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Bocconi Graduate Test study materials
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Bocconi Graduate Test practice questions

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Question 1Quantitative reasoning / Mathematics

A shirt is priced at €80. The price is reduced by 25%, and later the reduced price is increased by 20%. What is the final price?

  1. €72 Correct answer.
  2. €75
  3. €76
  4. €80

Explanation

A 25% reduction leaves 0.75 × 80 = €60, and a 20% increase gives 1.20 × 60 = €72. The two percentage changes apply to different bases, so they do not cancel out.

Question 2Quantitative reasoning / Mathematics

If 3x − 7 = 2x + 5, what is the value of x?

  1. 6
  2. −12
  3. 12 Correct answer.
  4. 2

Explanation

Subtracting 2x from both sides gives x − 7 = 5, and adding 7 gives x = 12. Substituting back: 3(12) − 7 = 29 and 2(12) + 5 = 29, so the solution checks.

Question 3Quantitative reasoning / Mathematics

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

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

Explanation

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

Question 4Quantitative reasoning / Mathematics

What is the value of log₂(32) + log₃(9)?

  1. 41
  2. 7 Correct answer.
  3. 6
  4. 11

Explanation

Since 2⁵ = 32, log₂(32) = 5; since 3² = 9, log₃(9) = 2. Their sum is 5 + 2 = 7.

Question 5Quantitative reasoning / Mathematics

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

  1. 13 Correct answer.
  2. √119
  3. 17
  4. 15

Explanation

By the Pythagorean theorem the hypotenuse satisfies c² = 5² + 12² = 25 + 144 = 169, so c = 13. This is the standard 5-12-13 triple.

Question 6Quantitative reasoning / Mathematics

The mean of five numbers is 12. If four of the numbers are 8, 10, 14 and 16, what is the fifth number?

  1. 18
  2. 10
  3. 14
  4. 12 Correct answer.

Explanation

A mean of 12 over five values requires a total of 5 × 12 = 60. The four known values sum to 48, so the missing value is 60 − 48 = 12.

Question 7Quantitative reasoning / Mathematics

Which of the following is the derivative of f(x) = 3x⁴ − 2x with respect to x?

  1. 3x³ − 2
  2. 12x⁴ − 2
  3. 12x³ − 2x
  4. 12x³ − 2 Correct answer.

Explanation

The power rule gives d/dx(3x⁴) = 12x³ and d/dx(2x) = 2, so f′(x) = 12x³ − 2. The constant multiple and difference rules are applied term by term.

Question 8Quantitative reasoning / Mathematics

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

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

Explanation

The first draw is red with probability 4/10 = 2/5, and then 3 of the remaining 9 balls are red. Multiplying gives (2/5)(3/9) = 6/45 = 2/15.

Question 9Quantitative reasoning / Mathematics

For which values of x is the expression (x − 3)/(x² − 9) undefined?

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

Explanation

The denominator factors as (x − 3)(x + 3), which vanishes at x = 3 and x = −3, so the expression is undefined at both. At x = 3 the numerator also vanishes, but division by zero is still undefined even after cancelling.

Question 10Quantitative reasoning / Mathematics

A geometric sequence has first term 3 and common ratio 2. What is the sum of its first 6 terms?

  1. 189 Correct answer.
  2. 126
  3. 63
  4. 192

Explanation

The sum of the first n terms is a(rⁿ − 1)/(r − 1) = 3(2⁶ − 1)/(2 − 1) = 3 × 63 = 189. Equivalently, 3 + 6 + 12 + 24 + 48 + 96 = 189.

Question 11Quantitative reasoning / Mathematics

A company's revenue grows by 10% in year one and by 20% in year two. By what single percentage is revenue higher at the end of year two than at the start?

  1. 15%
  2. 32% Correct answer.
  3. 30%
  4. 30.5%

Explanation

The growth factors multiply: 1.10 × 1.20 = 1.32, so revenue is 32% higher. Simple addition of the rates ignores the fact that the second increase applies to the already-increased amount.

Question 12Quantitative reasoning / Mathematics

The function f(x) = x² − 4x + 7 is defined for all real x. What is the minimum value of f?

  1. 7
  2. 4
  3. 3 Correct answer.
  4. −9

Explanation

Completing the square gives f(x) = (x − 2)² + 3, so the minimum is 3, attained at x = 2. The squared term is never negative.

Question 13Quantitative reasoning / Mathematics

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. What is P(A | B)?

  1. 0.4
  2. 0.25
  3. 0.5 Correct answer.
  4. 0.2

Explanation

From the addition rule, P(A ∩ B) = 0.5 + 0.4 − 0.7 = 0.2. Then P(A | B) = P(A ∩ B)/P(B) = 0.2/0.4 = 0.5.

Question 14Quantitative reasoning / Mathematics

What is the sum of all integers n with 1 ≤ n ≤ 100 that are divisible by 3?

  1. 1683 Correct answer.
  2. 1717
  3. 1515
  4. 5050

Explanation

The multiples of 3 up to 100 are 3, 6, …, 99, giving 33 terms. Their sum is 3(1 + 2 + … + 33) = 3 × 33 × 34/2 = 3 × 561 = 1683.

Question 15Quantitative reasoning / Mathematics

A line passes through the points (2, 5) and (6, 13). What is the equation of the line?

  1. y = 2x + 1 Correct answer.
  2. y = 2x − 1
  3. y = 0.5x + 4
  4. y = 4x − 3

Explanation

The slope is (13 − 5)/(6 − 2) = 8/4 = 2. Using the point (2, 5): 5 = 2(2) + b gives b = 1, so y = 2x + 1.

Question 16Quantitative reasoning / Mathematics

The function g(x) = x³ − 3x has derivative g′(x) = 3x² − 3. At which points does g have a local extremum?

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

Explanation

Setting g′(x) = 3(x² − 1) = 0 gives x = 1 and x = −1. The derivative changes sign at both points, so each is a local extremum (a minimum at x = 1 and a maximum at x = −1).

Question 17Quantitative reasoning / Mathematics

A circle has the equation x² + y² − 6x + 4y − 12 = 0. What are its centre and radius?

  1. Centre (3, −2), radius 25
  2. Centre (6, −4), radius 12
  3. Centre (−3, 2), radius 5
  4. Centre (3, −2), radius 5 Correct answer.

Explanation

Completing the square gives (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25, so the centre is (3, −2) and the radius is √25 = 5.

Question 18Quantitative reasoning / Mathematics

In how many ways can the letters of the word BANANA be arranged?

  1. 60 Correct answer.
  2. 120
  3. 720
  4. 180

Explanation

The word has 6 letters with A repeated 3 times and N repeated 2 times. The number of distinct arrangements is 6!/(3!2!) = 720/12 = 60.

Question 19Quantitative reasoning / Mathematics

For what values of the parameter k does the equation x² + kx + 9 = 0 have two distinct real solutions?

  1. k > 0
  2. −6 < k < 6
  3. k = 6 or k = −6
  4. k > 6 or k < −6 Correct answer.

Explanation

Two distinct real roots require a positive discriminant: k² − 36 > 0, i.e. k² > 36. This holds precisely when k > 6 or k < −6.

Question 20Quantitative reasoning / Mathematics

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

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

Explanation

There are 2⁴ = 16 equally likely sequences, and the number with exactly 2 heads is C(4,2) = 6. The probability is 6/16 = 3/8.

Question 21Quantitative reasoning / Mathematics

The function h(x) = x·e^(−x) is defined for x ≥ 0. At what value of x does h attain its maximum?

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

Explanation

Using the product rule, h′(x) = e^(−x) − x·e^(−x) = e^(−x)(1 − x). This is zero at x = 1 and changes from positive to negative there, so x = 1 gives the maximum.

Question 22Quantitative reasoning / Mathematics

A geometric series has first term 8 and common ratio −1/2. What is the sum of the infinite series?

  1. 16
  2. 8/3 Correct answer.
  3. 16/3
  4. 5⅓

Explanation

Since |−1/2| < 1, the sum is a/(1 − r) = 8/(1 + 1/2) = 8/(3/2) = 16/3, which simplifies to 5⅓. Note that 16/3 and 5⅓ are the same value, so this option is the answer.

Question 23Quantitative reasoning / Mathematics

For how many positive integers n does the inequality 2ⁿ < n² hold?

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

Explanation

Checking small values: n = 1 gives 2 > 1; n = 2 gives 4 = 4; n = 3 gives 8 < 9, which holds; n = 4 gives 16 = 16; n = 5 gives 32 > 25, and beyond that 2ⁿ grows faster. So the inequality holds only for n = 3, giving one value.

Question 24Quantitative reasoning / Mathematics

A matrix M is 2×3 and a matrix N is 3×2. Which of the following products is defined?

  1. M·N only
  2. M·N and N·M Correct answer.
  3. Neither product
  4. N·M only

Explanation

A product of matrices is defined when the number of columns of the first equals the number of rows of the second. M(2×3)·N(3×2) gives a 2×2 matrix, and N(3×2)·M(2×3) gives a 3×3 matrix, so both are defined.

Question 25Quantitative reasoning / Mathematics

A quantity decays exponentially so that it halves every 4 hours. Starting from 100 units, how many units remain after 12 hours?

  1. 12.5 Correct answer.
  2. 25
  3. 50
  4. 6.25

Explanation

Twelve hours is three half-lives (12/4 = 3), so the amount is 100 × (1/2)³ = 100/8 = 12.5 units.

Question 26Quantitative reasoning / Mathematics

A student's four test scores are 72, 80, 85 and 91. A fifth test is added and the mean of all five scores is exactly 84. What score was earned on the fifth test?

  1. 88
  2. 92 Correct answer.
  3. 96
  4. 90

Explanation

The total needed for five tests averaging 84 is 5 × 84 = 420. The four known scores sum to 72 + 80 + 85 + 91 = 328, so the fifth score is 420 − 328 = 92.

Question 27Quantitative reasoning / Mathematics

Simplify the expression (2x³y²)·(3x²y⁴) as far as possible.

  1. 6x⁶y⁸
  2. 5x⁵y⁶
  3. 6x⁵y⁸
  4. 6x⁵y⁶ Correct answer.

Explanation

Multiply the numerical coefficients: 2 × 3 = 6. For the powers of the same base, add the exponents: x³·x² = x⁵ and y²·y⁴ = y⁶. The simplified product is therefore 6x⁵y⁶.

Question 28Quantitative reasoning / Mathematics

The quadratic function f(x) = 2x² − 8x + 5 is defined for all real x. What is the minimum value of f?

  1. 5
  2. −3 Correct answer.
  3. −11
  4. 3

Explanation

The vertex of ax² + bx + c occurs at x = −b/(2a) = 8/4 = 2. Substituting gives f(2) = 2(4) − 16 + 5 = −3, and since a > 0 this vertex is the minimum.

Question 29Quantitative reasoning / Mathematics

Solve the equation log₂(x + 5) = 3 for x.

  1. 8
  2. 3 Correct answer.
  3. 2
  4. 11

Explanation

Rewrite the logarithmic equation in exponential form: x + 5 = 2³ = 8. Subtracting 5 gives x = 3, which is valid because x + 5 = 8 is positive.

Question 30Quantitative reasoning / Mathematics

An arithmetic sequence has first term 7 and common difference 4. What is the 20th term of the sequence?

  1. 87
  2. 83 Correct answer.
  3. 79
  4. 80

Explanation

The n-th term of an arithmetic sequence is a + (n − 1)d. Here that gives 7 + 19 × 4 = 7 + 76 = 83.

Question 31Quantitative reasoning / Mathematics

Which of the following is the derivative of f(x) = (x² + 1)/(x − 1) with respect to x?

  1. (x² + 2x − 1)/(x − 1)²
  2. (2x)/(x − 1)
  3. (2x)/(x − 1)²
  4. (x² − 2x − 1)/(x − 1)² Correct answer.

Explanation

Apply the quotient rule with u = x² + 1 and v = x − 1: f′(x) = (u′v − uv′)/v² = [2x(x − 1) − (x² + 1)]/(x − 1)². Expanding the numerator gives 2x² − 2x − x² − 1 = x² − 2x − 1.

Question 32Quantitative reasoning / Mathematics

The equation of a circle is x² + y² + 2x − 8y + 8 = 0. What is its radius?

  1. 3 Correct answer.
  2. 5
  3. 4
  4. 9

Explanation

Complete the squares: (x + 1)² − 1 + (y − 4)² − 16 + 8 = 0, so (x + 1)² + (y − 4)² = 9. The right side equals r², hence r = 3.

Question 33Quantitative reasoning / Mathematics

A quantity increases by 25% and then decreases by 20% from the new value. What is the net percentage change from the original value?

  1. A decrease of 5%
  2. An increase of 5%
  3. An increase of 45%
  4. No change Correct answer.

Explanation

Multiplying by 1.25 and then by 0.80 gives 1.25 × 0.80 = 1.00, so the final value equals the original. The percentage changes cancel exactly.

Question 34Quantitative reasoning / Mathematics

In how many distinct ways can three students be selected from a group of eight students to form a committee?

  1. 336
  2. 56 Correct answer.
  3. 24
  4. 512

Explanation

The order of selection does not matter, so this is a combination: C(8,3) = 8!/(3!5!) = (8 × 7 × 6)/(3 × 2 × 1) = 56.

Question 35Quantitative reasoning / Mathematics

The function f is defined by f(x) = x³ − 6x² + 9x + 1. At which value of x does f have a local maximum?

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

Explanation

Setting f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3) = 0 gives critical points x = 1 and x = 3. The second derivative is 6x − 12, which is negative at x = 1 and positive at x = 3, so the local maximum is at x = 1.

Question 36Quantitative reasoning / Mathematics

A fair coin is tossed five times. What is the probability of obtaining at least one head?

  1. 31/32 Correct answer.
  2. 1/32
  3. 1/2
  4. 5/32

Explanation

The complement of at least one head is no heads, whose probability is (1/2)⁵ = 1/32. Hence the required probability is 1 − 1/32 = 31/32.

Question 37Quantitative reasoning / Mathematics

The system of linear equations 2x + 3y = 7 and 4x + ky = 14 has infinitely many solutions. What is the value of k?

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

Explanation

Infinitely many solutions occur when the two equations are proportional. The second equation is exactly twice the first when k = 2 × 3 = 6, making it the same line.

Question 38Quantitative reasoning / Mathematics

The expression (x² − 4)/(x² − x − 6) is to be simplified for values of x where it is defined. Which simplified form is correct?

  1. (x − 2)/(x + 3)
  2. (x + 2)/(x + 3) Correct answer.
  3. (x − 2)/(x − 3)
  4. (x + 2)/(x − 3)

Explanation

Factor both parts: x² − 4 = (x − 2)(x + 2) and x² − x − 6 = (x − 3)(x + 2). Cancelling the common factor (x + 2), valid when x ≠ −2, leaves (x − 2)/(x − 3).

Question 39Quantitative reasoning / Mathematics

A geometric series has first term 12 and common ratio 1/3. What is the sum of the infinite series?

  1. 36
  2. 16
  3. 24
  4. 18 Correct answer.

Explanation

For |r| < 1 the infinite geometric sum is a/(1 − r). Here that is 12/(1 − 1/3) = 12/(2/3) = 18.

Question 40Quantitative reasoning / Mathematics

A right triangle has legs of length 5 and 12. What is the radius of the circle inscribed in the triangle?

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

Explanation

The hypotenuse is 13, so the area is (5 × 12)/2 = 30 and the semiperimeter is (5 + 12 + 13)/2 = 15. The inradius equals area divided by semiperimeter, giving 30/15 = 2.

Question 41Quantitative reasoning / Mathematics

Two events A and B are independent with P(A) = 0.6 and P(B) = 0.5. What is P(A ∪ B)?

  1. 0.3
  2. 0.5
  3. 1.1
  4. 0.8 Correct answer.

Explanation

Independence gives P(A ∩ B) = 0.6 × 0.5 = 0.3. The addition rule then gives P(A ∪ B) = 0.6 + 0.5 − 0.3 = 0.8.

Question 42Quantitative reasoning / Mathematics

The function f(x) = x² − 4x + 3 is restricted to the interval 0 ≤ x ≤ 5. What is the maximum value of f on this interval?

  1. −1
  2. 3
  3. 8 Correct answer.
  4. 15

Explanation

The vertex at x = 2 gives the minimum, f(2) = −1. Checking the endpoints: f(0) = 3 and f(5) = 25 − 20 + 3 = 8, so the maximum on the interval is 8.

Question 43Quantitative reasoning / Mathematics

The number of bacteria in a culture doubles every 3 hours. At the start there are 500 bacteria. How many bacteria are present after 18 hours?

  1. 16000
  2. 64000
  3. 3000
  4. 32000 Correct answer.

Explanation

In 18 hours the population doubles 18/3 = 6 times. The count is 500 × 2⁶ = 500 × 64 = 32000.

Question 44Quantitative reasoning / Mathematics

If the matrix A is 3×4 and the matrix B is 4×2, what are the dimensions of the product AB?

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

Explanation

A product AB is defined when the number of columns of A equals the number of rows of B, which holds here (4 = 4). The result then has the rows of A and the columns of B, so it is 3×2.

Question 45Quantitative reasoning / Mathematics

A polynomial p(x) has p(2) = 0 and p(−1) = 0. Which of the following must be a factor of p(x)?

  1. x² − 3x + 2
  2. x² + x − 2
  3. x² + x + 2
  4. x² − x − 2 Correct answer.

Explanation

By the factor theorem, p(2) = 0 means (x − 2) is a factor and p(−1) = 0 means (x + 1) is a factor. Their product (x − 2)(x + 1) = x² − x − 2 must therefore divide p(x).

Question 46Quantitative reasoning / Mathematics

The function f(x) = x·e^(−2x) is defined for x ≥ 0. At what value of x does f reach its maximum?

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

Explanation

By the product rule, f′(x) = e^(−2x) − 2x·e^(−2x) = e^(−2x)(1 − 2x). This is zero when x = 1/2, and it changes from positive to negative there, so the maximum occurs at x = 1/2.

Question 47Quantitative reasoning / Mathematics

A box contains 5 white and 3 black balls. Two balls are drawn at random without replacement. Given that at least one ball is white, what is the probability that both balls are white?

  1. 5/14
  2. 10/28
  3. 10/18 Correct answer.
  4. 5/8

Explanation

The probability of both white is (5/8)(4/7) = 20/56 = 5/14. The probability of at least one white is 1 minus both black, 1 − (3/8)(2/7) = 1 − 6/56 = 50/56. Dividing gives (20/56)/(50/56) = 20/50 = 2/5, which equals 10/18 after simplification.

Question 48Quantitative reasoning / Mathematics

For how many integer values of n does the inequality 2ⁿ > n² hold when n is a positive integer less than 10?

  1. 5
  2. 7
  3. 6 Correct answer.
  4. 4

Explanation

Checking n from 1 to 9: 2ⁿ > n² holds for n = 1 (2 > 1), fails at n = 2 (4 = 4), fails at n = 3 (8 < 9) and n = 4 (16 = 16), then holds for n = 5, 6, 7, 8, 9. That gives 1 + 5 = 6 values.

Question 49Quantitative reasoning / Mathematics

The sum of the first n terms of an arithmetic sequence is S(n) = n² + 3n. What is the 10th term of the sequence?

  1. 24
  2. 20
  3. 13
  4. 22 Correct answer.

Explanation

The n-th term equals S(n) − S(n − 1). Here S(10) = 100 + 30 = 130 and S(9) = 81 + 27 = 108, so the 10th term is 130 − 108 = 22.

Question 50Quantitative reasoning / Mathematics

A function f satisfies f′(x) = 3x² − 6x and f(2) = 5. What is f(0)?

  1. 1
  2. 9 Correct answer.
  3. 5
  4. 13

Explanation

Integrating gives f(x) = x³ − 3x² + C. Using f(2) = 8 − 12 + C = 5 gives C = 9, so f(x) = x³ − 3x² + 9 and f(0) = 9.