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The journey of every round
Every round-trip in the corpus is public. Open one to see the prompt, the zero-point index, the baseline response, the enhanced response, and how each grader read the difference. Leave a comment if a grader's reading lands — or doesn't.
State the formula for the volume of a sphere of radius r. Give the formula in exact form using pi.
Compute the limit of the sequence a_n = (5n^2 + 2) / (2n^2 - n + 4) as n approaches infinity. Give the exact value.
Compute the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... Give the exact value.
Is the repeating decimal 0.999... equal to 1? State the exact value and justify using a geometric-series argument.
How many distinct arrangements are there of the letters in the word "STATISTICS"? Give the exact integer.
Compute the number of permutations of 9 distinct objects taken 4 at a time, P(9, 4). Give the exact integer.
Compute 7! (seven factorial). Give the exact integer.
Compute the binomial coefficient C(10, 4). Give the exact integer.
State the smallest prime factor of 943. Give the exact integer.
Compute the greatest common divisor of 360 and 84 using the Euclidean algorithm. Give the exact integer.
A fair six-sided die is rolled twice. Compute the expected value of the sum of the two rolls. Give the exact value as a rational number.
A prior P(H) = 0.3 and likelihood P(E|H) = 0.9, with P(E|not H) = 0.2. Compute the posterior P(H|E) using Bayes' theorem. Give the exact value.
Compute the limit of (x^2 - 9) / (x - 3) as x approaches 3. Give the exact value.
Evaluate the definite integral of (4x - 3) from x = 1 to x = 5. Give the exact value.
Differentiate h(x) = cos(5x^3 - 2x) using the chain rule. Give the exact derivative.
Evaluate the indefinite integral of 5x^4 + 2e^x with respect to x. Give the antiderivative and include the constant of integration.
Differentiate f(x) = 7x^4 - 2x^3 + x - 9. Give the derivative f'(x) as a polynomial.
Simplify the rational expression (x^2 - 4) / (x^2 + 5x + 6) and state any restrictions on x. Give the simplified form.
Solve the linear system 4x - y = 7 and 2x + 3y = 7. Give the exact values of x and y.
Expand (3x + 1)(2x - 4) and give the result as a polynomial in standard form.
Factor the cubic polynomial x^3 - 4x^2 + x + 6 into linear factors over the rationals. Give the factored form.
State the quadratic formula and use it to solve 2x^2 + 5x - 3 = 0. Give the roots as exact rational numbers.
Given f(x) = 2x + 3, state the inverse function f^(-1)(x). Give the inverse in explicit form.
State the formula for the surface area of a sphere of radius r. Give the formula in exact form using pi.