Concours passerelle ENSAM Rabat — première année du cycle ingénieur (Morocco)

What is the limit of the indeterminate form (1 - cos(4x)) / x^2 as x approaches 0?

• A8
• B4
• C16
• D2

Consider the piecewise function f(x) defined by f(x) = sin(3x) / x for x != 0, and f(0) = k. For which value of k is f continuous at x = 0?

• A0
• B3
• C1
• D1/3

What is the first derivative of the function f(x) = x^2 * ln(x) for all x > 0?

• A2x * ln(x)
• Bx * (ln(x) + 1)
• Cx * (2*ln(x) + 1)
• D2x

What is the horizontal asymptote of the rational function f(x) = (3x^2 - 5x + 2) / (2x^2 + 7x - 1) as x approaches +infinity?

• Ay = 0
• By = 3
• Cy = 2/3
• Dy = 3/2

By the Mean Value Theorem applied to f(x) = x^3 on the closed interval [0, 3], there exists a number c in (0, 3) such that f'(c) = (f(3) - f(0)) / (3 - 0). What is the exact value of c?

• Asqrt(3)
• B3/2
• Csqrt(2)
• D2

What is the equation of the slant (oblique) asymptote of f(x) = (2x^2 + 5x + 1) / (x + 1) as x approaches +infinity?

• Ay = 2x + 5
• By = 2x + 3
• Cy = 2x - 3
• Dy = 2x + 1

What is the value of the definite integral I = int_0^1 x * e^(2x) dx?

• A(e^2 - 1) / 4
• Be^2 / 2
• C(e^2 + 1) / 4
• D(e^2 + 1) / 2

Evaluate the definite integral J = int_0^(pi/4) tan(x) * sec^2(x) dx.

• A1
• B1/4
• C2
• D1/2

What is the value of the integral int_0^1 (1 / (x^2 + 3x + 2)) dx?

• Aln(4/3)
• Bln(3/2)
• Cln(2)
• Dln(5/4)

For which real values of the parameter p does the improper integral int_1^(+infinity) (1 / x^p) dx converge?

• Ap >= 1
• Bp > 1
• Cp < 1
• Dp > 0

Solve the first-order linear differential equation y' + 3y = 6 subject to the initial condition y(0) = 4.

• Ay(x) = 2 + 4*e^(-3x)
• By(x) = 2 + 2*e^(-3x)
• Cy(x) = 3 + e^(-3x)
• Dy(x) = 4*e^(-3x)

What is the solution to the separable differential equation dy/dx = 2x * y for y > 0, given the initial condition y(0) = 3?

• Ay(x) = 3*e^(2x)
• By(x) = e^(x^2) + 2
• Cy(x) = 3*e^(x^2)
• Dy(x) = 3*x^2 + 3

What is the general solution of the second-order linear homogeneous differential equation y'' - 7y' + 12y = 0?

• Ay(x) = C1*e^(3x) + C2*e^(4x)
• By(x) = (C1 + C2*x)*e^(3x)
• Cy(x) = C1*e^(-3x) + C2*e^(-4x)
• Dy(x) = C1*cos(3x) + C2*sin(4x)

Find the general solution of the second-order differential equation y'' + 9y = 0.

• Ay(x) = C1*e^(3x) + C2*e^(-3x)
• By(x) = C1*cos(3x) + C2*sin(3x)
• Cy(x) = (C1 + C2*x)*e^(3x)
• Dy(x) = C1*cos(9x) + C2*sin(9x)

What is a particular solution y_p(x) of the non-homogeneous differential equation y'' - 4y = 8*e^(4x)?

• Ay_p = 2*e^(4x)
• By_p = (4/3)*e^(4x)
• Cy_p = (2/3)*e^(4x)
• Dy_p = (1/2)*e^(4x)

A sequence (u_n) is defined by u_0 = 1 and u_(n+1) = (1/2)*u_n + 3 for all n >= 0. If this sequence converges, what is its limit L?

• A6
• B3
• C2
• D4

What is the limit of the rational sequence a_n = (2n^2 + 3n - 1) / (5n^2 - 4) as n approaches infinity?

• A0
• B2/5
• C1
• D3/5

Calculate the sum of the infinite geometric series S = sum_(n=1)^infinity 5 * (1/3)^n.

• A5/3
• B5
• C5/2
• D10/3

Applying the D'Alembert ratio test to the series sum_(n=1)^infinity (n! / 3^n), what is the limit L = lim_(n->infinity) |a_(n+1) / a_n| and what is the convergence conclusion?

• AL = 0; the series converges
• BL = 1/3; the series converges
• CL = 1; the test is inconclusive
• DL = +infinity; the series diverges

What is the second-order Maclaurin polynomial P_2(x) of the function f(x) = ln(1 + 2x)?

• A2x - 2x^2
• B2x - 4x^2
• C2x + 2x^2
• Dx - x^2

Compute the determinant of the 2x2 matrix A = [[4, -2], [3, 5]].

• A14
• B26
• C-26
• D23

Evaluate the determinant of the 3x3 matrix M = [[1, 2, 3], [0, 4, 5], [1, 0, 2]].

• A8
• B-2
• C6
• D10

Given the 2x2 matrix A = [[2, 1], [0, 2]], compute the matrix power A^3.

• A[[8, 1], [0, 8]]
• B[[6, 3], [0, 6]]
• C[[8, 6], [0, 8]]
• D[[8, 12], [0, 8]]

For the linear transformation A = [[2, 1], [0, 3]], which vector is an eigenvector associated with eigenvalue 3?

• A(1, 1)
• B(1, 0)
• C(0, 1)
• D(2, 1)

For the matrix A = [[3, 1], [0, 2]], which of the following is an eigenvector corresponding to the eigenvalue lambda = 3?

• A[0, 1]^T
• B[1, 0]^T
• C[1, -1]^T
• D[1, 1]^T

A 3x3 real matrix A has trace tr(A) = 9 and determinant det(A) = 24. If two of its eigenvalues are lambda_1 = 2 and lambda_2 = 3, what is the third eigenvalue lambda_3?

• A3
• B2
• C4
• D6

Consider the three vectors v1 = (1, 0, 1), v2 = (0, 1, 1), and v3 = (1, 1, 0) in R^3. Do they form a basis of R^3?

• ANo, because v3 is a linear combination of v1 and v2
• BNo, because three vectors can never span R^3
• CYes, because their determinant is non-zero (det = -2)
• DYes, but only because they are pairwise orthogonal

Let T: R^4 -> R^3 be a linear transformation whose kernel (null space) has dimension 1. According to the Rank-Nullity Theorem, what is the dimension of the image (rank) of T?

• A2
• B1
• C4
• D3

In the Euclidean plane R^2, what is the standard matrix representing a counterclockwise rotation by an angle of pi/2 radians about the origin?

• A[[0, -1], [1, 0]]
• B[[0, 1], [-1, 0]]
• C[[1, 0], [0, -1]]
• D[[-1, 0], [0, -1]]

Given the vectors u = (2, -1, 3) and v = (1, 4, -2) in R^3, what is their dot product u . v?

• A-4
• B-8
• C4
• D12

What is the distance from the point P(2, -1, 3) to the plane defined by 2x - 2y + z + 5 = 0?

• A4
• B10/3
• C14/3
• D5

Let z = 2*(cos(pi/6) + i*sin(pi/6)). Using De Moivre's theorem, what is z^3?

• A8i
• B-8
• C4i
• D8

Using De Moivre's theorem, what is the exact value of (1 + i)^8?

• A16
• B16i
• C-16
• D8

What are the solutions of the quadratic equation z^2 - 2z + 5 = 0 in the complex field C?

• Az = 2 + 4i and z = 2 - 4i
• Bz = 1 + 2i and z = 1 - 2i
• Cz = -1 + 2i and z = -1 - 2i
• Dz = 1 + 4i and z = 1 - 4i

In the complex plane, which geometric transformation is represented by the complex mapping z -> z' = i*z + (1 - i)?

• AA translation by the vector (1, -1)
• BA rotation by an angle of pi around the origin
• CA rotation by an angle of pi/2 centered at the point omega = 1
• DA scaling (homothety) of ratio 1 centered at omega = i

A vehicle traveling at an initial velocity of v_0 = 30 m/s decelerates uniformly to rest over a braking distance of d = 75 m. What is the magnitude of the vehicle's deceleration?

• A4.0 m/s^2
• B6.0 m/s^2
• C8.0 m/s^2
• D12.0 m/s^2

A vehicle travels at 20 m/s around a frictionless banked curve of radius 40 m. Taking g = 10 m/s^2, what banking angle theta permits circular motion without lateral friction?

• A45 degrees
• B30 degrees
• C60 degrees
• Darctan(2)

A particle moves in uniform circular motion along a track of radius r = 0.5 m at a constant angular velocity omega = 10 rad/s. What is the magnitude of its centripetal acceleration?

• A20 m/s^2
• B5.0 m/s^2
• C50 m/s^2
• D100 m/s^2

A block of mass m = 4.0 kg slides down a frictionless plane inclined at an angle alpha = 30° to the horizontal. Taking g = 9.8 m/s^2, what is the acceleration of the block down the incline?

• A9.8 m/s^2
• B8.5 m/s^2
• C2.45 m/s^2
• D4.9 m/s^2

In an ideal Atwood machine, two masses m1 = 3.0 kg and m2 = 2.0 kg are connected by a light inextensible cord passing over a frictionless massless pulley. Taking g = 10 m/s^2, what is the acceleration a of the masses?

• A2.0 m/s^2
• B5.0 m/s^2
• C1.0 m/s^2
• D3.33 m/s^2

A spring with spring constant k = 200 N/m is compressed by x = 0.10 m from its equilibrium length. How much elastic potential energy is stored in the spring?

• A2.0 J
• B1.0 J
• C0.5 J
• D10 J

A roller coaster car enters a vertical circular loop of radius R = 2.5 m. Taking g = 10 m/s^2, what is the minimum speed v_top required at the highest point of the loop so that the car maintains contact with the track?

• A7.07 m/s
• B25 m/s
• C5.0 m/s
• D3.54 m/s

A ball of mass m1 = 2.0 kg moving along a straight line at v1 = 6.0 m/s collides head-on and elastically with a stationary ball of mass m2 = 4.0 kg. What is the velocity of the 4.0 kg ball immediately after the collision?

• A2.0 m/s
• B3.0 m/s
• C6.0 m/s
• D4.0 m/s

A flywheel with moment of inertia I = 0.40 kg*m^2 about its fixed axis is acted upon by a constant net torque of tau = 2.4 N*m. What is the angular acceleration alpha of the flywheel?

• A6.0 rad/s^2
• B0.96 rad/s^2
• C9.6 rad/s^2
• D1.67 rad/s^2

A mass m = 0.25 kg is attached to an ideal horizontal spring of stiffness k = 100 N/m. What is the natural period of oscillation T of this system?

• Api / 20 s
• Bpi / 10 s
• Cpi / 5 s
• D2*pi s

What is the length L of a simple pendulum that oscillates with a period of T = 2.0 s in a gravitational field where g = pi^2 m/s^2 (~9.87 m/s^2)?

• A0.50 m
• B2.0 m
• C1.0 m
• D4.0 m

The oscillation amplitude of a damped harmonic oscillator decreases from A_0 = 10.0 cm to A_5 = 5.0 cm over 5 complete periods. What is the logarithmic decrement delta per cycle (using ln(2) ~ 0.693)?

• A0.693
• B0.347
• C0.069
• D0.139

Three resistors with resistances R1 = 6.0 ohm, R2 = 3.0 ohm, and R3 = 2.0 ohm are connected in parallel. What is the equivalent resistance R_eq of the network?

• A1.0 ohm
• B11.0 ohm
• C0.5 ohm
• D1.5 ohm