Simplify the numerical expression: (2^3 * 4^(-1/2)) / 8^(2/3).
Algebra And Arithmetic
Question · Easy
Stats test: find the median
Short
Evaluate the radical expression: (sqrt(12) + sqrt(27)) / sqrt(3).
Compute the value of: log_2(32) - log_3(1/27) + log_5(1).
In an arithmetic progression (a_n), the first term is a_1 = 7 and the common difference is d = 4. What is the 15th term a_15?
Factor the quadratic polynomial 2x^2 - 5x - 3 completely over the integers.
Factor it
Simplify the rational expression: (x^2 - 9) / (2x^2 + 5x - 3) for all permissible values of x.
Question · Medium
Given that log_2(x) = 3 and log_2(y) = -2, determine the numerical value of log_2(x^2 * sqrt(y)).
In a geometric progression (b_n), the first term is b_1 = 3 and the common ratio is q = 2. What is the sum of the first 6 terms S_6?
A fair six-sided die is rolled once. What is the probability that the number rolled is greater than 4?
Probability And Statistics
Find the remainder when the polynomial P(x) = 3x^3 - 4x^2 + 2x - 5 is divided by (x - 2).
A class of 8 students must choose a 3-person committee in which all members have equal roles. How many different committees are possible?
Simplify the nested radical expression: sqrt(18 - 8*sqrt(2)).
If two real numbers satisfy a + b = 5 and a * b = 6, find the value of a^3 + b^3.
Given that log_a(b) = 4, evaluate the expression: log_(sqrt(a))(b^3).
A discrete random variable X takes the values 0, 1, and 2 with probabilities P(X = 0) = 0.2, P(X = 1) = 0.5, and P(X = 2) = 0.3. What is the expected value E(X)?
A random variable X takes the values 1, 2, and 4 with probabilities 0.5, 0.25, and 0.25. Using Var(X) = E(X^2) - (E(X))^2, find the variance of X.
What is the sum of the roots of the cubic equation x^3 - 6x^2 + 11x - 6 = 0?
Two archers shoot at a target independently. The first hits with probability 0.8 and the second with probability 0.7. What is the probability that at least one of them hits the target?
In an arithmetic progression, the sum of the first n terms is S_n = 2n^2 + 3n. What is the value of the 10th term a_10?
A factory has two production lines. Line A makes 60% of all items with a 2% defect rate; line B makes 40% of all items with a 5% defect rate. What is the probability that a randomly chosen item is defective?
Find all real solutions to the equation: (sqrt(3 + sqrt(8)))^x + (sqrt(3 - sqrt(8)))^x = 34.
Question · Hard
Given that log_2(3) = a, express log_12(18) in terms of a.
The sum of the first n terms of a sequence is given by S_n = 3^(n+1) - 3 for all positive integers n. Find the general term a_n of this sequence.
Factor the polynomial x^4 + 4 completely into irreducible quadratic polynomials over the real numbers.
A disease affects 1% of a population. A screening test is positive for 90% of people who have the disease and, by mistake, for 5% of people who do not. A randomly chosen person tests positive. What is the probability that the person has the disease?
Solve the linear equation: 3(2x - 4) - 2(x + 5) = 18.
Equations And Inequalities
Find all real roots of the quadratic equation: 2x^2 - 7x + 3 = 0.
Solve the rational equation: (2x - 1) / (x + 3) = 3 / 2.
Find the median of the data set: 7, 3, 9, 5, 12, 4.
Solve the quadratic inequality: x^2 - 4x - 5 < 0.
Solve the system of linear equations: { 2x + 3y = 13, 3x - y = 3 }.
Solve the irrational equation: sqrt(3x + 1) = x - 1.
Find all real solutions to the exponential equation: 4^x - 3 * 2^x - 4 = 0.
Solve the logarithmic equation: log_3(x - 2) + log_3(x + 6) = 2.
Find all solutions to the trigonometric equation 2*sin^2(x) - 3*sin(x) + 1 = 0 on the interval [0, 2*pi).
How many different 4-digit numbers can be formed from the digits 1, 2, 3, 4, and 5 if no digit may be repeated?
Solve the rational inequality: (x - 1) / (x + 2) <= 0.
Solve the irrational inequality: sqrt(2x + 6) < x + 1.
Solve the exponential inequality: (1/3)^(x^2 - 2x) >= 1/27.
Solve the logarithmic inequality: log_2(x - 3) + log_2(x - 1) <= 3.
Find all solutions to the equation cos(2x) = cos(x) on the interval [0, 2*pi).
A fair coin is tossed 4 times. What is the probability of getting exactly 2 heads?
A fair six-sided die is rolled twice. What is the probability of getting at least one 6?
Find all real values of k for which the quadratic equation x^2 - 2(k - 1)x + (k + 5) = 0 has two distinct real roots.
Solve the absolute value equation: |2x - 3| = 3x + 1.
Class A has 20 students with a mean test score of 70, and class B has 30 students with a mean test score of 80. What is the mean score of all 50 students combined?
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