Bocconi Maths Drill: 50 Questions in 75 Minutes, With Worked Solutions

How to use this

The real test is 50 questions in 75 minutes, of which 24 are mathematics. This drill is 50 maths questions in the same 75 minutes, so it is deliberately harder per minute than the exam — the point is to build fluency in the maths section specifically, not to simulate the whole paper. For the full picture of format, scoring and timing, see the official Bocconi test pages and our guide to the Bocconi maths section.

• Set a timer for 75 minutes and do not stop it. Running out of time is information.
• Score it properly. +1 correct, 0 blank, −0.2 wrong. Guessing blind across five options has a slightly negative expected value; guessing with two options eliminated is clearly worth it.
• Mark every question you guessed, even the ones you got right. A lucky guess is a gap.
• Read the solution for everything you did not solve cleanly, not only for what you got wrong.

The 50 questions

Answers and full solutions follow. Try to resist scrolling.

Algebra

1. Solve for x: 3(x − 2) − 4(2x − 5) = 7

   • Ax = 7/5
   • Bx = 11/5
   • Cx = −1
   • Dx = 3
   • Ex = 1

2. For which values of x is x² − 5x + 6 < 0 ?

   • A−3 < x < −2
   • BAll real x
   • Cx < 2 or x > 3
   • D2 < x < 3
   • Ex < −3 or x > −2

3. Solve |2x − 5| = 9

   • Ax = 7 only
   • Bx = −2 only
   • Cx = 7 or x = −2
   • Dx = 2 or x = −7
   • ENo solution

4. If x + y = 10 and xy = 21, what is x² + y² ?

   • A100
   • B42
   • C121
   • D58
   • E79

5. Simplify: (x² − 9) / (x² − x − 6), for x ≠ 3, x ≠ −2

   • A(x + 3)/(x + 2)
   • B(x − 3)/(x − 2)
   • C(x + 3)/(x − 2)
   • D(x − 3)/(x + 2)
   • Ex + 3

6. Solve the system: 2x + 3y = 12 and x − y = 1

   • Ax = 1, y = 0
   • Bx = 5, y = 4
   • Cx = 3, y = 2
   • Dx = 2, y = 1
   • Ex = 4, y = 3

7. The sum of the roots of 2x² − 7x + 3 = 0 is

   • A3
   • B7
   • C7/2
   • D3/2
   • E−7/2

8. If 3^(x+1) = 81, then x =

   • A27
   • B2
   • C3
   • D4
   • E1

9. For which k does x² + kx + 9 = 0 have exactly one real solution?

   • Ak = ±3
   • Bk = 9
   • Ck = 3 only
   • Dk = 6 only
   • Ek = ±6

10. Solve: (x − 1)/(x + 2) ≥ 0

    • A−2 < x < 1
    • Bx > −2
    • Cx ≥ 1
    • Dx ≤ −2 or x ≥ 1
    • Ex < −2 or x ≥ 1

Functions

11. If f(x) = 2x − 3 and g(x) = x², what is f(g(2)) ?

    • A1
    • B5
    • C8
    • D−1
    • E4

12. What is the domain of f(x) = √(x − 4) ?

    • Ax ≠ 4
    • Bx > 4
    • Cx ≥ 4
    • Dx ≤ 4
    • EAll real x

13. The inverse of f(x) = (x + 5)/3 is

    • Af⁻¹(x) = 3x − 5
    • Bf⁻¹(x) = 3x + 5
    • Cf⁻¹(x) = (x − 5)/3
    • Df⁻¹(x) = 3/(x + 5)
    • Ef⁻¹(x) = x/3 − 5

14. For f(x) = x² − 6x + 5, the minimum value of f is

    • A0
    • B−4
    • C5
    • D3
    • E−3

15. If f(x) = 5 − 2x, for which x is f(x) > 1 ?

    • Ax > 3
    • Bx < 3
    • Cx < −2
    • Dx > 2
    • Ex < 2

Analytical geometry

16. The line through (1, 2) and (5, 10) has equation

    • Ay = 2x − 1
    • By = 2x
    • Cy = 2x + 1
    • Dy = x + 1
    • Ey = 3x − 1

17. The distance between (−1, 2) and (3, 5) is

    • A7
    • B√29
    • C4
    • D5
    • E√7

18. The centre and radius of x² + y² − 6x + 4y − 12 = 0 are

    • Acentre (−3, 2), r = 5
    • Bcentre (3, −2), r = 25
    • Ccentre (6, −4), r = 5
    • Dcentre (−3, 2), r = 12
    • Ecentre (3, −2), r = 5

19. The vertex of the parabola y = x² − 4x + 7 is

    • A(4, 7)
    • B(1, 4)
    • C(2, 3)
    • D(−2, 3)
    • E(2, −3)

20. A line perpendicular to y = −(1/3)x + 4 has gradient

    • A−3
    • B3
    • C1/3
    • D−1/3
    • E−4

21. Where does y = 2x − 4 cross the x-axis?

    • A(0, −4)
    • B(2, 0)
    • C(4, 0)
    • D(−2, 0)
    • E(0, 2)

Trigonometry

22. The value of sin(30°) + cos(60°) is

    • A1
    • B1/2
    • C√3/2
    • D0
    • E√2

23. If sin(θ) = 3/5 and θ is acute, then cos(θ) =

    • A5/3
    • B4/5
    • C3/4
    • D5/4
    • E−4/5

24. In a right-angled triangle the hypotenuse is 10 and one angle is 30°. The side opposite that angle is

    • A10√3
    • B20
    • C5/2
    • D5
    • E5√3

25. Solve 2sin(x) = 1 for 0° ≤ x < 360°

    • A60° and 120°
    • B30° and 330°
    • C150° only
    • D30° only
    • E30° and 150°

Set theory

26. In a class of 30, 18 study French and 15 study German; 8 study both. How many study neither?

    • A5
    • B3
    • C7
    • D2
    • E0

27. If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A ∩ B has how many elements?

    • A4
    • B5
    • C1
    • D2
    • E3

28. A set has 5 elements. How many subsets does it have?

    • A10
    • B25
    • C32
    • D31
    • E120

Logarithms

29. log₂(32) =

    • A6
    • B16
    • C2
    • D4
    • E5

30. If log(x) + log(4) = log(20), then x =

    • A4
    • B5
    • C16
    • D80
    • E24

31. Solve 2^x = 8^(x−2)

    • Ax = 6
    • Bx = 4
    • Cx = 1
    • Dx = 2
    • Ex = 3

32. The solution of e^(2x) = 1 is

    • Ax = e
    • Bx = 1/2
    • CNo solution
    • Dx = 0
    • Ex = 1

33. log₃(9) + log₃(3) =

    • A27
    • B12
    • C2
    • D3
    • E4

Combinatorics

34. How many ways can 5 people be arranged in a row?

    • A20
    • B720
    • C25
    • D60
    • E120

35. How many ways can a committee of 3 be chosen from 8 people?

    • A112
    • B40
    • C24
    • D56
    • E336

36. How many 3-digit numbers can be formed from the digits 1–5 if repetition is allowed?

    • A15
    • B60
    • C125
    • D10
    • E243

37. In how many ways can the letters of the word LEVEL be arranged?

    • A30
    • B20
    • C24
    • D120
    • E60

38. A pizza shop offers 4 bases and 6 toppings. How many one-topping pizzas are possible?

    • A46
    • B64
    • C10
    • D24
    • E12

Numbers and percentages

39. A price rises by 20% and then falls by 20%. Compared with the original it is

    • Aunchanged
    • B4% lower
    • C4% higher
    • D2% lower
    • E20% lower

40. What is 15% of 240?

    • A36
    • B24
    • C3.6
    • D16
    • E40

41. After a 25% discount an item costs €90. What was the original price?

    • A€120
    • B€108
    • C€135
    • D€112.50
    • E€115

42. If a : b = 3 : 4 and b : c = 2 : 5, then a : c is

    • A6 : 20
    • B3 : 5
    • C8 : 15
    • D3 : 20
    • E3 : 10

43. A sum of €5,000 earns 4% simple interest per year. After 3 years the interest earned is

    • A€500
    • B€624.32
    • C€200
    • D€600
    • E€620

Probability

44. Two fair dice are rolled. What is the probability the total is 7?

    • A1/9
    • B5/36
    • C1/6
    • D1/12
    • E7/36

45. A bag holds 4 red and 6 blue balls. Two are drawn without replacement. P(both red) =

    • A2/5
    • B6/45
    • C2/15
    • D4/25
    • E1/6

46. P(A) = 0.6, P(B) = 0.5 and A and B are independent. P(A ∪ B) =

    • A0.3
    • B0.9
    • C0.7
    • D1.1
    • E0.8

47. A test is 90% accurate on the 5% of people who have a condition and gives a false positive 10% of the time. Given a positive result, the probability of actually having the condition is closest to

    • A90%
    • B32%
    • C50%
    • D5%
    • E10%

Statistics

48. The mean of 4, 7, 9, 12, 18 is

    • A10
    • B9.5
    • C11
    • D8
    • E9

49. The median of 3, 8, 2, 10, 5, 7 is

    • A5
    • B6
    • C7
    • D5.5
    • E8

50. Adding a constant of 5 to every value in a data set will

    • Aincrease the mean and leave the standard deviation unchanged
    • Bleave the mean unchanged and increase the standard deviation
    • Cleave both unchanged
    • Dincrease the mean and halve the standard deviation
    • Eincrease the mean and the standard deviation

Answer key

Score: (number correct) − 0.2 × (number wrong). Blanks count zero.

Worked solutions

1. 1. A — x = 7/5

   Expand: 3x − 6 − 8x + 20 = 7, so −5x + 14 = 7. Then −5x = −7 and x = 7/5.

2. 2. D — 2 < x < 3

   Factorise: (x − 2)(x − 3) < 0. An upward parabola is negative strictly between its roots, so 2 < x < 3.

3. 3. C — x = 7 or x = −2

   Either 2x − 5 = 9, giving x = 7, or 2x − 5 = −9, giving 2x = −4 and x = −2.

4. 4. D — 58

   x² + y² = (x + y)² − 2xy = 100 − 42 = 58.

5. 5. A — (x + 3)/(x + 2)

   Numerator (x − 3)(x + 3); denominator (x − 3)(x + 2). Cancel (x − 3) to get (x + 3)/(x + 2).

6. 6. C — x = 3, y = 2

   From the second equation x = y + 1. Substituting: 2(y + 1) + 3y = 12, so 5y = 10, y = 2 and x = 3.

7. 7. C — 7/2

   For ax² + bx + c = 0 the sum of the roots is −b/a = 7/2. (The roots are 3 and 1/2.)

8. 8. C — 3

   81 = 3⁴, so x + 1 = 4 and x = 3.

9. 9. E — k = ±6

   One real solution means the discriminant is zero: k² − 36 = 0, so k = 6 or k = −6.

10. 10. E — x < −2 or x ≥ 1

    The quotient is zero at x = 1 and undefined at x = −2. It is positive when numerator and denominator share a sign: x < −2 or x > 1. Include x = 1 (value 0) but never x = −2.

11. 11. B — 5

    g(2) = 4, then f(4) = 2·4 − 3 = 5.

12. 12. C — x ≥ 4

    A square root needs a non-negative argument: x − 4 ≥ 0, so x ≥ 4.

13. 13. A — f⁻¹(x) = 3x − 5

    Set y = (x + 5)/3, so 3y = x + 5 and x = 3y − 5. Swapping names, f⁻¹(x) = 3x − 5.

14. 14. B — −4

    The vertex is at x = −b/(2a) = 3, and f(3) = 9 − 18 + 5 = −4.

15. 15. E — x < 2

    5 − 2x > 1 gives −2x > −4. Dividing by −2 reverses the inequality: x < 2.

16. 16. B — y = 2x

    Gradient = (10 − 2)/(5 − 1) = 2. Then y − 2 = 2(x − 1) gives y = 2x.

17. 17. D — 5

    Δx = 4, Δy = 3, so the distance is √(16 + 9) = √25 = 5.

18. 18. E — centre (3, −2), r = 5

    Complete the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5.

19. 19. C — (2, 3)

    x = −b/(2a) = 4/2 = 2, and y = 4 − 8 + 7 = 3. Vertex (2, 3).

20. 20. B — 3

    Perpendicular gradients multiply to −1, so m = −1 ÷ (−1/3) = 3.

21. 21. B — (2, 0)

    Set y = 0: 2x − 4 = 0, so x = 2. The point is (2, 0).

22. 22. A — 1

    sin 30° = 1/2 and cos 60° = 1/2, so the sum is 1.

23. 23. B — 4/5

    cos²θ = 1 − 9/25 = 16/25. As θ is acute, cosθ = 4/5.

24. 24. D — 5

    opposite = hypotenuse × sin 30° = 10 × 1/2 = 5.

25. 25. E — 30° and 150°

    sin x = 1/2. In the given range sine is positive in the first and second quadrants, giving x = 30° and x = 180° − 30° = 150°.

26. 26. A — 5

    |F ∪ G| = 18 + 15 − 8 = 25. So 30 − 25 = 5 study neither.

27. 27. D — 2

    The common elements are 3 and 4, so |A ∩ B| = 2.

28. 28. C — 32

    A set with n elements has 2ⁿ subsets, so 2⁵ = 32 (including the empty set and the set itself).

29. 29. E — 5

    2⁵ = 32, so log₂32 = 5.

30. 30. B — 5

    log(4x) = log(20), so 4x = 20 and x = 5.

31. 31. E — x = 3

    8 = 2³, so 2^x = 2^(3x−6). Then x = 3x − 6, giving 2x = 6 and x = 3.

32. 32. D — x = 0

    e^(2x) = 1 = e⁰, so 2x = 0 and x = 0.

33. 33. D — 3

    log₃9 = 2 and log₃3 = 1, so the sum is 3. (Equivalently log₃27 = 3.)

34. 34. E — 120

    5! = 5 × 4 × 3 × 2 × 1 = 120.

35. 35. D — 56

    Order does not matter, so C(8,3) = 8!/(3!·5!) = 56.

36. 36. C — 125

    Each of the three positions has 5 choices: 5³ = 125.

37. 37. A — 30

    Five letters with L repeated twice and E repeated twice: 5!/(2!·2!) = 120/4 = 30.

38. 38. D — 24

    Independent choices multiply: 4 × 6 = 24.

39. 39. B — 4% lower

    Take 100. After +20% it is 120; after −20% it is 120 × 0.8 = 96. That is 4% below the original.

40. 40. A — 36

    0.15 × 240 = 36.

41. 41. A — €120

    90 is 75% of the original, so the original is 90 ÷ 0.75 = 120.

42. 42. E — 3 : 10

    Make b common: a : b = 3 : 4 = 6 : 8 and b : c = 2 : 5 = 8 : 20. So a : c = 6 : 20 = 3 : 10.

43. 43. D — €600

    Simple interest = 5000 × 0.04 × 3 = 600.

44. 44. C — 1/6

    Six of the 36 equally likely outcomes total 7, so the probability is 6/36 = 1/6.

45. 45. C — 2/15

    P = (4/10) × (3/9) = 12/90 = 2/15.

46. 46. E — 0.8

    For independent events P(A ∩ B) = 0.3, so P(A ∪ B) = 0.6 + 0.5 − 0.3 = 0.8.

47. 47. B — 32%

    By Bayes: P = (0.05·0.90)/(0.05·0.90 + 0.95·0.10) = 0.045/0.140 ≈ 0.321, so about 32%. This is the classic result that a rare condition keeps the posterior low even after a positive test.

48. 48. A — 10

    The sum is 50 and there are 5 values, so the mean is 10.

49. 49. B — 6

    Ordered: 2, 3, 5, 7, 8, 10. With six values the median is the mean of the third and fourth: (5 + 7)/2 = 6.

50. 50. A — increase the mean and leave the standard deviation unchanged

    A shift moves the centre but not the spread: the mean increases by 5 and the standard deviation is unchanged.

What your score means

This drill is harder per minute than the real maths section, so treat these as rough working bands rather than predictions of a Bocconi score:

• Below 20. The gap is content, not speed. Work through the syllabus areas where you could not start a question at all, then re-drill.
• 20 to 32. The mathematics is mostly there and the losses are pace and accuracy. Timed sets of ten questions are the most useful next step.
• Above 32. You are in good shape on content. The remaining marks are in the guess-versus-blank decision and in not losing easy questions to carelessness early in the paper.

If the same topic keeps costing you marks, that is exactly the thing one-to-one work fixes fastest — the Bocconi page sets out how the maths section is structured, and the SAT maths page covers the other route into Bocconi, since the university accepts SAT scores as an alternative.

Frequently asked questions

Are these real Bocconi past paper questions?

No. They are original practice questions written to cover the mathematics syllabus Bocconi publishes for its admissions test. Bocconi states there are no official preparation textbooks and provides one free simulation of its own; this drill exists because that simulation does not come with worked solutions.

How is the Bocconi test scored?

A correct answer is +1, a blank is 0 and a wrong answer is -0.2, or -0.33 on three-option critical-thinking items. Guessing blind among five options has a slightly negative expected value, so it is only worth answering once you can eliminate at least one option.

How much of the Bocconi test is mathematics?

24 of the 50 questions on the standard test, which is the largest single section. This drill uses 50 maths questions in the same 75 minutes so that the maths gets concentrated practice.

How long should I spend on each question?

The real test averages about 90 seconds a question. This drill gives you the same 75 minutes for 50 maths questions, so roughly the same per question, which is deliberately demanding. If you cannot see a route within about 30 seconds, mark it and move on.

Should I use a calculator on this drill?

Work the way you will sit the real test. Rules on calculators and permitted materials are set by Bocconi and can change between cycles, so confirm the current conditions on the official admissions pages before you practise under exam conditions.

What score should I be aiming for?

Bocconi publishes no cut-off beyond the floor of 17 on the real test, below which an application is not considered. Applicants to Mathematical and Computing Sciences for AI additionally need at least 11 out of 24 in the maths area. Treat any other number you see quoted as informal.

Can I retake the Bocconi test?

Yes, up to four attempts per test type per academic year, at a fee each time, and your best score is the one that counts. You cannot sit it on the same day or on consecutive days. Confirm current fees and dates on Bocconi's own pages before booking.