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"Convert the point $(0,3)$ in rectangular coordinates to polar coordinates. Enter your answer in th(...TRUNCATED) | [" Step 1: To convert from rectangular coordinates to polar coordinates, we use the formulas $r = \\(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,1,1,1(...TRUNCATED) |
"Define\n\\[p = \\sum_{k = 1}^\\infty \\frac{1}{k^2} \\quad \\text{and} \\quad q = \\sum_{k = 1}^\\i(...TRUNCATED) | [" Step 1: We can rewrite the sum as ки\nStep 2: \\begin{align*}\n\\sum_{j = 1}^\\infty \\sum_{k =(...TRUNCATED) | [1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0(...TRUNCATED) |
"If $f(x) = \\frac{3x-2}{x-2}$, what is the value of $f(-2) +f(-1)+f(0)$? Express your answer as a c(...TRUNCATED) | [" Step 1: To find $f(-2)$, we substitute $x=-2$ into the expression for $f(x)$:\n\\[f(-2) = \\frac{(...TRUNCATED) | [0,0,0,0,0,1,0,1,0,1,0,0,1,0,0,1,0,1,1,0,0,0,1,0,0,0,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0(...TRUNCATED) |
How many positive whole-number divisors does 196 have? | [" Step 1: To find the number of divisors of 196, we prime factorize it as $196=2^6\\cdot7^1$. ки\(...TRUNCATED) | [0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"The results of a cross-country team's training run are graphed below. Which student has the greates(...TRUNCATED) | [" Step 1: To find the average speed, we need to find the distance traveled and the time taken. ки(...TRUNCATED) | [0,0,0,1,0,1,0,0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0(...TRUNCATED) |
"A regular hexagon can be divided into six equilateral triangles. If the perimeter of one of the tri(...TRUNCATED) | [" Step 1: Each side of the equilateral triangle has length $21/3 = 7$ inches. ки\nStep 2: Since t(...TRUNCATED) | [1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,1,1,0,1,0,0,1,1,0,1(...TRUNCATED) |
"What is the smallest positive perfect cube that can be written as the sum of three consecutive inte(...TRUNCATED) | [" Step 1: Let the three consecutive integers be $n-1,$ $n,$ and $n+1.$ ки\nStep 2: We want $(n-1)(...TRUNCATED) | [0,0,1,0,0,0,0,1,0,1,0,0,0,1,0,0,1,1,1,1,1,1,1,0,0,0,1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,1,1,1,0,1,0,1,1(...TRUNCATED) |
"The set of points $(x,y,z)$ that satisfy\n\\[2x = 3y = -z\\]is a line.\n\nThe set of points $(x,y,z(...TRUNCATED) | [" Step 1: The first line is parameterized by $x = \\frac{3}{2} t$ and $y = t$, so it lies in the pl(...TRUNCATED) | [1,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0(...TRUNCATED) |
"What is the distance, in units, between the points $(2, -6)$ and $(-4, 3)$? Express your answer in (...TRUNCATED) | [" Step 1: Using the distance formula, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$(...TRUNCATED) | [1,1,0,0,0,0,1,1,0,1,0,0,1,1,1,0,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,1,1,0,1,0,1,1,0,1,1,1,0,0,1,1,1,0(...TRUNCATED) |
"The expression $2\\cdot 3 \\cdot 4\\cdot 5+1$ is equal to 121, since multiplication is carried out (...TRUNCATED) | [" Step 1: To obtain a different value, we need to change the order of operations, which means we ne(...TRUNCATED) | [0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
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