Review worksheet

Review and initial assessment worksheet (mathematics and computer science track)

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The PDF is in Romanian.

Mathematics and computer science track (M1)

Section A. Basic to medium level

  1. Show that the number $A = \sqrt{\left( 2\sqrt{3} - 4 \right)^{2}} - \sqrt{12} + |1 - \sqrt{3}|$ is a rational number.

  2. Solve in $\mathbb{R}$ the system of inequalities:

    $$\left\{ \begin{matrix} 2(x - 1) \geq 3x - 5 \\ \frac{x + 4}{2} > \frac{x - 1}{3} \end{matrix} \right.$$

  3. Find the set $S = \left\{ x \in \mathbb{Z} \mid \frac{3x + 1}{x - 2} \in \mathbb{Z} \right\}$.

  4. Consider the quadratic equation $x^{2} - (2m - 1)x + m^{2} - 1 = 0$, where $m \in \mathbb{R}$.

    a) Find $m$ so that the equation has distinct real roots.

    b) Find $m$ if the roots $x_{1},x_{2}$ satisfy the relation $x_{1}^{2} + x_{2}^{2} = 7$.

Section B. Medium to advanced level

  1. Consider the function $f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x) = - \frac{1}{2}x + 3$.

    a) Draw the graph of the function in a Cartesian coordinate system $xOy$.

    b) Find the area and the perimeter of the region bounded by the graph of the function and the coordinate axes.

  2. In triangle $ABC$, $m\left( \widehat{A} \right) = 90^{\circ}$, $AB = 6\sqrt{3}\text{~cm}$ and $AC = 6\text{~cm}$.

    a) Find the measures of angles $\widehat{B}$ and $\widehat{C}$ and the length of the altitude $AD$ ($D \in BC$).

    b) Calculate the value of the expression $E = \sin^{2}B + \sin^{2}C - 2\cos B \cdot \sin C$.

  3. In the Cartesian coordinate system $xOy$, consider the points $A( - 2,1)$, $B(4,5)$ and $C(1, - 2)$.

    a) Calculate the side lengths of triangle $ABC$ and show that it is isosceles.

    b) Find the coordinates of the midpoint of segment $\lbrack AB\rbrack$ and the length of the median from $C$.

Section C. Top-grade problems

  1. Find all natural numbers $n$ for which the number:

    $$P(n) = n^{4} + 4$$

    is a prime number.

  2. Find the number of real solutions of the equation:

    $$\left\lbrack \frac{x + 1}{3} \right\rbrack = \frac{x - 2}{2}$$

    where $\lbrack x\rbrack$ represents the integer part of the real number $x$.

  3. Consider a square $ABCD$ with side length $a$. Points $M \in (BC)$ and $N \in (CD)$ are such that $m\left( \widehat{MAN} \right) = 45^{\circ}$.

    Show that the perimeter of triangle $CMN$ is constant and independent of the positions of points $M$ and $N$, being equal to $2a$.