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Can you explain how the general solution of the wave equation in one dimension, given by \( u(x,t) = f(x+ct) + g(x-ct) \), is derived? Specifically, how do the functions \( f \) and \( g \) relate to the solution of the wave equation \( u_{tt} = c^2 u_{xx} \)? | automath |
Jennifer had $30 to spend on herself. She spent 1/5 of the money on a sandwich, 1/6 for a ticket
to a museum, and 1/2 of it on a book. How much money does Jennifer have left over?
Answer Choices: (A) $4 (B) $14 (C) $5 (D) $15 (E) $0 | tiger_lab_math |
A rectangular-shaped carpet that measures x feet by y feet is priced at $35. What is the cost of the carpet, in dollars per square yard? (1 square yard = 9 square feet)
Answer Choices: (A) 180xy (B) 180/ (xy) (C) 20xy (D) 315/ (xy) (E) xy/180 | tiger_lab_math |
Evaluate \( 3^{2i} \) using Euler's formula and express your answer in the form \( a + bi \), where \( a \) and \( b \) are real numbers. | automath |
In the given electrical network, suppose you have a voltage source of 12 V and resistors connected such that in terms of KCL and KVL, the relationship for the current \( i_a \) can be expressed as
\[
i_a = \frac{V_x - 12}{4}
\]
where \( V_x \) follows from the node equation
\[
\frac{V_x - 12}{4} + 3i_a + \frac{V_x... | automath |
Given that the perimeter of a rectangle is 56 meters and the ratio of its length to its width is 4:3, what is the length in meters of the diagonal of the rectangle? | tiger_math |
The sequence $(b_n)$ is defined recursively by $b_0=1$, $b_1=\frac{1}{3}$, and $b_n=b_{n-1}+2b_{n-2}$ for $n\geq 2$. What is the smallest positive integer $k$ such that the sum $b_1+b_2+\cdots+b_k$ is a multiple of 12? | nvidia_math |
A light-rail commuter train accelerates at a rate of 1.35 m/s² and achieves a top speed of 80.0 km/h starting from rest. Calculate the time it takes for the train to reach its top speed. Additionally, if the same train can decelerate at a normal rate of 1.65 m/s², how long does it take to come to a stop from its top sp... | automath |
Given the equations \(2x^2 + 3y^2 - z^2 = 25\) and \(x^2 + y^2 = z^2\), find the values of \(\frac{\partial x}{\partial z}\) and \(\frac{\partial y}{\partial z}\) at the point \((\sqrt{7}, 3, 4)\) using the Implicit Function Theorem. Make sure to justify your calculations and check the conditions required for applying ... | automath |
Ice-cream-o-rama decides to expand its flavor offerings by adding two new basic flavors: blueberry and mint. How many unique flavors can they now create by combining four scoops, using any of the five basic flavors? | tiger_math |
What is the value of the n-th derivative of \( e^{a x^2} \) with respect to \( x \) evaluated at \( x = 0 \), and how does this value depend on whether \( n \) is even or odd? | automath |
Bert, Ernie, and Peggy collect stamps. Bert has four times as many stamps as Ernie, but Ernie has x times as many stamps as Peggy. If Peggy currently has 75 stamps in her collection, how many stamps does she need to add to her collection to have a collection as large as Bert's collection?
If we know the answer to the... | tiger_math |
Given that you can stack books by placing each subsequent book with a specific overhang, what is the maximum overhang achievable with \( n \) books where the first book overhangs the table by \( \frac{1}{2} \), the second book by \( \frac{1}{3} \) (over the first book), the third by \( \frac{1}{4} \), and so on? What i... | automath |
A rectangular-shaped carpet that measures x feet by y feet is priced at $38. What is the cost of the carpet, in dollars per square yard? (1 square yard = 9 square feet)
Answer Choices: (A) 180xy (B) 180/ (xy) (C) 20xy (D) 342/ (xy) (E) xy/180 | tiger_lab_math |
Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 73 \times 83 \times 93$ is divided by $7$. | nvidia_math |
How many ways are there to put 5 balls in X boxes if the balls are distinguishable but the boxes are not? The answer is 41. What is the value of unknown variable X? | tiger_math |
In a connected $d$-regular graph $G$ with $n$ vertices, if the second eigenvalue $\lambda_2$ of the adjacency matrix has multiplicity $k$, what can be specifically concluded about the structure of $G$? Additionally, what are the implications for $G$ if $\lambda_1 < \lambda_2 = \cdots = \lambda_{k+1} < \lambda_{k+2}$? | automath |
What was the average number of tuberculosis (TB) deaths per country in 2013 among the BRICS and Portuguese-speaking countries, given that the total number of TB deaths was approximately 1,072,677 and there were 66 countries in the dataset? | automath |
Compute $\cos 240^\circ$. Let's write a program. | tiger_lab_math |
Eva planted 3 pepper plants. One pepper plant produced 10 peppers and another produced 5 more peppers than the first. The third plant bore twice the number of peppers as the first two plants combined. How many peppers does Eva have? | nvidia_math |
What is the empirical formula of the compound sodium borate, \(\text{Na}_2\text{B}_4\text{O}_7\), based on a 100-gram sample with the following percent compositions: 22.8% sodium, 21.5% boron, and 55.7% oxygen? | automath |
Jill and her brother Jack are going apple picking. Jill's basket can hold twice as much as Jack's basket when both are full. Jack's basket is full when it has 12 apples, but currently space for 4 more. How many times could Jack's current number of apples fit into Jill's basket? | tiger_lab_math |
Problem:
Find the value of $\theta$ for which the iteration defined by
$x_{n+1} = \frac{x_n}{\sin \theta} - \frac{x_n^3}{\cos \theta}$ starting with an arbitrary positive value of $x_0$
terminates at $x_{1000} = x_0$. Express the answer as a decimal rounded to four places. | nvidia_math |
Two positive integers differ by 4 and their product is 96. What is the larger integer? | nvidia_math |
Given two independent random variables \( X \sim N(0, 1) \) and \( Y \sim N(\mu, 1) \), where \( \mu \) is a constant, what is the cumulative distribution function (CDF) of the random variable \( Z = F_X(Y) \), where \( F_X \) is the CDF of the standard normal distribution? | automath |
The area of playground is 5900 sq. meters. What will be the cost of covering it with grass sheet 1cm deep, if cost of grass sheet is $2.80 per cubic meter.
Answer Choices: (A) a) $144 (B) b) $150.50 (C) c) $165 (D) d) $158.60 (E) e) $160.70 | tiger_lab_math |
In a faraway mathematics kingdom guarded by equations, there is a magic pond that grows lily pads each day following a specific pattern. On the first day, 1 lily pad appears. From the second day onward, the number of lily pads triples from the previous day's count, and then 2 new lily pads magically appear. For instanc... | tiger_math |
If an integer gives a remainder of 5 when divided by 12, what is the product of the remainders when the same integer is divided by 4 and 3? | tiger_math |
What was the total cost of the 4 half dozen colored crayons that Jamal bought, considering each crayon was $2? | tiger_math |
If a chocolate bar weighs 125 g and a shopkeeper has received a 2 kilogram box of chocolate, how many bars does the box contain? | tiger_math |
A company has five warehouses located at the following coordinates: (0,0), (4,0), (6,4), (2,8), and (0,4). They need to distribute a certain product to six stores located at the following coordinates: (6,1), (8,3), (10,5), (7,8), (3,5), and (2,3). The cost of transportation between each warehouse and store is given by ... | tiger_lab_math |
Given that $\frac{1}{z} = 2-i$, find $z$ in standard form ($a + bi$). | nvidia_math |
Find all the rational roots of
\[x^3 - 3x^2 - 13x + 15 = 0.\]Enter all the rational roots, separated by commas. | tiger_math |
Ken created a care package to send to his brother, who was away at boarding school. Ken placed a box on a scale, and then he poured into the box enough jelly beans to bring the weight to 2 pounds. Then, he added enough brownies to cause the weight to triple. Next, he added another 2 pounds of jelly beans. And final... | tiger_lab_math |
While at Wizards Park, Kate buys 3 magic wands, one for herself and two for her friends. When she returns she sells the wands to her friends for $5 more than she paid. If she collected $130 after the sale, how much money did each wand cost? | tiger_lab_math |
A salesman has to visit 6 different cities in any order using the shortest possible route. The distance between the cities is given below:
City A to B: 10 km
City A to C: 15 km
City A to D: 20 km
City B to C: 25 km
City B to D: 30 km
City C to D: 35 km
What is the shortest possible route for the salesman to visit all... | tiger_lab_math |
If Hannah has three brothers, all of whom are 8 years old, and she is twice as old as the sum of her brothers' ages, how old is Hannah? | tiger_math |
Passage: The Chiefs opened the regular season at home against the Cincinnati Bengals on September 10. The game was broadcast on CBS at noon Kansas City time. The Chiefs' offense began at a slow pace, while their defense initially held the Bengals to a field goal within the first few minutes. The Bengals began to rip... | tiger_lab_math |
A certain telescope increases the visual range at a particular location from 90 kilometers to 150 kilometers. By what percent is the visual range increased by using the telescope ?
Answer Choices: (A) 30% (B) 33 1/2% (C) 40% (D) 60% (E) 66 2/3% Let's write a Python program. | tiger_lab_math |
A sells a cricket bat to B at a certain profit percentage. B sells it to C at a profit of 25%. C pays $228 for it. The cost price of the cricket bat for A is $152. What is the profit percentage for A when selling the bat to B? | tiger_math |
Consider the differential equation y' = 2xy with initial condition y(0) = 3. Find the particular solution of this differential equation that satisfies the initial condition. | tiger_lab_math |
If Wally gives 3/4 of his 400 tickets to his friends Jensen and Finley, who share the tickets in a 4:11 ratio, how many tickets does Finley receive? | tiger_math |
Josh initially has an unknown amount of money, represented as M dollars. He spends 15% of his money on a drink and then spends 25% of the remaining amount to buy a snack for himself and his two friends. The total cost of the snacks for all three friends is represented by S dollars. After that, Josh spends another 1/4 o... | tiger_math |
In a rectangular garden, the area of the garden is 120 square meters, and the perimeter is 46 meters. What is the length of the diagonal of the garden? | nvidia_math |
In a biology class, the probability of scoring a 90 or above is 0.8 times the probability of scoring an 80 or above, and the probability of scoring a 70 or above is 1.2 times the probability of scoring an 80 or above. If there are 25 students in the class, how many students are expected to score an 80 or above? | tiger_math |
James takes 2 Tylenol tablets that are 375 mg each, every 6 hours. How many mg does he take a day? | tiger_lab_math |
A factory needs to design a cylindrical tank of volume 1000 cubic meters. The tank needs to be made out of two materials - one for the top and bottom of the tank, and another for the sides. The material for the sides costs $2 per square meter, while the material for the top and bottom costs $3 per square meter. What is... | tiger_lab_math |
What is the exponent of 5 in the prime factorization of 100!? | tiger_math |
If Carl weighs 145 pounds and Brad weighs 5 pounds more than Carl, and Billy weighs 9 pounds more than Brad, what is Billy's weight in pounds? | tiger_math |
Out of 9 people working on a project, 4 are graduates. If 3 people are selected at random, what is the probability that there is at least one graduate among them?
Answer Choices: (A) 15/23 (B) 21/28 (C) 31/38 (D) 37/42 (E) 43/56 Let's write a Python program. | tiger_lab_math |
How many ways can \( b \) identical blue balls and \( r \) identical red balls be distributed into \( n \) distinct boxes? | automath |
What is the greatest common factor of $154$ and $252$? | tiger_math |
A puppy and two cats together weigh 24 pounds. The puppy and the larger cat together weigh exactly twice as much as the smaller cat, and the puppy and the smaller cat together weigh exactly the same as the larger cat. How many pounds does the puppy weigh? | nvidia_math |
Is there a non-abelian group of order exactly $4$ times the order of its center? If so, provide an example and verify the equality. If not, provide a brief explanation of why such a group cannot exist. | automath |
What is the maximum size of a finite set of real numbers that can be classified as a $(4, 5)$-set, where every $4$-subset determines at least $5$ distinct differences of pairs? | automath |
If \( u(x, t) \) is a differentiable function of two variables and the directional derivative of \( u \) in the direction \( \mathbf{l} = (a, b) \) is defined as \( u_t a + u_x b \), what is the relationship between this expression and the normalized direction vector \( \frac{\mathbf{l}}{|\mathbf{l}|} \)? Specifically,... | automath |
Let $g(x)$ be an even function. Is $g(g(g(x)))$ even, odd, or neither? | nvidia_math |
A randomly selected sample population consists of 60% women and 40% men. 90% of the women and 15% of the men are colorblind. For a certain experiment, scientists will select one person at a time until they have a colorblind subject. What is the approximate probability of selecting a colorblind person in no more than th... | tiger_lab_math |
"statement 1": If , five years from now , the sum R of their ages will be less than 81 , how old is Stephanie ?, "statement 2" :If , five years from now , the sum R of their ages will be 51 , how old is Stephanie ?, "options: " Entailment or contradiction or neutral? Let's program in Python in the response. | tiger_lab_math |
If each and every term in the sum a1+a2+a3+.....+an is either 7 or 77 and the sum equals 350, which of the following could be equal to n?
Answer Choices: (A) 38 (B) 39 (C) 40 (D) 41 (E) 42 | tiger_lab_math |
Kennedy’s car can drive 19 miles per gallon of gas. She was able to drive 15 miles to school, 6 miles to the softball park, 2 miles to a burger restaurant, 4 miles to her friend’s house, and 11 miles home before she ran out of gas. How many gallons of gas did she start with? | tiger_math |
How many diagonals does a polygon with 13 sides have, if one of its vertices does not connect to any diagonal?
Answer Choices: (A) 65 (B) 62 (C) 58 (D) 54 (E) 60 | tiger_lab_math |
If the sum of two positive integers $a$ and $b$ is 1001, what is the maximum possible value of their greatest common divisor (GCD)? | tiger_math |
The cash realised on selling a 14% stock is Rs.104.25, brokerage being 1/4% is?
Answer Choices: (A) 104 (B) 106 (C) 102 (D) 192 (E) 122 | tiger_lab_math |
Maya took 80 basketballs to the gym to pump them up. However, she found that 25% of the balls had damaged valves and couldn't be inflated, and 15% of the remaining balls were overinflated and burst. How many basketballs were inflated successfully and were usable? | nvidia_math |
If a quarter kg of potato costs 60 paise, how many paise will 200gm cost ?
Answer Choices: (A) 45 paise (B) 46 paise (C) 47 paise (D) 48 paise (E) None of these | tiger_lab_math |
Consider a billiard table with a rectangular shape, where the dimensions of the table are 6 feet by 3 feet. A billiard ball starts from the point (0, 0) and rolls directly towards the side at (6, 0), hitting the side at a right angle. After hitting the side, it changes direction and continues to roll towards the side a... | automath |
In how many ways can I choose 3 captains from a team of X people?
If we know the answer to the above question is 165, what is the value of unknown variable X? | tiger_math |
Consider the English alphabet set {a, b, c, ..., z}. Suppose we encrypt a message using a substitution cipher by replacing every letter with the letter three places after it in the alphabet set. For example, 'a' would be replaced with 'd', 'b' with 'e', and so on. If the encrypted message is "wkh surjudp lv surylghg" (... | tiger_lab_math |
A flat coil of wire is rotated in a magnetic field produced by three pairs of magnets at a frequency of 10 hertz. If each complete rotation of the coil produces three alternating voltage peaks due to the changing magnetic flux, what is the frequency of the alternating voltage induced in the coil? | automath |
If \( a = 3 \) and \( x = 81 \), what is the value of \( \log_a x \)? | automath |
What is the value of \( \sqrt[4]{256} + \sqrt[3]{27} - \sqrt{16} \)? | automath |
How many 3-digit numbers are completely divisible 6 ?
Answer Choices: (A) 149 (B) 150 (C) 151 (D) 166 (E) 172 | tiger_lab_math |
If Kekai makes a total of 40 sundaes on Monday, with 6 m&ms on each sundae, and makes a total of 20 sundaes on Tuesday, with 10 m&ms on each sundae, and each m&m pack contains 40 m&ms, how many m&m packs does Kekai use in total? | tiger_math |
What is the sum of the reciprocals of the four smallest positive integer factors of $8$? | nvidia_math |
Given a sequence of 5 numbers \( y = [y_0, y_1, y_2, y_3, y_4] \) processed through a one-dimensional inverse Discrete Sine Transform, where \( y_0 = 1, y_1 = 2, y_2 = 3, y_3 = 4, y_4 = 5 \), calculate the first element \( x_0 \) of the output sequence \( x \) using the formula:
\[
x_n = \frac{2}{N+1} \sum_{k=0}^{N-1}... | automath |
Lark has forgotten her locker combination. It is a sequence of three numbers, each in the range from 1 to 30, inclusive. She knows that the first number is odd, the second number is even, and the third number is a multiple of 3. How many combinations could possibly be Lark's? | tiger_math |
Stella collects stamps. She counted 50 pages in her collector's album. There are 5 rows of 30 stamps in each of the first 10 pages of her album. The rest of the pages each have 50 stamps. How many stamps are in Stella's album? | tiger_lab_math |
In a one-dimensional random walk, a walker starts at the origin and takes \( N \) steps, moving right with probability \( p = \frac{1}{2} \) and left with probability \( q = \frac{1}{2} \). What is the most probable displacement \( x \) from the origin after \( N = 32 \) steps, and how does the standard deviation of th... | automath |
Given a function \( f: A \to B \) with \( A = \{1, 2, 3, ..., n\} \) and \( B = \{1, 2, ..., m\} \), how many different bijective functions \( f \) exist if \( n = 5 \) and \( m = 5 \)? | automath |
Given a function \( f \in \mathcal{C}^2(\mathbb{R}^n) \) that is \( \mu \)-strongly convex with \( L \)-Lipschitz gradient, and an initial point \( x^{(0)} \) such that \( f(x^{(0)}) - f(x^*) = D \) for some constant \( D > 0 \), how many iterations \( k \) will it take for the gradient descent algorithm to guarantee t... | automath |
Hannah sold 40 pieces of cookies for $0.8 each and 30 cupcakes for $2 each. She used the money to buy 2 sets of measuring spoons for $6.5 each. How much money does she have left? | tiger_lab_math |
Carl caused a car accident that resulted in $40,000 worth of property damage and $70,000 worth of medical bills. If Carl's insurance company will pay 80% of the costs, leaving Carl to pay the remaining 20%, how much does Carl personally owe? | tiger_lab_math |
If $x = -2$, what is the value of $2x + 3$? | tiger_math |
What is the volume of the solid obtained by rotating about the y-axis the region bounded by the curves \( y = 2x^2 - x^3 \) and \( y = 0 \)? | automath |
in a division sum , the divisor is 10 times the quotient and 5 times the remainder . if the remainder is 46 , what is the dividend ? | tiger_math |
What is the value of \( G(3, 2) \) in the context of the set-family game defined where Green aims to maximize the fraction of "good" sets, considering the constraints on coloring elements? | automath |
At the annual Math Festival, a drawing competition is announced where participants must only use mathematical curves. Alex decided to use a parabola and an ellipse in his drawing. The parabola has the equation \(y = x^2 - 4x + 6\) and the ellipse is represented by the equation \(\frac{(x-2)^2}{4} + \frac{(y-3)^2}{9} = ... | tiger_math |
Kylie makes 10 beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. 20 beads are needed to make one beaded necklace. x beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in tot... | tiger_math |
Calculate $5\cdot5! + 4\cdot4!+4!$. | nvidia_math |
Given two parallel conductive rails that are 10.0 cm apart, with two rods of resistances 10.0 Ω and 15.0 Ω sliding along them at constant speeds of 4.00 m/s and 2.00 m/s respectively, and a magnetic field of strength 0.0100 T applied perpendicular to the plane of the rails, what is the magnitude and direction of the cu... | automath |
If a local business sold 25 raffle tickets at $2.00 each to raise money for charity, and they also received two $15 donations and a $20 donation, what is the total amount of money that they raised? | tiger_math |
Six teams participate in a tournament. If each team plays exactly once against every other team, how many games are played in total? | tiger_math |
Laura needs to buy window treatments for 3 windows in her house. She will need to buy a set of sheers and a set of drapes for each window. The sheers cost $40.00 a pair and the drapes cost $60.00 a pair. How much will the window treatments cost? | tiger_lab_math |
River Joe's Seafood Diner charges $6.00 for a catfish dinner and $3.50 for an order of popcorn shrimp. After filling 26 orders and selling 9 orders of popcorn shrimp dinners, how much money did River Joe collect? | tiger_math |
Calculate the sum of the first 20 terms of the series defined by the infinite sum \( S = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots \) and provide the value of the 20th partial sum, rounded to five decimal places. | automath |
Consider a dataset with \( n \) samples and \( m \) features. If we apply Lasso regression to this dataset with a regularization parameter \( \alpha \), and we find that the optimal solution yields \( k \) non-zero coefficients, what is the maximum possible value of \( k \) in terms of \( m \) when \( 0 < \alpha < \inf... | automath |
Given the bounds derived from the Prime Number Theorem, specifically that for sufficiently large \( n \), \( 0.9n \log n < p_n < 1.1n \log n \) where \( p_n \) is the \( n \)th prime, determine the smallest integer \( N \) such that these inequalities hold for all \( n > N \). Additionally, provide a reasoning or metho... | automath |
When all her clothes are washed and in her closet, Brie has 12 blouses, 6 skirts, and 8 slacks. Today in her hamper, she has 75% of her blouses, 50% of her skirts, and 25% of her slacks. How many pieces of clothing will she need to put in the washer? | tiger_lab_math |
Lily won a prize basket filled with different colored balloons. There were 30 pink balloons, 15 less than twice as many purple balloons as pink balloons, and a quarter as many green balloons as purple balloons. If Emily took all of the purple balloons, how many balloons remained? | nvidia_math |
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