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2026 - KPSS - AGS Matematik Sorularla Genel Tekrar / TEK VİDEO ~ İlker Karabulut
Retro Yayıncılık
Overview
This video provides a comprehensive review of key mathematics topics for the KPSS and AGS exams, presented through a series of solved problems. It covers number properties (even/odd, factorials), logic, number systems, algebra (equations, inequalities, absolute values), set theory, and problem-solving strategies. The instructor emphasizes understanding core concepts and applying them to common exam question formats, offering practical tips and shortcuts for efficient problem-solving. The review aims to equip students with the knowledge and techniques needed to tackle a wide range of mathematical questions encountered in these competitive exams.
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Chapters
- Understand the properties of even and odd numbers in addition and multiplication.
- Determine the parity (even or odd) of expressions based on the parity of the variables.
- Apply these properties to solve equations where the result is either even or odd.
Mastering even and odd number properties is fundamental for solving algebraic equations and inequalities, as it provides a quick way to check the validity of solutions and understand the nature of results.
Given A * B + C is odd, determine which of the following expressions is always even. This involves analyzing cases for C (even or odd) and the resulting implications for A * B.
- Simplify equations by recognizing that coefficients of 1 in multiplication/division can be ignored.
- Deduce the parity of variables (A, B, C) based on given equations involving even and odd results.
- Use the deduced parities to determine the parity of other expressions.
This section demonstrates how to systematically determine the exact parity of unknown variables from given equations, which is crucial for solving more complex algebraic problems.
Given A - C is odd and 2B - (A - C) is even, determine the parity of A, B, and C. This leads to finding A is even, C is odd, and B is even.
- Represent regions on a number line as intervals defined by an 'a' value and a maximum distance 'b'.
- The interval is expressed as [a-b, a+b].
- Find the intersection of two such intervals by identifying the common range of numbers.
Understanding how to represent and find the intersection of intervals is essential for solving problems involving ranges of values, often seen in inequalities and real-world scenarios.
Find the common points for the regions defined by (7, 4) and (12, 5). This involves calculating the intervals [3, 11] and [7, 17] and finding their overlap, which is [7, 11].
- Simplify factorial expressions by reducing larger factorials to smaller ones (e.g., 9! = 9 * 8!).
- Use common factors to simplify complex expressions involving factorials.
- Perform basic arithmetic operations on the simplified terms to find the final result.
Factorial problems often test your ability to simplify and manipulate expressions efficiently, rather than direct calculation, which is key for time-constrained exams.
Simplify (10 * 9! + 9!) / (9 * 8! - 8!). This involves factoring out 9! and 8! to get (11 * 9!) / (8 * 8!) and further simplification.
- Convert multi-digit numbers into algebraic expressions based on place value (e.g., AB = 10A + B).
- Set up and solve linear equations involving these place-value expressions.
- Use the derived values of variables to find the required sum or other properties.
This skill is vital for solving problems involving numbers represented by digits, allowing you to translate word problems into solvable algebraic equations.
Given AB5 + 4AB = 13 * AB + 241, use place value to form the equation (100A + 10B + 5) + (400 + 10A + B) = 13(10A + B) + 241 and solve for A and B.
- Perform subtraction on multi-digit numbers, understanding the concept of borrowing from adjacent places.
- Recognize that if a digit in the subtrahend is larger than the digit in the minuend, borrowing is necessary.
- Apply this to solve equations where variables represent digits.
Understanding subtraction with borrowing is fundamental for arithmetic operations and solving problems involving digit manipulation, especially in number theory contexts.
In the subtraction C B A - 4 B 9 = 2 4 3, determine the values of A, B, and C by applying borrowing rules. This leads to A=9, B=4, C=1.
- Apply divisibility rules for numbers like 3, 5, and 15.
- For composite divisibility rules (like 15 = 3 * 5), prioritize the rule of the larger or more complex factor (3 in this case).
- Use divisibility rules to determine possible values for unknown digits and find the maximum possible value of a number.
Divisibility rules are shortcuts to determine if a number is divisible by another without performing division, crucial for number theory problems and optimization.
For a 5A37B number to be divisible by 15, it must be divisible by both 3 and 5. This means B must be 0 or 5, and the sum of digits must be divisible by 3. To maximize the number, B=5 and A=7.
- Perform arithmetic operations (addition, subtraction, multiplication, division) with decimal numbers.
- Understand and apply the difference of squares formula (a^2 - b^2 = (a-b)(a+b)) to simplify expressions.
- Simplify decimal expressions by canceling common terms or converting to fractions.
Proficiency in decimal arithmetic and algebraic simplification is essential for various mathematical contexts, including financial calculations and scientific measurements.
Simplify (0.6 * 0.8) / (1.4^2 - 0.6^2). This involves using the difference of squares for the denominator: (1.4 - 0.6)(1.4 + 0.6) = 0.8 * 2.0. The expression becomes (0.6 * 0.8) / (0.8 * 2.0), simplifying to 0.6 / 2.0 = 0.3.
- Simplify decimal division by shifting decimal points to make the divisor a whole number.
- When shifting the decimal in the divisor, the same shift must be applied to the dividend.
- Apply this technique to solve complex decimal division problems, especially those involving addition or subtraction in the numerator/denominator.
This technique simplifies decimal division, making calculations more manageable and reducing the chance of errors, particularly in multi-step problems.
Simplify (0.023 - 0.006) / (0.480 + 0.200) * (1/10). By shifting decimals three places to the right in the main fraction, it becomes (23 - 6) / (480 + 200) = 17 / 680. Multiplying by 1/10 gives 17 / 6800, which simplifies to 1/400. The example in the video actually simplifies (0.023 - 0.006) / (0.48 + 0.2) which is 17/680 = 1/40. The video example seems to have a slight calculation error or simplification path not fully shown.
- Express numbers as powers of their prime factors (e.g., 16 = 2^4, 27 = 3^3).
- Apply exponent rules: (a^m)^n = a^(m*n) and a^m * a^n = a^(m+n).
- Use common factors and the difference of squares formula (a^2 - b^2) to simplify complex expressions with exponents.
Understanding exponent rules is crucial for simplifying expressions involving large numbers and solving equations where variables are in the exponent.
Simplify (3^6 * 4^6 + 27^2) / (6^6 - 3^6). This involves rewriting terms like 4^6 as (2^2)^6 = 2^12 and 27^2 as (3^3)^2 = 3^6, then factoring out common terms like 3^6 and using the difference of squares on 2^12 - 1.
- Simplify square roots by factoring out perfect squares (e.g., sqrt(18) = sqrt(9*2) = 3*sqrt(2)).
- Rationalize the denominator of fractions containing square roots by multiplying by the conjugate.
- Combine like radical terms after simplification.
Simplifying radical expressions is a key algebraic skill, essential for solving equations involving roots and presenting answers in their simplest form.
Simplify 10/√5 + √10 - √2 / √2. Rationalize the first term to get 2√5. Factor √2 from the second part: √2(√5 - 1) / √2 = √5 - 1. The expression becomes 2√5 + (√5 - 1) + 1, simplifying to 3√5.
- Solve absolute value equations by setting the expression inside the absolute value equal to both the positive and negative values of the other side.
- If an equation is of the form |x| = 0, then x = 0.
- Use the solutions found for one variable to constrain the possible values of another variable and identify what is not possible.
Absolute value equations appear frequently and require understanding that the expression inside can be positive or negative, leading to multiple potential solutions.
Given |C+4|=0, |B-4|=2, and |A+B|=3, find which value A cannot be. From |C+4|=0, C=-4. From |B-4|=2, B=6 or B=2. If B=6, |A+6|=3 gives A=-3 or A=-9. If B=2, |A+2|=3 gives A=1 or A=-5. Thus, A cannot be 5.
- Simplify inequalities by removing even exponents (they don't change the sign).
- For odd exponents, remove them by taking the root (which preserves the sign).
- Determine the sign (+ or -) of variables based on the simplified inequalities and their relationships.
Analyzing the signs of variables in inequalities is crucial for ordering them and determining the truth of statements involving these variables.
Given x^2 * y < 0 and y - z < 0 and x > 0. Simplify to x > 0. Since x is positive, y must be negative (y < 0). Since y < z and y is negative, z must be positive (z > 0). The order is y < z < x.
- When asked for the simplified form of an algebraic expression with multiple variables, substitute simple numerical values for the variables.
- Choose values that avoid making denominators zero or causing other undefined operations.
- Substitute these values into the answer choices to find the one that matches the result.
This substitution method provides a quick way to find the correct simplified form of complex algebraic expressions, especially in multiple-choice formats.
To simplify (x^2 - xy - 2y^2) / (x^2 - 4y^2), substitute x=5 and y=2. The expression becomes (25 - 10 - 8) / (25 - 32) = 7 / -7 = -1. The correct answer choice will yield -1 when x=5 and y=2.
- Read word problems carefully, identifying key information and what is being asked.
- Translate sentences into mathematical expressions or equations, paying attention to keywords like 'directly proportional' or 'inversely proportional'.
- Set up equations based on the relationships described and solve them systematically.
Effective problem-solving hinges on accurately interpreting the problem statement and translating it into a solvable mathematical model.
If student numbers in branches A, B, C are proportional to 3, 4, 5 respectively, and a new branch D is added with the total students from A, B, C, and the average per branch becomes 72, find the number of students in A. This involves setting up A=3k, B=4k, C=5k, D=12k, and using the average to solve for k.
- Understand the relationship between total sum, average, and the number of items: Sum = Average * Count.
- Calculate the change in total sum based on changes in individual scores and the number of items.
- Use this to find the number of items (e.g., exams) based on changes in average.
Average problems are common and require understanding how changes in individual values affect the overall average and total sum.
If a student's average score increases from 72 to 75 when their score changes from 65 to 80, find the number of exams. The total score increase is 15, and the average increase is 3. The number of exams is Total Increase / Average Increase = 15 / 3 = 5.
- Model real-world situations involving choices or categories using set theory principles.
- Use Venn diagrams to visualize the relationships between different groups or choices.
- Apply the principle of inclusion-exclusion or simple addition/subtraction based on the problem's constraints (e.g., 'only two types', 'at least one').
Set theory provides a powerful framework for analyzing overlapping groups and making logical deductions in scenarios involving multiple categories.
In a survey of 150 people about metro and bus usage, given data on morning bus, evening metro, and different transport for round trip, find the number who used metro in the morning and bus in the evening. This involves setting up equations for combinations of travel (e.g., A-A, A-B, B-A, B-B) and solving them.
- Define variables for the number of days working at different rates (e.g., 4 questions/day, 5 questions/day).
- Set up equations based on the total number of questions produced over a period, considering different rates and rest days.
- Solve the system of equations to find the unknown number of days or total questions.
Work rate problems test your ability to manage multiple variables and rates, common in scenarios involving production, tasks, or time management.
A person works 4 or 5 questions a day for 2 months. If the first month produced 100 questions and the second 260, and the number of 4-question days in the first month is 1/3 of the second month, find the total number of days worked in the first month. This involves setting up equations like 4x + 5y = 100 and 12x + 5y = 260.
- Understand the concept of sharing items equally among a group.
- Model scenarios where items are consumed or distributed, leading to a difference in the number of items per person.
- Use the relationship between the number of people and the number of items per person to find possible total quantities.
This problem type tests logical deduction and the ability to work backward from a final state to an initial state, often involving factors and multiples.
A group of children share nuts equally. After eating 3 each, they give 1 nut each to a newcomer. The remaining nuts per person are now 4 less than the initial amount. Find the initial total number of nuts. This involves setting up equations like (y-3)*x = Total Nuts - x and finding pairs of factors with a difference of 4.
- Represent unknown quantities using variables, often multiples of denominators to avoid fractions.
- Set up equations based on the described changes in quantities (e.g., people moving between groups, items being added or removed).
- Solve the system of equations to find the required quantity.
Fraction and proportion problems require careful setup of equations to represent the relationships between different quantities accurately.
In a stadium, 'x' people are sitting and '3x' are standing. If 800 standing people sit, the stadium is 4/5 full. If 400 sitting people stand, it's 1/2 full. Find the initial number standing (3x). This involves setting up equations like x+800 = (4/5)*TotalSeats and x-400 = (1/2)*TotalSeats.
- Recognize that the age difference between two people remains constant over time.
- Set up equations based on the relationships between ages at different points in time (present, past, future).
- Use the constancy of age difference to solve for unknown ages.
Age problems test your ability to manage time shifts and relationships between different individuals' ages.
Ilker's uncle is 3 more than 5 times Ilker's age. When the uncle was Ilker's father's current age, Ilker was 3 years from birth. The sum of father's and uncle's ages 5 years ago was 65. Find the father's current age. This involves setting up equations for current ages and past ages, using age difference constancy.
- Represent the total quantity as 100x for percentage problems to simplify calculations.
- Calculate the number of errors or items in different segments based on given rates (e.g., 1 error per 6 questions).
- Sum the errors from all segments and equate it to the total given errors to solve for x.
Percentage problems are ubiquitous, requiring accurate calculation of parts of a whole and understanding how percentages relate to quantities.
A book has 30% solved with 1 error/6 questions, 50% solved with 1 error/10 questions, and 20% solved with 3 errors/20 questions. If total errors are 195, find the total number of questions. This involves calculating errors in each segment (5x, 5x, 3x) and solving 13x = 195.
- Represent actual weights using variables like 100x and 100y.
- Calculate the measured weights based on given percentage errors (e.g., +3% or -5%).
- Set up equations based on the sum of measured weights and the relationship between actual weights to find the required value.
Understanding percentage errors is crucial in practical applications involving measurements and estimations, where accuracy is key.
Two crates have a total actual weight of 72 kg. When weighed on faulty scales (one +3%, one -5%), the total measured weight is still 72 kg. Find the weight of the heavier crate. This involves setting up 100x + 100y = 72 and 103x + 95y = 72, then solving for x and y.
- Use the formula Distance = Speed * Time.
- Set up equations where the distance is constant but speed and time vary.
- Solve for the time variable and then calculate the required speed or distance.
Speed, distance, and time problems are fundamental in physics and everyday life, requiring logical application of the relationship between these three quantities.
To travel from Antalya to Ankara, going 120 km/h takes 2 hours longer than going 150 km/h. Find the speed needed to arrive 2 hours earlier than the slower time (i.e., 3 hours total travel time). This involves setting up 120(t+2) = 150t to find t, then calculating distance and the new required speed.
- Interpret set notation like 'B covers A' (B ⊇ A) and set differences (A C).
- Use Venn diagrams to represent the relationships between sets and their intersections/differences.
- Calculate the total number of elements in the union of sets based on the sizes of intersections and differences.
Set theory problems test your ability to visualize and manipulate relationships between different groups, often requiring careful interpretation of diagrams and notation.
Given B ⊇ A, B A = A ∩ B ∩ C, and A C = C B. If |A| = 30, find |A ∪ B ∪ C|. This involves drawing a Venn diagram that satisfies the conditions and calculating the total elements based on the given information.
- Model scenarios with limited choices using sets (e.g., tea, coffee, soda).
- Apply constraints such as 'only one type per person' or 'no one took two types' to simplify the set relationships.
- Use the number of remaining items to deduce the number of people who chose each item or combination.
This demonstrates how set theory can be applied to analyze consumer choices and resource allocation in practical situations.
In a group of 25 people, each chooses tea, coffee, or soda once. If 7 cups of tea, 8