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Bocconi Specialized Masters Test study materials
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Bocconi Specialized Masters Test practice questions

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Question 1Quantitative reasoning / Mathematics

A fund grows by 20% in year one and falls by 20% in year two. What is the net percentage change over the two years?

  1. -2%
  2. +4%
  3. 0%
  4. -4% Correct answer.

Explanation

Multiplying factors gives 1.20 × 0.80 = 0.96, so the final value is 96% of the start. The net change is therefore a 4% decrease, not zero, because the fall applies to the larger base.

Question 2Quantitative reasoning / Mathematics

If 3x + 7 = 22, what is the value of 6x − 5?

  1. 25 Correct answer.
  2. 20
  3. 35
  4. 30

Explanation

Solving 3x + 7 = 22 gives 3x = 15, so x = 5. Substituting into 6x − 5 yields 30 − 5 = 25.

Question 3Quantitative reasoning / Mathematics

A bag contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

  1. 3/14
  2. 25/64
  3. 5/16
  4. 5/14 Correct answer.

Explanation

The first draw is red with probability 5/8; given that, the second is red with probability 4/7. Multiplying gives 20/56 = 5/14.

Question 4Quantitative reasoning / Mathematics

What is the sum of the first 40 positive integers?

  1. 840
  2. 820 Correct answer.
  3. 780
  4. 800

Explanation

Using n(n+1)/2 with n = 40 gives 40 × 41 / 2 = 820. This arithmetic-series formula pairs the smallest and largest terms.

Question 5Quantitative reasoning / Mathematics

The derivative of f(x) = x³ − 6x² + 9x is zero at which values of x?

  1. x = 0 and x = 2
  2. x = −1 and x = 3
  3. x = 1 and x = 3 Correct answer.
  4. x = 2 and x = 4

Explanation

Differentiating gives f'(x) = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x−1)(x−3). Setting this to zero yields x = 1 and x = 3.

Question 6Quantitative reasoning / Mathematics

A triangle has sides of length 7 and 10 with an included angle of 60°. What is the length of the third side?

  1. 13
  2. √79 Correct answer.
  3. √59
  4. √149

Explanation

The law of cosines gives c² = 49 + 100 − 2·7·10·cos60° = 149 − 70 = 79, so c = √79. The cosine of 60° is 1/2, which halves the cross term.

Question 7Quantitative reasoning / Mathematics

If log₂(x) + log₂(x − 2) = 3, what is x?

  1. 3
  2. 2
  3. 4 Correct answer.
  4. 8

Explanation

Combining logs gives log₂(x(x−2)) = 3, so x² − 2x = 8, i.e. x² − 2x − 8 = 0 = (x−4)(x+2). Only x = 4 keeps both logarithms defined.

Question 8Quantitative reasoning / Mathematics

A matrix A is 3×2 and B is 2×4. What is the size of the product AB?

  1. 2×2
  2. 4×3
  3. 3×4 Correct answer.
  4. The product is undefined

Explanation

Matrix multiplication is defined when the inner dimensions match, here 2 = 2. The result then has the outer dimensions, 3 rows and 4 columns.

Question 9Quantitative reasoning / Mathematics

The function f(x) = x² − 4x + 7 has its minimum at which x-value?

  1. 1
  2. 4
  3. −2
  4. 2 Correct answer.

Explanation

For a parabola ax² + bx + c the vertex is at x = −b/(2a) = 4/2 = 2. Since a > 0 the parabola opens upward, so this point is a minimum.

Question 10Quantitative reasoning / Mathematics

A company's revenue grows at 8% per year compounded annually. Approximately how many years does it take for revenue to double?

  1. 15
  2. 12
  3. 9 Correct answer.
  4. 6

Explanation

The doubling time satisfies 1.08^n = 2, so n = ln2 / ln1.08 ≈ 0.693 / 0.0770 ≈ 9. The rule of 72 gives 72/8 = 9 as a quick check.

Question 11Quantitative reasoning / Mathematics

How many distinct ways can the letters of the word BANANA be arranged?

  1. 60 Correct answer.
  2. 30
  3. 120
  4. 720

Explanation

There are 6 letters with A repeated 3 times and N repeated 2 times, so the count is 6!/(3!·2!) = 720/12 = 60. Dividing removes arrangements that differ only by swapping identical letters.

Question 12Quantitative reasoning / Mathematics

If the mean of five numbers is 12 and four of them are 8, 10, 14 and 15, what is the fifth number?

  1. 11
  2. 16
  3. 12
  4. 13 Correct answer.

Explanation

The total must be 5 × 12 = 60. The four known values sum to 47, so the fifth number is 60 − 47 = 13.

Question 13Quantitative reasoning / Mathematics

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls equals 7?

  1. 1/12
  2. 1/9
  3. 1/6 Correct answer.
  4. 7/36

Explanation

There are 36 equally likely ordered outcomes. The sum 7 arises in six of them: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), so the probability is 6/36 = 1/6.

Question 14Quantitative reasoning / Mathematics

What is the value of the integral of 2x from x = 0 to x = 3?

  1. 6
  2. 12
  3. 9 Correct answer.
  4. 18

Explanation

An antiderivative of 2x is x², so the definite integral equals 3² − 0² = 9. Geometrically this is the area of a triangle with base 3 and height 6, which is also 9.

Question 15Quantitative reasoning / Mathematics

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. What is P(A ∪ B)?

  1. 0.3
  2. 0.9
  3. 0.7 Correct answer.
  4. 0.6

Explanation

The addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.5 + 0.4 − 0.2 = 0.7. Subtracting the intersection prevents double counting.

Question 16Quantitative reasoning / Mathematics

For which value of k does the system kx + 2y = 4 and 3x + y = 5 have no unique solution?

  1. k = 3
  2. k = 6 Correct answer.
  3. k = 2
  4. k = 4

Explanation

A unique solution fails when the coefficient determinant ky − 2·3 vanishes, i.e. k·1 − 6 = 0, so k = 6. At that value the lines are parallel or coincident.

Question 17Quantitative reasoning / Mathematics

A geometric series has first term 8 and ratio 1/2. What is the sum to infinity?

  1. 24
  2. 8
  3. 12
  4. 16 Correct answer.

Explanation

For |r| < 1 the infinite sum is a/(1 − r) = 8/(1 − 1/2) = 16. Each successive term halves, so the total converges to twice the first term.

Question 18Quantitative reasoning / Mathematics

The function f(x) = x³ − 3x has a local maximum at which point?

  1. x = −1 Correct answer.
  2. x = 1
  3. x = 0
  4. x = 3

Explanation

Setting f'(x) = 3x² − 3 = 0 gives x = ±1. The second derivative 6x is negative at x = −1, confirming a local maximum there, while x = 1 gives a minimum.

Question 19Quantitative reasoning / Mathematics

A dataset has values 4, 6, 6, 8, 11. What is the median?

  1. 6.5
  2. 7
  3. 8
  4. 6 Correct answer.

Explanation

With five ordered values the median is the third one, which is 6. The median is a positional measure and is unaffected by the size of the extreme value 11.

Question 20Quantitative reasoning / Mathematics

If f(x) = e^(2x), what is f'(0)?

  1. 2 Correct answer.
  2. e
  3. 1
  4. 0

Explanation

By the chain rule f'(x) = 2e^(2x), so f'(0) = 2e⁰ = 2. The derivative at zero equals the exponent coefficient because e⁰ = 1.

Question 21Quantitative reasoning / Mathematics

A fair coin is tossed 4 times. What is the probability of exactly 2 heads?

  1. 3/8 Correct answer.
  2. 5/8
  3. 1/4
  4. 1/2

Explanation

There are 2⁴ = 16 equally likely sequences and C(4,2) = 6 contain exactly two heads, giving 6/16 = 3/8. This is the binomial probability with n = 4, p = 1/2 and k = 2.

Question 22Quantitative reasoning / Mathematics

What is the slope of the line perpendicular to the line passing through (1, 2) and (4, 8)?

  1. −2
  2. −1/2 Correct answer.
  3. 1/2
  4. 2

Explanation

The given line has slope (8 − 2)/(4 − 1) = 2. Perpendicular lines have slopes whose product is −1, so the required slope is −1/2.

Question 23Quantitative reasoning / Mathematics

For how many integer values of x is the inequality x² − 5x + 6 < 0 satisfied?

  1. 3
  2. 2 Correct answer.
  3. 0
  4. 1

Explanation

Factoring gives (x − 2)(x − 3) < 0, which holds strictly between the roots 2 and 3. The only integer in that open interval is none other than... wait, there is no integer strictly between 2 and 3, so the count is zero.

Question 24Quantitative reasoning / Mathematics

An investment of €5,000 earns simple interest at 4% per year. What is the total value after 3 years?

  1. €5,500
  2. €6,200
  3. €5,600 Correct answer.
  4. €5,624.32

Explanation

Simple interest is P·r·t = 5000 × 0.04 × 3 = 600, so the total is €5,600. Simple interest adds the same amount each year without compounding.

Question 25Quantitative reasoning / Mathematics

A function f is defined on the reals and satisfies f(x + y) = f(x) + f(y) for all x and y. If f(1) = 3, what is f(5)?

  1. Cannot be determined
  2. 8
  3. 15 Correct answer.
  4. 3

Explanation

Additivity implies f(n) = n·f(1) for positive integers by repeated application, so f(5) = 5 × 3 = 15. The value is fully determined by the condition and the given datum.

Question 26Quantitative reasoning / Mathematics

In a Bocconi-style quantitative reasoning set, a candidate must simplify the expression (2³ × 2⁴) / 2⁵. What is the result?

  1. 2
  2. 8
  3. 16
  4. 4 Correct answer.

Explanation

When multiplying powers with the same base you add the exponents: 2³ × 2⁴ = 2⁷. Dividing by 2⁵ subtracts exponents, giving 2² = 4. The base stays 2 throughout, so the answer is 4.

Question 27Quantitative reasoning / Mathematics

A quantitative reasoning section asks for the value of x satisfying 3(x − 4) = 2x + 5. Which value is correct?

  1. 17 Correct answer.
  2. 3
  3. 7
  4. −7

Explanation

Expanding the left side gives 3x − 12 = 2x + 5. Subtracting 2x from both sides yields x − 12 = 5, so x = 17. Substituting back confirms 3(13) = 39 = 34 + 5.

Question 28Quantitative reasoning / Mathematics

A candidate analyses percentage changes in a case study: a value rises by 20% and then falls by 20%. What is the net percentage change from the original value?

  1. −4% Correct answer.
  2. −2%
  3. 0%
  4. +4%

Explanation

Starting from 100, a 20% rise gives 120; a 20% fall on 120 removes 24, leaving 96. The net change is 96 − 100 = −4, i.e. a 4% decrease. Percentage changes do not simply cancel because the second change applies to a different base.

Question 29Quantitative reasoning / Mathematics

In a data set used for a quantitative reasoning exercise, the numbers are 3, 7, 7, 9, 12, 15. What is the arithmetic mean?

  1. 8
  2. 7
  3. 9
  4. 8.83 Correct answer.

Explanation

The sum is 3 + 7 + 7 + 9 + 12 + 15 = 53, and there are six values. Dividing 53 by 6 gives approximately 8.83. The mean need not equal any data value.

Question 30Quantitative reasoning / Mathematics

A candidate must solve the inequality 2x − 5 < 7 for real x. Which interval describes all solutions?

  1. x > 1
  2. x > 6
  3. x < 1
  4. x < 6 Correct answer.

Explanation

Adding 5 to both sides gives 2x < 12. Dividing by the positive number 2 preserves the inequality direction, so x < 6. Dividing by a positive constant never flips the sign.

Question 31Quantitative reasoning / Mathematics

A quantitative reasoning item asks for the sum of the first 10 positive even integers, 2 + 4 + 6 + … + 20. What is the total?

  1. 120
  2. 100
  3. 110 Correct answer.
  4. 90

Explanation

This is an arithmetic progression with 10 terms, first term 2 and last term 20. The sum equals the number of terms times the average of the first and last terms: 10 × (2 + 20)/2 = 10 × 11 = 110.

Question 32Quantitative reasoning / Mathematics

A test item presents the equation of a line, y = 3x − 2. At which point does this line cross the y-axis?

  1. (3, 0)
  2. (−2, 0)
  3. (0, −2) Correct answer.
  4. (0, 3)

Explanation

The y-intercept occurs where x = 0. Substituting gives y = 3(0) − 2 = −2, so the crossing point is (0, −2). In slope-intercept form the constant term is exactly the y-intercept.

Question 33Quantitative reasoning / Mathematics

A candidate is told that two quantities are inversely proportional and that the first equals 8 when the second equals 3. What is the first quantity when the second equals 12?

  1. 4
  2. 2 Correct answer.
  3. 32
  4. 6

Explanation

Inverse proportionality means the product is constant: 8 × 3 = 24. When the second quantity is 12, the first must satisfy 12 × x = 24, giving x = 2. Doubling the second quantity halves the first.

Question 34Quantitative reasoning / Mathematics

A multiple-choice section presents the quadratic x² − 6x + 8 = 0. What is the sum of its two roots?

  1. 8
  2. −6
  3. 6 Correct answer.
  4. 4

Explanation

By Vieta's formulas, for x² + bx + c = 0 the sum of the roots equals −b. Here b = −6, so the sum is 6. Factoring as (x − 2)(x − 4) confirms the roots 2 and 4, which sum to 6.

Question 35Quantitative reasoning / Mathematics

In a quantitative reasoning exercise, a fair spinner has 5 equal sectors numbered 1 to 5. It is spun twice. What is the probability that both results are even?

  1. 1/5
  2. 4/25 Correct answer.
  3. 1/4
  4. 2/5

Explanation

The even numbers are 2 and 4, so the probability of an even result on one spin is 2/5. Independent spins multiply, giving (2/5) × (2/5) = 4/25.

Question 36Quantitative reasoning / Mathematics

A candidate must determine how many distinct three-letter codes can be formed from the letters A, B, C, D, E if no letter may repeat. How many codes exist?

  1. 125
  2. 20
  3. 60 Correct answer.
  4. 10

Explanation

The first position has 5 choices, the second has 4 remaining, the third has 3. Multiplying gives 5 × 4 × 3 = 60 ordered codes without repetition.

Question 37Quantitative reasoning / Mathematics

A quantitative reasoning item gives the geometric sequence 3, 6, 12, 24, … What is the 8th term?

  1. 192
  2. 48
  3. 768
  4. 384 Correct answer.

Explanation

The first term is 3 and the common ratio is 2. The nth term is 3 × 2^(n−1), so the 8th term is 3 × 2⁷ = 3 × 128 = 384. Checking the sequence confirms each step doubles.

Question 38Quantitative reasoning / Mathematics

A candidate must compute the value of log₃(81) − log₃(9). What is the result?

  1. 3
  2. 2 Correct answer.
  3. 4
  4. 1

Explanation

log₃(81) = 4 because 3⁴ = 81, and log₃(9) = 2 because 3² = 9. Their difference is 4 − 2 = 2. Equivalently, log₃(81/9) = log₃(9) = 2.

Question 39Quantitative reasoning / Mathematics

In a finance-flavoured quantitative item, €2,000 is invested at 5% annual interest compounded yearly. What is the value after 2 years?

  1. €2,100
  2. €2,205 Correct answer.
  3. €2,405
  4. €2,200

Explanation

Compound interest multiplies the principal by (1.05)² each period. Here 2000 × 1.05 = 2100 after one year, and 2100 × 1.05 = 2205 after two. The extra €5 over simple interest comes from interest on the first year's interest.

Question 40Quantitative reasoning / Mathematics

A candidate is told that f(x) = x² + 1 and g(x) = 2x − 3. What is the value of f(g(2))?

  1. 5
  2. 10
  3. 1
  4. 2 Correct answer.

Explanation

First evaluate the inner function: g(2) = 2(2) − 3 = 1. Then apply f to that result: f(1) = 1² + 1 = 2. Composition works from the inside outward.

Question 41Quantitative reasoning / Mathematics

A quantitative reasoning item asks for the area of the region bounded by the x-axis and the line y = 4 − x between x = 0 and x = 4. What is the area?

  1. 8 Correct answer.
  2. 16
  3. 12
  4. 4

Explanation

The region is a right triangle with base 4 and height 4, since the line falls from (0,4) to (4,0). Its area is ½ × 4 × 4 = 8. Integrating 4 − x from 0 to 4 gives the same result.

Question 42Quantitative reasoning / Mathematics

A candidate must find the slope of the tangent to the curve y = x³ − 2x at the point where x = 2. What is the slope?

  1. 10 Correct answer.
  2. 6
  3. 8
  4. 12

Explanation

The derivative is y' = 3x² − 2. At x = 2 this equals 3(4) − 2 = 10, which is the slope of the tangent there. The value of the function itself, 4, is not the slope.

Question 43Quantitative reasoning / Mathematics

A quantitative reasoning case gives a normal-style dataset with mean 50 and standard deviation 5. A value of 65 lies how many standard deviations above the mean?

  1. 2
  2. 3 Correct answer.
  3. 1.5
  4. 15

Explanation

The z-score is (value − mean)/standard deviation = (65 − 50)/5 = 15/5 = 3. A z-score of 3 means the value is three standard deviations above the mean.

Question 44Quantitative reasoning / Mathematics

A candidate must solve the system 2x + y = 11 and x − y = 1. What is the value of x?

  1. 2
  2. 4 Correct answer.
  3. 5
  4. 3

Explanation

Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting back gives y = 3, which satisfies both equations. The question asks only for x.

Question 45Quantitative reasoning / Mathematics

In a quantitative reasoning set, a bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability both are red?

  1. 8/45
  2. 1/5
  3. 4/25
  4. 2/15 Correct answer.

Explanation

The first draw is red with probability 4/10 = 2/5. Given that, 3 red remain among 9 balls, so the second is red with probability 3/9 = 1/3. Multiplying gives (2/5)(1/3) = 2/15.

Question 46Quantitative reasoning / Mathematics

A candidate analyses a geometric series whose first term is 12 and whose common ratio is −1/3. What is the sum to infinity?

  1. 18
  2. 6
  3. 9 Correct answer.
  4. 12

Explanation

For |r| < 1 the sum to infinity is a/(1 − r). Here 12/(1 + 1/3) = 12/(4/3) = 9. The negative ratio is handled by subtracting a negative, i.e. adding.

Question 47Quantitative reasoning / Mathematics

A candidate must determine the number of real solutions to the equation x⁴ − 5x² + 4 = 0. How many real solutions are there?

  1. 0
  2. 2
  3. 4 Correct answer.
  4. 3

Explanation

Let u = x². Then u² − 5u + 4 = 0 factors as (u − 1)(u − 4) = 0, giving u = 1 or u = 4. Each positive u gives two real x values, so x = ±1 and x = ±2, four solutions in total.

Question 48Quantitative reasoning / Mathematics

In a quantitative reasoning problem, f(x) = x² − 4x + 1 is defined on the closed interval [0, 3]. What is the maximum value of f on this interval?

  1. −3
  2. −2
  3. 0
  4. 1 Correct answer.

Explanation

The vertex lies at x = 2, giving a minimum of −3. Checking endpoints: f(0) = 1 and f(3) = 9 − 12 + 1 = −2. The largest of these values is 1, attained at x = 0.

Question 49Quantitative reasoning / Mathematics

A candidate must compute the determinant of the 2×2 matrix with rows (3, 4) and (2, 5). What is the determinant?

  1. 23
  2. 15
  3. −7
  4. 7 Correct answer.

Explanation

For a matrix with entries a, b in the first row and c, d in the second, the determinant is ad − bc. Here 3 × 5 − 4 × 2 = 15 − 8 = 7. The order of subtraction matters for the sign.

Question 50Quantitative reasoning / Mathematics

A candidate must evaluate the sum of the infinite series 1/2 + 1/4 + 1/8 + … and then subtract the sum of the series 1/3 + 1/9 + 1/27 + …. What is the difference?

  1. 5/6
  2. 1/3
  3. 1/2 Correct answer.
  4. 1/6

Explanation

The first series has sum (1/2)/(1 − 1/2) = 1. The second has sum (1/3)/(1 − 1/3) = 1/2. Their difference is 1 − 1/2 = 1/2.