1−2x)⁻²?

• 12
• 4
• −12
• 6

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For y′−2y=3$e^{2x}$, which particular trial can work?

• Kx$e^{2x}$
• K$e^{2x}$
• K$e^{-2x}$
• Kx

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For y″−4y′+4y=0, which is the full complementary function?

• (A+Bx)$e^{2x}$
• A$e^{2x}$
• A$e^{2x}$+B$e^{-2x}$
• A+Bx

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For k>0, when is r=k+(9/k)cos θ nonnegative for every θ?

• k≥3
• k>0
• k≤3
• k≥9

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For r=sin(3θ), what does ½∫₀²π$r^{2}$dθ count?

• Twice the enclosed area
• The enclosed area once
• The area of one petal
• Zero because r changes sign

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A line is r=(1,2,0)+t(1,−1,0), and a plane is x+y=3. What is their relationship?

• The whole line lies in the plane
• They meet only at t=0
• They are disjoint
• They are perpendicular

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A line is r=(1,1,0)+t(1,−1,0), and a plane is x+y=3. What is their relationship?

• They are parallel and disjoint
• The whole line lies in the plane
• There is one intersection
• The origin is the intersection

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A proof establishes P(1) and P(k)⇒P(k+1) for each integer k≥1. What has it established?

• P(n) for every integer n≥1
• P(n) for every integer n≥0
• Only P(1) and P(2)
• P(n) for every real n≥1

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Under $H_{0}$, X has Bin(4,$\frac{1}{2}$). At nominal 10%, which upper-tail critical region is admissible?

• X≥4
• X≥3
• X≥2
• X≥1

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Particles approach with signed velocities $u_{1}$=4 and $u_{2}$=−2 m/s. What is their approach speed?

• 6 m/s
• 2 m/s
• −6 m/s
• −2 m/s

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In an isolated collision with positive masses, nonzero approach speed and 0≤e<1, what is conserved?

• Momentum, while kinetic energy decreases
• Both momentum and kinetic energy
• Kinetic energy, while momentum decreases
• Neither momentum nor kinetic energy

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A network has a negative edge weight. What may be concluded about the usual Dijkstra settled-node rule?

• Its usual correctness guarantee no longer applies
• It is guaranteed correct if the graph is connected
• A shortest route must have negative total weight
• The first settled node must be wrong

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An activity has total float 2. Its duration increases by 3, with all other data fixed and no resource constraints. What is the project delay?

• 1 time unit
• 3 time units
• 2 time units
• No delay

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What is the remainder when $7^{6}$ is divided by 7?

• 0
• 1
• 6
• There is no remainder because Fermat’s theorem fails

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