Review worksheet

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

Published · by

Open PDF The PDF is in Romanian.

Summary

The mathematics-computer-science track starts with a more technical initial test: functions, trigonometry and analytic geometry, on top of the algebra base. The worksheet reviews middle-school content at the math-track level, stressing exact computation and proof. It is the yardstick for measuring your start in grade 9.

Press “Check” next to an exercise. Your answers stay on this device.

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 $2a$.

Frequently asked questions

What does the math-CS track mean for the initial assessment?

That the test demands more than basic computation: fully studied functions, exact trigonometric values and analytic-geometry problems. The items check rigour, not just the result, so solutions need justifications. The worksheet reproduces this level so test day brings no surprises.

How do you review trigonometry for the initial test?

Starting from the unit circle: the values for standard angles, the signs per quadrant and the basic identities. Then train simple equations and right-triangle applications, where they appear most often in the initial test. Remember that the math track asks for the exact value, not the decimal approximation.