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Panhellenic Economic & Computer Studies (Οικονομικών & Πληροφορικής) Track (Greece)

Practice Questions100 questions

Exam Review

Key Facts: Panhellenic Exact Sciences Exam

3 Subjects

Orientation Papers Examined

Ministry of Education (ΥΠΑΙΘΑ)

180 min

Duration Per Subject Paper

Central Examination Committee (ΚΕΕ)

Field 2

Target Scientific Field

Law 4186/2013 / Law 4777/2021

€0

Examination Fee

Greek Public Education System

The Panhellenic Exact Sciences track is the university admissions pathway for engineering, physics, math, and polytechnic programs in Greece. Administered by ΥΠΑΙΘΑ, candidates sit three 3-hour national papers in Advanced Mathematics, Physics, and Chemistry. Testing is tuition-free for public school candidates, with admissions determined by departmental Minimum Admission Base (EBE) cut-offs.

Sample Panhellenic Exact Sciences Practice Questions

Try these sample questions to review concepts for the Panhellenic Exact Sciences exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Calculate the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 3x + 2}$.

A.4

B.2

C.0

D.The limit does not exist

Explanation: Factoring both the numerator and the denominator gives $\frac{x^2 - 4}{x^2 - 3x + 2} = \frac{(x - 2)(x + 2)}{(x - 2)(x - 1)}$. For $x \neq 2$, we can cancel the common factor $(x - 2)$ to obtain $\frac{x + 2}{x - 1}$. Taking the limit as $x \to 2$ yields $\frac{2 + 2}{2 - 1} = \frac{4}{1} = 4$.

2Compute the trigonometric limit: $\lim_{x \to 0} \frac{1 - \cos(4x)}{x \sin(2x)}$.

A.2

B.4

C.8

D.1

Explanation: Recall the half-angle identity $1 - \cos(4x) = 2\sin^2(2x)$. Substituting this into the limit gives $\lim_{x \to 0} \frac{2\sin^2(2x)}{x\sin(2x)} = \lim_{x \to 0} \frac{2\sin(2x)}{x}$. Since $\lim_{x \to 0} \frac{\sin(2x)}{2x} = 1$, we have $\lim_{x \to 0} 2 \cdot 2 \cdot \frac{\sin(2x)}{2x} = 4 \cdot 1 = 4$. Alternatively, applying L'Hôpital's rule: $\frac{4\sin(4x)}{\sin(2x) + 2x\cos(2x)} \to \frac{16\cos(4x)}{2\cos(2x) + 2\cos(2x) - 4x\sin(2x)} = \frac{16}{4} = 4$.

3Determine the limit at infinity: $\lim_{x \to +\infty} \left(\sqrt{x^2 + 6x + 5} - x\right)$.

A.0

B.3

C.6

D.+\infty

Explanation: Multiplying and dividing by the conjugate expression $\sqrt{x^2 + 6x + 5} + x$ yields $\frac{(x^2 + 6x + 5) - x^2}{\sqrt{x^2 + 6x + 5} + x} = \frac{6x + 5}{\sqrt{x^2(1 + 6/x + 5/x^2)} + x}$. For $x > 0$, $\sqrt{x^2} = x$, so we can factor $x$ from the denominator: $\frac{x(6 + 5/x)}{x(\sqrt{1 + 6/x + 5/x^2} + 1)} = \frac{6 + 5/x}{\sqrt{1 + 6/x + 5/x^2} + 1}$. As $x \to +\infty$, this approaches $\frac{6 + 0}{\sqrt{1 + 0 + 0} + 1} = \frac{6}{2} = 3$.

4If $2 - x^2 \le f(x) \le 2 + x^2$ for all $x \in \mathbb{R} \setminus \{0\}$, what is $\lim_{x \to 0} f(x)$?

A.0

B.2

C.4

D.The limit cannot be determined

Explanation: We evaluate the limits of the bounding functions as $x \to 0$: $\lim_{x \to 0} (2 - x^2) = 2 - 0 = 2$ and $\lim_{x \to 0} (2 + x^2) = 2 + 0 = 2$. Since both bounds converge to the same value 2, by the Squeeze Theorem (Κριτήριο Παρεμβολής), $\lim_{x \to 0} f(x) = 2$.

5Find the real constants $a$ and $b$ such that $\lim_{x \to 1} \frac{x^2 + ax + b}{x - 1} = 5$.

A.$a = 3,\ b = -4$

B.$a = 5,\ b = -6$

C.$a = 2,\ b = -3$

D.$a = -3,\ b = 2$

Explanation: For the finite limit to exist while the denominator approaches $1 - 1 = 0$, the numerator must also vanish at $x = 1$: $1^2 + a(1) + b = 0 \implies b = -a - 1$. Substituting $b$: $x^2 + ax - (a + 1) = (x - 1)(x + a + 1)$. The limit becomes $\lim_{x \to 1} \frac{(x - 1)(x + a + 1)}{x - 1} = \lim_{x \to 1} (x + a + 1) = 1 + a + 1 = a + 2$. Setting $a + 2 = 5$ gives $a = 3$. Then $b = -3 - 1 = -4$.

6For what positive value of $k$ is the function $f(x) = \begin{cases} \frac{\sin(3x)}{x}, & x \neq 0 \\ k^2 - 1, & x = 0 \end{cases}$ continuous at $x = 0$?

A.$k = 1$

B.$k = 2$

C.$k = 3$

D.$k = 4$

Explanation: For $f$ to be continuous at $x = 0$, we must have $\lim_{x \to 0} f(x) = f(0)$. The limit is $\lim_{x \to 0} \frac{\sin(3x)}{x} = 3 \lim_{x \to 0} \frac{\sin(3x)}{3x} = 3(1) = 3$. Therefore, $k^2 - 1 = 3 \implies k^2 = 4$. Since $k > 0$, we have $k = 2$.

7Consider the polynomial $P(x) = x^3 - 3x + 1$ on the interval $[0, 1]$. Which statement correctly applies Bolzano's Theorem?

A.$P(0)P(1) < 0$, guaranteeing at least one root $x_0 \in (0, 1)$

B.$P(0)P(1) > 0$, proving no roots exist in $(0, 1)$

C.$P'(0)P'(1) < 0$, guaranteeing a local extremum in $(0, 1)$

D.Bolzano's theorem does not apply because $P(x)$ is not monotonic on $[0, 1]$

Explanation: $P(x)$ is continuous on $[0, 1]$ as a polynomial function. We evaluate the endpoints: $P(0) = 0^3 - 3(0) + 1 = 1 > 0$ and $P(1) = 1^3 - 3(1) + 1 = -1 < 0$. Since $P(0)P(1) = 1(-1) = -1 < 0$, by Bolzano's Theorem there exists at least one $x_0 \in (0, 1)$ such that $P(x_0) = 0$.

8Let $f: [a, b] \to [a, b]$ be a continuous function. Which mathematical argument rigorously proves that $f$ has a fixed point, i.e., there exists $c \in [a, b]$ such that $f(c) = c$?

A.Define $g(x) = f(x) - x$; then $g(a) \ge 0$ and $g(b) \le 0$, and Bolzano's theorem applies on $[a, b]$

B.Apply Rolle's theorem to $f(x)$ on $[a, b]$ because $f(a) = f(b)$

C.Apply Fermat's theorem to find where $f'(x) = 0$

D.The Intermediate Value Theorem requires $f$ to be strictly increasing, which is not guaranteed

Explanation: Define the auxiliary function $g(x) = f(x) - x$ on $[a, b]$. Since $f$ and $x$ are continuous on $[a, b]$, $g$ is continuous on $[a, b]$. Because $f([a, b]) \subseteq [a, b]$, we have $f(a) \ge a \implies g(a) = f(a) - a \ge 0$, and $f(b) \le b \implies g(b) = f(b) - b \le 0$. If $g(a) = 0$, then $c = a$; if $g(b) = 0$, then $c = b$. If $g(a) > 0$ and $g(b) < 0$, then $g(a)g(b) < 0$, and by Bolzano's Theorem there exists $c \in (a, b)$ such that $g(c) = 0 \iff f(c) = c$.

9Let $f$ be continuous on $[1, 5]$ with $f(1) = 2$ and $f(5) = 10$. Which value is guaranteed by the Intermediate Value Theorem to be attained by $f(x)$ for some $x \in (1, 5)$?

A.1

B.6

C.12

D.-2

Explanation: The Intermediate Value Theorem (Θεώρημα Ενδιάμεσων Τιμών) states that if $f$ is continuous on $[a, b]$, it takes every value between $f(a)$ and $f(b)$ at least once. Here $f(1) = 2$ and $f(5) = 10$, so $f$ must attain every value $\eta \in (2, 10)$. Among the choices, only 6 lies strictly between 2 and 10.

10Compute the derivative of $f(x) = x^2 e^{3x}$ evaluated at $x = 1$.

A.$2e^3$

B.$3e^3$

C.$5e^3$

D.$6e^3$

Explanation: Using the product rule and the chain rule: $f'(x) = \frac{d}{dx}[x^2] \cdot e^{3x} + x^2 \cdot \frac{d}{dx}[e^{3x}] = 2x e^{3x} + x^2 (3e^{3x}) = (2x + 3x^2)e^{3x}$. Evaluating at $x = 1$: $f'(1) = (2(1) + 3(1)^2)e^{3(1)} = (2 + 3)e^3 = 5e^3$.

About the Panhellenic Exact Sciences Exam

The Panhellenic Exact Sciences (Θετικών Επιστημών) track governs university admissions into Greek polytechnics, schools of engineering, physical sciences, mathematics, architecture, and technology faculties (2nd Scientific Field / 2ο Επιστημονικό Πεδίο). Candidates take three rigorous orientation examinations: Advanced Mathematics, Physics, and Chemistry, in addition to the common Modern Greek core. Our practice bank offers an English-language MCQ study adaptation delivering worked calculation problems, conceptual mechanics questions, and chemical reaction analyses for independent exam study.

Exam sponsor: Ministry of Education, Religious Affairs and Sports (Υπουργείο Παιδείας, Θρησκευμάτων και Αθλητισμού - ΥΠΑΙΘΑ). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Questions

12 questions

Time Limit

180 minutes per paper (3 papers: Mathematics, Physics, Chemistry)

Passing Score

Departmental Minimum Admission Base (EBE / ΕΒΕ) benchmark multiplied by candidate scores on a 20,000-point scale

Exam / Certification Fees

€0 (Free for Greek public secondary students and registered candidates)

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Official sources

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

35%

Advanced Calculus & Real Functions

Function monotonicity, inflection points, l'Hôpital's rule, definite integrals, and optimization modeling.

35%

Oscillations, Waves, Collisions, Rigid Body and Electromagnetism

Simple harmonic motion, damping and resonance, standing waves, elastic and inelastic collisions, torque and rigid-body equilibrium, angular momentum, the magnetic field and the Laplace force, electromagnetic induction and alternating current.

30%

Redox, Thermochemistry, Kinetics, Equilibria and Organic Chemistry

Oxidation numbers and electrolysis, enthalpy and Hess's law, rate laws, Le Chatelier's principle and Kc, buffer pH and titration, atomic orbitals and periodic trends, and organic reaction types.

Preparing for the Panhellenic Exact Sciences Exam

What You Need to Know

Passing score: Departmental Minimum Admission Base (EBE / ΕΒΕ) benchmark multiplied by candidate scores on a 20,000-point scale
Exam length: 12 questions
Time limit: 180 minutes per paper (3 papers: Mathematics, Physics, Chemistry)
Exam / certification fees: €0 (Free for Greek public secondary students and registered candidates) Official sources

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Panhellenic Exact Sciences: Suggested Study Strategy

1Master the differential calculus theorems that the paper returns to every year: Bolzano, the intermediate value theorem, Rolle's theorem, the mean value theorem and Fermat's theorem

2Work through elastic and perfectly inelastic collisions along a straight line, including the special cases of equal masses and of a very massive target

3Practise magnetic-field and induction problems: the field of a straight wire and a solenoid, the Laplace force, motional emf, Lenz's law and effective values in alternating current

4Calculate equilibrium constants Kc and buffer pH with ICE tables and weak acid-base dissociation constants, and be able to assign oxidation numbers in redox and electrolysis problems

Frequently Asked Questions

Which university programs accept the Positive Sciences track?

The 2nd Scientific Field grants admission to all engineering and polytechnic faculties (NTUA, AUTH, Patras, etc.), physics, chemistry, mathematics, computer engineering, and earth science departments.

What subjects are examined in the Panhellenic Sciences track?

Candidates are examined in Advanced Mathematics, Physics, and Chemistry, alongside the mandatory common core paper in Modern Greek Language & Literature.

Does this track include quantitative problem solving?

Yes. Real Panhellenic exams in Mathematics, Physics, and Chemistry are heavily computational. Our practice bank includes genuine quantitative questions with numerical calculations and step-by-step solutions.

How are admission scores calculated for engineering schools?

Each engineering school weights Mathematics and Physics heavily (frequently 30–35% each), while Chemistry and Modern Greek carry the remainder. Candidates must satisfy the Minimum Admission Base (EBE) for both the subject and the department.