100+ Free Cuba 12th Grade Math Final Exam Practice Questions

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2026 Statistics

Key Facts: Cuba 12th Grade Math Final Exam Exam

50% course weight

The final paper carries 50 of 100 points of the subject's final course grade (Art. 40, RM 238/2014).

Resolución Ministerial 238/2014 (MINED), Artículo 40

30 / 100 pass floor

Candidates must achieve at least 30 points on the final paper and 60 points overall in the subject.

Resolución Ministerial 238/2014 (MINED), Artículo 26

4 hours duration

The examination lasts 4 hours, and collection of completed papers may begin after 2 hours.

Resolución Ministerial 238/2014 (MINED), Artículo 24 c

Centralized authorship

12th-grade final papers are prepared centrally by national subject methodologists of MINED.

Resolución Ministerial 238/2014 (MINED), Artículo 15

1.00 pt deduction

Orthographic deductions on 12th-grade final examinations are 1.00 full point per spelling error.

Resolución Ministerial 238/2014 (MINED), Anexo 1, Segundo

3 written finals

Matemática, Español-Literatura, and Historia are the only three 12th-grade subjects with written finals.

Resolución Ministerial 238/2014 (MINED), Anexo 6, §1

Cuba's national 12th-grade Mathematics final is a 4-hour written paper set centrally by MINED methodologists, worth 50% of the final course grade. Passing requires 30/100 on the exam and 60/100 overall under RM 238/2014. The official programme has four units: induction and sequences, statistics and combinatorics, complex numbers, and space geometry. This 100-question English MCQ bank gives worked practice across all four.

Sample Cuba 12th Grade Math Final Exam Practice Questions

Try these sample questions to test your Cuba 12th Grade Math Final Exam exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1In an arithmetic progression, the first term is $a_1 = 7$ and the common difference is $d = 4$. What is the 15th term ($a_{15}$) of this sequence?

A.63

B.67

C.59

D.71

Explanation: The general formula for the $n$-th term of an arithmetic progression is $a_n = a_1 + (n - 1)d$. Substituting the given values: $a_{15} = 7 + (15 - 1) \cdot 4 = 7 + 14 \cdot 4 = 7 + 56 = 63$.

2What is the sum of the first 20 terms of the arithmetic progression $3, 8, 13, 18, \dots$?

A.1010

B.1000

C.980

D.1040

Explanation: Here $a_1 = 3$ and the common difference is $d = 8 - 3 = 5$. The 20th term is $a_{20} = 3 + 19 \cdot 5 = 3 + 95 = 98$. The sum formula is $S_n = \frac{n}{2}(a_1 + a_n)$. Thus, $S_{20} = \frac{20}{2}(3 + 98) = 10 \cdot 101 = 1010$.

3In a geometric progression, the first term is $a_1 = 5$ and the common ratio is $r = 2$. What is the sum of the first 6 terms ($S_6$)?

A.315

B.320

C.310

D.160

Explanation: The sum of the first $n$ terms of a geometric progression is $S_n = \frac{a_1(r^n - 1)}{r - 1}$. For $n = 6$: $S_6 = \frac{5(2^6 - 1)}{2 - 1} = 5(64 - 1) = 5 \cdot 63 = 315$.

4What is the sum of the infinite convergent geometric series $12 + 6 + 3 + \frac{3}{2} + \dots$?

A.24

B.18

C.36

D.21

Explanation: The first term is $a_1 = 12$ and the common ratio is $r = \frac{6}{12} = \frac{1}{2}$. Since $|r| < 1$, the series converges and its sum is $S_\infty = \frac{a_1}{1 - r} = \frac{12}{1 - 1/2} = \frac{12}{1/2} = 24$.

5In a mathematical induction proof of the statement $P(n): 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ for all positive integers $n$, what is the inductive step assuming $P(k)$ is true?

A.Prove that $P(k+1)$ holds: $[1 + 2 + \dots + k] + (k+1) = \frac{(k+1)(k+2)}{2}$

B.Verify that $P(1)$ is true by checking $\frac{1(2)}{2} = 1$

C.Assume that $P(k+1)$ is false and deduce a contradiction

D.Substitute $n = \infty$ to evaluate the limit

Explanation: In mathematical induction, after verifying the base case $P(1)$, the inductive hypothesis assumes that $P(k)$ holds: $1 + 2 + \dots + k = \frac{k(k+1)}{2}$. The inductive step requires proving that $P(k+1)$ holds: $[1 + 2 + \dots + k] + (k+1) = \frac{k(k+1)}{2} + (k+1) = (k+1)(\frac{k}{2} + 1) = \frac{(k+1)(k+2)}{2}$.

6For which natural numbers $n \ge 1$ is the algebraic expression $A(n) = 7^n - 1$ divisible by 6?

A.For all natural numbers $n \ge 1$

B.Only for even natural numbers $n$

C.Only for odd natural numbers $n$

D.Only for prime numbers $n$

Explanation: For $n = 1$, $7^1 - 1 = 6$, which is divisible by 6. If $7^k - 1 = 6m$, then for $k+1$: $7^{k+1} - 1 = 7 \cdot 7^k - 1 = 7(6m + 1) - 1 = 42m + 7 - 1 = 42m + 6 = 6(7m + 1)$, which is also divisible by 6. Alternatively, $7 \equiv 1 \pmod 6 \implies 7^n - 1 \equiv 1^n - 1 = 0 \pmod 6$ for all $n \ge 1$.

7The imaginary unit is defined by $i = \sqrt{-1}$, so that $i^2 = -1$. What is the value of $i^{2026}$?

A.$-1$

B.$1$

C.$i$

D.$-i$

Explanation: The powers of $i$ repeat in a cycle of length four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. To evaluate $i^n$, divide the exponent by 4 and use the remainder. Here $2026 = 4 \cdot 506 + 2$, so the remainder is 2 and $i^{2026} = i^2 = -1$.

8A complex number written in binomic (rectangular) form is $z = a + bi$, where $a$ is its real part $\text{Re}(z)$ and $b$ its imaginary part $\text{Im}(z)$. For $z = -4 + 9i$, what are $\text{Re}(z)$ and $\text{Im}(z)$?

A.$\text{Re}(z) = -4$ and $\text{Im}(z) = 9$

B.$\text{Re}(z) = -4$ and $\text{Im}(z) = 9i$

C.$\text{Re}(z) = 9$ and $\text{Im}(z) = -4$

D.$\text{Re}(z) = 4$ and $\text{Im}(z) = -9$

Explanation: In the binomic form $z = a + bi$ the coefficients $a$ and $b$ are both real numbers. For $z = -4 + 9i$ the real part is $a = -4$ and the imaginary part is $b = 9$. The imaginary part is the real coefficient that multiplies $i$, not the product $9i$ itself.

9What is the product $(3 + 2i)(4 - 5i)$ expressed in binomic form?

A.$22 - 7i$

B.$2 - 7i$

C.$12 - 10i$

D.$22 + 7i$

Explanation: Multiply as ordinary binomials and then replace $i^2$ by $-1$: $(3 + 2i)(4 - 5i) = 12 - 15i + 8i - 10i^2$. Combining the imaginary terms gives $12 - 7i - 10i^2$, and since $i^2 = -1$ the last term becomes $+10$. Therefore the product is $12 + 10 - 7i = 22 - 7i$.

10What is the quotient $\dfrac{2 + 3i}{1 - i}$ expressed in binomic form?

A.$-\frac{1}{2} + \frac{5}{2}i$

B.$\frac{5}{2} - \frac{1}{2}i$

C.$2 + 3i$

D.$-\frac{1}{2} - \frac{5}{2}i$

Explanation: Multiply numerator and denominator by the conjugate of the denominator, $1 + i$. The denominator becomes $(1 - i)(1 + i) = 1^2 + 1^2 = 2$, a real number. The numerator becomes $(2 + 3i)(1 + i) = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i$. Hence the quotient is $\frac{-1 + 5i}{2} = -\frac{1}{2} + \frac{5}{2}i$.

About the Cuba 12th Grade Math Final Exam Exam

The Prueba Final de Duodécimo Grado en Matemática is the national school-leaving examination in Mathematics administered across Cuba by the Ministerio de Educación (MINED). Formally governed by MINED Resolución Ministerial 238/2014, the paper is prepared directly by national subject methodologists and dispatched to all pre-university institutes. It is one of three compulsory written finals required for high school graduation. Scored out of 100 points, the paper accounts for 50 of the 100 points of the student's final course mark (evaluación final = 50%, parciales = 40%, sistemáticas = 10%). A student must achieve at least 30 out of 100 points on the exam and an overall subject average of 60 to pass and obtain the Título de Bachiller. The official 12th-grade programme, set out in MINED's textbook Matemática 12mo. Grado. Perfeccionamiento (Editorial Pueblo y Educación, 2025), comprises exactly four units: complete induction and numerical sequences; statistics, probability and combinatorics; complex numbers; and space geometry. MINED's own revision broadcasts, Repaso Prueba Final 12mo grado, follow the same four blocks. This 100-question practice bank is an English-language MCQ study adaptation weighted to those four units and featuring authentic worked calculations and geometric problems.

Assessment

Centrally drafted by MINED national subject methodologists and sat in a single 4-hour session. Candidates sit in semigrupos, one per desk, invigilated by two teachers who do not teach mathematics. Papers are sealed and dispatched under provincial custody. Students may submit completed papers starting at 2 hours.

Time Limit

4 hours (collection may begin after 2 hours)

Passing Score

30 / 100 on final exam and 60 / 100 overall

Exam Fee

Free (public education in Cuba is state-funded with no examination or certification fees) (Ministerio de Educación (MINED), Dirección Nacional de Preuniversitario)

Cuba 12th Grade Math Final Exam Exam Content Outline

18 practice questions

Inducción completa. Sucesiones numéricas (Unit 1)

Complete induction in divisibility, summation and inequality proofs; arithmetic and geometric progressions, nth-term formulas, means, finite sums, convergent series, recursive sequences and limits.

26 practice questions

Estadística y probabilidades. Combinatoria (Unit 2)

Mean, weighted mean, median, mode, variance and standard deviation; Laplace's rule, complementary and conditional probability; permutations, variations, combinations, Pascal's rule, Binomial theorem and inclusion-exclusion.

26 practice questions

Números complejos (Unit 3)

Powers of i, binomic form, conjugate, modulus and equality; the four operations and division by the conjugate; Argand plane and loci; trigonometric form, argument, De Moivre's formula, n-th roots and equations in the complex domain.

30 practice questions

Geometría del espacio (Unit 4)

Relative positions of lines and planes, skew lines, three-perpendiculars theorem, dihedral angles; area and volume of prisms, pyramids, frustums, cylinders, cones, spheres and regular polyhedra; equations of planes and spheres, distances, and the trigonometry used to solve them.

How to Pass the Cuba 12th Grade Math Final Exam Exam

What You Need to Know

Passing score: 30 / 100 on final exam and 60 / 100 overall
Assessment: Centrally drafted by MINED national subject methodologists and sat in a single 4-hour session. Candidates sit in semigrupos, one per desk, invigilated by two teachers who do not teach mathematics. Papers are sealed and dispatched under provincial custody. Students may submit completed papers starting at 2 hours.
Time limit: 4 hours (collection may begin after 2 hours)
Exam fee: Free (public education in Cuba is state-funded with no examination or certification fees)

Keys to Passing

Work through all 100 available questions
Review every answer and explanation
Track weak areas and revisit them
Use our AI tutor for tough concepts

Cuba 12th Grade Math Final Exam Study Tips from Top Performers

1Master combinatorial formulas: distinguish permutations (order matters, all elements used), variations (order matters, subset of elements), and combinations (order does not matter). Practice Pascal's triangle and the Binomial theorem.

2Work fluently with complex numbers in both forms: binomic form for sums, products and division by the conjugate, and trigonometric form for powers and n-th roots via De Moivre. Fix the argument from the quadrant, never from the arctangent alone.

3Practice mathematical induction: establish the base step for n = 1, assume true for n = k (inductive hypothesis), and prove true for n = k + 1.

4Memorize solid geometry formulas: lateral surface area, total surface area, and volume for right and oblique prisms, regular pyramids, cylinders, cones, and spheres.

5Practise the statistics block as well as the counting block: mean from a frequency table, median of an even-sized set, variance and standard deviation, and the complement rule for 'at least one' probability problems.

6Work through the trigonometry you actually need here: double-angle and angle-addition formulas and the laws of sines and cosines, used to solve triangles inside solids and to convert complex numbers to trigonometric form.

Frequently Asked Questions

What is the Prueba Final de Duodécimo Grado in Mathematics?

It is the standardized national school-leaving examination in Mathematics administered across Cuba by the Ministerio de Educación (MINED). It evaluates pre-university graduates on the four units of the official 12th-grade programme — complete induction and numerical sequences, statistics, probability and combinatorics, complex numbers, and space geometry — forming part of the requirements for the Bachiller diploma.

How does this exam differ from the university entrance exam (cu-ingreso-matematica)?

The Prueba Final de 12mo Grado is administered by MINED in March/April under RM 238/2014, awards the Título de Bachiller, and focuses specifically on 12th-grade coursework (such as mathematical induction, combinatorics, and solid geometry). In contrast, the Examen de Ingreso is administered by MES in May/June under MES Res. 129/2018 for university admissions and tests all pre-university mathematics.

Who designs the examination papers?

Under Artículo 15 of MINED RM 238/2014, 12th-grade final examinations are prepared centrally by national subject methodologists of the Ministerio de Educación, rather than by individual classroom teachers.

What is the passing score and course weighting?

The paper is scored out of 100 points and accounts for 50% of the final course grade. Under Artículo 26 of RM 238/2014, a student must obtain a minimum of 30 points on the final paper and achieve an overall course composite mark of at least 60 points to pass.

Are spelling mistakes penalized in mathematics?

Yes. Under Anexo 1 of RM 238/2014, spelling mistakes on written examinations in all subjects, including Mathematics, are subject to a 1.00-point deduction per error in 12th grade.

Why is this practice bank in English?

The official paper is written and sat in Spanish. This bank is an English-language multiple-choice study adaptation of it — not an official translation, and not a simulation of a paper on which you must show full handwritten working, proofs and geometric constructions. It exists so that people who study and search in English, including Cuban students revising abroad, can practise the same four units of content, with worked calculations and step-by-step reasoning explained in English.