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

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Question 1Advanced analysis, convergence and multivariable reasoning

A geometric sequence $(u_n)$ of positive terms satisfies $u_2 = 12$ and $u_5 = 96$. What is its common ratio $q$ and its first term $u_0$?

  1. q = 2 and u_0 = 3 Correct answer.
  2. q = 2 and u_0 = 6
  3. q = 4 and u_0 = 3
  4. q = 8 and u_0 = 1.5

Explanation

For a geometric sequence, $u_5 = u_2 \cdot q^{3}$, so $q^3 = \frac{u_5}{u_2} = \frac{96}{12} = 8$ and therefore $q = 2$. Working back from $u_2 = u_0 \cdot q^2$: $12 = u_0 \times 4$, so $u_0 = 3$. Checking: the sequence runs $3, 6, 12, 24, 48, 96$.

Question 2Advanced analysis, convergence and multivariable reasoning

What is the cross product $\vec{u} \times \vec{v}$ of vectors $\vec{u} = (1, 2, 3)$ and $\vec{v} = (4, 5, 6)$ in $\mathbb{R}^3$?

  1. (-3, 6, -3) Correct answer.
  2. (3, -6, 3)
  3. (32, 32, 32)
  4. (0, 0, 0)

Explanation

Using determinant expansion: $\vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} = \vec{i}(12 - 15) - \vec{j}(6 - 12) + \vec{k}(5 - 8) = -3\vec{i} + 6\vec{j} - 3\vec{k} = (-3, 6, -3)$.

Question 3Advanced analysis, convergence and multivariable reasoning

What is the distance from point $P(2, -1, 3)$ to the plane $\mathcal{P}: 2x - y + 2z - 5 = 0$ in 3D Euclidean space?

  1. 2.0 (d = |2(2) - (-1) + 2(3) - 5| / √(2² + (-1)² + 2²)) Correct answer.
  2. 6.0
  3. 1.5
  4. 3.0

Explanation

The distance formula is $d = \frac{|A x_0 + B y_0 + C z_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$. Substituting: $d = \frac{|2(2) - (-1) + 2(3) - 5|}{\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{|4 + 1 + 6 - 5|}{\sqrt{4 + 1 + 4}} = \frac{|6|}{\sqrt{9}} = \frac{6}{3} = 2.0$.

Question 4Advanced analysis, convergence and multivariable reasoning

A committee of 4 students must be chosen from a class of 10 students. How many different committees are possible if the order of selection does not matter?

  1. 210 Correct answer.
  2. 5040
  3. 40
  4. 24

Explanation

Because the committee is an unordered selection, the count is the combination $\binom{10}{4} = \frac{10!}{4! \, 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210$.

Question 5Advanced analysis, convergence and multivariable reasoning

What is the remainder $R(X)$ in the Euclidean division of polynomial $P(X) = X^4 - 3X^3 + 2X^2 - 5X + 7$ by $(X - 2)$?

  1. -11
  2. -3 Correct answer.
  3. +5
  4. 0

Explanation

By the Remainder Theorem, the remainder of dividing $P(X)$ by $(X - a)$ is $R = P(a)$. Evaluating at $a = 2$: $P(2) = 2^4 - 3(2^3) + 2(2^2) - 5(2) + 7 = 16 - 3(8) + 2(4) - 10 + 7 = 16 - 24 + 8 - 10 + 7 = 0 - 10 + 7 = -3$.

Question 6Advanced analysis, convergence and multivariable reasoning

Solve the system of 3 linear equations using Gaussian elimination: $\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases}$. What is $(x, y, z)$?

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

Explanation

Adding eq 1 and eq 3: $(x+y+z) + (x+2y-z) = 6 + 2 \implies 2x + 3y = 8$. Subtracting eq 1 from eq 2: $(2x-y+z) - (x+y+z) = 3 - 6 \implies x - 2y = -3 \implies x = 2y - 3$. Substitute into $2(2y-3) + 3y = 8 \implies 4y - 6 + 3y = 8 \implies 7y = 14 \implies y = 2$. Then $x = 2(2) - 3 = 1$. From eq 1: $1 + 2 + z = 6 \implies z = 3$. The unique solution is $(x, y, z) = (1, 2, 3)$.

Question 7Advanced analysis, convergence and multivariable reasoning

Compute the sum of the roots of the polynomial $P(X) = 2X^3 - 6X^2 + 8X - 10$ using Vieta's formulas (relations entre coefficients et racines).

  1. 3.0 Correct answer.
  2. -3.0
  3. 4.0
  4. 5.0

Explanation

By Vieta's formulas for a cubic polynomial $a_3 X^3 + a_2 X^2 + a_1 X + a_0$, the sum of roots is $\sum x_i = -\frac{a_2}{a_3}$. Here $a_3 = 2$ and $a_2 = -6$, so $\sum x_i = -\frac{-6}{2} = 3.0$.

Question 8Advanced analysis, convergence and multivariable reasoning

In the plane, $A$, $B$ and $C$ are three points. What are the coordinates of the barycentre $G$ of the weighted system $\{(A, 2), (B, 1), (C, 1)\}$ when $A(1, 4)$, $B(3, 0)$ and $C(-2, 2)$?

  1. G(0.75, 2.5) Correct answer.
  2. G(0.667, 2)
  3. G(3, 10)
  4. G(1, 3.33)

Explanation

The barycentre of $\{(A, \alpha), (B, \beta), (C, \gamma)\}$ with $\alpha + \beta + \gamma \ne 0$ has coordinates $\frac{\alpha x_A + \beta x_B + \gamma x_C}{\alpha + \beta + \gamma}$, and likewise for $y$. The total weight is $2 + 1 + 1 = 4$, so $x_G = \frac{2(1) + 1(3) + 1(-2)}{4} = \frac{3}{4} = 0.75$ and $y_G = \frac{2(4) + 1(0) + 1(2)}{4} = \frac{10}{4} = 2.5$.

Question 9Advanced analysis, convergence and multivariable reasoning

Find the general solution in $\mathbb{C}$ to the equation $z^3 = -8i$.

  1. z_k = 2 e^{i(\pi/2 + 2k\pi/3)} for k ∈ {0, 1, 2} Correct answer.
  2. z_k = 8 e^{i(-\pi/2 + 2k\pi/3)}
  3. z_k = 2 e^{i(2k\pi/3)}
  4. z_k = -2i only

Explanation

Express $-8i$ in exponential form: $|-8i| = 8$, and $\arg(-8i) = -\pi/2 = 3\pi/2$. The cube roots are $z_k = \sqrt[3]{8} e^{i(3\pi/2 + 2k\pi)/3} = 2 e^{i(\pi/2 + 2k\pi/3)}$ for $k \in \{0, 1, 2\}$. For $k = 0$: $z_0 = 2e^{i\pi/2} = 2i$ (check: $(2i)^3 = 8i^3 = -8i$). For $k = 1$: $z_1 = 2e^{i 7\pi/6} = -\sqrt{3} - i$. For $k = 2$: $z_2 = 2e^{i 11\pi/6} = \sqrt{3} - i$.

Question 10Advanced analysis, convergence and multivariable reasoning

Let $P(X) = (X^2 + 1)^2 (X - 3)^3 (X + 2)$. What is the degree of $P(X)$ and the multiplicity of root $x = 3$?

  1. Degree = 8; Multiplicity of root 3 = 3 Correct answer.
  2. Degree = 7; Multiplicity of root 3 = 3
  3. Degree = 6; Multiplicity of root 3 = 2
  4. Degree = 8; Multiplicity of root 3 = 1

Explanation

The degree is the sum of degrees of factors: $(2 \times 2) + (1 \times 3) + (1 \times 1) = 4 + 3 + 1 = 8$. The root $x = 3$ appears in the term $(X - 3)^3$, so its algebraic multiplicity is 3.

Question 11Advanced analysis, convergence and multivariable reasoning

What is the equation of the tangent line to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ at point $M_0(2, \frac{3\sqrt{3}}{2})$?

  1. \frac{x}{8} + \frac{y\sqrt{3}}{6} = 1 Correct answer.
  2. \frac{x}{16} + \frac{y}{9} = 1
  3. 2x + 3y = 12
  4. \frac{x}{4} + \frac{y}{3} = 1

Explanation

The equation of the tangent to an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at point $(x_0, y_0)$ is $\frac{x x_0}{a^2} + \frac{y y_0}{b^2} = 1$. Substituting $(x_0, y_0) = (2, \frac{3\sqrt{3}}{2})$: $\frac{2x}{16} + \frac{y(3\sqrt{3}/2)}{9} = 1 \implies \frac{x}{8} + \frac{y\sqrt{3}}{6} = 1$.

Question 12Advanced analysis, convergence and multivariable reasoning

What is the remainder of the Euclidean division of $7^{2026}$ by 5?

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

Explanation

Work modulo 5. Since $7 \equiv 2 \pmod 5$, we need $2^{2026} \bmod 5$. The powers of 2 cycle with period 4: $2^1 \equiv 2$, $2^2 \equiv 4$, $2^3 \equiv 3$, $2^4 \equiv 1 \pmod 5$. Because $2026 = 4 \times 506 + 2$, we get $2^{2026} \equiv 2^2 \equiv 4 \pmod 5$, so the remainder is 4.

Question 13Advanced analysis, convergence and multivariable reasoning

In an orthonormal frame of space, what is the Cartesian equation of the sphere of centre $\Omega(1, -2, 3)$ and radius $R = 5$?

  1. (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25 Correct answer.
  2. (x + 1)^2 + (y - 2)^2 + (z + 3)^2 = 25
  3. (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 5
  4. x^2 + y^2 + z^2 = 25

Explanation

A point $M(x, y, z)$ lies on the sphere when $\Omega M = R$, that is $\Omega M^2 = R^2$. Writing the squared distance from $\Omega(x_0, y_0, z_0)$ gives $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2$. Substituting the centre $(1, -2, 3)$ and $R = 5$ yields $(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25$.

Question 14Advanced analysis, convergence and multivariable reasoning

The plane transformation $S$ associates with each point of affix $z$ the point of affix $z' = 2i\,z + 3$. What kind of transformation is $S$, and what are its ratio and angle?

  1. A direct similitude of ratio 2 and angle π/2 Correct answer.
  2. A rotation of angle π/2 about the point of affix 3
  3. A translation by the vector of affix 3
  4. A direct similitude of ratio 2i and angle π

Explanation

Any map $z' = az + b$ with $a \in \mathbb{C}^*$ and $a \ne 1$ is a direct similitude whose ratio is $|a|$ and whose angle is $\arg(a)$. Here $a = 2i$, so the ratio is $|2i| = 2$ and the angle is $\arg(2i) = \frac{\pi}{2}$. Its centre is the fixed point $\omega$ solving $\omega = 2i\omega + 3$, giving $\omega = \frac{3}{1 - 2i} = \frac{3(1 + 2i)}{5}$.

Question 15Advanced analysis, convergence and multivariable reasoning

Evaluate the limit: $L = \lim_{x \to 0} \frac{\sin(x) - x + \frac{x^3}{6}}{x^5}$.

  1. 1/120 Correct answer.
  2. -1/120
  3. 1/24
  4. 0

Explanation

Using Taylor series expansion around $x = 0$: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + o(x^5) = x - \frac{x^3}{6} + \frac{x^5}{120} + o(x^5)$. Substituting into the numerator: $\sin(x) - x + \frac{x^3}{6} = \frac{x^5}{120} + o(x^5)$. Dividing by $x^5$: $L = \lim_{x \to 0} \left(\frac{1}{120} + \frac{o(x^5)}{x^5}\right) = \frac{1}{120}$.

Question 16Advanced analysis, convergence and multivariable reasoning

Evaluate the definite integral: $I = \int_0^{\pi/2} x \cos(x) \, dx$ using integration by parts.

  1. \frac{\pi}{2} - 1 Correct answer.
  2. \frac{\pi}{2} + 1
  3. 1
  4. 0

Explanation

Let $u = x \implies du = dx$, and $dv = \cos(x)\,dx \implies v = \sin(x)$. $I = [x \sin(x)]_0^{\pi/2} - \int_0^{\pi/2} \sin(x)\,dx = \left(\frac{\pi}{2}\sin(\frac{\pi}{2}) - 0\right) - [-\cos(x)]_0^{\pi/2} = \frac{\pi}{2}(1) + (\cos(\pi/2) - \cos(0)) = \frac{\pi}{2} + (0 - 1) = \frac{\pi}{2} - 1$.

Question 17Advanced analysis, convergence and multivariable reasoning

What is the general solution $y(x)$ to the second-order linear differential equation $y'' - 4y' + 4y = 0$?

  1. y(x) = (C_1 + C_2 x) e^{2x} Correct answer.
  2. y(x) = C_1 e^{2x} + C_2 e^{-2x}
  3. y(x) = C_1 \cos(2x) + C_2 \sin(2x)
  4. y(x) = C_1 e^{4x} + C_2

Explanation

Characteristic equation: $r^2 - 4r + 4 = 0 \implies (r - 2)^2 = 0$. This has a double root $r = 2$ of multiplicity 2. The general solution is $y(x) = (C_1 + C_2 x) e^{2x}$.

Question 18Advanced analysis, convergence and multivariable reasoning

A sequence is defined by $u_0 = 1$ and $u_{n+1} = \frac{1}{2}u_n + 3$ for all $n \ge 0$. Assuming it converges, what is its limit $\ell$?

  1. 6 Correct answer.
  2. 3
  3. 1
  4. It diverges to +∞

Explanation

If $(u_n)$ converges to $\ell$, then passing to the limit in the recurrence gives $\ell = \frac{1}{2}\ell + 3$, hence $\frac{1}{2}\ell = 3$ and $\ell = 6$. The convergence itself is confirmed by setting $v_n = u_n - 6$: then $v_{n+1} = \frac{1}{2}v_n$, so $v_n = -5 \left(\frac{1}{2}\right)^n \to 0$ and $u_n = 6 - 5\left(\frac{1}{2}\right)^n \to 6$.

Question 19Advanced analysis, convergence and multivariable reasoning

Evaluate the improper integral: $I = \int_0^\infty x e^{-2x} \, dx$.

  1. 1/4 Correct answer.
  2. 1/2
  3. 1
  4. \infty (divergent)

Explanation

Using integration by parts with $u = x, dv = e^{-2x} dx \implies v = -\frac{1}{2}e^{-2x}$: $I = \left[ -\frac{x}{2}e^{-2x} \right]_0^\infty + \int_0^\infty \frac{1}{2}e^{-2x} dx = 0 + \left[ -\frac{1}{4}e^{-2x} \right]_0^\infty = 0 - \left(-\frac{1}{4}\right) = \frac{1}{4}$. (Alternatively, from Laplace transform $\mathcal{L}\{t\} = 1/s^2$ at $s = 2$, $I = 1/2^2 = 1/4$).

Question 20Advanced analysis, convergence and multivariable reasoning

What is the sum of the convergent numerical series $S = \sum_{n=0}^\infty \frac{2^n}{n!}$?

  1. e² Correct answer.
  2. 2e
  3. e
  4. \infty

Explanation

From the Taylor series expansion of the exponential function $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. Evaluating at $x = 2$: $S = \sum_{n=0}^\infty \frac{2^n}{n!} = e^2$.

Question 21Advanced analysis, convergence and multivariable reasoning

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of the function $f(x, y) = x^3 y^2 + \sin(x y)$.

  1. \frac{\partial f}{\partial x} = 3x^2 y^2 + y \cos(xy) \quad \text{and} \quad \frac{\partial f}{\partial y} = 2x^3 y + x \cos(xy) Correct answer.
  2. \frac{\partial f}{\partial x} = 3x^2 y^2 + \cos(xy) \quad \text{and} \quad \frac{\partial f}{\partial y} = 2x^3 y + \cos(xy)
  3. \frac{\partial f}{\partial x} = 3x^2 + y \cos(xy) \quad \text{and} \quad \frac{\partial f}{\partial y} = 2y + x \cos(xy)
  4. \frac{\partial f}{\partial x} = 3x^2 y^2 - y \cos(xy) \quad \text{and} \quad \frac{\partial f}{\partial y} = 2x^3 y - x \cos(xy)

Explanation

Differentiating with respect to $x$ (treating $y$ as constant): $\frac{\partial f}{\partial x} = 3x^2 y^2 + \frac{\partial(xy)}{\partial x}\cos(xy) = 3x^2 y^2 + y\cos(xy)$. Differentiating with respect to $y$ (treating $x$ as constant): $\frac{\partial f}{\partial y} = x^3(2y) + \frac{\partial(xy)}{\partial y}\cos(xy) = 2x^3 y + x\cos(xy)$.

Question 22Advanced analysis, convergence and multivariable reasoning

Evaluate the double integral $I = \iint_D (x + y) \, dx \, dy$ over the rectangle $D = [0, 2] \times [0, 3]$.

  1. 15 Correct answer.
  2. 30
  3. 7.5
  4. 6

Explanation

$I = \int_0^2 \left( \int_0^3 (x + y) \, dy \right) dx = \int_0^2 \left[ x y + \frac{y^2}{2} \right]_0^3 dx = \int_0^2 \left( 3x + \frac{9}{2} \right) dx = \left[ \frac{3x^2}{2} + \frac{9x}{2} \right]_0^2 = \frac{3(4)}{2} + \frac{9(2)}{2} = 6 + 9 = 15$.

Question 23Advanced analysis, convergence and multivariable reasoning

Evaluate the limit $L = \lim_{x \to +\infty} \frac{\ln(x)}{x}$.

  1. 0 Correct answer.
  2. 1
  3. +∞
  4. The limit does not exist

Explanation

This is the standard comparative-growth result (croissance comparée): the exponential dominates the power, which dominates the logarithm. Formally $\lim_{x \to +\infty} \frac{\ln x}{x} = 0$, because $\ln x$ grows without bound far more slowly than $x$ does. The same conclusion follows from L'Hôpital's rule on the indeterminate form $\frac{\infty}{\infty}$: the ratio of derivatives is $\frac{1/x}{1} = \frac{1}{x} \to 0$.

Question 24Advanced analysis, convergence and multivariable reasoning

Evaluate the integral $I = \int_0^1 \frac{1}{1 + x^2} \, dx$.

  1. \pi/4 Correct answer.
  2. \pi/2
  3. 1
  4. \ln(2)

Explanation

The standard antiderivative is $\arctan(x)$: $I = [\arctan(x)]_0^1 = \arctan(1) - \arctan(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

Question 25Advanced analysis, convergence and multivariable reasoning

Find the general solution of the non-homogeneous ODE $y' + 2y = 4x$.

  1. y(x) = C e^{-2x} + 2x - 1 Correct answer.
  2. y(x) = C e^{2x} + 2x - 1
  3. y(x) = C e^{-2x} + 4x
  4. y(x) = C e^{-2x} + 2x + 1

Explanation

Homogeneous solution: $y_h = C e^{-2x}$. Particular solution form: $y_p = A x + B \implies y_p' = A$. Substituting: $A + 2(Ax + B) = 4x \implies 2Ax + (A + 2B) = 4x$. Equating coefficients: $2A = 4 \implies A = 2$; $A + 2B = 0 \implies 2 + 2B = 0 \implies B = -1$. Thus, $y_p = 2x - 1$, giving total solution $y(x) = C e^{-2x} + 2x - 1$.

Question 26Advanced analysis, convergence and multivariable reasoning

What is the gradient vector $\nabla f$ of the scalar field $f(x, y, z) = x^2 y + y z^3$ at the point $P(1, 2, -1)$?

  1. (4, 0, 6) Correct answer.
  2. (4, 2, -6)
  3. (2, 1, 3)
  4. (4, 0, -6)

Explanation

The gradient vector is $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) = (2xy, x^2 + z^3, 3yz^2)$. Evaluating at $P(1, 2, -1)$: $\frac{\partial f}{\partial x} = 2(1)(2) = 4$, $\frac{\partial f}{\partial y} = 1^2 + (-1)^3 = 1 - 1 = 0$, and $\frac{\partial f}{\partial z} = 3(2)(-1)^2 = 6$. Therefore, $\nabla f(1, 2, -1) = (4, 0, 6)$.

Question 27Advanced analysis, convergence and multivariable reasoning

Compute the curl (rotationnel $\vec{\nabla} \times \vec{F}$) of the 3D vector field $\vec{F}(x, y, z) = (y z, x z, x y)$.

  1. (0, 0, 0) (the vector field is conservative / irrotationnel) Correct answer.
  2. (x, y, z)
  3. (2z, 2y, 2x)
  4. (1, 1, 1)

Explanation

Curl components: $\text{curl}(\vec{F}) = \left( \frac{\partial(xy)}{\partial y} - \frac{\partial(xz)}{\partial z}, \frac{\partial(yz)}{\partial z} - \frac{\partial(xy)}{\partial x}, \frac{\partial(xz)}{\partial x} - \frac{\partial(yz)}{\partial y} \right) = (x - x, y - y, z - z) = (0, 0, 0)$. Indeed, $\vec{F} = \vec{\nabla}(x y z)$ is the exact gradient of scalar potential $\Phi = x y z$.

Question 28Advanced analysis, convergence and multivariable reasoning

What is the divergence (divergence $\vec{\nabla} \cdot \vec{v}$) of the vector field $\vec{v}(x, y, z) = (x^2, y^2, z^2)$ at point $M(1, 2, 3)$?

  1. 12 Correct answer.
  2. 6
  3. 14
  4. 36

Explanation

Divergence is $\vec{\nabla} \cdot \vec{v} = \frac{\partial(x^2)}{\partial x} + \frac{\partial(y^2)}{\partial y} + \frac{\partial(z^2)}{\partial z} = 2x + 2y + 2z = 2(x + y + z)$. At $M(1, 2, 3)$: $\text{div}(\vec{v}) = 2(1 + 2 + 3) = 2(6) = 12$.

Question 29Advanced analysis, convergence and multivariable reasoning

The region bounded by the curve $y = \sqrt{x}$, the $x$-axis and the lines $x = 0$ and $x = 4$ is rotated one full turn about the $x$-axis. What is the volume $V$ of the solid generated?

  1. 8π cubic units Correct answer.
  2. 16π/3 cubic units
  3. 4π cubic units
  4. 16π cubic units

Explanation

A solid of revolution about the $x$-axis has volume $V = \pi \int_a^b [f(x)]^2 \, dx$. Here $[f(x)]^2 = (\sqrt{x})^2 = x$, so $V = \pi \int_0^4 x \, dx = \pi \left[\frac{x^2}{2}\right]_0^4 = \pi \times \frac{16}{2} = 8\pi$.

Question 30Advanced analysis, convergence and multivariable reasoning

What is the derivative of $f(x) = \ln\left(\tan\left(\frac{x}{2}\right)\right)$ for $x \in (0, \pi)$?

  1. f'(x) = \frac{1}{\sin(x)} Correct answer.
  2. f'(x) = \frac{1}{\cos(x)}
  3. f'(x) = \frac{1}{2\sin(x)}
  4. f'(x) = \tan(x)

Explanation

Chain rule: $f'(x) = \frac{\frac{d}{dx}(\tan(x/2))}{\tan(x/2)} = \frac{\frac{1}{2}\frac{1}{\cos^2(x/2)}}{\frac{\sin(x/2)}{\cos(x/2)}} = \frac{1}{2 \sin(x/2) \cos(x/2)}$. Using the double-angle identity $2\sin(u)\cos(u) = \sin(2u)$, the denominator simplifies to $\sin(x)$, so $f'(x) = \frac{1}{\sin(x)}$.

Question 31Advanced analysis, convergence and multivariable reasoning

Evaluate the integral $I = \int_0^1 \frac{x}{\sqrt{1 - x^2}} \, dx$.

  1. 1 Correct answer.
  2. \pi/2
  3. \pi/4
  4. \infty

Explanation

Substitute $u = 1 - x^2 \implies du = -2x\,dx \implies x\,dx = -\frac{du}{2}$. When $x = 0, u = 1$; when $x = 1, u = 0$. $I = \int_1^0 \frac{-du/2}{\sqrt{u}} = \frac{1}{2} \int_0^1 u^{-1/2} du = \frac{1}{2} [2 u^{1/2}]_0^1 = [\sqrt{u}]_0^1 = 1 - 0 = 1$.

Question 32Advanced analysis, convergence and multivariable reasoning

What is the general solution of the differential equation $x y' - 2y = x^3$ for $x > 0$?

  1. y(x) = C x^2 + x^3 Correct answer.
  2. y(x) = C x^2 - x^3
  3. y(x) = C x^3 + x^2
  4. y(x) = C e^{2x} + x^3

Explanation

Divide by $x$: $y' - \frac{2}{x}y = x^2$. Integrating factor is $\mu(x) = e^{\int -2/x \, dx} = e^{-2\ln x} = \frac{1}{x^2}$. Multiplying the ODE: $\frac{d}{dx}\left(\frac{y}{x^2}\right) = \frac{x^2}{x^2} = 1$. Integrating both sides: $\frac{y}{x^2} = x + C \implies y(x) = x^3 + C x^2$.

Question 33Advanced analysis, convergence and multivariable reasoning

Find the Taylor series expansion of $f(x) = \frac{1}{1 - 2x}$ around $x = 0$ for $|x| < 1/2$.

  1. \sum_{n=0}^\infty 2^n x^n = 1 + 2x + 4x^2 + 8x^3 + \dots Correct answer.
  2. \sum_{n=0}^\infty (-2)^n x^n = 1 - 2x + 4x^2 - 8x^3 + \dots
  3. \sum_{n=0}^\infty \frac{2^n}{n!} x^n
  4. \sum_{n=0}^\infty 2 x^n

Explanation

From the geometric series $\frac{1}{1 - u} = \sum_{n=0}^\infty u^n$ for $|u| < 1$. Substituting $u = 2x$: $\frac{1}{1 - 2x} = \sum_{n=0}^\infty (2x)^n = \sum_{n=0}^\infty 2^n x^n = 1 + 2x + 4x^2 + 8x^3 + \dots$.

Question 34Advanced analysis, convergence and multivariable reasoning

What is the value of the Wallis integral $W_4 = \int_0^{\pi/2} \sin^4(x) \, dx$?

  1. \frac{3\pi}{16} Correct answer.
  2. \frac{3\pi}{8}
  3. \frac{1}{16}
  4. \frac{\pi}{4}

Explanation

Using the Wallis recurrence relation $W_n = \frac{n-1}{n} W_{n-2}$: $W_0 = \int_0^{\pi/2} 1\,dx = \frac{\pi}{2}$. $W_2 = \frac{1}{2} W_0 = \frac{\pi}{4}$. $W_4 = \frac{3}{4} W_2 = \frac{3}{4} \times \frac{\pi}{4} = \frac{3\pi}{16}$.

Question 35Advanced analysis, convergence and multivariable reasoning

What is the line integral $\oint_C \vec{F} \cdot d\vec{r}$ of $\vec{F} = (2x y, x^2)$ around any closed counter-clockwise curve $C$ in $\mathbb{R}^2$?

  1. 0 (because curl(F) = 0 and the vector field is conservative) Correct answer.
  2. \pi R^2
  3. 2\pi
  4. Area of C

Explanation

By Green's theorem: $\oint_C (P dx + Q dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx dy$. Here $P = 2xy \implies \frac{\partial P}{\partial y} = 2x$, and $Q = x^2 \implies \frac{\partial Q}{\partial x} = 2x$. The 2D curl is $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2x - 2x = 0$. Since $\vec{F} = \vec{\nabla}(x^2 y)$ is a conservative gradient field, the closed circulation integral is identically zero.

Question 36Advanced analysis, convergence and multivariable reasoning

What is the value of the Gaussian integral $I = \int_{-\infty}^{+\infty} e^{-x^2} \, dx$?

  1. \sqrt{\pi} Correct answer.
  2. \pi
  3. \sqrt{\pi/2}
  4. 1

Explanation

Squaring the integral and converting to polar coordinates: $I^2 = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2)} \, dx \, dy = \int_0^{2\pi} d\theta \int_0^\infty e^{-r^2} r \, dr = 2\pi \left[ -\frac{1}{2}e^{-r^2} \right]_0^\infty = 2\pi \left(0 - \left(-\frac{1}{2}\right)\right) = \pi$. Taking the square root: $I = \sqrt{\pi}$.

Question 37Advanced analysis, convergence and multivariable reasoning

Evaluate the line integral $I = \int_C (y \, dx + x \, dy)$ along the parabolic path $y = x^2$ from $(0, 0)$ to $(1, 1)$.

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

Explanation

Notice that the differential form is exact: $y\,dx + x\,dy = d(x y)$. Therefore, the integral is path-independent and depends only on the endpoints: $I = [x y]_{(0,0)}^{(1,1)} = (1)(1) - (0)(0) = 1$. (Direct substitution on $y=x^2, dy=2x dx$: $\int_0^1 (x^2 dx + x(2x dx)) = \int_0^1 3x^2 dx = [x^3]_0^1 = 1$).

Question 38Inner 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 39Inner 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 40Inner 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 41Inner 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 42Inner 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 43Inner 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 44Inner 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 45Advanced 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 46Advanced 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 47Advanced 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 48Advanced 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 49Advanced 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 50Analytical 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 $\alpha = 45^\circ$).