BStat BMath UGA 2024 Quiz

Challenge yourself with this 15-question multiple-choice quiz covering essential math concepts, probability, algebra, geometry, and number theory from the BStat-BMath-UGA-2024 syllabus. Each question is uniquely designed to test your problem-solving and analytical skills for competitive exams.

1. Algebraic Evaluation

   If x = 1 + √2 + 2^{1/2} + 2^{1/4} + 2^{1/8}, what is the value of (1 + x/2)^{30}?

   1. 5
   2. 2
   3. 32
   4. 64

2. Probability of Random Selection

   A number j is chosen at random from the set {1, 2, ..., 2024}. What is the probability that j is divisible by both 9 and 15?

   1. 1/23
   2. 1/44
   3. 1/46
   4. 1/253

3. Combinatorics on Digits

   Let Sₙ be the set of all n-digit numbers whose digits are only 1 or 2, with no consecutive 2's allowed (for example, 112 is allowed but 221 is not). How many numbers are there in S₁₀?

   1. 256
   2. 144
   3. 512
   4. 89

4. Probabilities in Examinations

   A student knows the answers to 20 out of 30 True/False questions and guesses the remaining 10. What is the probability that the student gets exactly 24 correct answers?

   1. 105/2^{10}
   2. 105/2^9
   3. 4/2^{10}
   4. 105/2^8

5. Triangle Properties

   For a right-angled triangle whose side lengths are in a geometric progression, what is the maximum number of sides that can have integer lengths?

   1. 1
   2. 0
   3. 3
   4. 2

6. Optimization with Square Roots

   Let x₁, x₂, ..., xₙ be non-negative real numbers such that their sum is 1. What is the maximum possible value of the sum of their square roots: √x₁ + √x₂ + ... + √xₙ?

   1. 1
   2. n^{3/4}
   3. √n
   4. n

7. Monotonicity of Logarithmic Functions

   On which interval is the function f(x) = log_{1/2}(x² - 2x - 3) monotonically decreasing?

   1. (-∞, -1)
   2. (3, ∞)
   3. (-∞, 1)
   4. (1, ∞)

8. Geometry of Circles

   What is the angle subtended at the origin by the common chord of the circles x² + y² - 6x - 6y = 0 and x² + y² = 36?

   1. π/3
   2. π/2
   3. 2π/3
   4. π/4

9. Triangle Geometry

   In triangle ABC, CD is the median, BE is the altitude, and CD = BE. What is the value of the angle ∠ACD?

   1. π/5
   2. π/4
   3. π/3
   4. π/6

10. Complex Numbers on Circles

    If z₁ and z₂ are points on the circles |z|=2 and |z|=3 respectively, and the angle between them is 60°, what is the value of |(z₁+z₂)/(z₁-z₂)|?

    1. √19
    2. √(19/7)
    3. √(7/19)
    4. √7

11. Maximal Set of Primes

    Let n ≥ 1. What is the maximum number of prime numbers in the set {n+6, n+7, ..., n+35}?

    1. 12
    2. 8
    3. 7
    4. 13

12. Balls and Boxes Counting

    Forty distinguishable balls, 10 defective and 30 non-defective, are distributed into four boxes with each box getting exactly 10. How many ways can this be done if all defective balls go into the first two boxes?

    1. 30!.10!/(10!)^4
    2. 30!.20!/(10!)^5
    3. 40!/(10!)^4
    4. 20!.20!/(10!)^5

13. Solution Set Cardinality

    How many elements are in the set {x : 0 ≤ x ≤ 2, |x − x²| = |x³ − x⁰|}?

    1. 4
    2. 3
    3. 2
    4. 5

14. Probability and Limits

    In a room with n ≥ 2 people, where each pair shakes hands with independent probability, if pₙ is the probability that there are at most one handshake, what is limₙ→∞ pₙ?

    1. 1
    2. e^{-1}
    3. 0
    4. 2e^{-1}

15. Equation with Trigonometric and Logarithmic Terms

    How many positive solutions does the equation e·sin(z) = log(x) + e√x + 2 have?

    1. 2
    2. 0
    3. ∞
    4. 1