Sample · 01-Mathematics
Chapter 1 Overview
Bocconi Online Test · CHAPTER 1: Mathematics
The Mathematics area is the largest part of the Online Bocconi Test, with 24 of the 50 questions. This study book works through every topic on the official syllabus: algebra, functions, plane geometry, analytical geometry, trigonometry, sets, logarithms and exponentials, discrete mathematics, numbers, probability, statistics, and applied problem solving. For each topic you will find a short explanation, a box of the key formulas you must know, a worked example, and a set of practice questions with fully explained solutions. Remember the test scoring: a correct answer earns \( +1 \), a blank earns \( 0 \), and a wrong answer costs \( -0.2 \), so guess only when you can eliminate options. Applicants to the Mathematical and Computing Sciences for Artificial Intelligence bachelor must also score at least \( 11 \) out of \( 24 \) in this area.
1.1 Algebra
1.1 Algebra
Algebra is the backbone of the mathematics section. You must be comfortable manipulating expressions with powers and radicals, and solving first- and second-degree equations, inequalities, systems, and expressions involving absolute values. Most questions reward careful, systematic work rather than clever tricks.
The power laws let you combine exponents: \( a^m\cdot a^n=a^{m+n} \), \( \dfrac{a^m}{a^n}=a^{m-n} \), and \( (a^m)^n=a^{mn} \). For quadratics, factoring is fastest when it works; otherwise use the quadratic formula. The sign of the discriminant \( b^2-4ac \) tells you how many real roots exist.
📋 Key formulas
| Tool | Formula |
|---|---|
| Quadratic formula | \( x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \) |
| Difference of squares | \( a^2-b^2=(a+b)(a-b) \) |
| Discriminant | \( \Delta=b^2-4ac \) (two roots if \( \Delta>0 \), one if \( \Delta=0 \), none if \( \Delta<0 \)) |
| Absolute value | \( |x|<a \Leftrightarrow -a<x<a \); \( |x|>a \Leftrightarrow x<-a \) or \( x>a \) |
⚠️ Common traps
When you multiply or divide an inequality by a negative number, reverse the inequality sign. With absolute-value equations, remember there are usually two cases. When factoring a quadratic, check that your two numbers give both the correct product and the correct sum.
Worked Example — Solving a quadratic
Question: Solve \( x^2-x-12=0 \).
Working: Look for two numbers with product \( -12 \) and sum \( -1 \); these are \( -4 \) and \( 3 \). So \( (x-4)(x+3)=0 \).
📋 Exam-level techniques
| Situation | Method |
|---|---|
| Parameter and real roots | \( ax^2+bx+c=0 \) has real solutions when \( \Delta=b^2-4ac\ge 0 \). Solve this for the parameter, e.g. \( x^2+6x+c=0 \) has real roots iff \( 36-4c\ge 0 \), i.e. \( c\le 9 \). |
| Number of solutions of a linear system | Two lines: one solution if not proportional; no solution if the coefficients are proportional but the constants are not; infinitely many if the whole equations are proportional (the same line). |
| Reading a formula's coefficients | In \( 60+12nh \), the constant \( 60 \) is a fixed fee and \( 12 \) multiplies the worker-hours \( nh \), so it is the cost per worker per hour. |
| Nested (continued) fractions | Simplify from the innermost fraction outward, combining over a common denominator at each step. |
Worked Example — Nested fraction & a dependent system
Question: Simplify \( \dfrac{1}{2+\dfrac{3}{4+\frac{5}{x}}} \).
Working: Innermost: \( 4+\dfrac{5}{x}=\dfrac{4x+5}{x} \). Then \( \dfrac{3}{(4x+5)/x}=\dfrac{3x}{4x+5} \), and \( 2+\dfrac{3x}{4x+5}=\dfrac{11x+10}{4x+5} \).
Continue reading — 1 chapters, worked examples and practice.
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