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Concours CCINP - independent subject practice study materials
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Concours CCINP - independent subject practice practice questions

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Question 1Advanced analysis, convergence and multivariable reasoning

In R^3 equipped with the standard dot product, apply the Gram-Schmidt orthonormalisation process to the vectors v_1 = (1, 1, 0) and v_2 = (1, 0, 1). What is the resulting unit vector e_2?

  1. e_2 = (1/sqrt(2), 0, 1/sqrt(2))
  2. e_2 = (1/sqrt(3), -1/sqrt(3), 1/sqrt(3))
  3. e_2 = (-1/sqrt(6), 1/sqrt(6), 2/sqrt(6))
  4. e_2 = (1/sqrt(6), -1/sqrt(6), 2/sqrt(6)) Correct answer.

Explanation

First, e_1 = v_1 / ||v_1|| = (1/sqrt(2), 1/sqrt(2), 0). Next, the orthogonal component is u_2 = v_2 - <v_2, e_1> e_1. Here <v_2, e_1> = 1/sqrt(2), so u_2 = (1, 0, 1) - (1/2, 1/2, 0) = (1/2, -1/2, 1). Its norm is ||u_2|| = sqrt(1/4 + 1/4 + 1) = sqrt(6)/2. Normalising yields e_2 = u_2 / ||u_2|| = (1/sqrt(6), -1/sqrt(6), 2/sqrt(6)).

Question 2Advanced analysis, convergence and multivariable reasoning

Consider the sequence of functions f_n(x) = x^n on the closed interval [0, 1]. Does (f_n) converge uniformly on [0, 1]?

  1. Yes, because the limit function f(x) = 0 is continuous on [0, 1]
  2. No, because the pointwise limit function is discontinuous at x = 1, whereas each f_n is continuous Correct answer.
  3. Yes, because sup_{x in [0, 1]} |f_n(x)| tends to 0 as n approaches infinity
  4. No, because f_n is not differentiable at x = 1

Explanation

The pointwise limit of f_n(x) is f(x) = 0 for x in [0, 1[ and f(1) = 1. Since each f_n is continuous on [0, 1] but the pointwise limit f is discontinuous at x = 1, the uniform limit theorem implies that the convergence cannot be uniform on [0, 1].

Question 3Advanced analysis, convergence and multivariable reasoning

Let sum_{n=1}^inf u_n(x) be a series of functions defined on an interval I. In the CPGE analysis syllabus, what does normal convergence (convergence normale) on I guarantee?

  1. Pointwise convergence only, without uniform convergence
  2. Uniform convergence of the derivatives u'_n(x) on I
  3. Uniform convergence on I, and if each u_n is continuous on I, continuity of the sum function S(x) on I Correct answer.
  4. Convergence of the series of integrals to a strictly positive constant

Explanation

Normal convergence on I means that sum ||u_n||_{infty, I} < infinity. By the Weierstrass M-test (criterion of normal convergence), normal convergence implies uniform convergence on I. Consequently, if each term u_n is continuous on I, the sum function S(x) = sum u_n(x) is continuous on I.

Question 4Advanced analysis, convergence and multivariable reasoning

To differentiate a series of functions S(x) = sum_{n=0}^inf u_n(x) term-by-term on an open interval I, which set of hypotheses from the CPGE theorem is sufficient?

  1. Each u_n is of class C^1 on I and the series sum u_n(x) converges normally on I
  2. Each u_n is continuous and bounded on I
  3. The series sum u_n(x) converges uniformly on I
  4. Each u_n is of class C^1 on I, the series sum u_n(x_0) converges at least at one point x_0 in I, and the series of derivatives sum u'_n converges uniformly on every compact subinterval of I Correct answer.

Explanation

The standard term-by-term differentiation theorem requires: (1) each u_n is C^1 on I, (2) sum u_n(x_0) converges at least at one point x_0 in I, and (3) the series of derivatives sum u'_n converges uniformly on every compact segment [a, b] subset I. Then S is of class C^1 on I and S'(x) = sum u'_n(x).

Question 5Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence R of the complex power series sum_{n=1}^inf ((n + 1) / 4^n) z^(2n)?

  1. R = 2 Correct answer.
  2. R = 4
  3. R = 1/2
  4. R = 16

Explanation

Letting w = z^2, the series becomes sum_{n=1}^inf a_n w^n with a_n = (n + 1) / 4^n. Applying d'Alembert's ratio test to a_n gives lim |a_{n+1}/a_n| = lim ((n+2)/(n+1)) * (1/4) = 1/4, so the radius in w is R_w = 4. Since |w| = |z|^2 < 4 is equivalent to |z| < 2, the radius of convergence in z is R = sqrt(4) = 2.

Question 6Advanced analysis, convergence and multivariable reasoning

What is the exact value of the infinite sum S = sum_{n=1}^inf (1 / (n * 2^n))?

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

Explanation

The standard power series expansion for -ln(1 - x) is sum_{n=1}^inf x^n / n for |x| < 1. Evaluating this series at x = 1/2 gives sum_{n=1}^inf (1/2)^n / n = -ln(1 - 1/2) = -ln(1/2) = ln(2).

Question 7Advanced analysis, convergence and multivariable reasoning

What is the standard power series expansion of f(x) = 1 / (1 + x^2) around x = 0, and what is its radius of convergence R?

  1. sum_{n=0}^inf x^(2n) with R = infinity
  2. sum_{n=0}^inf (-1)^n x^n with R = 1
  3. sum_{n=0}^inf (-1)^n x^(2n) with R = 1 Correct answer.
  4. sum_{n=1}^inf (-1)^n x^(2n) / (2n) with R = 1

Explanation

Using the geometric series formula 1 / (1 - u) = sum_{n=0}^inf u^n valid for |u| < 1 with u = -x^2, we obtain 1 / (1 + x^2) = sum_{n=0}^inf (-x^2)^n = sum_{n=0}^inf (-1)^n x^(2n). The convergence condition |-x^2| < 1 translates to |x| < 1, so the radius of convergence is R = 1.

Question 8Advanced analysis, convergence and multivariable reasoning

Let f be the 2pi-periodic function defined on [-pi, pi[ by f(t) = t. What is the value of the Fourier cosine coefficient a_n(f) for all integers n >= 0?

  1. a_0 = pi/2 and a_n = 2 / (pi n^2) for n >= 1
  2. a_n = (-1)^n / n for all n >= 1
  3. a_n = 2/n for all n >= 1
  4. a_n = 0 for all n >= 0, because f is an odd function Correct answer.

Explanation

On the symmetric interval [-pi, pi], f(t) = t is an odd function. For any integer n >= 0, the integrand t * cos(n t) is the product of an odd function and an even function, which is an odd function. The integral of any odd function over a symmetric interval [-pi, pi] is identically zero, so a_n(f) = (1/pi) int_{-pi}^pi t cos(nt) dt = 0 for all n >= 0.

Question 9Advanced analysis, convergence and multivariable reasoning

According to Dirichlet's theorem on Fourier series, if a 2pi-periodic piecewise C^1 function f has a jump discontinuity at t_0, what value does its Fourier series converge to at t_0?

  1. (f(t_0^+) + f(t_0^-)) / 2 Correct answer.
  2. f(t_0^+)
  3. 0
  4. f(t_0^-)

Explanation

Dirichlet's theorem states that for any 2pi-periodic function that is piecewise C^1, the Fourier series converges at every point t_0 to the regularised value (f(t_0^+) + f(t_0^-)) / 2, which is the arithmetic mean of the right-hand and left-hand limits at the discontinuity.

Question 10Advanced analysis, convergence and multivariable reasoning

The Fourier series of the 2pi-periodic even function f(t) = t^2 on [-pi, pi] is f(t) = pi^2/3 + 4 sum_{n=1}^inf ((-1)^n / n^2) cos(nt). Applying Parseval's identity, what is the exact sum S = sum_{n=1}^inf (1 / n^4)?

  1. pi^4 / 45
  2. pi^4 / 90 Correct answer.
  3. pi^4 / 120
  4. pi^4 / 72

Explanation

Parseval's identity states (1/(2pi)) int_{-pi}^pi |f(t)|^2 dt = a_0^2 + (1/2) sum_{n=1}^inf a_n^2. Here (1/(2pi)) int_{-pi}^pi t^4 dt = pi^4 / 5. The Fourier coefficients are a_0 = pi^2 / 3 and a_n = 4 (-1)^n / n^2. Thus pi^4 / 5 = (pi^4 / 9) + (1/2) * 16 sum_{n=1}^inf (1 / n^4). This yields 8 sum (1 / n^4) = pi^4 (1/5 - 1/9) = 4 pi^4 / 45, so sum_{n=1}^inf (1 / n^4) = pi^4 / 90.

Question 11Advanced analysis, convergence and multivariable reasoning

What is the gradient vector grad f(1, 2) of the scalar field f(x, y) = 3 x^2 y - y^3 + 5 at the point (1, 2)?

  1. (6, -9)
  2. (12, 1)
  3. (12, -9) Correct answer.
  4. (6, 3)

Explanation

The partial derivatives are partial f / partial x = 6 x y and partial f / partial y = 3 x^2 - 3 y^2. Evaluating at (x, y) = (1, 2) yields partial f / partial x = 6(1)(2) = 12, and partial f / partial y = 3(1)^2 - 3(2)^2 = 3 - 12 = -9. Therefore, grad f(1, 2) = (12, -9).

Question 12Advanced analysis, convergence and multivariable reasoning

Let f: R^2 -> R be a C^1 function and let g(r, theta) = f(r cos(theta), r sin(theta)) be its expression in polar coordinates. Using the chain rule, which formula gives partial g / partial theta?

  1. cos(theta) * (partial f / partial x) + sin(theta) * (partial f / partial y)
  2. r cos(theta) * (partial f / partial x) - r sin(theta) * (partial f / partial y)
  3. -sin(theta) * (partial f / partial x) + cos(theta) * (partial f / partial y)
  4. -r sin(theta) * (partial f / partial x) + r cos(theta) * (partial f / partial y) Correct answer.

Explanation

Let x(r, theta) = r cos(theta) and y(r, theta) = r sin(theta). By the multivariate chain rule, partial g / partial theta = (partial f / partial x) * (partial x / partial theta) + (partial f / partial y) * (partial y / partial theta). Since partial x / partial theta = -r sin(theta) and partial y / partial theta = r cos(theta), we obtain -r sin(theta) (partial f / partial x) + r cos(theta) (partial f / partial y).

Question 13Advanced analysis, convergence and multivariable reasoning

What is the maximum value of f(x, y) = x y subject to the constraint x^2 + y^2 = 8 on R^2?

  1. 8
  2. 4 Correct answer.
  3. 2
  4. 4 sqrt(2)

Explanation

Using the arithmetic-geometric mean inequality (or Lagrange multipliers), for any real numbers x and y we have x y <= (x^2 + y^2) / 2. Given x^2 + y^2 = 8, this yields x y <= 8 / 2 = 4. Equality holds when x = y = 2 (or x = y = -2), where 2^2 + 2^2 = 8 and f(2, 2) = 4.

Question 14Inner products, spectral reasoning and matrix structure

Consider the 3x3 real matrix A = [[3, 0, 1], [0, 2, 0], [1, 0, 3]]. What are the eigenvalues of A and their respective algebraic multiplicities?

  1. lambda = 2 with multiplicity 2, and lambda = 4 with multiplicity 1 Correct answer.
  2. lambda = 3 with multiplicity 2, and lambda = 2 with multiplicity 1
  3. lambda = 4 with multiplicity 2, and lambda = 2 with multiplicity 1
  4. lambda = 1, lambda = 2, and lambda = 3 each with multiplicity 1

Explanation

Expanding the characteristic polynomial det(A - lambda I) along the second row gives (2 - lambda) * [(3 - lambda)^2 - 1] = (2 - lambda)(2 - lambda)(4 - lambda) = (2 - lambda)^2 (4 - lambda). The roots are lambda = 2 with algebraic multiplicity 2, and lambda = 4 with algebraic multiplicity 1.

Question 15Inner products, spectral reasoning and matrix structure

In the CPGE mathematics syllabus, what is the necessary and sufficient condition for an endomorphism u of a finite-dimensional vector space E to be diagonalisable over the base field K?

  1. The characteristic polynomial Chi_u is irreducible over K
  2. The characteristic polynomial Chi_u is split (scindé) over K and for each eigenvalue, its geometric multiplicity equals its algebraic multiplicity Correct answer.
  3. All eigenvalues of u have algebraic multiplicity strictly greater than 1
  4. The trace of the matrix representing u is equal to the dimension of E

Explanation

An endomorphism u is diagonalisable over K if and only if its characteristic polynomial is split over K and the dimension of each eigenspace E_lambda(u) equals the algebraic multiplicity of the eigenvalue lambda (dim(E_lambda) = mult(lambda)). This guarantees that E decomposes into a direct sum of eigenspaces.

Question 16Inner products, spectral reasoning and matrix structure

Let A = [[1, 2], [3, 4]] in M_2(R). Using the Cayley-Hamilton theorem, which expression represents the matrix inverse A^(-1)?

  1. A^(-1) = (5 I_2 - A) / 2
  2. A^(-1) = (A + 5 I_2) / 2
  3. A^(-1) = (A - 5 I_2) / 2 Correct answer.
  4. A^(-1) = (A - 2 I_2) / 5

Explanation

The characteristic polynomial of A is Chi_A(X) = X^2 - tr(A) X + det(A) = X^2 - 5X - 2. By the Cayley-Hamilton theorem, A^2 - 5A - 2 I_2 = 0. Multiplying by A^(-1) gives A - 5 I_2 - 2 A^(-1) = 0, which rearranges to A^(-1) = (A - 5 I_2) / 2.

Question 17Inner products, spectral reasoning and matrix structure

Let N in M_3(C) be a nilpotent matrix such that N^2 != 0 and N^3 = 0. Which statement regarding N and its Jordan reduction is correct?

  1. N is diagonalisable over C with unique eigenvalue 0
  2. The eigenspace ker(N) has dimension 2
  3. The minimal polynomial of N is mu_N(X) = X^2
  4. The Jordan canonical form of N consists of a single Jordan block of size 3 with eigenvalue 0, and dim(ker(N)) = 1 Correct answer.

Explanation

Since N^2 != 0 and N^3 = 0, the minimal polynomial is mu_N(X) = X^3. In M_3(C), the size of the largest Jordan block equals the degree of the minimal polynomial, so there is exactly one Jordan block of size 3. Consequently, the geometric multiplicity dim(ker(N)) equals the number of Jordan blocks, which is 1.

Question 18Inner products, spectral reasoning and matrix structure

Let u and v be two commuting endomorphisms (u o v = v o u) of a finite-dimensional complex vector space E. Which property is guaranteed?

  1. u and v possess at least one common eigenvector in E Correct answer.
  2. If u is diagonalisable, then v must also be diagonalisable
  3. u and v have identical characteristic polynomials
  4. ker(u) and ker(v) must have trivial intersection {0}

Explanation

Because u and v commute, every eigenspace E_lambda(u) of u is stable under v. Over C, the restriction of v to E_lambda(u) has at least one eigenvalue and therefore a non-zero eigenvector in E_lambda(u), which is simultaneously an eigenvector of u and v.

Question 19Inner products, spectral reasoning and matrix structure

Given the upper triangular matrix M = [[2, 1], [0, 3]] in M_2(R), what is the entry in the first row, second column of M^n for any positive integer n?

  1. 3^n + 2^n
  2. 3^n - 2^n Correct answer.
  3. 2 * 3^n - 2^n
  4. 3^(n-1) - 2^(n-1)

Explanation

M has distinct eigenvalues 2 and 3 with eigenvectors (1, 0)^T and (1, 1)^T respectively. Diagonalising M = P D P^(-1) with P = [[1, 1], [0, 1]] and P^(-1) = [[1, -1], [0, 1]] yields M^n = P [[2^n, 0], [0, 3^n]] P^(-1) = [[2^n, 3^n - 2^n], [0, 3^n]]. The (1, 2) entry is 3^n - 2^n.

Question 20Inner products, spectral reasoning and matrix structure

According to the CPGE syllabus, an endomorphism u of a finite-dimensional vector space E is diagonalisable if and only if its minimal polynomial mu_u(X):

  1. Is irreducible over the base field K
  2. Has degree equal to the dimension of E
  3. Is split with simple roots (scindé à racines simples) over the base field K Correct answer.
  4. Has only non-zero roots in K

Explanation

A central theorem of endomorphism reduction states that u is diagonalisable over K if and only if there exists a non-zero annihilating polynomial that is split with simple roots over K, which is equivalent to the minimal polynomial mu_u(X) being split with simple roots over K.

Question 21Inner products, spectral reasoning and matrix structure

In R^3 equipped with the standard Euclidean norm, what is the distance from the point A(1, 2, 3) to the affine plane P with Cartesian equation 2x - y + 2z - 9 = 0?

  1. 1 Correct answer.
  2. 2
  3. 3
  4. sqrt(3)

Explanation

The distance from a point A(x_0, y_0, z_0) to a plane ax + by + cz + d = 0 is d(A, P) = |a x_0 + b y_0 + c z_0 + d| / sqrt(a^2 + b^2 + c^2). Substituting the coordinates gives |2(1) - 1(2) + 2(3) - 9| / sqrt(2^2 + (-1)^2 + 2^2) = |2 - 2 + 6 - 9| / sqrt(9) = |-3| / 3 = 1.

Question 22Inner products, spectral reasoning and matrix structure

According to the spectral theorem (théorème spectral) for real Euclidean vector spaces, every real symmetric matrix S in S_n(R):

  1. Is invertible with strictly positive determinant
  2. Is diagonalisable in an orthonormal basis of R^n, and all of its eigenvalues are real Correct answer.
  3. Has strictly positive eigenvalues and zero trace
  4. Can be triangularised over R but not necessarily diagonalised

Explanation

The spectral theorem asserts that any real symmetric matrix S is orthogonally diagonalisable: there exists an orthogonal matrix P in O_n(R) such that P^T S P is a diagonal matrix with all diagonal entries being real eigenvalues.

Question 23Inner products, spectral reasoning and matrix structure

For which real values of parameter m is the quadratic form q(x, y) = x^2 + 2m x y + 4 y^2 positive definite on R^2?

  1. m > 4 or m < -4
  2. -4 < m < 4
  3. -2 < m < 2 Correct answer.
  4. m > 2

Explanation

The matrix associated with q is M = [[1, m], [m, 4]]. By Sylvester's criterion, M is positive definite if and only if all leading principal minors are strictly positive: Delta_1 = 1 > 0 and Delta_2 = det(M) = 4 - m^2 > 0. Solving 4 - m^2 > 0 yields m^2 < 4, which means -2 < m < 2.

Question 24Inner products, spectral reasoning and matrix structure

Let E be a real Euclidean space and p in L(E) a projection (p^2 = p). Which condition is necessary and sufficient for p to be an orthogonal projection?

  1. tr(p) = dim(E)
  2. ker(p) = {0}
  3. ||p(x)|| = ||x|| for all x in E
  4. p is a self-adjoint endomorphism (p* = p) Correct answer.

Explanation

A projection p is orthogonal if and only if ker(p) is orthogonal to im(p). In an inner product space, this orthogonality of image and kernel is equivalent to p being self-adjoint (symmetric), meaning <p(x), y> = <x, p(y)> for all x, y in E, or p* = p.

Question 25Inner products, spectral reasoning and matrix structure

Let R in SO(3) be the 3D rotation matrix given by R = [[0, 0, 1], [1, 0, 0], [0, 1, 0]]. What are the angle theta (with 0 <= theta <= pi) and the axis of rotation of R?

  1. theta = 2pi/3 around the line directed by (1, 1, 1) Correct answer.
  2. theta = pi/3 around the line directed by (1, 0, 1)
  3. theta = pi/2 around the line directed by (1, -1, 0)
  4. theta = 3pi/4 around the line directed by (0, 0, 1)

Explanation

In SO(3), the trace of a rotation matrix satisfies tr(R) = 1 + 2 cos(theta). Here tr(R) = 0 + 0 + 0 = 0, so 1 + 2 cos(theta) = 0, yielding cos(theta) = -1/2, which gives theta = 2pi/3. The axis of rotation is the eigenspace ker(R - I): R (x, y, z)^T = (z, x, y)^T = (x, y, z)^T requires x = y = z, directing the axis along (1, 1, 1).

Question 26Inner products, spectral reasoning and matrix structure

Consider the function f(x, y) = x^3 + y^3 - 3xy on R^2. At the critical point (1, 1), what is the nature of the extremum according to the Hessian matrix?

  1. Strict local minimum, because det(H) = 27 > 0 and partial^2 f / partial x^2 = 6 > 0 Correct answer.
  2. Strict local maximum, because det(H) = 27 > 0 and partial^2 f / partial x^2 = -6 < 0
  3. Saddle point (point-col), because det(H) = -27 < 0
  4. Inconclusive, because det(H) = 0

Explanation

The first partial derivatives are f_x = 3x^2 - 3y and f_y = 3y^2 - 3x. The second derivatives are f_xx = 6x, f_yy = 6y, and f_xy = -3. At (1, 1), the Hessian matrix is H = [[6, -3], [-3, 6]]. The determinant is det(H) = 36 - 9 = 27 > 0. Since det(H) > 0 and f_xx = 6 > 0, the Hessian is positive definite, confirming a strict local minimum.

Question 27Advanced probability, moments and conditional models

Let X follow a geometric distribution with parameter p in ]0, 1[ on N* (P(X = k) = p (1 - p)^(k - 1) for k >= 1). What are the expectation E(X) and the memoryless property of X?

  1. E(X) = 1/p^2 and P(X > s + t | X > s) = P(X > t)
  2. E(X) = (1 - p)/p and P(X = s + t | X > s) = P(X = t)
  3. E(X) = 1/p and P(X > s + t | X > s) = P(X > t) for all integers s, t >= 1 Correct answer.
  4. E(X) = p and P(X > s + t) = P(X > s) P(X > t)

Explanation

For a geometric distribution supported on N*, the expectation is E(X) = 1/p. The memoryless property states that P(X > s + t | X > s) = P(X > s + t) / P(X > s) = (1 - p)^(s + t) / (1 - p)^s = (1 - p)^t = P(X > t) for all integers s, t in N*.

Question 28Advanced probability, moments and conditional models

Let X follow a Poisson distribution with parameter lambda = 3. What is the conditional probability P(X = 2 | X >= 1)?

  1. (9 / 2) e^(-3)
  2. (9 / 2) / (1 - e^(-3))
  3. (9 / 2) e^(-3) / (1 - 4 e^(-3))
  4. (9 / 2) e^(-3) / (1 - e^(-3)) Correct answer.

Explanation

By Bayes' formula, P(X = 2 | X >= 1) = P(X = 2 and X >= 1) / P(X >= 1) = P(X = 2) / (1 - P(X = 0)). For a Poisson variable with parameter 3, P(X = 2) = (3^2 / 2!) e^(-3) = (9/2) e^(-3) and P(X = 0) = e^(-3). Thus P(X = 2 | X >= 1) = (9/2) e^(-3) / (1 - e^(-3)).

Question 29Advanced probability, moments and conditional models

What is the probability generating function (fonction génératrice) G_X(t) = E(t^X) of a random variable X following the Binomial distribution B(n, p)?

  1. G_X(t) = (1 - p + p t)^n Correct answer.
  2. G_X(t) = (p + (1 - p) t)^n
  3. G_X(t) = e^(n p (t - 1))
  4. G_X(t) = (p t)^n

Explanation

By definition, G_X(t) = sum_{k=0}^n (n choose k) p^k (1 - p)^(n - k) t^k = sum_{k=0}^n (n choose k) (p t)^k (1 - p)^(n - k). Applying the binomial theorem directly yields ((1 - p) + p t)^n.

Question 30Advanced probability, moments and conditional models

A random variable X has expectation E(X) = 20 and variance V(X) = 4. Using the Bienaymé-Chebyshev inequality, what is the best guaranteed upper bound on P(|X - 20| >= 6)?

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

Explanation

The Bienaymé-Chebyshev inequality states P(|X - E(X)| >= a) <= V(X) / a^2 for any a > 0. Setting E(X) = 20, V(X) = 4, and a = 6 gives P(|X - 20| >= 6) <= 4 / 6^2 = 4 / 36 = 1/9.

Question 31Advanced probability, moments and conditional models

A workshop has two machines. Machine A produces 60% of the parts and 3% of its output is defective; machine B produces the remaining 40% and 5% of its output is defective. A part drawn at random is found to be defective. What is the probability that it came from machine A?

  1. 9/19 ≈ 0.474 Correct answer.
  2. 3/100 = 0.030
  3. 3/5 = 0.600
  4. 10/19 ≈ 0.526

Explanation

Total probability of a defective part: $P(D) = P(A)P(D|A) + P(B)P(D|B) = 0.60 \times 0.03 + 0.40 \times 0.05 = 0.018 + 0.020 = 0.038$. Bayes' rule then gives $P(A|D) = \frac{P(A)P(D|A)}{P(D)} = \frac{0.018}{0.038} = \frac{18}{38} = \frac{9}{19} \approx 0.474$.

Question 32Analytical mechanics, oscillation and model reduction

An electromagnetic wave propagates in a collisionless plasma with dispersion relation omega^2 = omega_p^2 + c^2 k^2, where omega_p is the plasma frequency. What is the group velocity v_g of the wave for frequencies omega > omega_p?

  1. v_g = c^2 / v_phi = c sqrt(1 - omega_p^2 / omega^2) Correct answer.
  2. v_g = c / sqrt(1 - omega_p^2 / omega^2)
  3. v_g = c (1 + omega_p^2 / omega^2)
  4. v_g = c^2 k / omega_p

Explanation

Differentiating the dispersion relation omega^2 = omega_p^2 + c^2 k^2 with respect to k gives 2 omega (d omega / dk) = 2 c^2 k, so the group velocity is v_g = d omega / dk = c^2 k / omega. Since phase velocity is v_phi = omega / k, we have v_g * v_phi = c^2. Using k = (omega / c) sqrt(1 - omega_p^2 / omega^2), this gives v_g = c sqrt(1 - omega_p^2 / omega^2) < c, which respects relativistic causality.

Question 33Analytical mechanics, oscillation and model reduction

In one-dimensional stationary wave mechanics, for a potential V(x) displaying a finite step discontinuity at x = 0, what boundary conditions must the stationary wave function psi(x) satisfy at x = 0?

  1. psi(x) is continuous, but d psi / dx is necessarily discontinuous
  2. psi(x) is discontinuous, but d psi / dx is continuous
  3. Both psi(x) and its first spatial derivative d psi / dx are continuous across x = 0 Correct answer.
  4. psi(x) must vanish identically at x = 0

Explanation

Integrating the time-independent Schrödinger equation -(hbar^2 / 2m) (d^2 psi / dx^2) + V(x) psi = E psi across [-epsilon, +epsilon] as epsilon -> 0 shows that since (E - V(x)) psi remains bounded across a finite step, the integral vanishes, ensuring d psi / dx is continuous. Integrating once more guarantees that psi(x) itself is continuous.

Question 34Analytical mechanics, oscillation and model reduction

A point projectile is launched with speed v_0 at angle α above horizontal and lands at the same height under uniform gravity g, with no drag. What is its horizontal range for this given angle?

  1. X_{max} = \frac{v_0^2 \sin(2\alpha)}{g} Correct answer.
  2. X_{max} = \frac{v_0^2 \cos(2\alpha)}{g}
  3. X_{max} = \frac{2 v_0 \sin(\alpha)}{g}
  4. X_{max} = \frac{v_0^2 \sin^2(\alpha)}{2g}

Explanation

Parametric equations: $x(t) = v_0 \cos(\alpha) t$, $y(t) = -\frac{1}{2}g t^2 + v_0 \sin(\alpha) t$. Flight time when landing ($y = 0$): $t_{flight} = \frac{2 v_0 \sin(\alpha)}{g}$. Substituting into $x(t)$: $X_{max} = v_0 \cos(\alpha) \left(\frac{2 v_0 \sin(\alpha)}{g}\right) = \frac{v_0^2 (2 \sin\alpha \cos\alpha)}{g} = \frac{v_0^2 \sin(2\alpha)}{g}$. (Maximum range occurs at $\alpha = 45^\circ$).

Question 35Analytical mechanics, oscillation and model reduction

A simple pendulum of length $L = 0.981\text{ m}$ swings with small angular oscillations in a gravitational field $g = 9.81\text{ m/s}^2$. What is its period of oscillation $T_0$?

  1. 1.99 s (~2.0 s, T_0 = 2π √(L/g)) Correct answer.
  2. 1.00 s
  3. 6.28 s
  4. 0.31 s

Explanation

For small oscillations ($\sin\theta \approx \theta$), the period is $T_0 = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.981}{9.81}} = 2\pi \sqrt{0.1} = 2\pi \times 0.316227 \approx 1.9869\text{ s} \approx 2.0\text{ seconds}$.

Question 36Analytical mechanics, oscillation and model reduction

What is the moment of inertia $J_{\Delta}$ of a solid homogeneous cylinder of mass $M$ and radius $R$ about its central longitudinal axis of symmetry?

  1. J_{\Delta} = \frac{1}{2} M R^2 Correct answer.
  2. J_{\Delta} = M R^2
  3. J_{\Delta} = \frac{2}{5} M R^2
  4. J_{\Delta} = \frac{1}{12} M R^2

Explanation

Integrating concentric thin cylindrical shells: $J_{\Delta} = \int_0^R r^2 \, dm = \int_0^R r^2 (\rho 2\pi r L \, dr) = 2\pi \rho L \left[ \frac{r^4}{4} \right]_0^R = \frac{1}{2} (\rho \pi R^2 L) R^2 = \frac{1}{2} M R^2$.

Question 37Analytical mechanics, oscillation and model reduction

According to Bernoulli's equation for steady, incompressible, frictionless streamline fluid flow with density $\rho$, what quantity remains constant along a streamline?

  1. P + \frac{1}{2} \rho v^2 + \rho g z = \text{constant} Correct answer.
  2. P + \rho v + g z = \text{constant}
  3. \frac{P}{\rho} + v^2 + g z = \text{constant}
  4. P · v · z = \text{constant}

Explanation

Bernoulli's theorem represents conservation of mechanical energy per unit volume for an inviscid, incompressible fluid in steady flow: Static pressure ($P$) + Dynamic pressure ($\frac{1}{2}\rho v^2$) + Hydrostatic potential pressure ($\rho g z$) = constant along any streamline.

Question 38Analytical mechanics, oscillation and model reduction

A venturi tube with inlet cross-section $S_1 = 100\text{ cm}^2$ and throat constriction $S_2 = 25\text{ cm}^2$ carries water ($\rho = 1000\text{ kg/m}^3$). If the measured pressure difference is $\Delta P = P_1 - P_2 = 30\text{ kPa}$ ($30000\text{ Pa}$), what is the volume flow rate $Q$?

  1. 20.0 L/s (0.020 m³/s) Correct answer.
  2. 40.0 L/s
  3. 10.0 L/s
  4. 80.0 L/s

Explanation

By continuity: $v_1 S_1 = v_2 S_2 \implies v_2 = v_1 \frac{S_1}{S_2} = 4 v_1$. From Bernoulli: $P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2) = \frac{1}{2}\rho(16 v_1^2 - v_1^2) = \frac{15}{2}\rho v_1^2$. Thus, $30000 = 7.5 \times 1000 \times v_1^2 = 7500 v_1^2 \implies v_1^2 = \frac{30000}{7500} = 4.0 \implies v_1 = 2.0\text{ m/s}$. Volume flow rate is $Q = S_1 \cdot v_1 = (100 \times 10^{-4}\text{ m}^2) \times 2.0\text{ m/s} = 0.020\text{ m}^3/\text{s} = 20.0\text{ L/s}$.

Question 39Analytical mechanics, oscillation and model reduction

A mass $m = 0.5\text{ kg}$ on a horizontal frictionless table is attached to an ideal spring with spring constant $k = 50\text{ N/m}$. The mass is displaced by $x_0 = 0.10\text{ m}$ from equilibrium and released from rest. What is the maximum velocity $v_{max}$ of the mass?

  1. 1.0 m/s (v_max = x_0 · √(k/m) = 0.10 · √(50/0.5) = 0.10 · 10 = 1.0 m/s) Correct answer.
  2. 10.0 m/s
  3. 0.10 m/s
  4. 5.0 m/s

Explanation

By conservation of mechanical energy: $\frac{1}{2} k x_0^2 = \frac{1}{2} m v_{max}^2 \implies v_{max} = x_0 \sqrt{\frac{k}{m}} = 0.10 \sqrt{\frac{50}{0.5}} = 0.10 \sqrt{100} = 0.10 \times 10 = 1.0\text{ m/s}$.

Question 40Analytical mechanics, oscillation and model reduction

What is the escape velocity $v_{esc}$ from the surface of a spherical planet of mass $M$ and radius $R$?

  1. v_{esc} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g_0 R} Correct answer.
  2. v_{esc} = \sqrt{\frac{G M}{R}}
  3. v_{esc} = \frac{2 G M}{R}
  4. v_{esc} = \sqrt{\frac{G M}{2 R}}

Explanation

To escape the gravitational potential well to infinity with zero residual kinetic energy: $E_m = E_k + E_p = \frac{1}{2} m v_{esc}^2 - \frac{G M m}{R} = 0 \implies \frac{1}{2} v_{esc}^2 = \frac{G M}{R} \implies v_{esc} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g_0 R}$. (For Earth, $v_{esc} \approx 11.2\text{ km/s}$).

Question 41Analytical mechanics, oscillation and model reduction

What is the Reynolds number ($Re$) of fluid flow with density $\rho = 1000\text{ kg/m}^3$, dynamic viscosity $\mu = 10^{-3}\text{ Pa}\cdot\text{s}$, velocity $v = 0.5\text{ m/s}$ in a pipe of diameter $D = 0.04\text{ m}$ ($4\text{ cm}$), and what is the flow regime?

  1. Re = 20,000 (Turbulent flow / régime turbulent car Re > 4000) Correct answer.
  2. Re = 2,000 (Laminar flow)
  3. Re = 200 (Creeping flow)
  4. Re = 200,000

Explanation

Reynolds number is $Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{1000 \times 0.5 \times 0.04}{10^{-3}} = \frac{20}{10^{-3}} = 20000$. Since $Re > 4000$, the flow is fully turbulent (laminar flow occurs for $Re < 2300$).

Question 42Analytical mechanics, oscillation and model reduction

In the presence of quadratic fluid drag force $\vec{F}_d = -\frac{1}{2} C_d \rho A v^2 \vec{u}_v$, what is the terminal falling velocity $v_t$ (vitesse limite de chute) of a spherical body of mass $m$ falling in air under gravity $g$?

  1. v_t = \sqrt{\frac{2 m g}{C_d \rho A}} Correct answer.
  2. v_t = \frac{m g}{C_d \rho A}
  3. v_t = \sqrt{\frac{m g}{2 C_d \rho A}}
  4. v_t = \frac{2 m g}{C_d \rho A}

Explanation

At terminal velocity, downward weight equals upward aerodynamic drag force ($a = 0$): $m g = \frac{1}{2} C_d \rho A v_t^2 \implies v_t^2 = \frac{2 m g}{C_d \rho A} \implies v_t = \sqrt{\frac{2 m g}{C_d \rho A}}$.

Question 43Analytical mechanics, oscillation and model reduction

What is the angular momentum $\vec{L}_O$ of a point particle of mass $m = 2\text{ kg}$ located at position $\vec{r} = (1, 3, 0)\text{ m}$ moving with velocity $\vec{v} = (4, -2, 0)\text{ m/s}$ relative to the origin $O$?

  1. \vec{L}_O = (0, 0, -28) \text{ kg}\cdot\text{m}^2\text{/s} \quad (-28 \vec{k}) Correct answer.
  2. \vec{L}_O = (0, 0, +28) \text{ kg}\cdot\text{m}^2\text{/s}
  3. \vec{L}_O = (0, 0, -14) \text{ kg}\cdot\text{m}^2\text{/s}
  4. \vec{L}_O = (0, 0, 0)

Explanation

Angular momentum is $\vec{L}_O = \vec{r} \times \vec{p} = m (\vec{r} \times \vec{v})$. $\vec{r} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 3 & 0 \\ 4 & -2 & 0 \end{vmatrix} = \vec{k} \big((1)(-2) - (3)(4)\big) = \vec{k}(-2 - 12) = -14 \vec{k}$. Multiplying by mass $m = 2\text{ kg}$: $\vec{L}_O = 2 \times (-14 \vec{k}) = -28 \vec{k} = (0, 0, -28)\text{ kg}\cdot\text{m}^2\text{/s}$.

Question 44Analytical mechanics, oscillation and model reduction

A uniform solid cylinder of mass $M$ and radius $R$ rotates about its central axis with angular velocity $\omega_0$. What is its total kinetic energy of rotation $E_k$?

  1. E_k = \frac{1}{4} M R^2 \omega_0^2 Correct answer.
  2. E_k = \frac{1}{2} M R^2 \omega_0^2
  3. E_k = M R^2 \omega_0^2
  4. E_k = \frac{1}{2} M \omega_0^2

Explanation

Rotational kinetic energy is $E_k = \frac{1}{2} J \omega_0^2$. Since $J = \frac{1}{2} M R^2$ for a solid cylinder: $E_k = \frac{1}{2} \left(\frac{1}{2} M R^2\right) \omega_0^2 = \frac{1}{4} M R^2 \omega_0^2$.

Question 45Analytical mechanics, oscillation and model reduction

In a damped harmonic oscillator governed by $\ddot{x} + 2\lambda \dot{x} + \omega_0^2 x = 0$ with underdamping ($\lambda < \omega_0$), what is the pseudo-frequency $\omega_p$ of the oscillations?

  1. \omega_p = \sqrt{\omega_0^2 - \lambda^2} Correct answer.
  2. \omega_p = \omega_0 + \lambda
  3. \omega_p = \sqrt{\omega_0^2 + \lambda^2}
  4. \omega_p = \omega_0 - \lambda

Explanation

Characteristic equation: $r^2 + 2\lambda r + \omega_0^2 = 0 \implies r = -\lambda \pm \sqrt{\lambda^2 - \omega_0^2} = -\lambda \pm i \sqrt{\omega_0^2 - \lambda^2}$. The pseudo-frequency of the decaying sinusoidal response $x(t) = X_0 e^{-\lambda t} \cos(\omega_p t + \phi)$ is $\omega_p = \sqrt{\omega_0^2 - \lambda^2}$.

Question 46Analytical mechanics, oscillation and model reduction

What is the logarithmic decrement $\delta = \ln\left(\frac{x(t)}{x(t + T_p)}\right)$ of the damped oscillator in terms of damping coefficient $\lambda$ and pseudo-period $T_p$?

  1. \delta = \lambda · T_p Correct answer.
  2. \delta = \frac{\lambda}{T_p}
  3. \delta = \frac{2\pi}{\lambda T_p}
  4. \delta = \lambda^2 T_p

Explanation

The amplitude envelope decays as $A(t) = X_0 e^{-\lambda t}$. The ratio of two successive peak amplitudes separated by one pseudo-period $T_p$ is $\frac{x(t)}{x(t + T_p)} = \frac{e^{-\lambda t}}{e^{-\lambda(t + T_p)}} = e^{\lambda T_p}$. Taking natural logarithm: $\delta = \ln(e^{\lambda T_p}) = \lambda T_p$.

Question 47Analytical mechanics, oscillation and model reduction

What is the center of mass $G(\bar{x}, \bar{y})$ of a uniform thin semicircular wire of radius $R$ positioned in the upper half-plane $y \ge 0$ centered at the origin?

  1. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{2R}{\pi} Correct answer.
  2. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{4R}{3\pi}
  3. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{R}{2}
  4. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{R}{\pi}

Explanation

By symmetry, $\bar{x} = 0$. For a 1D thin wire of length $L = \pi R$: $\bar{y} = \frac{1}{L} \int y \, dl = \frac{1}{\pi R} \int_0^\pi (R \sin\theta) (R \, d\theta) = \frac{R}{\pi} [-\cos\theta]_0^\pi = \frac{R}{\pi} (1 - (-1)) = \frac{2R}{\pi} \approx 0.637 R$.

Question 48Electromagnetic fields, thermal physics and wave models

A capacitor with capacitance C = 10 microF is charged to a potential difference V = 200 V. What is the total electrostatic energy W_e stored in its electric field?

  1. 0.20 J Correct answer.
  2. 0.40 J
  3. 2.0 mJ
  4. 20 J

Explanation

The electrostatic energy stored in a capacitor is W_e = (1/2) C V^2. Substituting C = 10 * 10^(-6) F and V = 200 V yields W_e = 0.5 * (10 * 10^(-6)) * (200)^2 = 0.5 * 10^(-5) * 40,000 = 0.20 J.

Question 49Electromagnetic fields, thermal physics and wave models

An ideal long solenoid of length L, cross-sectional area S, and total number of turns N is placed in vacuum. What is its self-inductance L_ind?

  1. L_ind = mu_0 N S / L
  2. L_ind = mu_0 N^2 S L
  3. L_ind = mu_0 N^2 S / L Correct answer.
  4. L_ind = mu_0 N S / (2 L)

Explanation

The magnetic field inside an ideal solenoid is uniform and equal to B = mu_0 (N / L) I. The magnetic flux through a single turn is Phi_1 = B S = mu_0 (N / L) I S. The total flux through all N turns is Phi = N Phi_1 = mu_0 (N^2 S / L) I. Since L_ind = Phi / I, we obtain L_ind = mu_0 N^2 S / L.

Question 50Electromagnetic fields, thermal physics and wave models

In a monochromatic plane electromagnetic wave propagating in vacuum along +u_z, the electric field is E(z, t) = E_0 cos(omega t - k z) u_x. What is the time-averaged Poynting vector <Pi>?

  1. <Pi> = (E_0^2 / (mu_0 c)) u_z
  2. <Pi> = (E_0^2 / (4 mu_0 c)) u_z
  3. <Pi> = (epsilon_0 c E_0^2) u_x
  4. <Pi> = (E_0^2 / (2 mu_0 c)) u_z Correct answer.

Explanation

The magnetic field associated with the wave is B(z, t) = (E_0 / c) cos(omega t - k z) u_y. The instantaneous Poynting vector is Pi = (E x B) / mu_0 = (E_0^2 / (mu_0 c)) cos^2(omega t - k z) u_z. Taking the time average yields <cos^2(omega t - k z)> = 1/2, giving <Pi> = (E_0^2 / (2 mu_0 c)) u_z.