Baccalaureate Mathematics Exam 2018 - Detailed Notes

Exam Information

• Examination: Baccalaureate Mathematics

• Duration: 3 hours

• Session: 2018

• Section: Experimental Sciences

• Coefficient: 3

• Pages: 4 numbered pages (1/4 to 4/4)

Exercise 1 (5 points)

• 1. Given the plane Q defined by the equation:
  $x + y + \sqrt{2}z - 2 = 0$

  • a. Show that the plane Q intersects the axes at the points:

  • A$(2, 0, 0)$

  • B$(0, 2, 0)$

  • C$(0, 0, \sqrt{2})$

• 2. The sphere (S) defined by:
  $x^2 + y^2 + z^2 = 1$

  • Find where the plane Q is tangent to the sphere and determine the point of tangency.

• 3. For a strictly positive real number $a$, consider the points M$(a, 0, 0)$ and N$(0, 1, 0)$.

  • Determine components of the vector CMN in terms of $a$.

• 4. a) Show that an equation of the plane (CMN) is
  $4x + a^2y + 2a\sqrt{2}z - 4a = 0$

  b) Distance from point O to plane (CMN):
  $d = 1 - a + 4$

  c) Determine the value of $a$ for which distance $d$ is maximized.

  • 5. a) Show that the volume of the tetrahedron OCMN is $2\sqrt{2}$ for all $a > 0$.

  b) Show that the area of triangle CMN is at least $2 - \sqrt{2}$ for all $a > 0$.

  c) Identify M and N for which area of triangle CMN is exactly $2\sqrt{2}$.

Exercise 2 (3 points)

• Game Description: A customer plays a game with a fair cubic die, faces marked G (1), R (2), D (3).

  • Payoffs:

  • G: 100 DT and game stops

  • R: 0 DT and game stops

  • D: Second roll

    • G: receives 50 DT

    • R/D: 0 DT and game stops

• Events:

  • G₁ : Receives 100 DT

  • G₂ : Receives 50 DT

• 1. a) Calculate $p(G_1)$ (probability of event G₁).

  • b) Show $p(G_2) = \frac{1}{2}$.

  • c) Determine probability of receiving a non-zero amount.

• 2. Define random variable X (amount received).

  • a) Determine probability distribution of X.

  • b) Calculate the expected value E(X).

• 3. If 200 clients participate, define Y (clients receiving a non-zero amount). Determine E(Y).

• 4. Assess if the manager's total amount of 1200 DT is correctly estimated.

Exercise 3 (5 points)

• 1. Solve the equation
  $(E): z^2 - i\sqrt{3}z - 1 = 0$ in $C$.

• 2. Define polynomial
  $P(z) = 3z - 71\sqrt{3}z - 18z^2 + 7i\sqrt{3}z + 3$

  • a) Show that $P(i\sqrt{3}) = 0$ and $P(e^{3}) = 0$.

  • b) Show that $P(7\sqrt{3}) - 1 = P(z)$ for all non-zero $z$.

  • c) Conclude that both $e$ and $e^{3}$ are solutions to the equation $P(z) = 0$.

Exercise 4 (7 points)

• Graph representation: Curve (r) represents the function: $u(x) = x - 1 - 4\ln x$

  • Asymptotes and intersections:

  • y-axis is a vertical asymptote

  • Line y=x is an asymptote at $+\infty$

  • Unique horizontal tangent at $x = 4$

  • Crosses the y-axis at abscissas 1 and a

• A)

  • 1. Find values of $u(1)$, $u(a)$, $u'(4)$, limits of $u(x)$ as $x \to \infty$.

  • 2. Determine signs of $u(x)$ and $u'(x)$.

• B) Define function: $f(x) = (x - 1) + 4\ln(x)$

  • 1. a) Show that $f(x) = u(x) - u(x)$

  • b) Calculate $f(a)$.

  • c) Analyze limits as $x \to 0^+, x \to \infty$.

  • d) Identify branches of curve (C).

• 2 a) Check that $f'(x) = u'(x)(e^x - 1)$.

  • b) Show that f'(x) > 0 iff $x \in [1, 4] \cup [a, +\infty)$.

  • c) Create a variation table for f.

• 3 a) Prove that e^x - 2x > 0 for all $x$.

  • b) Determine the position relative to the curves (C) and (r).

  • c) Sketch curve (C) on provided annex.

• 4 Define area A, the area limited by (C), and area A' for curve (r).

  • a) Show $A = 20\ln(5) - 12\ln(3) - 14$.

  • b) Show A' < A < 2f(4) and deduce 5 < A < 5.25.

Final Notes

• Fill in personal information on the last page: Name, Birthdate, Registration Number, Series, Supervisors' Signatures.