Special Course for the University Proficiency Examination – Mathematics I
Overall unit structure
01 Exponentials and logarithms: n-th roots, laws of exponents, definition and properties of logarithms, change of base, common (base‑10) logarithms, calculations and inequalities involving exponential and logarithmic expressions
02 Exponential and logarithmic functions: graphs and properties, translations, symmetries, maxima and minima, use in equations and inequalities
03 Meaning of trigonometric functions and their graphs: directed angle, radian measure, definition, relations and graphs of trigonometric functions, properties, equations and inequalities
04 Sine rule and cosine rule: sides, angles and area of a triangle, determining the shape of a triangle, problems involving circumscribed and inscribed circles
05 Arithmetic and geometric sequences: meaning, general term and sum, arithmetic mean and geometric mean, recursive definition, applications to past exam questions
06 Sum of sequences and the principle of mathematical induction: Σ notation and its properties, sums of powers, sums of fractions, recursively defined sequences, proofs by induction
01 Exponentials and logarithms
n-th roots: denote the real solution of \(x^n=a\) by \( \sqrt[n]{a} \), and study existence conditions and properties according to whether the exponent is even or odd
Exponents: definition of integer, rational and real exponents, extension of exponent laws such as \(a^m a^n = a^{m+n}\), \((a^m)^n=a^{mn}\), \((ab)^n=a^n b^n\)
Definition of the logarithm: \(a^x=N \Leftrightarrow x=\log_a N\), definition conditions \(a>0,a\neq1,N>0\)
Change of base: \(\log_a b = \dfrac{\log_c b}{\log_c a}\), \(\log_a b = \dfrac1{\log_b a}\), \(a^{\log_b c}=c^{\log_b a}\)
Common (decimal) logarithms: \(\log_{10} N=\log N\), use of common logarithm tables, finding approximations by writing \(N\) in the form \(N=a\cdot 10^n\)
Past questions: computing values using n-th roots and logarithm properties, conditions for obtaining a natural number, problems combining Σ and logarithms
02 Exponential and logarithmic functions
Exponential function \(y=a^x (a>0,a\neq1)\)
Domain: set of real numbers; range: \(y>0\); it passes through the point (0,1) and has the asymptote \(y=0\)
\(a>1\): increasing function; \(0<a<1\): decreasing function
Exponential equations and inequalities: interpret them by comparing exponents according to the size of the base
Logarithmic equations and inequalities: first check the conditions on the arguments, then use monotonicity for comparison
Past questions: intersection points of graphs, areas, combinations with triangles and circles, problems where the unknown is found after transforming exponential or logarithmic expressions
03 Meaning of trigonometric functions and graphs
Directed angle and radian measure
Directed angle: fix the initial side and express the rotation of the moving ray in the form \(360^\circ n + \alpha^\circ\)
Radian measure: \(1 \text{ rad}=\dfrac{180^\circ}{\pi},\ 1^\circ=\dfrac{\pi}{180}\) rad
Circular sector: arc length \(l=r\theta\), area \(S=\frac12 r^2\theta\)
Definition of trigonometric functions
For a point \(P(x,y)\) on a circle of radius \(r\) and an angle \(\theta\): \(\sin\theta=\dfrac y r,\ \cos\theta=\dfrac x r,\ \tan\theta=\dfrac y x\)
Signs: arrangement of the signs of sin, cos and tan by quadrant; angles \(\theta = \frac{\pi}{2}+k\pi\) for which \(\tan\theta\) is not defined
Quadrilateral: if the lengths of the two diagonals are \(p,q\) and the angle between them is \(\theta\), then \(S=\frac12 pq\sin\theta\)
Area combined with inscribed and circumscribed circles; many past questions using inscribed or circumscribed triangles
05 Arithmetic and geometric sequences
Arithmetic sequence
General term: if the first term is \(a\) and the common difference is \(d\), then \(a_n=a+(n-1)d\)
Sum: \(S_n=\dfrac{n(a_1+a_n)}{2}=\dfrac{n\{2a_1+(n-1)d\}}{2}\); \(S_n\) is a quadratic expression in \(n\)
Arithmetic mean: if \(a,b,c\) form an arithmetic sequence, then \(b=\dfrac{a+c}{2}\)
Geometric sequence
General term: if the first term is \(a\) and the common ratio is \(r\), then \(a_n=ar^{n-1}\)
Sum: if \(r\neq1\), \(S_n=a\dfrac{1-r^n}{1-r}=a\dfrac{r^n-1}{r-1}\); if \(r=1\), \(S_n=na\)
Geometric mean: if \(a,b,c\) form a geometric sequence, then \(b^2=ac\)
Recursive definition
Arithmetic sequence: \(a_1=a,\ a_{n+1}=a_n+d\)
Geometric sequence: \(a_1=a,\ a_{n+1}=ra_n\)
Deciding whether a sequence is arithmetic or geometric from conditions such as \(2a_{n+1}=a_n+a_{n+2}\), \(a_{n+1}^2=a_na_{n+2}\)
Past questions: superposition of arithmetic and geometric sequences, determining particular terms, the common difference or the common ratio using sum formulas, identifying the type from given conditions
06 Sum of sequences and the principle of mathematical induction
Summation symbol Σ
\(\displaystyle \sum_{k=1}^n a_k = a_1+a_2+\cdots +a_n\), sum over an interval: \(\sum_{k=m}^n a_k = \sum_{k=1}^n a_k-\sum_{k=1}^{m-1}a_k\)
Properties: linearity (term‑by‑term sum, constant factor, sum of a constant)
Derivation using differences of identical equalities, then direct application to various summation problems
Sum of fractional terms
Partial fraction decomposition: for example \(\dfrac{1}{k(k+1)}=\dfrac1k-\dfrac1{k+1}\) to obtain a telescoping sum
Rationalize the denominator of an irrational expression and then transform it into a telescoping structure: \(\dfrac{1}{\sqrt{k+16}+\sqrt{k}}=\sqrt{k+16}-\sqrt{k}\)
Sequences defined recursively
Substitute \(n=1,2,3,\dots\) into the recurrence relation to understand the first term and the rule, then check whether the sequence is arithmetic or geometric
Combine recurrence relations and Σ to determine particular terms, partial sums and indices satisfying given conditions
Principle of mathematical induction
First step: verify that the statement is true for \(n=1\) (or \(n=m\))
Second step: assume the statement is true for \(n=k\) and prove that it is also true for \(n=k+1\)
Scope of application: statements about the set of natural numbers, such as sum formulas, inequalities, product structures, etc.
