Romania Clasa a X-a Mathematics (common core, 2 h)
Chapters: 4
1. Sets of numbers
Real powers and radicals · Logarithms
- Radicals and nth Roots – The nth root of a number a is the number that, multiplied by itself n times, gives a. We write it ⁿ√a, and it is the same as a^(1/n). Radicals can be multiplied, divided and simplified by pulling out perfect powers, and a root in a denominator can be removed by rationalising.
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
2. Functions and equations
Power and radical functions · Exponential and logarithmic functions · Equations solved using function properties · Reading function properties from graphs
- Power Functions: y = a·xⁿ – A power function has the form y = a·xⁿ, where a is a number and n is a fixed exponent. Even whole powers (x², x⁴) make U shapes that are symmetric about the y-axis. Odd whole powers (x³, x⁵) make S shapes that are symmetric about the origin. All y = xⁿ with n > 0 pass through (0, 0) and (1, 1). Negative powers (x⁻¹ = 1/x) have asymptotes and are not defined at x = 0. Fractional powers are roots: x^(1/2) = √x, x^(1/3) = ∛x. The root function is the inverse of the matching power.
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- Exponential Equations and Inequalities – An exponential equation has the unknown in the power, like 2^x = 8. Solve it by making the bases equal, by substitution when it hides a quadratic, or by taking logarithms. For inequalities, keep the sign when the base is bigger than 1 and flip it when the base is between 0 and 1.
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
3. Financial mathematics
Counting problems · Financial calculations · Statistics · Probability
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- Smart Ways to Manage Your Finances – Inflation makes prices rise, so the same money buys less. Money kept in a bank earns interest: simple interest is paid only on the original amount, while compound interest also earns interest on earlier interest, so it grows faster over time. A budget plans income into needs, wants and savings. Savings are kept safe; investments can grow but carry risk, and higher possible returns mean higher risk. Insurance shares risk among many people. Income tax is paid on income above a limit, at rates that rise with income.
- Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.
4. Geometry
Analytic geometry of the plane
- Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).