Comparison of Convergence Tests
Compare the ratio test, root test, and integral test for convergence of an infinite series. Apply each test to determine whether the following series converge: (a) sum_{n=1}^{∞} n^2 / 2^n, (b) sum_{n=2}^{∞} 1/(n ln n), and (c) sum_{n=1}^{∞} (-1)^n / n. For each series, state which test is most appropriate and justify your choice.
Define Big-O, Big-Theta, and Big-Omega notation. Classify the time complexity of linear search, binary search, bubble sort, and merge sort, and explain why each holds.
Provide formal definitions for Big-O, Big-Theta, and Big-Omega notations. Then, for each of the following algorithms: linear search, binary search, bubble sort, and merge sort, state its best-case, average-case, and worst-case time complexity using the appropriate asymptotic notation. Justify each classification with a brief explanation of the algorithm's behavior.
Describe the phases of a neuronal action potential and the roles of voltage-gated Na⁺ and K⁺ channels and the Na⁺/K⁺-ATPase pump.
Outline the sequence of events during a neuronal action potential, including the resting state, depolarization, repolarization, and hyperpolarization. Explain how voltage-gated Na⁺ and K⁺ channels contribute to each phase, and describe the function of the Na⁺/K⁺-ATPase pump in restoring ionic gradients.
Mendel's Laws and Dihybrid Cross
State Mendel's laws of segregation and independent assortment. Then, work a dihybrid cross between two heterozygous pea plants (RrYy × RrYy), where R = round seeds (dominant), r = wrinkled seeds (recessive), Y = yellow seeds (dominant), y = green seeds (recessive). Predict the phenotypic ratio in the F1 generation.
Compare C3 and C4 photosynthesis pathways, explaining the role of PEP carboxylase and bundle sheath cells in C4 plants, and why C4 plants outperform C3 plants in hot, dry environments.
In a comparative analysis, describe the key differences between C3 and C4 photosynthesis pathways. Specifically, explain the function of PEP carboxylase and the significance of bundle sheath cells in C4 plants. Additionally, discuss the physiological reasons why C4 plants are more efficient than C3 plants under conditions of high temperature and drought.
Describe the four phases of bacterial population growth in batch culture and explain the factors driving transitions between phases.
In a typical batch culture, bacteria exhibit a characteristic growth curve with four distinct phases. Outline each phase and discuss the metabolic and environmental factors that cause the population to move from one phase to the next.
Comparison of Simply Supported and Cantilever Beams under Uniformly Distributed Load
For a simply supported beam of length L carrying a uniformly distributed load w per unit length, derive the expression for the maximum bending moment and state its location. Repeat for a cantilever beam of the same length and loading. Compare the magnitudes of the maximum bending moments.
State Kirchhoff's current law and voltage law. Apply them to a two-loop circuit with two batteries and three resistors to solve for all branch currents.
Consider a two-loop circuit with two batteries: V1 = 12 V and V2 = 6 V, and three resistors: R1 = 2 Ω, R2 = 4 Ω, and R3 = 6 Ω. The circuit is arranged as follows: Battery V1 is in the left loop with R1 and R2; battery V2 is in the right loop with R2 and R3; R2 is shared between the two loops. State Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL). Then, using these laws, set up the equations and solve for all branch currents (I1 through R1, I2 through R2, and I3 through R3). Assume all currents flow clockwise in each loop.
State Bernoulli's equation for steady, incompressible, frictionless fluid flow along a streamline. Apply it to derive Torricelli's theorem for the exit velocity of a tank draining through a small hole at depth h.
Consider a large open tank filled with water to a depth h above a small hole in the side. The tank is open to the atmosphere, and the hole is small enough that the water surface in the tank can be assumed to be stationary. Assuming steady, incompressible, frictionless flow, state Bernoulli's equation and use it to derive an expression for the velocity of the water exiting the hole (Torricelli's theorem).
Compare BFS and DFS on Graphs
Compare breadth-first search (BFS) and depth-first search (DFS) on a graph. Discuss the data structures they use, their time and space complexity, and the kinds of problems each is best suited for, such as shortest path in unweighted graphs, cycle detection, and topological sort.
Hash Table Collisions: Separate Chaining vs. Open Addressing
Explain what a hash table collision is. Compare separate chaining and open addressing (including linear probing, quadratic probing, and double hashing) as collision resolution strategies. Discuss how the load factor affects the average lookup time for each strategy.
Compare Quicksort and Mergesort
Compare quicksort and mergesort in terms of average-case and worst-case time complexity, space complexity, and stability. Also, describe one pivot-selection strategy that improves quicksort's worst-case performance. Finally, state when each algorithm is preferable in practice.
Derive least-squares estimators for slope and intercept in simple linear regression and interpret R².
Consider the simple linear regression model y = β₀ + β₁x + ε, where ε are independent errors with mean 0 and constant variance. Derive the least-squares estimators for β₀ and β₁. Then, define and interpret the coefficient of determination R² in terms of the proportion of variance explained.
Compare Normal and Binomial Distributions with Approximation
Compare the normal and binomial distributions. State the conditions under which a binomial distribution is well-approximated by a normal distribution, and apply the approximation to find P(X ≥ 60) for X ~ Binomial(n=100, p=0.5).
Hypothesis Testing: Type I and Type II Errors with a One-Tailed Z-Test
Define Type I and Type II errors in hypothesis testing. Then, conduct a one-tailed z-test for a sample mean given the following: sample size n = 36, sample mean = 52, hypothesized population mean = 50, population standard deviation σ = 6, and significance level α = 0.05. State the null and alternative hypotheses, compute the test statistic, determine the critical value, and state the decision regarding the null hypothesis.
Solve a first-order separable ODE and a first-order linear ODE.
Consider the initial value problem: dy/dx = xy, with y(0) = 2. Solve this separable ODE. Then, solve the first-order linear ODE: dy/dx + 2y = e^x. Provide the general solution for each and, for the first, the particular solution satisfying the initial condition.
Define specific heat capacity and solve a calorimetry problem.
Define specific heat capacity. A 200 g iron block at 100 °C is dropped into 500 g of water at 20 °C. Assuming no heat loss to the surroundings, find the equilibrium temperature. (Specific heat capacity of iron = 450 J/kg·°C, water = 4186 J/kg·°C.)
Explain wave-particle duality with reference to the photoelectric effect (Einstein) and de Broglie wavelength. Calculate the de Broglie wavelength for an electron with kinetic energy 100 eV.
Wave-particle duality is a fundamental concept in quantum mechanics. Using Einstein's explanation of the photoelectric effect, discuss how light exhibits particle-like behavior. Then, using de Broglie's hypothesis, explain how matter exhibits wave-like behavior. Finally, calculate the de Broglie wavelength for an electron with a kinetic energy of 100 eV.
Faraday's Law, Lenz's Law, and EMF in a Rotating Loop
State Faraday's law of electromagnetic induction and Lenz's law. A circular loop of radius r and N turns rotates with uniform angular velocity ω in a uniform magnetic field B perpendicular to the axis of rotation. Derive an expression for the induced EMF in the loop and identify the device as a generator.
Newton's Laws and the Inclined Plane
State Newton's three laws of motion. Then, consider a block of mass m on an inclined plane with angle θ relative to the horizontal. The coefficient of kinetic friction between the block and the plane is μk. Derive an expression for the acceleration of the block as it slides down the plane in terms of m, g, θ, and μk.
Gibbs Free Energy and Spontaneity
Define Gibbs free energy (G) and explain how the sign of ΔG predicts the spontaneity of a chemical reaction. For an endothermic reaction (ΔH > 0) that increases entropy (ΔS > 0), derive the expression ΔG = ΔH − TΔS and discuss how temperature affects spontaneity. Determine the temperature range (in terms of ΔH and ΔS) for which the reaction is spontaneous.
Compare galvanic and electrolytic cells, define standard electrode potential, cell EMF, and the Nernst equation, using the Daniell cell as a worked example.
In electrochemistry, galvanic and electrolytic cells represent two fundamental types of electrochemical cells. (a) Compare and contrast galvanic and electrolytic cells in terms of spontaneity, electrode polarity, and energy conversion. (b) Define standard electrode potential, cell electromotive force (EMF), and the Nernst equation. (c) For the Daniell cell (Zn|Zn²⁺(1 M)||Cu²⁺(1 M)|Cu), calculate the standard cell EMF using standard reduction potentials (E°(Zn²⁺/Zn) = -0.76 V, E°(Cu²⁺/Cu) = +0.34 V). Then, using the Nernst equation, determine the cell EMF when [Zn²⁺] = 0.1 M and [Cu²⁺] = 2.0 M at 298 K.
Equilibrium Constant and Le Chatelier's Principle in the Haber Process
For the Haber process: N2(g) + 3H2(g) ⇌ 2NH3(g), the equilibrium constant Kc is defined. (a) Write the expression for Kc. (b) Predict the direction of shift in equilibrium when: (i) pressure is increased, (ii) temperature is increased (the forward reaction is exothermic), (iii) concentration of N2 is increased. Explain each prediction using Le Chatelier's principle.