6-Month Math Exam Study Plan — From 5 Points to 9 Points (Grade 12)

Why do you need a 6-month study plan for the national math exam?

The most common question grade 12 students ask every year at the start of the second semester is: "Do I have enough time to prepare for the national high school math exam?" The answer is always: yes — if you study with a systematic plan.

The harsh reality is that most students study in an "ad hoc" way — today working on function problems, tomorrow reviewing integral theory, next week going back to trigonometry because they feel weak there. The result is that 3 months pass without any solid foundation, and when they walk into the exam room and hit a hard question, they panic.

Furthermore, the current national high school math exam has 50 multiple-choice questions in 90 minutes — averaging 1.8 minutes per question. Many students know the material well but still can't finish in time because they haven't practiced speed. And the last 5 questions (high-level application) are enough to push scores from 8 to 10 — if you know the right test-taking strategy.

The 6-month study plan below is designed on the principle: consolidate foundations first, advance later, practice exams continuously. Each month focuses on one group of topics, ensuring the entire curriculum is reviewed before moving to comprehensive practice exams.

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Month 1 — Consolidate Basic Knowledge (Target: 6/10 points)

Month 1 Goals

Solidify the grade 10–11 foundation that still appears in the exam: linear functions, quadratic functions, systems of equations, inequalities. This section accounts for 8–10 questions in the exam — skipping it means losing easy points.

Topics to Review

Weeks 1–2: Functions and Basic Graphs

• Linear function y = ax + b: monotonicity, axis intersections.
• Quadratic function y = ax² + bx + c: vertex of parabola, axis of symmetry, roots.
• Practice: 20 questions on identifying quadratic function graphs each day.

Week 3: Systems of Equations

• Substitution, elimination, and basic matrix methods.
• Word problems (sharing, ages, filling tanks).

Week 4: Inequalities and Absolute Value

• Linear and quadratic inequalities.
• |f(x)| ≤ k and |f(x)| ≥ k.

Time Allocation

8 hours/week (average students) or 12 hours/week (students aiming for 9+).

• 2 hours learning theory and noting formulas.
• 4 hours solving practice problems by type.
• 2 hours reviewing incorrectly answered problems.

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Month 2 — Trigonometry and Exponential-Logarithmic Functions

Month 2 Goals

This is the section where average students lose the most points because there are many formulas and it's easy to confuse them. This month focuses on systematizing formulas and practicing trigonometry problem types in the exam.

Trigonometry Topics (Weeks 1–2)

Must memorize:

• Special trigonometric values table (30°, 45°, 60°, 90°).
• Addition formulas: sin(a ± b), cos(a ± b), tan(a ± b).
• Double angle formulas: sin 2a, cos 2a, tan 2a.
• Half-angle reduction, product-to-sum formulas.

Practice problem types:

• Simplify trigonometric expressions (very common in questions 1–10).
• Basic trigonometric equations: sin x = a, cos x = a.
• Derived trigonometric equations: sin x = sin α, cos x = cos α.

Exponential-Logarithmic Function Topics (Weeks 3–4)

Must understand:

• Properties of y = a^x (increasing when a > 1, decreasing when 0 < a < 1).
• Properties of logarithmic function y = log_a(x).
• Solving exponential equations: a^f(x) = a^g(x) if and only if f(x) = g(x).
• Solving logarithmic equations: log_a(f(x)) = log_a(g(x)) if and only if f(x) = g(x) (with domain condition).
• Substitution technique for exponential equations.

Daily practice: 15 exponential-logarithmic questions + 10 trigonometry questions.

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Month 3 — Curve Sketching and Analysis of Functions

Month 3 Goals

This is the topic that average grade 12 students are weakest in and it also accounts for the most questions in the national exam (around 10–15 related questions). Month 3 requires more time investment than other months.

6-Step Function Analysis Process (Required)

Step	Content	Note
1	Domain	Exclude values that make denominators zero
2	Monotonicity	Calculate f'(x), solve f'(x) = 0
3	Extrema	Use sign chart
4	Asymptotes	Vertical (x = a), horizontal (y = L)
5	Sign chart	Present in correct format
6	Draw graph	Mark special points

Function Types to Practice

• Cubic functions: y = ax³ + bx² + cx + d — always has 2 extrema or none.
• Linear-over-linear rational functions: y = (ax+b)/(cx+d) — has vertical and horizontal asymptotes.
• Quadratic-over-linear rational functions: y = (ax²+bx+c)/(dx+e) — has oblique asymptote.
• Square root functions: y = √f(x) — note the condition f(x) ≥ 0.

Recommended Practice for Month 3

Every day, solve at least 1 complete curve sketching problem (domain → draw graph) and 10 multiple-choice questions on reading function graphs.

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Month 4 — Integrals and Applications

Month 4 Goals

Integrals account for around 8–10 questions in the national exam. This includes: finding antiderivatives, indefinite integrals, definite integrals, and applications such as calculating areas and volumes.

Antiderivative Table to Memorize

This is essentially the derivatives table read in reverse. Students who know the derivatives table well will find this section easier.

Function	Antiderivative	Condition
x^n (n ≠ −1)	x^(n+1)/(n+1) + C	—
1/x	ln	x
e^x	e^x + C	—
a^x	a^x / ln(a) + C	a > 0, a ≠ 1
sin(x)	−cos(x) + C	—
cos(x)	sin(x) + C	—
1/cos²(x)	tan(x) + C	—

Full formula:

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$

$\int \frac{1}{x}dx=\ln |x|+C$

Advanced Integration Techniques

• Substitution method: Let u = g(x), compute du = g'(x)dx, change limits if definite integral.
• Integration by parts: ∫u dv = uv − ∫v du — used when the integrand is a product of a polynomial and an exponential/trigonometric function.

Applications of Integrals

Area of region bounded by y = f(x) and the x-axis:

$S={\int }_a^b|f(x)|dx$

Volume of solid of revolution:

$V=\pi {\int }_a^b[f(x){]}^{2}dx$

Practice: 3 area problems and 2 volume problems per week — these are hard problem types but commonly appear in exams.

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Month 5 — Complex Numbers and Solid Geometry

Month 5 Goals

These two topics account for 6–8 questions in the exam. Students often skip them because they seem hard, but in reality both have template problem types that can be learned.

Complex Numbers (Weeks 1–2)

Key knowledge:

• Algebraic form: z = a + bi, i² = −1.
• Modulus: |z| = √(a² + b²).
• Complex conjugate: z̄ = a − bi.
• Operations: addition, subtraction, multiplication, division of complex numbers.
• Square roots of complex numbers.
• Representing complex numbers on the Argand plane.

Common problem types:

• Calculate z^n for large n (using trigonometric form).
• Find complex number satisfying given conditions.
• Solve quadratic equations when Delta < 0 (complex roots).

Solid Geometry (Weeks 3–4)

Priority topics:

• Regular pyramids, right prisms — calculating volume and lateral surface area.
• Relative positions of lines and planes (parallel, perpendicular, intersecting).
• Angle between a line and a plane; angle between two planes.
• Distance from a point to a plane.

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Month 6 — Comprehensive Practice Exams and Test Strategy

Month 6 Goals

This is the decisive month. No new theory — focus 100% on full practice exams and increasing speed.

Practice Exam Schedule

Weeks 1–2: Every day, complete 1 practice exam (50 questions, 90 minutes). After each exam, spend 30 minutes analyzing mistakes: wrong because of not knowing the formula, computational errors, or not reading the question carefully?

Weeks 3–4: Focus on real past exams from 2022–2025. Pay special attention to question types you got wrong repeatedly in months 1–2.

Time Allocation in the Exam Room (90 minutes for 50 questions)

Section	Questions	Time	Strategy
Questions 1–30 (easy–medium)	30	35 min	Work quickly, don't spend more than 1 min/question
Questions 31–45 (application)	15	40 min	Watch for traps, read carefully
Questions 46–50 (high application)	5	15 min	Carefully choose which ones you can solve

Strategy: never leave a question blank — multiple choice has no penalty for wrong answers, so always select an answer even if you have to guess.

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5 Effective Multiple-Choice Test-Taking Tips

Tip 1: Easy questions first, hard questions last Quickly scan all 50 questions in the first 3 minutes. Mark easy questions (you immediately recognize the formula) and solve them first. Leave hard questions for the end.

Tip 2: Don't spend more than 2 minutes on one question If after 2 minutes you haven't found a solution approach, skip and move to the next question. Wasting time on one question may cause you to miss 5–10 easy questions later.

Tip 3: Use the Casio calculator smartly

• Calculate complex numbers quickly using CMPLX mode.
• Calculate numerical integrals using INTEGRAL mode (to check answers).
• Solve quadratic and cubic equations using EQN mode.

Tip 4: Eliminate wrong answers to narrow choices Many multiple-choice questions allow you to immediately eliminate 2 wrong answers using estimation or trying special values. With 2 remaining answers, your probability of being correct is already 50%.

Tip 5: Check by substituting your answer back For equation-solving problems, substitute your answer back into the original equation to verify. Takes less than 30 seconds but prevents costly mistakes.

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Frequently Asked Questions

Q1: How long does it take to go from 5 points to 9 points in math?

With a 6-month study plan following the schedule correctly (8–12 hours/week, covering all 6 topic groups, regular practice exams), students at the 5-point level can realistically target 7.5–8.5 points. Achieving 9+ truly requires studying all the way to the final month with 24+ real practice exams and completing all 5 high-level application questions. More importantly: don't compare yourself to others — compare yourself to who you were last week.

Q2: Is it necessary to attend tutoring to score well on the national math exam?

Not required. Many students self-study successfully with quality materials, practice exams from reputable high schools, and consistent study discipline. Tutoring can be helpful if you need someone to explain difficult problems directly — but if you can research independently and have good support tools (like Witza for instant problem solving when needed), you can absolutely self-study effectively.

Q3: Where should I find practice exams for the national math exam?

Quality sources include: official exams from the Ministry of Education 2020–2025 (on the ministry website), practice exams from major provincial education departments (Hanoi, Ho Chi Minh City), and exams from prestigious specialized high schools. Additionally, the Witza app provides topic-specific multiple-choice questions — you can practice 20 questions in 20 minutes anytime without needing to sit at a desk.