Free Bocconi Test Exam Flashcards

Difference of squares: a²−b²=(a−b)(a+b). Difference of cubes: a³−b³=(a−b)(a²+ab+b²). Sum of cubes: a³+b³=(a+b)(a²−ab+b²). Expand the products to check signs. The quadratic factors have no further real linear factorisation when the other variable is a fixed nonzero real number; special zero cases need separate treatment.

Roots: x = (−b ± √(b² − 4ac)) / (2a). Discriminant Δ = b² − 4ac: If Δ > 0, there are two distinct real roots (parabola crosses x-axis twice); if Δ = 0, there is exactly one real root of multiplicity 2 (parabola vertex touches x-axis); if Δ < 0, there are no real roots (parabola does not intersect x-axis).

If x₁ and x₂ are the roots: Sum: x₁ + x₂ = −b / a. Product: x₁ · x₂ = c / a. Useful application: Calculate symmetric expressions like x₁² + x₂² = (x₁ + x₂)² − 2x₁x₂ = (b²/a²) − 2(c/a) directly without computing individual square roots.

Fractional exponent: a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m (for a > 0). Power of a power: (a^m)^n = a^(m·n). Product of powers: a^m · a^n = a^(m+n). Critical absolute value rule: √(a²) = |a|, not simply a, because the principal square root is non-negative for every real number a.

For a > 0, the parabola opens upward. The quadratic expression ax² + bx + c > 0 is positive outside the roots: x < x₁ or x > x₂ (interval notation: (−∞, x₁) ∪ (x₂, +∞)). Conversely, ax² + bx + c < 0 is negative strictly between the roots: x₁ < x < x₂ (interval (x₁, x₂)).

Find all real zeros of numerator P(x) and denominator Q(x). Mark them as boundary points on a real line, partitioning it into test intervals. Determine the sign of each factor per interval. Essential rule: Exclude all points where Q(x) = 0 from the solution set (using strict open brackets), because division by zero is undefined, even when the inequality includes '≥ 0'.

For constant c > 0: 1. |f(x)| = c ⇔ f(x) = c or f(x) = −c. 2. |f(x)| ≤ c ⇔ −c ≤ f(x) ≤ c (intersection of intervals). 3. |f(x)| ≥ c ⇔ f(x) ≤ −c or f(x) ≥ c (union of intervals). If c < 0, |f(x)| ≤ c has no real solutions, and |f(x)| ≥ c is satisfied for all x in the domain.

Compute D=a₁b₂−a₂b₁. If D≠0, there is one solution. If D=0, compare the ranks of the coefficient and augmented matrices: equal ranks below two give infinitely many solutions; unequal ranks give none. Ratio shortcuts fail when coefficients are zero, so check equations directly, including contradictions such as 0=1.

A function f is invertible if and only if it is bijective (both injective / one-to-one and surjective / onto). Injective: f(x₁) = f(x₂) ⇒ x₁ = x₂ (passes the Horizontal Line Test). The inverse f⁻¹: B → A satisfies f⁻¹(f(x)) = x for all x ∈ A and f(f⁻¹(y)) = y for all y ∈ B. Geometrically, the graph of y = f⁻¹(x) is the reflection of y = f(x) across the line y = x.

(g ∘ f)(x) applies f first, then applies g to the output: g(f(x)). The composite function exists only for x in the domain of f such that f(x) lies within the domain of g: Domain(g ∘ f) = {x ∈ Domain(f) | f(x) ∈ Domain(g)}. Note that function composition is associative ((h ∘ g) ∘ f = h ∘ (g ∘ f)) but generally not commutative (g ∘ f ≠ f ∘ g).

For y=f(x)+c, shift vertically by c; for y=f(x−c), shift right by c. Multiplying f by a scales vertical distances by |a| and reflects across the x-axis if a<0. Replacing x with kx scales horizontal distances by 1/|k| and also reflects across the y-axis when k<0; assume k≠0.

1. Radical √(f(x)): Requires f(x) ≥ 0 (non-negative radicand). 2. Rational fraction g(x)/f(x): Requires f(x) ≠ 0 (non-zero denominator). 3. Logarithm log_b(f(x)): Requires f(x) > 0 (strictly positive argument), base b > 0, and b ≠ 1. For compound expressions, the total domain is the intersection of all simultaneous constraints.

If base a > 1: Both y = aˣ and y = log_a(x) are strictly increasing. Thus, aᵘ > aᵛ ⇔ u > v, and log_a(u) > log_a(v) ⇔ u > v. If base 0 < a < 1: Both functions are strictly decreasing. Therefore, taking powers or logarithms with base in (0, 1) reverses the inequality sign: aᵘ > aᵛ ⇔ u < v, and log_a(u) > log_a(v) ⇔ u < v.

For x, y > 0 and base b > 0, b ≠ 1: 1. log_b(xy) = log_b(x) + log_b(y). 2. log_b(x/y) = log_b(x) − log_b(y). 3. log_b(xᵏ) = k·log_b(x). 4. Change of base formula: log_b(x) = log_c(x) / log_c(b) = ln(x) / ln(b). In particular, log_b(a) = 1 / log_a(b).

A necessary assumption is a premise the argument needs. Negate its exact quantifiers and test whether the reasoning still supports the conclusion. For example, a plan to reach some applicants solely online requires at least one intended applicant to have online access; its negation is that none can access that channel.

From all A being B and all B being C, every A must be C. The converse need not hold: some C may not be A. Universal statements alone do not establish that any A exists. A conclusion beginning with some therefore needs an existence premise, such as at least one A.

Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²), derived directly from the Pythagorean theorem. Midpoint M coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). The perpendicular distance from a point (x₀, y₀) to a line Ax + By + C = 0 is d = |Ax₀ + By₀ + C| / √(A² + B²).

Parallel lines: Have identical slopes m₁ = m₂ (and different y-intercepts q₁ ≠ q₂). Perpendicular lines: Slopes are negative reciprocals: m₁ · m₂ = −1, or m₂ = −1/m₁. A horizontal line (slope m = 0, equation y = k) is perpendicular to a vertical line (undefined slope, equation x = c).

Standard form: (x − h)² + (y − k)² = r². From general expanded form x² + y² + ax + by + c = 0: Center (h, k) = (−a/2, −b/2). Radius r = √((a/2)² + (b/2)² − c). A real circle exists if and only if the radicand (a/2)² + (b/2)² − c > 0.

Vertex coordinates: V = (−b / (2a), −Δ / (4a)), where Δ = b² − 4ac. Alternatively, evaluate the x-coordinate x_v = −b / (2a) and substitute into y = f(x_v). Axis of symmetry: Vertical line x = −b / (2a). If a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and is a maximum.

Fundamental identity: sin²(θ) + cos²(θ) = 1 for all real angles θ. Dividing by cos²(θ) yields tan²(θ) + 1 = sec²(θ) (for cos θ ≠ 0). Dividing by sin²(θ) yields 1 + cot²(θ) = csc²(θ) (for sin θ ≠ 0). Definitions: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ), sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ).

Some means at least one; it does not logically exclude all. Not all means at least one exception. No A are B excludes every overlap, whereas some A are not B establishes only a nonempty outside subset. Translate these quantities before deciding whether a proposed conclusion is entailed, contradicted, or undetermined.

Permutations (order matters): Ordered selections of k items from n distinct items: P(n, k) = n! / (n − k)!. Combinations (order does NOT matter): Unordered subsets of k items chosen from n distinct items: C(n, k) = (n choose k) = n! / [k!(n − k)!]. Relation: P(n, k) = k! · C(n, k).

An inclusive or permits either or both alternatives. Exactly one excludes both together. From A or B and not A, infer B; from A or B and A, nothing follows about B unless exclusivity is stated. Follow the wording of the passage rather than assuming everyday either/or always means exclusive choice.

General Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtracting P(A ∩ B) prevents double-counting the overlap. For mutually exclusive (disjoint) events where A ∩ B = ∅, P(A ∩ B) = 0, so the formula simplifies to P(A ∪ B) = P(A) + P(B).

For P(B)>0, conditional probability is P(A|B)=P(A∩B)/P(B), so P(A∩B)=P(B)P(A|B). The symmetric formula uses P(A)>0. Independence means P(A∩B)=P(A)P(B), including zero-probability cases; when P(B)>0, this is equivalent to P(A|B)=P(A). Independence and mutual exclusivity are different conditions.

Bayes' Theorem: P(A_i|B) = [P(B|A_i) · P(A_i)] / P(B). By the Law of Total Probability across a partition of mutually exclusive, exhaustive events A₁,...,A_n: P(B) = ∑ [P(B|A_j) · P(A_j)]. Thus: P(A_i|B) = [P(B|A_i) · P(A_i)] / [∑ P(B|A_j)·P(A_j)]. It updates prior probability P(A_i) to posterior probability P(A_i|B) after observing evidence B.

Measures of central tendency shift and scale: Mean(ax + b) = a·Mean(x) + b; Median(ax + b) = a·Median(x) + b. Measures of dispersion scale by magnitude but are completely invariant to constant shifts b: Standard Deviation(ax + b) = |a|·SD(x); Variance(ax + b) = a²·Var(x). Adding a constant shifts the distribution without altering its spread.

Synthesize the author's primary argument across the passage as a whole rather than choosing a factually true but secondary supporting detail. Key strategy: Focus on the introductory thesis hook, topic sentences of body paragraphs, and transition pivots (however, yet, nonetheless) where the author introduces their perspective against conventional views.

Matching words do not establish matching meaning. An option can repeat the passage while reversing a relationship, changing a condition, or attributing a claim to the wrong person. Equally, a correct option may quote the wording exactly. Check the whole proposition against the relevant sentences; do not reward either quotation or paraphrase automatically.

For a question asking what must follow, select only an inference entailed by the passage without added assumptions. For a question asking what is best supported, match that stated evidentiary standard. In either case, reject outside facts, unsupported generalisations, and causal claims stronger than the text permits.

Locate the author’s own evaluations, modal verbs, and responses to competing arguments. Tone may be cautious, supportive, sceptical, or strongly critical when the language supports it. Do not select a moderate answer merely because it sounds academic, or reject a strong answer solely for intensity; the passage must determine the judgement.

Never select the most common dictionary definition without testing context. Substitute each answer option directly into the original sentence in place of the target word. The correct choice must preserve the author's specific syntactic role, technical nuance, and tone within the surrounding economic or logical argument.

Contrast markers signal a difference between claims, not necessarily the author’s final thesis. Concessions acknowledge a point without necessarily rejecting it later. Causal markers present a claimed consequence, whose logical force depends on the premises. Map what each clause contributes before choosing a conclusion; connective vocabulary alone cannot establish that an argument is valid.

Ask 'Why did the author include this paragraph?' rather than 'What does it say?' Common paragraph roles: 1. Introducing a puzzle or conventional economic model; 2. Presenting empirical counter-evidence; 3. Qualifying an earlier claim; 4. Reconciling conflicting viewpoints; 5. Outlining policy implications.

Separate amounts, rates, and the bases to which rates apply. A larger percentage can represent a smaller quantity when its base is smaller. Growth slowing from positive 9% to positive 4% still means expansion; a falling rate that crosses below zero means contraction. Inspect the actual numbers before inferring whether output rose or fell.

Track attribution framing: 'Proponents argue...', 'Traditional models assume...', or 'Critics contend...' introduce external viewpoints. The author's view is signalled by subsequent evaluative framing: 'However, empirical findings show...', 'This assumption overlooks...', or 'More compellingly...'.

The standard interface shows three questions per screen and does not allow returning after Next. The overall budget averages 90 seconds per item across 75 minutes, but this is a planning average, not a per-screen timer or required reading protocol. Use practice to vary time by difficulty while checking each screen before advancing.

Words such as always, never, and solely make strong claims that require correspondingly strong textual support. They are not automatic distractors: a passage may explicitly establish an absolute rule. Compare quantifiers and exceptions precisely. A moderate-sounding alternative can also be wrong if it weakens, shifts, or contradicts what the author actually says.

Percentage Change = [(New Value − Old Value) / Old Value] × 100%. Base Reversal Trap: If a quantity drops by 20%, it requires a 25% increase to return to the original level, NOT 20%. Calculation: 100 × 0.80 = 80; to reach 100 from 80 requires +20/80 = +25%. The denominator must always be the original baseline value.

Profit margin=(price−cost)/price; markup=(price−cost)/cost, each multiplied by 100%. With positive price and nonnegative cost, margin cannot exceed 100%; markup may exceed it. For cost100 and price125, profit25 gives margin20% and markup25%. Check the specified base and assumptions rather than treating the two percentages as interchangeable.

If Entity A completes a task in T_A hours and Entity B in T_B hours, their combined rate is R_comb = (1/T_A) + (1/T_B) = (T_A + T_B) / (T_A · T_B) tasks/hour. The combined completion time is the reciprocal: T_comb = (T_A · T_B) / (T_A + T_B). Example: Rates of 3 hours and 6 hours combine to (3 × 6)/(3 + 6) = 18/9 = 2 hours.

Weighted Average: X_w = (n₁·x₁ + n₂·x₂) / (n₁ + n₂). Alligation Lever Rule: The ratio of group sizes is inversely proportional to their distances from the combined mean: n₁ / n₂ = (x₂ − X_w) / (X_w − x₁). Example: Mixing a 10% solution and a 25% solution to achieve 15% requires: n₁/n₂ = (25 − 15)/(15 − 10) = 10/5 = 2:1.

1. Orient: Read the chart title, axis labels, units (e.g., thousands vs millions), and legend. 2. Filter: Identify exactly what the question asks and ignore superfluous columns/series. 3. Round: Round numbers to 2–3 significant figures to estimate quickly. 4. Check options: Check option spacing before calculating; wide spacing permits aggressive estimation.

Final Value = Initial Value × ∏(1 + r_t). For successive percentage changes +p% and +q%, the net multiplier is (1 + p/100)(1 + q/100) = 1 + (p + q + pq/100)/100. Net change = p + q + (p·q)/100 %. Example: A +20% gain followed by a −10% loss yields: 20 − 10 + (20 × −10)/100 = 10 − 2 = +8% net change (multiplier 1.20 × 0.90 = 1.08).

True: Follows necessarily and conclusively from the provided text alone without additional assumptions. False: Directly contradicted, refuted, or made logically impossible by the text. Cannot Be Deduced: The text lacks sufficient evidence to prove or disprove the statement conclusively. Outside facts or real-world plausibility must never be used.

Given conditional P → Q: Modus Ponens (valid): P is affirmed, therefore Q necessarily follows. Fallacy of Affirming the Consequent (invalid): Q is observed, and one concludes P must have occurred. Example: 'If it rains (P), the pitch gets wet (Q).' Given the pitch is wet (Q), concluding it rained is a fallacy (sprinklers could have caused it).

Given conditional P → Q: Modus Tollens (valid): ¬Q (not Q) is true, therefore ¬P (not P) necessarily follows (the contrapositive ¬Q → ¬P is logically equivalent). Fallacy of Denying the Antecedent (invalid): ¬P is observed, and one concludes ¬Q must be true. Example: 'If it rains, pitch gets wet.' It did not rain (¬P); concluding the pitch is not wet is invalid.

An assumption-negation test examines whether a proposed premise is necessary for the argument’s support. Negate the exact proposition, not an exaggerated alternative. If the reasoning no longer justifies the conclusion, the premise may be necessary. This does not mean the conclusion must become factually false; a conclusion can remain true for other reasons.

For a guess with probability p of being correct and wrong-answer penalty c, expected points are p−c(1−p), positive when p>c/(1+c). Uniform blind guessing gives +0.04 for five options at −0.20 and about +0.113 for three at −0.33. These are expected-score calculations, not guaranteed gains or an official requirement to guess.