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ENSTP Access Contest - independent subject practice study materials
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ENSTP Access Contest - independent subject practice practice questions

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Question 1Advanced analysis, convergence and multivariable reasoning

What is the value of the double integral ∬_D (x^2 + y^2) dA over the circular disk D: x^2 + y^2 <= 4?

  1. 8π Correct answer.
  2. 4π
  3. 16π
  4. 2π

Explanation

Convert to polar coordinates: x^2 + y^2 = r^2 and dA = r dr dθ. The region is 0 <= r <= 2 and 0 <= θ <= 2π. Compute ∬_D (x^2 + y^2) dA = ∫_0^{2π} dθ ∫_0^2 r^2 * r dr = 2π ∫_0^2 r^3 dr = 2π [r^4 / 4]_0^2 = 2π (16 / 4) = 8π.

Question 2Advanced analysis, convergence and multivariable reasoning

Using Green's Theorem, what is the value of the line integral ∮_C (x dy - y dx) along the positively oriented boundary C of an ellipse x^2/a^2 + y^2/b^2 = 1?

  1. 2πab Correct answer.
  2. πab
  3. 0
  4. 4πab

Explanation

Here P(x, y) = -y and Q(x, y) = x. By Green's Theorem, ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x - ∂P/∂y) dA. Compute ∂Q/∂x = 1 and ∂P/∂y = -1, so the integrand is 1 - (-1) = 2. Thus ∮_C (x dy - y dx) = 2 ∬_D dA = 2 * Area(D). Since the area of the ellipse is π a b, the integral evaluates to 2π a b.

Question 3Advanced analysis, convergence and multivariable reasoning

What does the ratio test determine regarding the convergence of the series ∑_{n=1}^∞ (n^2 + 1) / 2^n?

  1. It converges absolutely because lim |a_{n+1} / a_n| = 1/2 < 1 Correct answer.
  2. It diverges because lim |a_{n+1} / a_n| = 2 > 1
  3. The ratio test is inconclusive because the limit is 1
  4. It converges conditionally by the alternating series test

Explanation

Compute the ratio of consecutive terms: a_{n+1} / a_n = [((n+1)^2 + 1) / 2^{n+1}] * [2^n / (n^2 + 1)] = (1/2) * [(n^2 + 2n + 2) / (n^2 + 1)]. Taking the limit as n → ∞: lim_{n→∞} |a_{n+1} / a_n| = (1/2) * 1 = 1/2. Since L = 1/2 < 1, the series converges absolutely by d'Alembert's ratio test.

Question 4Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence R of the power series ∑_{n=1}^∞ (x - 2)^n / (n * 4^n)?

  1. 4 Correct answer.
  2. 2
  3. 1/4
  4. 1

Explanation

Let c_n = 1 / (n * 4^n). Compute the ratio limit: lim_{n→∞} |c_n / c_{n+1}| = lim_{n→∞} [(n+1) * 4^{n+1}] / [n * 4^n] = 4 * lim_{n→∞} (n+1)/n = 4. Therefore, the radius of convergence is R = 4. The series converges for |x - 2| < 4.

Question 5Advanced analysis, convergence and multivariable reasoning

What is the divergence of the vector field F(x, y, z) = (x y, y z, z x) evaluated at the point P(2, 3, 4)?

  1. 9 Correct answer.
  2. 24
  3. 14
  4. 0

Explanation

Compute the divergence: div(F) = ∂(xy)/∂x + ∂(yz)/∂y + ∂(zx)/∂z = y + z + x. At P(2, 3, 4), div(F) = 3 + 4 + 2 = 9.

Question 6Advanced analysis, convergence and multivariable reasoning

Consider the vector field F(x, y, z) = (2xy + z^3, x^2 + 2y, 3xz^2). Which statement correctly describes its curl and conservation property on ℝ^3?

  1. curl(F) = (0, 0, 0); F is conservative with scalar potential φ(x, y, z) = x^2 y + y^2 + x z^3 Correct answer.
  2. curl(F) = (2x, 3z^2, 2y); F is non-conservative
  3. curl(F) = (0, 0, 0); but F cannot possess a potential function
  4. curl(F) = (0, 0, 2xy); F is rotational

Explanation

Compute curl(F) = ∇ × F: Component 1: ∂(3xz^2)/∂y - ∂(x^2 + 2y)/∂z = 0 - 0 = 0. Component 2: ∂(2xy + z^3)/∂z - ∂(3xz^2)/∂x = 3z^2 - 3z^2 = 0. Component 3: ∂(x^2 + 2y)/∂x - ∂(2xy + z^3)/∂y = 2x - 2x = 0. Since curl(F) = (0, 0, 0) on the simply connected domain ℝ^3, F is conservative. Integrating yields φ = x^2 y + y^2 + x z^3 + C.

Question 7Advanced analysis, convergence and multivariable reasoning

What is the total surface area of a sphere of radius a, computed via the surface integral ∬_S 1 dS?

  1. 4π a^2 Correct answer.
  2. (4/3)π a^3
  3. 2π a^2
  4. π a^2

Explanation

In spherical coordinates on the surface r = a, the surface area element is dS = a^2 sin(θ) dθ dφ for 0 <= θ <= π and 0 <= φ <= 2π. The integral is ∬_S dS = ∫_0^{2π} dφ ∫_0^π a^2 sin(θ) dθ = (2π) * a^2 * [-cos(θ)]_0^π = 2π a^2 (1 - (-1)) = 4π a^2.

Question 8Advanced analysis, convergence and multivariable reasoning

Which set of conditions (Dirichlet conditions) guarantees that the Fourier series of a 2L-periodic function f(x) converges to f(x) at all points of continuity and to [f(x^+) + f(x^-)] / 2 at jump discontinuities?

  1. f(x) is piecewise continuous on [-L, L], has a finite number of finite discontinuities, and has a finite number of local extrema Correct answer.
  2. f(x) is infinitely differentiable on ℝ
  3. f(x) is strictly positive and monotonically increasing everywhere
  4. f(x) is a polynomial of degree at most 2

Explanation

Dirichlet's theorem states that if a periodic function f(x) is piecewise continuous on [-L, L], possesses only a finite number of finite jump discontinuities in any single period, and has a finite number of local maxima and minima, its Fourier series converges everywhere. At points of continuity, it converges to f(x); at jump discontinuities, it converges to the arithmetic mean of the one-sided limits [f(x^+) + f(x^-)]/2.

Question 9Advanced analysis, convergence and multivariable reasoning

What is the Fourier transform F(ω) = ∫_{-∞}^∞ f(t) e^{-iωt} dt of the Gaussian function f(t) = e^{-a t^2} (a > 0)?

  1. √(π / a) e^{-ω^2 / (4a)} Correct answer.
  2. √(a / π) e^{-a ω^2}
  3. 1 / (a^2 + ω^2)
  4. 2a / (a^2 + ω^2)

Explanation

Compute the integral by completing the square in the exponent: -a t^2 - i ω t = -a (t + i ω / (2a))^2 - ω^2 / (4a). Substituting u = √a (t + i ω / (2a)) and using the known Gaussian integral ∫_{-∞}^∞ e^{-u^2} du = √π yields F(ω) = e^{-ω^2 / (4a)} * (1 / √a) ∫_{-∞}^∞ e^{-u^2} du = √(π / a) e^{-ω^2 / (4a)}. A Gaussian in time transforms into a Gaussian in frequency.

Question 10Advanced analysis, convergence and multivariable reasoning

For any harmonic function u(x, y) satisfying Laplace's equation ∇^2 u = 0 in an open domain Ω, what does the Mean Value Property state for any disk D(z_0, R) ⊂ Ω centered at z_0 = (x_0, y_0)?

  1. u(x_0, y_0) equals both the average of u over the boundary circle and the average of u over the interior of the disk Correct answer.
  2. u(x_0, y_0) must be identically zero
  3. u(x_0, y_0) equals the maximum value of u on the boundary circle
  4. The gradient ∇u must vanish everywhere inside the disk

Explanation

The Mean Value Theorem for harmonic functions states that the value of u at the center of any ball/disk equals the average of u over the boundary sphere/circle: u(z_0) = (1 / 2π) ∫_0^{2π} u(x_0 + R cos θ, y_0 + R sin θ) dθ. Integrating with respect to radius r shows that u(z_0) also equals the area average over the interior of the disk: u(z_0) = (1 / (π R^2)) ∬_D u dA.

Question 11Advanced analysis, convergence and multivariable reasoning

Applying the Weierstrass M-test to the series of functions ∑_{n=1}^∞ sin(nx) / n^3 on ℝ, what conclusion can be drawn?

  1. It converges uniformly and absolutely on ℝ because |sin(nx) / n^3| <= 1/n^3 and ∑ 1/n^3 converges Correct answer.
  2. It converges pointwise on ℝ but fails to converge uniformly
  3. It diverges everywhere because sin(nx) oscillates indefinitely
  4. The Weierstrass M-test is inconclusive because sin(nx) can be negative

Explanation

For all x ∈ ℝ and all n >= 1, |sin(nx)| <= 1, so |sin(nx) / n^3| <= 1 / n^3 = M_n. The series ∑_{n=1}^∞ M_n = ∑_{n=1}^∞ 1 / n^3 is a p-series with p = 3 > 1, which converges. By the Weierstrass M-test, the series of functions converges uniformly and absolutely on the entire real line ℝ.

Question 12Advanced analysis, convergence and multivariable reasoning

Let each f_n be continuous on [a, b], and suppose the series ∑_{n=1}^{∞} f_n converges uniformly there to S. Which interchange is guaranteed?

  1. ∫_a^b S(x) dx = ∑_{n=1}^{∞} ∫_a^b f_n(x) dx Correct answer.
  2. S'(x) = ∑_{n=1}^{∞} f_n'(x) without any additional assumptions
  3. The same integration identity follows from pointwise convergence alone
  4. Every integral ∫_a^b f_n(x) dx must equal zero

Explanation

Uniform convergence controls the remainder uniformly: if S_N = ∑_{n=1}^N f_n, then |∫(S-S_N)| <= (b-a) sup|S-S_N|, which tends to zero. Since each finite sum may be integrated term by term, passing to the limit gives ∫_a^b S = ∑_{n=1}^{∞} ∫_a^b f_n.

Question 13Advanced analysis, convergence and multivariable reasoning

What is the fundamental solution (heat kernel) K(x, t) for the one-dimensional heat equation u_t = α u_{xx} on -∞ < x < ∞ for t > 0?

  1. K(x, t) = (1 / √(4π α t)) e^{-x^2 / (4 α t)} Correct answer.
  2. K(x, t) = (1 / √(2π α t)) e^{-x^2 / (2 α t)}
  3. K(x, t) = e^{-x^2 / (4 α t)}
  4. K(x, t) = (1 / (4π α t)) e^{-x / (2 α t)}

Explanation

The fundamental solution to u_t = α u_{xx} corresponds to the response to a spatial Dirac delta initial condition u(x, 0) = δ(x). Applying the Fourier transform in x gives U_t(ω, t) = -α ω^2 U(ω, t), so U(ω, t) = e^{-α ω^2 t}. Taking the inverse Fourier transform yields the Gaussian kernel K(x, t) = (1 / √(4π α t)) e^{-x^2 / (4 α t)}.

Question 14Advanced analysis, convergence and multivariable reasoning

For u_t = αu_{xx} on 0 < x < L with u(0,t) = u(L,t) = 0 and u(x,0) = sin(2πx/L), which separated solution satisfies the problem?

  1. u(x,t) = e^{-4π^2αt/L^2} sin(2πx/L) Correct answer.
  2. u(x,t) = e^{-2παt/L} sin(2πx/L)
  3. u(x,t) = e^{+4π^2αt/L^2} sin(2πx/L)
  4. u(x,t) = e^{-4π^2αt/L^2} cos(2πx/L)

Explanation

The spatial eigenfunction sin(nπx/L) has X'' = -(nπ/L)^2X. Here n = 2, so separation gives T' = -α(2π/L)^2T and T(t) = exp[-4π^2αt/L^2]. Multiplying by the initial spatial mode produces the stated solution, which also vanishes at both boundaries.

Question 15Advanced analysis, convergence and multivariable reasoning

What is the value of the triple integral ∭_V z dV over the upper hemisphere V: x^2 + y^2 + z^2 <= 1 with z >= 0?

  1. π / 4 Correct answer.
  2. π / 2
  3. 2π / 3
  4. π / 8

Explanation

In spherical coordinates, z = r cos(θ) and dV = r^2 sin(θ) dr dθ dφ. The limits are 0 <= r <= 1, 0 <= θ <= π/2 (upper hemisphere), and 0 <= φ <= 2π. Compute ∭_V z dV = (∫_0^{2π} dφ) * (∫_0^{π/2} cos θ sin θ dθ) * (∫_0^1 r^3 dr) = (2π) * [sin^2(θ) / 2]_0^{π/2} * [r^4 / 4]_0^1 = (2π) * (1/2) * (1/4) = π / 4.

Question 16Advanced analysis, convergence and multivariable reasoning

What is the exact sum of the alternating harmonic series ∑_{n=1}^∞ (-1)^{n-1} / n = 1 - 1/2 + 1/3 - 1/4 + ...?

  1. ln(2) Correct answer.
  2. 1
  3. π / 4
  4. e - 1

Explanation

The Taylor series for ln(1 + x) is ∑_{n=1}^∞ (-1)^{n-1} x^n / n for |x| < 1. By Abel's theorem, the series also converges at the right endpoint x = 1: ln(1 + 1) = ln(2) = ∑_{n=1}^∞ (-1)^{n-1} / n ≈ 0.69315.

Question 17Advanced analysis, convergence and multivariable reasoning

What is the work done by the conservative force field F(x, y) = (3x^2 y, x^3) in moving a particle from the origin (0, 0) to the point (1, 2)?

  1. 2 Correct answer.
  2. 6
  3. 3
  4. 1

Explanation

Check conservation: ∂(x^3)/∂x = 3x^2 and ∂(3x^2 y)/∂y = 3x^2. The field is conservative with potential function φ(x, y) = x^3 y. The work is path-independent: W = φ(1, 2) - φ(0, 0) = (1^3)(2) - 0 = 2.

Question 18Advanced analysis, convergence and multivariable reasoning

Using a geometric-series expansion about x = 0, what is the radius of convergence of the real power series for f(x) = 1 / (x^2 + 9)?

  1. 3 Correct answer.
  2. 9
  3. 1/3
  4. Infinity

Explanation

Rewrite f(x) = (1/9)[1/(1 + x^2/9)] = (1/9)∑_{n=0}^{∞}(-x^2/9)^n. A geometric series converges when the magnitude of its ratio is less than one, so |x^2/9| < 1, equivalently |x| < 3. Therefore, the radius of convergence is 3.

Question 19Advanced analysis, convergence and multivariable reasoning

What is the classification of the second-order partial differential equation u_{xx} - 2 u_{xy} + u_{yy} = 0?

  1. Parabolic Correct answer.
  2. Hyperbolic
  3. Elliptic
  4. Non-linear

Explanation

For A u_{xx} + B u_{xy} + C u_{yy} = 0, compute the discriminant Δ = B^2 - 4AC. Here A = 1, B = -2, and C = 1. Thus Δ = (-2)^2 - 4(1)(1) = 4 - 4 = 0. Since Δ = 0, the equation is parabolic. Note that the operator can be factored as (∂/∂x - ∂/∂y)^2 u = 0.

Question 20Advanced analysis, convergence and multivariable reasoning

If S_1 and S_2 are two different smooth oriented surfaces that share the exact same boundary curve C = ∂S_1 = ∂S_2, what does Stokes' Theorem imply about the flux of curl(F) across S_1 versus S_2?

  1. ∬_{S_1} (curl F) · dS = ∬_{S_2} (curl F) · dS because both equal the line integral ∮_C F · dr Correct answer.
  2. The two flux values are completely unrelated
  3. The flux across S_1 must be the negative of the flux across S_2
  4. Both flux integrals must equal zero

Explanation

By Stokes' Theorem, ∬_S (curl F) · dS = ∮_{∂S} F · dr. Since both surfaces have the exact same positively oriented boundary curve C, ∬_{S_1} (curl F) · dS = ∮_C F · dr and ∬_{S_2} (curl F) · dS = ∮_C F · dr. Therefore, the two flux integrals are strictly equal. (Equivalently, the union of S_1 and -S_2 forms a closed surface S, and by the Divergence Theorem, the flux of curl F over any closed surface is zero because div(curl F) = 0).

Question 21Advanced analysis, convergence and multivariable reasoning

What is the value of the line integral ∮_C (y dx + x dy) around any closed, piecewise smooth simple curve C?

  1. 0 Correct answer.
  2. Area enclosed by C
  3. 2 * Area enclosed by C
  4. Perimeter of C

Explanation

The differential form is y dx + x dy = d(xy), which is exact. By the fundamental theorem of line integrals (or by Green's Theorem with ∂Q/∂x - ∂P/∂y = ∂x/∂x - ∂y/∂y = 1 - 1 = 0), the line integral of an exact differential around any closed loop is identically 0.

Question 22Advanced analysis, convergence and multivariable reasoning

For any smooth vector field F, what is the outward flux of curl(F) across any closed orientable surface S?

  1. 0 Correct answer.
  2. Volume enclosed by S
  3. Surface area of S
  4. 4π

Explanation

By the Divergence Theorem, ∯_S (curl F) · dS = ∭_V div(curl F) dV. Because the divergence of the curl of any twice continuously differentiable vector field is identically zero (div(curl F) = ∇ · (∇ × F) = 0), the volume integral vanishes: ∭_V 0 dV = 0.

Question 23Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence of the series ∑_{n=1}^∞ (n! / n^n) x^n?

  1. e Correct answer.
  2. 1/e
  3. 1
  4. Infinity

Explanation

Let c_n = n! / n^n. The ratio of consecutive coefficients is c_{n+1}/c_n = [(n+1)! / (n+1)^{n+1}] * [n^n / n!] = [n / (n+1)]^n = 1 / (1 + 1/n)^n. Taking the limit as n → ∞ gives lim_{n→∞} |c_{n+1} / c_n| = 1/e. The radius of convergence is R = 1 / L = 1 / (1/e) = e ≈ 2.718.

Question 24Advanced analysis, convergence and multivariable reasoning

For the even function f(x) = x^2 defined on [-π, π] and extended with period 2π, what is the constant term a_0 / 2 in its Fourier series expansion?

  1. π^2 / 3 Correct answer.
  2. 2π^2 / 3
  3. π^2 / 6
  4. π^2

Explanation

The coefficient a_0 is a_0 = (1/π) ∫_{-π}^π x^2 dx = (2/π) ∫_0^π x^2 dx = (2/π) [x^3 / 3]_0^π = (2/π) (π^3 / 3) = 2π^2 / 3. The constant term in the Fourier expansion f(x) = a_0 / 2 + ∑ a_n cos(nx) is a_0 / 2 = (1/2) * (2π^2 / 3) = π^2 / 3.

Question 25Advanced analysis, convergence and multivariable reasoning

What is the gradient vector of the scalar field f(x, y) = ln(x^2 + y^2) evaluated at the point (1, 1)?

  1. (1, 1) Correct answer.
  2. (2, 2)
  3. (1/2, 1/2)
  4. (0, 0)

Explanation

Compute the partial derivatives: ∂f/∂x = 2x / (x^2 + y^2) and ∂f/∂y = 2y / (x^2 + y^2). At the point (x, y) = (1, 1): x^2 + y^2 = 1^2 + 1^2 = 2. Thus ∂f/∂x = 2(1) / 2 = 1 and ∂f/∂y = 2(1) / 2 = 1. The gradient is ∇f(1, 1) = (1, 1).

Question 26Advanced analysis, convergence and multivariable reasoning

By Cauchy's root test (Cauchy-Hadamard theorem), how is the radius of convergence R of a power series ∑_{n=0}^∞ a_n x^n expressed?

  1. 1 / R = lim sup_{n→∞} |a_n|^{1/n} Correct answer.
  2. R = lim sup_{n→∞} |a_n|^{1/n}
  3. 1 / R = lim_{n→∞} |a_n / a_{n+1}|
  4. R = lim_{n→∞} |a_n|^n

Explanation

The Cauchy-Hadamard theorem states that for any power series ∑ a_n (x - x_0)^n, the reciprocal of the radius of convergence is given by 1 / R = lim sup_{n→∞} |a_n|^{1/n} (with conventions 1/0 = ∞ and 1/∞ = 0). This formula is universal and holds even when the ratio limit lim |a_{n+1}/a_n| does not exist.

Question 27Advanced analysis, convergence and multivariable reasoning

In cylindrical coordinates (r, θ, z) where x = r cos θ, y = r sin θ, and z = z, what is the volume differential element dV?

  1. r dr dθ dz Correct answer.
  2. dr dθ dz
  3. r^2 dr dθ dz
  4. r sin θ dr dθ dz

Explanation

The transformation from (r, θ, z) to (x, y, z) has Jacobian matrix with determinant J = r. Therefore, the infinitesimal volume element is dV = r dr dθ dz.

Question 28Advanced analysis, convergence and multivariable reasoning

What is the exact value of the improper Gaussian integral I = ∫_0^∞ e^{-x^2} dx?

  1. √π / 2 Correct answer.
  2. √π
  3. π / 2
  4. 1

Explanation

By symmetry, I = (1/2) ∫_{-∞}^∞ e^{-x^2} dx. Squaring the full integral gives (2I)^2 = (∫_{-∞}^∞ e^{-x^2} dx) (∫_{-∞}^∞ e^{-y^2} dy) = ∬_{ℝ^2} e^{-(x^2 + y^2)} dx dy = ∫_0^{2π} dθ ∫_0^∞ e^{-r^2} r dr = 2π [-e^{-r^2}/2]_0^∞ = 2π(1/2) = π. Therefore 2I = √π, which gives I = √π / 2.

Question 29Advanced analysis, convergence and multivariable reasoning

What are the Fourier sine series coefficients b_n on [0, L] for the constant function f(x) = 1 in the expansion f(x) = ∑_{n=1}^∞ b_n sin(nπx / L)?

  1. b_n = 4 / (nπ) for odd n, and b_n = 0 for even n Correct answer.
  2. b_n = 2 / (nπ) for all n
  3. b_n = 4 / (n^2 π^2) for odd n
  4. b_n = (-1)^{n+1} / n

Explanation

The Fourier sine coefficients are b_n = (2/L) ∫_0^L 1 * sin(nπx/L) dx = (2/L) [ -L/(nπ) cos(nπx/L) ]_0^L = (2/(nπ)) [1 - cos(nπ)] = (2/(nπ)) [1 - (-1)^n]. When n is even, (-1)^n = 1, so b_n = 0. When n is odd, (-1)^n = -1, so b_n = 2/(nπ) * (2) = 4 / (nπ).

Question 30Inner products, spectral reasoning and matrix structure

Let $z_1 = 1 + i\sqrt{3}$ and $z_2 = 1 - i$. What is the trigonometric/polar form of the quotient $Z = \frac{z_1}{z_2}$?

  1. Z = \sqrt{2} \left(\cos\left(\frac{7\pi}{12}\right) + i \sin\left(\frac{7\pi}{12}\right)\right) Correct answer.
  2. Z = \sqrt{2} \left(\cos\left(\frac{\pi}{12}\right) + i \sin\left(\frac{\pi}{12}\right)\right)
  3. Z = 2 \left(\cos\left(\frac{7\pi}{12}\right) + i \sin\left(\frac{7\pi}{12}\right)\right)
  4. Z = 1 \left(\cos\left(\frac{5\pi}{12}\right) + i \sin\left(\frac{5\pi}{12}\right)\right)

Explanation

$|z_1| = \sqrt{1 + 3} = 2$ and $\arg(z_1) = \frac{\pi}{3}$. $|z_2| = \sqrt{1 + 1} = \sqrt{2}$ and $\arg(z_2) = -\frac{\pi}{4}$. Thus, $|Z| = \frac{|z_1|}{|z_2|} = \frac{2}{\sqrt{2}} = \sqrt{2}$. The argument is $\arg(Z) = \arg(z_1) - \arg(z_2) = \frac{\pi}{3} - \left(-\frac{\pi}{4}\right) = \frac{4\pi + 3\pi}{12} = \frac{7\pi}{12}$. Therefore, $Z = \sqrt{2} e^{i 7\pi/12}$.

Question 31Inner products, spectral reasoning and matrix structure

What is the determinant of the $3 \times 3$ real matrix $A = \begin{pmatrix} 2 & 1 & 3 \\ 0 & 4 & -1 \\ 0 & 0 & 5 \end{pmatrix}$?

  1. 40 Correct answer.
  2. 0
  3. 20
  4. 11

Explanation

Matrix $A$ is upper triangular. The determinant of any triangular matrix is simply the product of its main diagonal elements: $\det(A) = 2 \times 4 \times 5 = 40$.

Question 32Inner products, spectral reasoning and matrix structure

Find the eigenvalues $\lambda$ of the matrix $M = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$.

  1. λ₁ = 4 and λ₂ = 1 Correct answer.
  2. λ₁ = 3 and λ₂ = 2
  3. λ₁ = 5 and λ₂ = 0
  4. λ₁ = 2 and λ₂ = -1

Explanation

Characteristic polynomial: $P(\lambda) = \det(M - \lambda I) = (3 - \lambda)(2 - \lambda) - (1)(2) = \lambda^2 - 5\lambda + 6 - 2 = \lambda^2 - 5\lambda + 4 = 0$. Factoring gives $(\lambda - 4)(\lambda - 1) = 0 \implies \lambda_1 = 4, \lambda_2 = 1$. (Check: $\text{Tr}(M) = 3 + 2 = 5 = 4 + 1$; $\det(M) = 6 - 2 = 4 = 4 \times 1$).

Question 33Inner products, spectral reasoning and matrix structure

What is the inverse $A^{-1}$ of the $2 \times 2$ matrix $A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$?

  1. \begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix} Correct answer.
  2. \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix}
  3. \begin{pmatrix} 0.4 & -0.2 \\ -0.7 & 0.6 \end{pmatrix}
  4. \begin{pmatrix} -4 & -7 \\ -2 & -6 \end{pmatrix}

Explanation

Determinant: $\det(A) = (4)(6) - (7)(2) = 24 - 14 = 10$. The inverse formula is $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{10} \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix}$.

Question 34Inner products, spectral reasoning and matrix structure

What is the rank (rang) of the real matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$?

  1. 1 Correct answer.
  2. 3
  3. 2
  4. 0

Explanation

Row 2 is $2 \times \text{Row } 1$, and Row 3 is $3 \times \text{Row } 1$. All rows are linearly dependent scalar multiples of the first row $(1, 2, 3)$. Thus, there is only 1 linearly independent row/column, so $\text{rank}(A) = 1$.

Question 35Inner products, spectral reasoning and matrix structure

In matrix algebra, what is the Cayley-Hamilton theorem?

  1. Every square matrix satisfies its own characteristic equation: P_A(A) = 0 Correct answer.
  2. Every matrix has non-zero determinant
  3. The transpose of a matrix equals its inverse
  4. Matrix multiplication is always commutative: A · B = B · A

Explanation

The Cayley-Hamilton theorem states that if $P_A(\lambda) = \det(A - \lambda I) = (-1)^n \lambda^n + c_{n-1} \lambda^{n-1} + \dots + c_0$ is the characteristic polynomial of $A$, then substituting matrix $A$ yields the zero matrix: $P_A(A) = (-1)^n A^n + c_{n-1} A^{n-1} + \dots + c_0 I = 0$, widely used to compute matrix powers $A^k$ and inverses $A^{-1}$.

Question 36Inner products, spectral reasoning and matrix structure

Using the Cayley-Hamilton theorem for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ with characteristic polynomial $P(\lambda) = \lambda^2 - 5\lambda - 2$, express $A^2$ in terms of $A$ and the identity matrix $I$.

  1. A^2 = 5A + 2I Correct answer.
  2. A^2 = 5A - 2I
  3. A^2 = 2A + 5I
  4. A^2 = A + I

Explanation

By Cayley-Hamilton: $A^2 - 5A - 2I = 0 \implies A^2 = 5A + 2I$. Check directly: $A^2 = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 15 & 22 \end{pmatrix}$. $5A + 2I = \begin{pmatrix} 5 & 10 \\ 15 & 20 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 15 & 22 \end{pmatrix}$. Match is exact.

Question 37Numerical roots, linear systems, quadrature and ODE stability

What is the primary operational purpose of implementing partial pivoting (row swapping) during Gaussian elimination for solving A x = b?

  1. To avoid division by zero and minimize round-off error amplification by selecting the largest available pivot element in the current column Correct answer.
  2. To transform the matrix directly into diagonal form in a single step
  3. To guarantee that the matrix determinant becomes strictly positive
  4. To ensure that all computed eigenvalues are real numbers

Explanation

Partial pivoting selects the available row with largest absolute pivot in the current column and swaps it into place. This avoids a zero pivot when a usable one exists and keeps elimination multipliers at most one in magnitude. It can control important error sources, but does not guarantee the smallest global round-off error or cure an ill-conditioned system.

Question 38Numerical roots, linear systems, quadrature and ODE stability

For the 2×2 matrix A = [[3, 1], [2, 4]], what is its condition number κ_1(A) in the 1-norm (maximum absolute column sum)?

  1. 3.0 Correct answer.
  2. 5.0
  3. 0.6
  4. 10.0

Explanation

The 1-norm ||A||_1 is the maximum column sum: Column 1 sum = |3| + |2| = 5; Column 2 sum = |1| + |4| = 5; so ||A||_1 = 5. Compute det(A) = 3(4) - 1(2) = 10. The inverse is A^{-1} = (1/10) [[4, -1], [-2, 3]] = [[0.4, -0.1], [-0.2, 0.3]]. The column sums of A^{-1} are: Col 1 = |0.4| + |-0.2| = 0.6; Col 2 = |-0.1| + |0.3| = 0.4. Thus ||A^{-1}||_1 = 0.6. The condition number is κ_1(A) = ||A||_1 * ||A^{-1}||_1 = 5 * 0.6 = 3.0.

Question 39Numerical roots, linear systems, quadrature and ODE stability

Under what necessary and sufficient condition on the iteration matrix B does the linear iterative method x^{(k+1)} = B x^{(k)} + c converge to the unique solution for ANY starting vector x^{(0)}?

  1. The spectral radius ρ(B) is strictly less than 1 (ρ(B) < 1) Correct answer.
  2. The trace of B must be zero (tr(B) = 0)
  3. All entries of B must be positive
  4. B must be a symmetric matrix

Explanation

The error vector satisfies e^{(k)} = B^k e^{(0)}. A matrix power B^k converges to the zero matrix as k → ∞ for all initial vectors if and only if every eigenvalue λ of B satisfies |λ| < 1. The maximum absolute eigenvalue is the spectral radius ρ(B) = max_i |λ_i|. Therefore, ρ(B) < 1 is the necessary and sufficient condition for convergence.

Question 40Numerical roots, linear systems, quadrature and ODE stability

What does the Ostrowski-Reich theorem establish regarding the convergence of the Gauss-Seidel method for solving A x = b?

  1. If A is symmetric and positive-definite, the Gauss-Seidel method converges for any initial guess x^{(0)} Correct answer.
  2. Gauss-Seidel converges only if A is triangular
  3. Gauss-Seidel converges if and only if A has all positive entries
  4. Gauss-Seidel always diverges for symmetric matrices

Explanation

The Ostrowski-Reich theorem is a fundamental result in numerical linear algebra: for any symmetric positive-definite (SPD) matrix A, the successive over-relaxation (SOR) method converges for any relaxation parameter ω ∈ (0, 2). Since Gauss-Seidel corresponds to ω = 1, it is guaranteed to converge for every symmetric positive-definite matrix regardless of the initial starting vector.

Question 41Numerical roots, linear systems, quadrature and ODE stability

Why is relying solely on the residual stopping criterion |f(x_k)| < ε potentially hazardous when finding roots of nonlinear equations f(x) = 0?

  1. If the derivative f'(x) is very small near the root (flat function), |f(x_k)| can be extremely small even when x_k is far from the true root Correct answer.
  2. Residual stopping criteria only work for polynomial equations
  3. |f(x_k)| is always larger than the step size |x_{k+1} - x_k|
  4. Computers cannot evaluate absolute values of floating-point numbers

Explanation

By Taylor approximation, f(x_k) ≈ f'(r) (x_k - r). Therefore, the actual error in the root is |x_k - r| ≈ |f(x_k)| / |f'(r)|. If |f'(r)| << 1 (the graph is nearly horizontal near the root), |f(x_k)| may easily satisfy a small tolerance ε while the iterate x_k remains unacceptably far from the true root r. Robust stopping criteria therefore combine residual checks with step-size checks |x_{k+1} - x_k| < ε.

Question 42Numerical roots, linear systems, quadrature and ODE stability

Given the data points (1, 2), (2, 5), and (4, 17), what is the second-order divided difference f[1, 2, 4], and what is the interpolated value P(3)?

  1. f[1, 2, 4] = 1; P(3) = 10 Correct answer.
  2. f[1, 2, 4] = 3; P(3) = 12
  3. f[1, 2, 4] = 2; P(3) = 11
  4. f[1, 2, 4] = 0; P(3) = 8

Explanation

First divided differences: f[1, 2] = (5 - 2)/(2 - 1) = 3; f[2, 4] = (17 - 5)/(4 - 2) = 12/2 = 6. Second divided difference: f[1, 2, 4] = (f[2, 4] - f[1, 2]) / (4 - 1) = (6 - 3) / 3 = 1. The Newton interpolating polynomial is P(x) = f[1] + f[1, 2](x - 1) + f[1, 2, 4](x - 1)(x - 2) = 2 + 3(x - 1) + 1(x - 1)(x - 2) = x^2 + 1. At x = 3: P(3) = 3^2 + 1 = 10.

Question 43Numerical roots, linear systems, quadrature and ODE stability

For a function f(x) that is strictly concave downward (f''(x) < 0) on the interval [a, b], how does the trapezoidal rule numerical approximation T compare to the exact integral I = ∫_a^b f(x) dx?

  1. T underestimates the exact integral (T < I) Correct answer.
  2. T overestimates the exact integral (T > I)
  3. T equals the exact integral (T = I)
  4. T oscillates unpredictably depending on step size

Explanation

The error of the trapezoidal rule is E_T = I - T = -[(b - a)/12] h^2 f''(ξ). If f''(x) < 0 (concave downward), the chord connecting (a, f(a)) and (b, f(b)) lies entirely below the curve of f(x). Consequently, the trapezoid area is smaller than the true area under the curve, so T < I (underestimate, or positive error E_T > 0).

Question 44Numerical roots, linear systems, quadrature and ODE stability

What is Runge's phenomenon in polynomial interpolation, and how can it be mitigated when interpolating functions on [-1, 1]?

  1. Severe oscillations near the interval boundaries when using high-degree polynomials with equidistant nodes; mitigated by using Chebyshev nodes Correct answer.
  2. Divergence caused by round-off errors; mitigated by using single precision
  3. The loss of continuity in spline functions; mitigated by increasing polynomial degree to 10
  4. Failure of Newton-Raphson to converge; mitigated by bisection

Explanation

Interpolating some smooth functions, such as f(x) = 1/(1 + 25x^2), with one high-degree polynomial at equally spaced nodes can produce growing endpoint oscillations. Chebyshev nodes cluster near the interval ends and greatly reduce the maximum interpolation error and Lebesgue-constant growth. They mitigate Runge's phenomenon, although convergence still depends on the regularity of the function being interpolated.

Question 45Numerical roots, linear systems, quadrature and ODE stability

What continuity property distinguishes a cubic spline interpolant S(x) across its internal knot points x_1, x_2, ..., x_{n-1}?

  1. S(x), S'(x), and S''(x) are all continuous across every knot (C^2 continuity) Correct answer.
  2. Only S(x) is continuous; derivatives are discontinuous
  3. S(x) and S'(x) are continuous, but S''(x) is discontinuous (C^1 continuity)
  4. All derivatives up to S^{(4)}(x) are continuous

Explanation

A cubic spline is defined as a piecewise cubic polynomial on each subinterval [x_i, x_{i+1}]. At each interior knot x_i, the pieces are matched such that the function values, the first derivatives (slopes), and the second derivatives (curvatures) are all continuous. Thus S(x) ∈ C^2[a, b], providing a visually smooth curve that minimizes bending energy.

Question 46Numerical roots, linear systems, quadrature and ODE stability

Why is the implicit (backward) Euler method y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}) preferred over the explicit Euler method when solving stiff systems of ordinary differential equations?

  1. It is unconditionally A-stable, allowing large step sizes without numerical instability Correct answer.
  2. It requires zero computational effort per time step
  3. It has fourth-order global accuracy O(h^4)
  4. It avoids solving nonlinear algebraic equations

Explanation

Stiff systems possess disparate time scales, with fast decaying modes that force explicit methods to use impractically tiny step sizes strictly for numerical stability (e.g. h < 2/λ_{max}). For the test problem y' = λ y with Re(λ) < 0, backward Euler has amplification factor σ = 1 / (1 - h λ). Since |1 - h λ| >= 1 for all Re(h λ) <= 0, |σ| <= 1 for all h > 0. It is unconditionally A-stable, allowing step sizes dictated by accuracy rather than stability.

Question 47Numerical roots, linear systems, quadrature and ODE stability

Applying Heun's method (improved Euler predictor-corrector) with step size h = 0.1 to y' = 2t y with y(0) = 1, what is the computed value of y_1 at t_1 = 0.1?

  1. 1.01 Correct answer.
  2. 1.00
  3. 1.02
  4. 1.10

Explanation

Here f(t, y) = 2t y. Initial state: t_0 = 0, y_0 = 1. Predictor step (explicit Euler): ỹ_1 = y_0 + h f(t_0, y_0) = 1 + 0.1 * (2 * 0 * 1) = 1 + 0 = 1.0. Slope at end of interval: f(t_1, ỹ_1) = 2(0.1)(1.0) = 0.2. Corrector step (trapezoidal average): y_1 = y_0 + (h / 2) [f(t_0, y_0) + f(t_1, ỹ_1)] = 1 + (0.1 / 2) [0 + 0.2] = 1 + 0.05 * 0.2 = 1 + 0.01 = 1.01.

Question 48Numerical roots, linear systems, quadrature and ODE stability

In the Successive Over-Relaxation (SOR) method, what is the necessary range of the relaxation parameter ω for the method to converge?

  1. 0 < ω < 2 Correct answer.
  2. 1 < ω < ∞
  3. -1 < ω < 1
  4. ω >= 2

Explanation

By the Kahan theorem, the spectral radius of the SOR iteration matrix satisfies ρ(L_ω) >= |ω - 1|. For convergence, we must have ρ(L_ω) < 1, which requires |ω - 1| < 1 => -1 < ω - 1 < 1 => 0 < ω < 2. When 1 < ω < 2, the method is over-relaxed to accelerate convergence; when ω = 1, it reduces to standard Gauss-Seidel.

Question 49Numerical roots, linear systems, quadrature and ODE stability

For a diagonalizable matrix with one eigenvalue whose magnitude is strictly larger than all others, and an initial vector with a nonzero component in its eigenvector direction, what does the Power Method approximate?

  1. The unique dominant eigenvalue and its associated eigenvector Correct answer.
  2. All eigenvalues simultaneously
  3. The inverse of the matrix in O(n) operations
  4. The smallest singular value of a rectangular matrix

Explanation

The Power Method repeatedly forms and normalizes Av_k. Expressing the initial vector in the eigenvector basis shows that the component associated with the eigenvalue of strictly largest magnitude dominates, so the directions converge to its eigenvector up to sign. A scaling ratio or Rayleigh quotient then estimates the corresponding dominant eigenvalue.

Question 50Numerical roots, linear systems, quadrature and ODE stability

A two-point Gauss-Legendre rule on [-1, 1] uses nodes x = ±1/√3 with weight 1 at each node. What value does it return for ∫_{-1}^{1} x^2 dx?

  1. 2/3 Correct answer.
  2. 1/3
  3. 1
  4. 2/√3

Explanation

Evaluate x^2 at both nodes: (1/√3)^2 = (-1/√3)^2 = 1/3. With both weights equal to one, the quadrature sum is 1/3 + 1/3 = 2/3. This equals the exact integral because a two-point Gauss-Legendre rule is exact for every polynomial through degree three.