C877 Mathematical Modeling and Applications
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Free C877 Mathematical Modeling and Applications Questions
In the Rosenzweig-MacArthur predator-prey model with Holling type II functional response, the predator isocline is:
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Vertical line
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Hump-shaped
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Horizontal line
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Diagonal line
Explanation
Predator equation: dy/dt = y (e a x/(1 + a h x) − d). Set to zero → e a x/(1 + a h x) = d → x = d(1 + a h x)/(e a). Solving gives a hump-shaped (inverted U) isocline in (x,y) phase space. This hump creates the paradox of enrichment: increasing prey carrying capacity destabilizes the equilibrium, leading to large-amplitude cycles—explains plankton blooms and lynx-hare crashes.
Correct Answer Is:
Hump-shaped
In the PageRank algorithm, the damping factor d (usually 0.85) represents:
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The probability of following a link vs jumping randomly
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The probability of staying on the same page
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The convergence rate
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The number of iterations
Explanation
PageRank models a random surfer who with probability d follows a random outgoing link, and with probability 1−d teleports to any page uniformly. This irreducible aperiodic Markov chain guarantees a unique stationary distribution (the PageRank vector). d = 0.85 is the sweet spot: high enough to respect link structure, low enough to handle dead-ends and speed convergence.
Correct Answer Is:
The probability of following a link vs jumping randomly
The period of a simple pendulum is measured for different lengths. Variables: T (period), L (length), g (gravity), m (mass). Using dimensional analysis, which variable does NOT affect the period?
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T
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L
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G
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m
Explanation
Dimensions: [T] = T, [L] = L, [g] = L T^{-2}, [m] = M. There are 4 variables and 3 fundamental dimensions (M, L, T), so 4−3 = 1 dimensionless group. The only possible Π is T √(g/L), meaning T = constant × √(L/g). Mass M appears nowhere in the final Π group, proving theoretically that pendulum period is independent of mass (Galileo’s famous insight).
Correct Answer Is:
m
Which model is most appropriate for predicting the spread of a rumor in a fixed population where each person tells 3 others on average, but slows as fewer remain unaware?
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Logistic growth
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Exponential growth
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Linear growth
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Gompertz growth
Explanation
Rumor spread is density-dependent: initial rapid transmission (like exponential) slows as the pool of unaware individuals shrinks, mirroring logistic growth where the growth rate is proportional to both current spreaders and remaining susceptibles, i.e., dR/dt = k R (M - R), with M total population. This S-shaped curve fits bounded phenomena like epidemics (SIR simplified), technology adoption, or information diffusion, unlike unbounded exponential or linear models.
Correct Answer Is:
Logistic growth
In goal programming, if we have multiple goals with priorities P1 > P2 > P3, the method used is:
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Preemptive goal programming
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Non-preemptive (weighted) goal programming
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Lexicographic ordering
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Archimedean goal programming
Explanation
When goals have strict priority levels (P1 must be satisfied before even considering P2), we use preemptive (lexicographic) goal programming. The algorithm minimizes deviation from the highest priority first, then fixes that value and moves to the next priority level hierarchical objectives like “first minimize fatalities, then minimize cost”.
Correct Answer Is:
Preemptive goal programming
In dimensional analysis, the Buckingham Pi theorem states that if there are n variables with k independent dimensions, the number of dimensionless Pi groups is:
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n − k
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n + k
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n × k
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k − n
Explanation
Buckingham’s Π-theorem is the cornerstone of dimensional analysis. If a physical problem involves n variables (including units) that can be described by k fundamental dimensions (M, L, T, etc.), then the phenomenon can be fully described by exactly n − k independent dimensionless parameters (Π groups). This reduction dramatically simplifies modeling, enabling scale-model testing and derivation of empirical formulas.
Correct Answer Is:
n − k
In Markov chains, what does the transition matrix P represent?
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Probability of moving from state i to state j
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Initial state vector
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Steady-state probabilities
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Absorbing states
Explanation
The (i,j)-th entry P_{ij} of the transition matrix P is the probability of going from state i to state j in one step. The rows sum to 1. Multiplying the current state vector π by P gives the next-step distribution, and π P^n gives the distribution after n steps. This matrix fully defines the stochastic process.
Correct Answer Is:
Probability of moving from state i to state j
The Gauss-Seidel method differs from Jacobi because:
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It uses newly computed values immediately
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It uses over-relaxation
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It is explicit
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It only works for tridiagonal matrices
Explanation
Gauss-Seidel updates each point using the most recent values available (left and upper neighbors already updated in the same sweep), roughly halving the number of iterations needed compared to Jacobi. Successive over-relaxation (SOR) takes this further with a relaxation parameter ω > 1.
Correct Answer Is:
It uses newly computed values immediately
A cup of coffee cools from 90°C to 60°C in 10 minutes in a 20°C room. How long to cool to 25°C?
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30 minutes
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40 minutes
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50 minutes
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60 minutes
Explanation
Using Newton’s law: T(t) = 20 + 70e^{-kt}. At t=10, 60 = 20 + 70e^{-10k} → 40 = 70e^{-10k} → e^{-10k} = 4/7 → −10k = ln(4/7) → k = −(1/10)ln(4/7). For T=25: 25 = 20 + 70e^{-kt} → 5/70 = e^{-kt} → e^{-kt} = 1/14 → −kt = ln(1/14) = −ln14 → t = ln14 / k = ln14 × 10 / ln(7/4) ≈ 2.639 × 10 / 0.5596 ≈ 47.1 minutes ≈ 50 minutes
Correct Answer Is:
50 minutes
In game theory, a dominant strategy is:
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A strategy that is best no matter what the opponent does
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A strategy that maximizes joint payoff
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A strategy that is chosen in Nash equilibrium
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A strategy that is Pareto optimal
Explanation
A strategy is dominant if it gives a higher payoff than any other strategy regardless of what the opponent plays. For example, in Prisoner’s Dilemma, “confess” strictly dominates “stay silent” for both players. Dominant strategies are the strongest solution concept—if one exists, rational players always play it, and the outcome is predictable even without communication.
Correct Answer Is:
A strategy that is best no matter what the opponent does
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