Actuaries Exam: Mathematics and Logical Reasoning

• This is a mathematics and statistics exam for the Institute of Actuaries of India, ACET December 2023
• The exam includes logical reasoning and English questions

Mathematics

• The solutions to the equation |2x + 23| = |x + 11| are -34/3 and 4.
• If f(x) = 4(x − 3) and g(x) = x² – 7x + 12, the value of x for which the ratio g(x) / f(x) is undefined is 3.
• For any two sets A and B, if A ∩ B = A U B, then A = B.
• If f(x) = 2x² + 5x − 24 and g(x) = 3x + 2 (x real), then the roots of f(x) + g(x) are real and distinct.
• The value of (√3 + 1)⁴ – (√3 – 1)⁴ is 32√3.
• The value of cos(cos⁻¹(4/5) + tan⁻¹(1/7)) is π/4 assuming angles are in the first quadrant.
• The value of Σ(j=1 to n) (3j − 1)² is (n/2) * [6n² - 3n + 1].
• The conjugate of ((1+2i)(2-i)) / ((3-2i)(2+3i)) is 63/169 - (16/169)i.
• Given the function f(x) defined piecewise: f(x) = 7 if x ≤ 3, f(x) = mx + n if 3 < x < 12, and f(x) = 18 if x ≥ 12, for f(x) to be continuous everywhere, (m, n) must be (11/9, 28/3).
• If x³ + y³ = cos x + y, then dy/dx = (3x² + sinx) / (1 - 3y²).
• Let f(x) = −2x³ − 9x² − 12x + 1 be a real function; f(x) is increasing in x in the interval (-2, -1).
• The value of the definite integral ∫(from -π/5 to π/5) (7 sin x) / (5 + cos x) dx is 0.
• The area bounded by the curve y = x²/2, the lines y = 0, x = 1 and x = 3 is 4 1/3.
• The value of the integral ∫ x log 3x dx is (x²/2) log 3x − x²/4 + C.
• If M = | 1 cosθ sinθ ; -cosθ -1 1 ; -sinθ 1 1 |, then |M| = 0.
• If | xy 6 ; z+4 x+y 11 ; 3 2 5 | = | 16 a 1 ; -8 12 b ; c 1 d |, then x + y + z = 16.
• The approximate positive root of x² − 2 = 0 after three iterations by the Newton-Raphson method, starting from x₀ = 1, is 1.41426.
• Given a table of values for x and f(x), the value of f(54) by Newton's forward interpolation formula is 207.12.
• If Ā = x + y + z, B = −x + y − z and Ĉ = y − z, the area of the triangle ∆ABC is 1.
• Let Ā = 2x + 4y + z, B = 3x − y + 2z , the angle between A and B is cos⁻¹(4 / (7√6)).

Statistics

• The mean of 20 observations is 25; two observations were wrongly recorded as 15 and 17, the correct values being 25 and 27, the correct mean is 26.
• The weighted arithmetic mean of 12, 16, and 20 with weights w₁, w₂, and w₃ is 18; if the weights are changed to 3w₁, 3w₂, and 3w₃, the weighted arithmetic mean remains 18.
• The standard deviation of 20 observations is 2.1; each observation is multiplied by -0.5 and then 1.5 is added; the standard deviation of the new observations is 1.05.
• If the relationship between x and y is 3x – 5y = 4 and the mode of x is 13, the mode of y is 7.
• For a symmetrical distribution, the first quartile is 30, and the median is 45, so the third quartile is 60.
• Given the letters of the word “ENGINEERING” are written down at random, the probability that all the Es occur together is 3/55.
• If the probabilities of solving a specific problem independently by students S₁ and S₂ are 1/3 and 1/5, the probability that exactly one of them solves the problem is 5/6.
• From a bag containing 5 white, 6 red, and 4 black balls, three balls are drawn one by one with replacement; the probability that at least one ball is red is 98/125.
• From a box of 15 chips, 4 of which are defective, 3 chips are selected at random without replacement, the probability that all three chips are defective is 4/455.
• Given error probabilities for three engineers, if a bid results in an error, the probability that the error was made by engineer 1 is 3/19.
• If X is uniformly distributed on {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, then E(|X|) is 3.
• If the number of accidents per month at a busy intersection follows a Poisson distribution with an average of λ = 5 and each accident costs Rs. 15,000, the average accident costs to the local government over a year is Rs. 900,000.
• If X follows a Poisson distribution with a mean of 8, the value of k such that P(X = k) = P(X = k + 1) is 7.
• If thickness X is a Uniform random variable from 150 to 200 millimeters, the probability that a steel sheet produced by this machine must be scrapped is 0.20.
• Given a random variable X with probability density function f(x) = e⁻ˣ for x > 0, the expected value of e^(0.75X) is 4.
• If X ~ N(μ, σ²), then P(X ≥ μ) = 0.50.
• Given a random variable X with a probability density function f(x) = θe⁻θx, where x > 0 and θ > 0 and if the median of the distribution is M, then the mean of the distribution is M / ln(0.5).
• If X₁ ~ Binomial (n₁, p₁) is independent of X₂ ~ Binomial (n₂, p₂), then P(X₁ + X₂ = 1) equals p₁(1 - p₂)ⁿ² + p₂(1 − p₁)ⁿ¹.
• If X ~ N(5, 2²) is independent of Y ~ N(0, 1), then the correlation between Y and XY is 5/√29.
• If 2u = x + 5 and 6v = 2y – 7, and the regression coefficient of x on y is 4, then the regression coefficient of u on v is 4/3.

Data Interpretation

• Based on a frequency distribution of marks of 200 students, the minimum possible value of the mean of the actual marks is 49.8.
• The percentage of students scoring less than 80% but not less than 60% is 23.
• Based on a frequency distribution of marks of 200 students, if the pass mark is 40%, the percentage of students who failed the subject is 17.
• The countries that won the maximum number of bronze medals is kazakhstan
• Among Uzbekistan, Kazakhstan, Chinese Taipei, and Thailand, the country that won the maximum number of medals is Uzbekistan.
• The lifetime distribution of light bulbs of M1 is positively skewed.
• The median lifetime of light bulbs of M1 is greater than that of M2.
• The correlation coefficient between x and y in Plot 3 is nearly zero.
• The correlation coefficient between x and y is positive in Plot 4.
• Based on a table of vehicle arrivals, the percentage of time intervals with at least 5 arriving vehicles is 72.7.
• The median of the distribution of the number of vehicles is 5.

English Exam

• A synonym for "Benevolent" is Kindhearted.
• A synonym for "Eloquent" is Fluent.
• The antonym of the word "Humble" is Arrogant.
• The antonym of the word "Voracious" is Undesirous.
• The phrase "to take responsibility for one's actions" is Own up.
• The phrase meaning "to reveal a secret" is Let the cat out of the bag.
• The term for the act of speaking or writing in an evasive or indirect manner is Circumlocution.
• The term for the quality of being easily broken or damaged is Fragility.
• Correct sentence: Her dedication to work and the quality of her performance is truly remarkable.
• Correct sentence: His acumen in deciphering complex algorithms and applying it in practical scenarios has earned him numerous accolades.
• Central Park is unique because of its proximity to skyscrapers, is not mentioned as an activity in Central Park, and role does is play is primarily as a sanctuary to nature enthusiasts.

Logical Reasoning

• The number of days between January 15, 2024, and March 05, 2024, inclusive, is 51.
• In a cube of side n painted red on all sides, and cut into n³ identical cubes, the number of smaller cubes with three sides painted is 8.
• Statements: 1) Depositors lost money, 2) The bank went bankrupt: Statement 2 (bankrupt) is the cause, and Statement 1 the effect.
• Hardik introduces Axar as the son of the only brother of his father's wife, Axar is Hardik's cousin.
• A watch gains 5 seconds every 3 minutes and was set right at 8 AM, at 10 PM it will show 10 hours 23 minutes 20 seconds.
• The set of three statements where the third can be logically derived from the first two: 4, 2, 5.
• Nine persons watch a movie, B is at one end, and given certain seating conditions, the position of I from the left is 3.
• In reliability theory, components can be in series or parallel; entry into an email account is a system in which the username and password are two components in series.