Concours National ENSA — première année (Morocco)

Evaluate the limit: L = lim_{x -> 0} (e^(2x) - 1 - 2x) / x^2.

• A0
• B1
• C2
• D4

What is the value of the limit L = lim_{x -> +infinity} x * (sqrt(x^2 + 4) - x)?

• A0
• B1
• C2
• D+infinity

Let f(x) = (ln(1 + 3x)) / sin(2x) for x != 0, and f(0) = c. For what value of the real constant c is f continuous at x = 0?

• A1/2
• B1
• C3/2
• D2/3

Consider the equation x^3 + 3x - 5 = 0. According to the Intermediate Value Theorem (Théorème des Valeurs Intermédiaires), on which interval does this equation admit a unique real root?

• A[-2, -1]
• B[-1, 0]
• C[0, 1]
• D[1, 2]

What is the equation of the tangent line to the curve y = f(x) = x * e^(-x) at the point with abscissa x = 0?

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

Find the inflection point of the curve defined by f(x) = x^3 - 6x^2 + 9x + 2.

• A(1, 6)
• B(2, 4)
• C(3, 2)
• D(0, 2)

Let f(x) = x^3 + 2x - 1. The function f is strictly increasing on R. If g is the inverse function f^(-1), what is the value of g'(2)?

• A1/5
• B1/4
• C1/2
• D5

Evaluate the limit: L = lim_{x -> 0} (1 - cos(x)) / (x * sin(x)).

• A0
• B1/2
• C1
• D2

Find the oblique asymptote (asymptote oblique) of the curve y = f(x) = (2x^2 + 5x - 1) / (x + 1) as x -> +infinity.

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

What is the value of the limit L = lim_{x -> 0+} x^(sqrt(x))?

• A0
• B1/e
• C1
• D+infinity

Let f(x) = |x^2 - 4|. At how many points in R is the function f NOT differentiable?

• A0
• B1
• C2
• D3

Evaluate the limit: L = lim_{x -> +infinity} ((x + 3) / (x - 1))^x.

• A1
• Be^2
• Ce^3
• De^4

Calculate the definite integral: I = integral_{0}^{1} x * e^(2x) dx.

• A(e^2 + 1) / 4
• B(e^2 - 1) / 4
• C(3e^2 + 1) / 4
• De^2 / 2

Evaluate the integral: J = integral_{1}^{e} (ln(x))^2 dx.

• Ae - 2
• Be - 1
• C2e - 1
• De

What is the general solution of the first-order differential equation y' + 2y = 4x?

• Ay(x) = C * e^(-2x) + 2x - 1
• By(x) = C * e^(2x) + 2x - 1
• Cy(x) = C * e^(-2x) + 2x + 1
• Dy(x) = C * e^(-2x) + 4x - 2

Find the particular solution to the second-order differential equation y'' + 4y = 0 satisfying the initial conditions y(0) = 3 and y'(0) = 6.

• Ay(x) = 3 cos(2x) + 6 sin(2x)
• By(x) = 3 cos(4x) + 3 sin(4x)
• Cy(x) = 3 cos(2x) + 3 sin(2x)
• Dy(x) = 3 e^(2x) + 3 e^(-2x)

Compute the integral: K = integral_{0}^{pi/4} (tan(x)) / (cos^2(x)) dx.

• A1/4
• B1/2
• C1
• Dln(2)

What is the area of the domain bounded by the curves y = x^2 and y = 2x - x^2?

• A1/6
• B1/3
• C1/2
• D2/3

Evaluate the limit of the Riemann sum: S = lim_{n -> +infinity} sum_{k=1}^{n} (n / (n^2 + k^2)).

• Api / 6
• Bpi / 4
• Cpi / 2
• Dln(2)

Compute the integral: I = integral_{0}^{1} (2x + 1) / (x^2 + x + 1) dx.

• Aln(2)
• Bln(3)
• C2 ln(3)
• Dpi / 3

Determine the value of the improper integral I = integral_{1}^{+infinity} 1 / (x * sqrt(x)) dx.

• A1
• B2
• C3/2
• D+infinity (diverges)

Solve the initial value problem y' - 3y = 6 with y(0) = 1. What is the value of y(ln(2))?

• A6
• B10
• C22
• D24

What is the value of the integral I = integral_{-1}^{1} (x^3 * cos(x) + x^4) dx?

• A0
• B2/5
• C1/5
• D4/5

Solve in R the inequality: ln(2x - 1) <= ln(x + 3).

• A(1/2, 4]
• B[4, +infinity)
• C(-3, 4]
• D(1/2, +infinity)

Determine the number of real solutions to the equation e^(2x) - 5 e^x + 6 = 0.

• A0
• B1
• C2
• D3

What is the derivative of the function f(x) = ln(x + sqrt(x^2 + 1)) on R?

• A1 / (x + sqrt(x^2 + 1))
• B1 / sqrt(x^2 + 1)
• Cx / sqrt(x^2 + 1)
• D2 / sqrt(x^2 + 1)

Evaluate the limit: L = lim_{x -> +infinity} (ln(x))^3 / x.

• A0
• B1
• C3
• D+infinity

Solve in R the exponential equation: 2^(x+1) + 4^x = 8.

• A0
• B1
• C2
• Dln(2)

What is the product of the real roots of the equation (ln(x))^2 - 3 ln(x) + 2 = 0?

• Ae
• Be^2
• C2e
• De^3

Find the slope of the tangent line to the curve y = f(x) = x^x at the point where x = 1.

• A0
• B1
• Ce
• D2

A radioactive sample decays according to N(t) = N_0 * e^(-lambda * t). If the half-life of the sample is 10 days, what fraction of the original quantity N_0 remains after 30 days?

• A1/2
• B1/4
• C1/6
• D1/8

Calculate the exact value of the complex expression: (1 + i)^10.

• A-32
• B32
• C32i
• D-32i

Find the modulus |z| and principal argument arg(z) of the complex number z = (sqrt(3) - i) / (1 + i).

• A|z| = sqrt(2), arg(z) = -5pi / 12
• B|z| = 2, arg(z) = -pi / 12
• C|z| = sqrt(2), arg(z) = 5pi / 12
• D|z| = 1, arg(z) = -7pi / 12

For theta in the interval (0, pi), find the solutions in C of the equation z^2 - 2z * cos(theta) + 1 = 0.

• Acos(theta) +- sin(theta)
• B2 cos(theta) +- 2i sin(theta)
• Ce^(i*theta) and e^(-i*theta)
• De^(i*(theta + pi/2)) and e^(-i*(theta + pi/2))

In the complex plane, what is the geometric nature of the transformation f defined by z' = i*z + 1 - i?

• ARotation of angle pi/2 centered at the point of affix omega = 1
• BRotation of angle pi/2 centered at the origin
• CHomothety of ratio -1 centered at omega = 1 - i
• DTranslation of vector u with affix 1 - i

Let A, B, and C be three points in the complex plane with affixes z_A = 1, z_B = 2 + i, and z_C = 1 + 2i. What is the geometric nature of the triangle ABC?

• AEquilateral triangle
• BRight-angled triangle at A, with AB = AC
• CTriangle with an angle of pi/4 at A and AC = sqrt(2) * AB
• DRight-angled triangle at B

Let j = e^(2i*pi / 3). What is the value of the sum S = 1 + j + j^2 + j^3 + j^4 + j^5?

• A0
• B1
• C-1
• D3

Using Euler's formulas, what is the linearization of cos^3(x)?

• A(1/2) cos(3x) + (1/2) cos(x)
• B(1/4) cos(3x) + (1/4) cos(x)
• C(3/4) cos(3x) + (1/4) cos(x)
• D(1/4) cos(3x) + (3/4) cos(x)

Factor the polynomial P(z) = z^4 + 4 into irreducible quadratic factors with real coefficients.

• A(z^2 + 2)(z^2 - 2)
• B(z^2 + 2)^2
• C(z^2 - 2z + 2)(z^2 + 2z + 2)
• D(z^2 - 4z + 4)(z^2 + 4z + 4)

What is the geometric set of points M of affix z such that (z - 1) / (z + 1) is a purely imaginary number?

• AThe circle of center O and radius 1, deprived of the point (-1, 0)
• BThe real axis deprived of the point (-1, 0)
• CThe imaginary axis deprived of the origin
• DThe straight line of equation x = 1

Let (u_n) be a sequence with u_0 = 1 and u_{n+1} = (u_n + 6) / (u_n + 2) for all n >= 0. Given that (u_n) converges to a positive real limit L, what is the value of L?

• A1
• B3/2
• C2
• D3

Evaluate the limit: L = lim_{n -> +infinity} sum_{k=1}^{n} (1 / (k * (k + 1))).

• A1/2
• B1
• C2
• D+infinity

Evaluate the limit: L = lim_{n -> +infinity} sum_{k=1}^{n} (1 / sqrt(n^2 + k)).

• A0
• B1
• C2
• D+infinity

A sequence is defined by u_0 = 5 and u_{n+1} = 2*u_n - 3 for all n >= 0. What is the explicit value of u_5?

• A35
• B41
• C61
• D67

Which of the following conditions characterizes two real sequences (u_n) and (v_n) as adjacent (suites adjacentes)?

• A(u_n) and (v_n) are both strictly increasing and lim (u_n + v_n) = 0
• B(u_n) and (v_n) are geometric sequences with the same common ratio
• C(u_n) is increasing, (v_n) is decreasing, u_n <= v_n for all n, and lim (v_n - u_n) = 0
• D(u_n) and (v_n) converge to distinct limits L_1 != L_2

Consider the sequence defined by u_0 = 0 and u_{n+1} = sqrt(2 + u_n). To what value does (u_n) converge as n -> +infinity?

• Asqrt(2)
• Bsqrt(2) + 1
• C2
• D4

A committee of 3 students is to be chosen from a group consisting of 5 boys and 4 girls. How many different committees can be formed that include at least one girl?

• A60
• B70
• C80
• D74

An urn contains 4 red balls and 6 green balls. Two balls are drawn simultaneously at random. What is the probability that the two balls drawn are of different colors?

• A4/15
• B7/15
• C2/5
• D8/15