benchmarks.wiki / Public workspace

MATH / Tasks

Choose a problem to read its original content, examples and discussion.

Option counts cover the whole public benchmark.

Clear filters

1–50 / 200 tasks

algebra / test
Given the functions f(x)=(x+5)/(3) and g(x)=(1)/(f^(−1)(x)+1), find the value of g(3).
Answer published by the source
Identifiers
task_id
00cc7bda-6478-5134-880f-ee85c86b454d
task_key
algebra--test--00cc7bda-6478-5134-880f-ee85c86b454d
algebra / test
The difference of the roots of the quadratic equation x²+bx+c=0 is |b−2c|. If c≠0, then find c in terms of b.
Answer published by the source
Identifiers
task_id
014cec44-3541-523d-85d5-bfbb7cf7c09a
task_key
algebra--test--014cec44-3541-523d-85d5-bfbb7cf7c09a
algebra / test
Given that f(x) is a function such that f(1)=2, f(4)=3, f(7)=4, and f^(−1)(x) is the inverse of f(x), what is f^(−1)(f^(−1)(3))?
Answer published by the source
Identifiers
task_id
02b4a4a5-a183-59f2-9ff3-ea1426323b23
task_key
algebra--test--02b4a4a5-a183-59f2-9ff3-ea1426323b23
algebra / test
A sequence starts with the term 2222. Each succeeding term is found by adding…
Answer published by the source
Identifiers
task_id
034b00be-a8a2-5255-9b06-60cc90db3b47
task_key
algebra--test--034b00be-a8a2-5255-9b06-60cc90db3b47
algebra / test
A bookstore is deciding what price it should charge for a certain book. After…
Answer published by the source
Identifiers
task_id
048ea14d-c916-5aae-91cf-3e50009ddca8
task_key
algebra--test--048ea14d-c916-5aae-91cf-3e50009ddca8
algebra / test
Let f(x)=x+1 and g(x)=2x. Also denote the inverses to these functions as f^(−1) and g^(−1). Compute f(g^(−1)(f^(−1)(f^(−1)(g(f(5)))))).
Answer published by the source
Identifiers
task_id
07bf5a45-d2ca-5272-b1bf-add1cbbf5b8d
task_key
algebra--test--07bf5a45-d2ca-5272-b1bf-add1cbbf5b8d
algebra / test
The vertex of the parabola described by the equation y=2x²−4x+4 is (m,n). What is m?
Answer published by the source
Identifiers
task_id
08a4fe74-fe8d-5416-8cd0-cc756254a905
task_key
algebra--test--08a4fe74-fe8d-5416-8cd0-cc756254a905
algebra / test
Let m and n satisfy mn=7 and m+n=8. What is |m−n|?
Answer published by the source
Identifiers
task_id
09a4f2e5-75bc-58cd-b7c6-92e6c353fc9c
task_key
algebra--test--09a4f2e5-75bc-58cd-b7c6-92e6c353fc9c
algebra / test
Of the four points (2,2), (9,11), (5,7), and (11,17), three lie on the same line. Which point is not on the line?
Answer published by the source
Identifiers
task_id
0a325108-7464-56e1-abda-ad23c9da4511
task_key
algebra--test--0a325108-7464-56e1-abda-ad23c9da4511
algebra / test
For what real value of k is (13−√(131))/(4) a root of 2x²−13x+k?
Answer published by the source
Identifiers
task_id
0b578636-f57c-5f15-9686-47b1d0cd24c5
task_key
algebra--test--0b578636-f57c-5f15-9686-47b1d0cd24c5
algebra / test
Find the domain of √(6−x−x²).
Answer published by the source
Identifiers
task_id
0e1f9f7a-a493-5025-adcb-c74e538d50e6
task_key
algebra--test--0e1f9f7a-a493-5025-adcb-c74e538d50e6
algebra / test
If a and b are the solutions to the equation x²−5x+9=0, what is the value of (a−1)(b−1)?
Answer published by the source
Identifiers
task_id
0f4040c3-15e6-5af0-abce-81b22f2a0f20
task_key
algebra--test--0f4040c3-15e6-5af0-abce-81b22f2a0f20
algebra / test
Two consecutive positive even numbers are each squared. The difference of the…
Answer published by the source
Identifiers
task_id
0f83eaab-81ca-5d8d-859f-e3e2a78b52e5
task_key
algebra--test--0f83eaab-81ca-5d8d-859f-e3e2a78b52e5
algebra / test
What is the value of x in the equation (17^6-17^5)div16=17^x?
Answer published by the source
Identifiers
task_id
0f97960a-e3bb-5905-8932-b07c40166ab5
task_key
algebra--test--0f97960a-e3bb-5905-8932-b07c40166ab5
algebra / test
If m is a real number and 2x²+mx+8 has two distinct real roots, then what are the possible values of m? Express your answer in interval notation.
Answer published by the source
Identifiers
task_id
122e4612-3dff-5a12-a0b2-7752371c1b2c
task_key
algebra--test--122e4612-3dff-5a12-a0b2-7752371c1b2c
algebra / test
Suppose that f is a function and f^(−1) is the inverse of f. If f(1)=2, f(2)=6, and f(3)=5, then what is f^(−1)(f^(−1)(6))?
Answer published by the source
Identifiers
task_id
149278f1-a0c9-5d87-978b-66e325514536
task_key
algebra--test--149278f1-a0c9-5d87-978b-66e325514536
algebra / test
Let a be a real number for which there exists a unique value of b such that the quadratic equation x²+2bx+(a−b)=0 has one real solution. Find a.
Answer published by the source
Identifiers
task_id
15c4df1d-2a19-50eb-8864-18c6f33dfdde
task_key
algebra--test--15c4df1d-2a19-50eb-8864-18c6f33dfdde
algebra / test
The points (x,y) represented in this table lie on a straight line. The point (13,q) lies on the same line. What is the value of p + q? Express your answer as a decimal to the neare…
Answer published by the source
Identifiers
task_id
168eae69-c44f-522f-99e3-df868f1d89c4
task_key
algebra--test--168eae69-c44f-522f-99e3-df868f1d89c4
algebra / test
A line ℓ passes through the points B(7,−1) and C(−1,7). The equation of this line can be written in the form y=mx+b; compute m+b.
Answer published by the source
Identifiers
task_id
18575da8-8060-5465-be85-a06f390fd055
task_key
algebra--test--18575da8-8060-5465-be85-a06f390fd055
algebra / test
If a=8, what is the value of (16root[3](a²))^((1)/(3))?
Answer published by the source
Identifiers
task_id
185fbf57-4dff-58e1-a1dd-8d5c539efd34
task_key
algebra--test--185fbf57-4dff-58e1-a1dd-8d5c539efd34
algebra / test
What is the largest integer n such that (1 + 2 + 3 + cdots+ n)^2 < 1^3 + 2^3 + cdots+ 7^3?
Answer published by the source
Identifiers
task_id
18c910cf-4e4c-5797-8881-f94e714db00b
task_key
algebra--test--18c910cf-4e4c-5797-8881-f94e714db00b
algebra / test
Consider the function g(x)=3. Find g(2).
Answer published by the source
Identifiers
task_id
19b1a4eb-d892-5b32-b05d-22a438118ad0
task_key
algebra--test--19b1a4eb-d892-5b32-b05d-22a438118ad0
algebra / test
Let f(x)=2^(x). Find √(f(f(f(f(1))))).
Answer published by the source
Identifiers
task_id
1a6a5622-e029-5b9f-8db5-f6d1a8654a21
task_key
algebra--test--1a6a5622-e029-5b9f-8db5-f6d1a8654a21
algebra / test
Evaluate log₂64.
Answer published by the source
Identifiers
task_id
1bbf4e47-638e-5c61-be93-1932a2c04c40
task_key
algebra--test--1bbf4e47-638e-5c61-be93-1932a2c04c40
algebra / test
Define f(x)=(x²+2x+3)/(x) and g(x)=x³+2. Evaluate g(f(x)) when x=−3.
Answer published by the source
Identifiers
task_id
1fb52372-8cdf-54ac-958a-ccefcb4e0222
task_key
algebra--test--1fb52372-8cdf-54ac-958a-ccefcb4e0222
algebra / test
The function f(x) is graphed below. Each small box has width and height 1.
Answer published by the source
Identifiers
task_id
22c6097d-dc3f-5d4b-9d56-3d61305f70cd
task_key
algebra--test--22c6097d-dc3f-5d4b-9d56-3d61305f70cd
algebra / test
The function f(n)=n²+n+17 for 0≤n≤15 generates prime numbers. What is the value of f(10)−f(9)?
Answer published by the source
Identifiers
task_id
2327bf0e-24d2-5ddd-ad08-604b4c5a7ce9
task_key
algebra--test--2327bf0e-24d2-5ddd-ad08-604b4c5a7ce9
algebra / test
The height (in meters) of a shot cannonball follows a trajectory given by h(t)=−4.9t²+14t−0.4 at time t (in seconds). As an improper fraction, for how long is the cannonball above …
Answer published by the source
Identifiers
task_id
23621cb4-8ebf-5a81-bc98-92a32b66b346
task_key
algebra--test--23621cb4-8ebf-5a81-bc98-92a32b66b346
algebra / test
Let f(x)=7x+5 and g(x)=x−1. If h(x)=f(g(x)), then what is the inverse of h(x)?
Answer published by the source
Identifiers
task_id
23fb8592-4136-58a7-a9fd-34721469f9f9
task_key
algebra--test--23fb8592-4136-58a7-a9fd-34721469f9f9
algebra / test
Given that the point (8,8) is on the graph of y=(1)/(4)f((1)/(2)x), there is one point that must be on the graph of y=f(x). What is the sum of coordinates of that point?
Answer published by the source
Identifiers
task_id
2555d514-3ad1-57f0-9838-d1b45cb0a0ac
task_key
algebra--test--2555d514-3ad1-57f0-9838-d1b45cb0a0ac
algebra / test
Solve for x: (√(12x)+12)(√(3x)−6)=4(x+3)+x−34
Answer published by the source
Identifiers
task_id
260c1785-616d-5a65-82f4-e53ee6576f6a
task_key
algebra--test--260c1785-616d-5a65-82f4-e53ee6576f6a
algebra / test
Let a be the smallest integer satisfying the inequality x²−15<2x, and let b be the largest integer satisfying the same inequality. What is b−a?
Answer published by the source
Identifiers
task_id
276dd654-3756-5df0-9061-d3c87be6d0d5
task_key
algebra--test--276dd654-3756-5df0-9061-d3c87be6d0d5
algebra / test
Eleven pencils cost as much as three pens. If seven pens cost 9.24, what is the cost, in cents, of one pencil?
Answer published by the source
Identifiers
task_id
29427b32-44ef-5b77-9110-a64d8fdf37aa
task_key
algebra--test--29427b32-44ef-5b77-9110-a64d8fdf37aa
algebra / test
In a 8 fluid ounce bottle of Vitamin Water, there are 125 calories. How many…
Answer published by the source
Identifiers
task_id
2a37605a-4c99-599c-92f1-e6ee9f67ef01
task_key
algebra--test--2a37605a-4c99-599c-92f1-e6ee9f67ef01
algebra / test
Let f(x) = begin{cases} k(x) &text{if }x>3, x^2-6x+12&text{if }xleq3. end{cases} Find the function k(x) such that f is its own inverse.
Answer published by the source
Identifiers
task_id
2abc351a-7da4-581b-9152-9d4f5c042010
task_key
algebra--test--2abc351a-7da4-581b-9152-9d4f5c042010
algebra / test
Find the radius of the circle with equation x²+8x+y²−6y=0.
Answer published by the source
Identifiers
task_id
2b511c26-351c-5f27-a24c-455706a297da
task_key
algebra--test--2b511c26-351c-5f27-a24c-455706a297da
algebra / test
Let f(x) = begin{cases} 3x + 5 &text{if }x<-3, 7-4x&text{if }xge -3. end{cases} Find f(−10).
Answer published by the source
Identifiers
task_id
2baed2ba-67f3-58f5-a7cf-244912fe7e2a
task_key
algebra--test--2baed2ba-67f3-58f5-a7cf-244912fe7e2a
algebra / test
The solutions to the equation (x+1)(x+2)=x+3 can be written in the form m+√(n) and m−√(n), where m and n are integers. What is m+n?
Answer published by the source
Identifiers
task_id
2c3a74d0-f1db-5747-8e0e-177b441d8399
task_key
algebra--test--2c3a74d0-f1db-5747-8e0e-177b441d8399
algebra / test
The solutions of x(x−3)=1 may be expressed in the form (a+√(b))/(c) and (a−√(b))/(c), where a, b, and c are prime numbers. Find abc.
Answer published by the source
Identifiers
task_id
2e2a52ab-660c-50c9-9ccc-915324139ed1
task_key
algebra--test--2e2a52ab-660c-50c9-9ccc-915324139ed1
algebra / test
The expression 24x²−19x−35 can be written as (Ax−5)(2Bx+C), where A, B, and C are positive numbers. Find AB−3C.
Answer published by the source
Identifiers
task_id
317c06eb-396c-56f5-91b0-4aeff77de529
task_key
algebra--test--317c06eb-396c-56f5-91b0-4aeff77de529
algebra / test
What is the sum of all of the multiples of 3 between 100 and 200?
Answer published by the source
Identifiers
task_id
31b826e1-b9b2-515f-8b1a-3a0bcbb327a7
task_key
algebra--test--31b826e1-b9b2-515f-8b1a-3a0bcbb327a7
algebra / test
Find (1)/(a−1)+(1)/(b−1), where a and b are the roots of the quadratic equation 2x²−7x+2=0.
Answer published by the source
Identifiers
task_id
320739f0-b686-56cc-8814-2baa8bb536b4
task_key
algebra--test--320739f0-b686-56cc-8814-2baa8bb536b4
algebra / test
Express 0.6(333)¯ as a common fraction.
Answer published by the source
Identifiers
task_id
33ea37be-2883-5cf0-81e2-d578a7d73e53
task_key
algebra--test--33ea37be-2883-5cf0-81e2-d578a7d73e53
algebra / test
Eight pounds of feathers and two ounces of gold together cost 932. Fourteen pounds of feathers and three ounces of gold together cost 1402. What is the cost of five pounds of feath…
Answer published by the source
Identifiers
task_id
36a3210f-bc94-58f9-b326-4861ef50add7
task_key
algebra--test--36a3210f-bc94-58f9-b326-4861ef50add7
algebra / test
Simplify (3−i)(6+2i).
Answer published by the source
Identifiers
task_id
3cd6cbe1-1e79-5d3a-a81f-85388fd91210
task_key
algebra--test--3cd6cbe1-1e79-5d3a-a81f-85388fd91210
algebra / test
Given positive integers x and y such that 2x²y³+4y³=149+3x², what is the value of x+y?
Answer published by the source
Identifiers
task_id
3cec749e-2511-5456-8fad-d4ee4e2bb347
task_key
algebra--test--3cec749e-2511-5456-8fad-d4ee4e2bb347
algebra / test
A horse 64 feet from the center of a merry-go-round makes 27 revolutions. In…
Answer published by the source
Identifiers
task_id
3e644b55-edfd-5c7a-afb2-e085a34e9e72
task_key
algebra--test--3e644b55-edfd-5c7a-afb2-e085a34e9e72
algebra / test
For the arithmetic sequence 1000, ~987, ~974, ~961, ldots , what is the least positive integer in the sequence?
Answer published by the source
Identifiers
task_id
406ac1ae-1c8a-587b-9f61-1f63d08aa8f1
task_key
algebra--test--406ac1ae-1c8a-587b-9f61-1f63d08aa8f1
algebra / test
The point (a,b) lies on the line with the equation 3x+2y=12. When a=4, what is the value of b?
Answer published by the source
Identifiers
task_id
42443402-723b-5c60-a9dd-61da6537eada
task_key
algebra--test--42443402-723b-5c60-a9dd-61da6537eada
algebra / test
Evaluate root[3](12)×root[3](20)×root[3](15)×root[3](60).
Answer published by the source
Identifiers
task_id
424551ea-7dcf-5d18-8a78-32ca0cbd769a
task_key
algebra--test--424551ea-7dcf-5d18-8a78-32ca0cbd769a