Given the 2 × 2 matrix A = [[4, 3], [2, 5]], what is the determinant of matrix A?
Mathematiques Appliquees Et Statistiques
Question · Easy
Under what necessary and sufficient condition is an n × n square matrix A invertible?
Calculate the determinant of the 3 × 3 matrix M = [[2, 1, 0], [1, 3, 2], [0, 1, 4]].
Question · Medium
Consider a 3 × 3 matrix A whose third row is exactly equal to 2 × (Row 1) + 3 × (Row 2). What is the maximum possible rank of matrix A?
What are the eigenvalues of the matrix B = [[5, 2], [2, 2]]?
Question · Hard
Which of the following statements is a fundamental property of real symmetric matrices in linear algebra?
Using Cramer's rule, what is the value of x in the system of linear equations: 2x + 3y = 13 and x - y = -1?
For the multivariable function f(x, y) = 3x²y + 4y³ - 5x, what is the first-order partial derivative with respect to x, ∂f/∂x?
According to Schwarz's Theorem (Clairaut's Theorem) on multivariable functions, under what condition are the mixed second-order partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x equal?
Let (x₀, y₀) be a critical point of a twice continuously differentiable function f(x, y). If the Hessian determinant det(H) = f_xx · f_yy - (f_xy)² > 0 and f_xx > 0, what is the nature of the critical point?
A production manager maximizes output Q(x, y) = x · y subject to the cost budget constraint 2x + y = 20. Using the method of Lagrange multipliers, what is the maximum output Q achievable?
In constrained economic optimization, what does the Lagrange multiplier λ* represent at the optimal solution?
Which Hessian condition is sufficient to guarantee that a twice continuously differentiable function f(x, y) is strictly concave on a convex open set?
A demand model is D(p) = 1000e^(-0.02p), where p is price in MAD. What does D'(p)/D(p) imply about an infinitesimal 1 MAD increase in price?
A business in Casablanca invests 10,000 MAD in a term deposit account earning an annual interest rate of 5% compounded annually. What is the future value of the investment at the end of 3 years?
Comptabilite Et Gestion Financiere
A treasury fund invests 20,000 MAD at a nominal annual interest rate of 6% with continuous compounding. What is the accrued value of the investment after 2 years? (Use e^(0.12) ≈ 1.127497)
An investor is promised an ordinary annuity of 5,000 MAD paid at the end of each year for 3 consecutive years. If the annual discount rate is 10%, what is the present value (PV) of this annuity stream?
In loan amortization with constant periodic annuity payments (annuités constantes), how do the principal repayment component and the interest component evolve over time?
For a unimodal continuous distribution that is strongly skewed to the right (positive skewness), what is the typical relationship between the mean, median, and mode?
A commercial portfolio has a sample mean return of 8% and a sample variance of 16 (%²). What is the coefficient of variation (CV) of this portfolio?
Two financial variables X and Y have Cov(X, Y) = 18, Var(X) = 25, and Var(Y) = 16. What is the Pearson correlation coefficient r between X and Y?
In a simple Ordinary Least Squares (OLS) regression model Y = α + βX + ε, sample data yields Cov(X, Y) = 24, Var(X) = 8, mean(X) = 5, and mean(Y) = 20. What are the estimated OLS parameters β and α?
In an econometric model, the Total Sum of Squares (SST) is 500 and the Sum of Squared Residuals (SSE) is 125. What is the coefficient of determination R² of the regression?
An audit firm determines that 40% of loan portfolios exhibit high credit exposure (Event A), and among those with high exposure, 80% require enhanced impairment provisioning (Event B|A). What is the probability that a randomly audited loan exhibits both high credit exposure and requires enhanced provisioning, P(A ∩ B)?
A quality control test is conducted on n = 4 independent components, each with a probability p = 0.5 of passing inspection. Under a binomial distribution B(4, 0.5), what is the probability that exactly 2 components pass inspection?
Incoming transaction queries at an electronic payment platform follow a Poisson distribution with an average arrival rate of λ = 3 queries per minute. What is the probability that zero queries arrive in a given minute? (Use e^(-3) ≈ 0.0498)
In a sales outreach campaign, the probability of closing an enterprise contract on each independent pitch is p = 0.20. Following a geometric distribution, what is the expected number of pitches required to secure the first closed contract?
A continuous random variable X is uniformly distributed over the interval [10, 50]. What is the probability that X falls between 20 and 35, P(20 ≤ X ≤ 35)?
Which of the following distributions is uniquely characterized among continuous distributions by the 'memoryless property', satisfying P(X > s + t | X > s) = P(X > t) for all s, t ≥ 0?
A standardized entrance score X is normally distributed with mean μ = 100 and standard deviation σ = 15. A candidate scores X = 130. What is the candidate's z-score and the approximate percentile rank, P(X ≤ 130)? (Given standard normal CDF Φ(2.0) ≈ 0.9772)
In neoclassical consumer theory, which fundamental axiom ensures that consumer preference orderings are logically consistent and do not produce circular, contradictory choices (i.e., if A ≿ B and B ≿ C, then A ≿ C)?
Economie Generale Et Analyse Economique
A consumer with utility U(x, y) faces prices P_x = 4 MAD and P_y = 2 MAD. At their current consumption bundle, their marginal utility of x is MU_x = 24 and marginal utility of y is MU_y = 8. How should the consumer adjust consumption to maximize utility within their fixed budget?
When the price of a consumer good rises from 40 MAD to 50 MAD (+25%), the quantity demanded falls from 200 units to 160 units (-20%). What is the price elasticity of demand (using standard percentage changes) and how is the good characterized?
If the cross-price elasticity of demand between two products A and B is strictly positive (e_AB = %ΔQ_A / %ΔP_B > 0), what is the economic relationship between products A and B?
An econometric survey in an urban market finds that a 10% increase in household income leads to a 16% increase in expenditures on premium organic food products. How is this category of goods classified based on income elasticity?
What is the core premise of the classical Law of Diminishing Marginal Returns in short-run production theory?
At the cost-minimizing combination of labor (L) and capital (K) for a given output level, what condition must hold regarding the Marginal Rate of Technical Substitution (MRTS_LK) and input prices (wage w and capital rental rate r)?
In the short run, under what pricing condition should a perfectly competitive firm operating at an economic loss continue to produce rather than shut down immediately?
What is the economic term for the lowest level of output at which a firm's Long-Run Average Cost (LRAC) curve reaches its minimum?
A monopolist faces the linear market demand curve P = 100 - 2Q, with marginal revenue MR = 100 - 4Q. The firm operates with constant marginal cost MC = 20 MAD and zero fixed costs. What are the profit-maximizing output Q* and price P*?
Which form of price discrimination occurs when a seller charges each individual customer their exact maximum willingness to pay, thereby extracting the entirety of consumer surplus?
When a chemical factory emits unregulated pollutants into a local river, its Marginal Social Cost (MSC) exceeds its Marginal Private Cost (MPC). What corrective policy instrument restores allocative efficiency according to Arthur Pigou?
Two identical firms compete in quantities in a Cournot duopoly facing market demand P = 120 - Q, where Q = q₁ + q₂. Both firms produce with zero marginal cost (MC = 0). What are the Cournot-Nash equilibrium outputs q₁*, q₂* and the resulting market price P*?
In a Stackelberg duopoly facing inverse demand P = 120 - (q₁ + q₂) with MC = 0, Firm 1 acts as the market leader and Firm 2 as the follower. Knowing Firm 2's reaction function q₂ = (120 - q₁) / 2, what output does Firm 1 choose and what is Firm 2's resulting output?
According to public economics, what two defining criteria classify a good or service as a 'pure public good' (such as national defense or lighthouses)?
In a national economy, macroeconomic aggregates show: Household Final Consumption (C) = 600 billion MAD, Gross Fixed Capital Formation & Investment (I) = 200 billion MAD, Government Consumption (G) = 150 billion MAD, Exports (X) = 180 billion MAD, and Imports (M) = 210 billion MAD. What is the Gross Domestic Product (GDP) using the expenditure approach?
In a hypothetical 2026 scenario, an economy records nominal GDP of 1,320 billion MAD and real GDP of 1,200 billion MAD in base-year prices. What is the GDP deflator, and what does it indicate relative to the base year?
What is a major methodological difference between the Consumer Price Index (CPI) and the GDP Deflator in measuring price changes across an economy?
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