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Consider the damped spring-mass system described by the ordinary differential equations:
\[
\frac{dx_1}{dt} = x_2,
\]
\[
\frac{dx_2}{dt} = -k x_1 - b x_2,
\]
where \(x_1(t)\) is the position, \(x_2(t)\) is the velocity, \(k\) is the spring constant, and \(b\) is the damping coefficient. If the initial conditions are... | automath |
The average of 12, X and $x$ is 18. The value of $x$ is 21. What is the value of unknown variable X? | tiger_math |
In what time will a train 100 m long cross an electric pole, it its speed be 144 km/hr?
Answer Choices: (A) 2.5 sec (B) 4.25 sec (C) 5 sec (D) 12.5 sec (E) 12 sec | tiger_lab_math |
In a preordered set \( P \) with \( n \) elements, where the preorder is represented by a relation \( \preceq \), how many distinct reflexive and transitive relations can be defined on the set, given that \( f(n) \) represents the number of such relations for size \( n \)? Calculate \( f(3) \) and \( f(4) \). | automath |
Given the polynomial function \( f(x) = -4.1261 + 0.99213x - 0.562172x^2 - 3.7120x^3 + 3.06508x^4 \), find the value of the first derivative \( f'(0) \) at \( x = 0 \). What is the coefficient of \( x \) in the Chebyshev expansion of \( f(x) \)? Lastly, calculate the average value of \( f(x) \) over the interval \([-1,... | automath |
A flagpole stands vertically on level ground, and the distance from the flagpole to a point on the ground is 15 meters. The angle of elevation from the ground to the top of the flagpole is 60 degrees. How tall is the flagpole? | tiger_lab_math |
Compute $16\left (\frac{125}{2}+\frac{25}{4}+\frac{9}{16}+1\right)$. Let's write a Python program. | tiger_lab_math |
How many cubic feet are in one cubic yard? One yard is equal to three feet.
[asy]import three;
currentprojection=orthographic(1/2,1/2,1);
draw((0,0,0)--(10,0,0)--(10,-10,0)--(0,-10,0)--cycle);
draw((0,0,10)--(10,0,10)--(10,-10,10)--(0,-10,10)--cycle);
draw((0,0,0)--(0,0,10));
draw((10,0,0)--(10,0,10));
draw((10,-10,0)... | tiger_math |
Let \( A = \{1\} \) and \( B = \{1.1\} \) be two finite subsets of the real numbers. Define the distance \( d(A, B) \) between sets \( A \) and \( B \) using the Hausdorff distance. Calculate \( d(A, B) \) and determine whether \( d(A, B) \) is less than, greater than, or equal to \( d(A, \{2\}) \). What is the conclus... | automath |
A recipe calls for $4 \frac{1}{2}$ cups of flour. If you want to double the recipe, then how many cups of flour do you need? | tiger_math |
Holly needs to take 2 insulin pills per day, 3 blood pressure pills per day, and twice as many anticonvulsants as blood pressure pills each day. How many pills does Holly take in a week? Let's write a Python program to solve it. | tiger_lab_math |
each month a retailer sells 100 identical items . on each item he makes a profit of $ 30 that constitutes 10 % of the item ' s price to the retailer . if the retailer contemplates giving a 5 % discount on the items he sells , what is the least number of items he will have to sell each month to justify the policy of the... | tiger_lab_math |
Michelle took a taxi from her office to her home, which is a distance of 4 miles. The taxi has a ride fee of $2 as soon as you enter, and a charge of $2.5 per mile. What is the total amount that Michelle paid for her ride? | tiger_math |
A cistern can be filled by a tap in 3 hours while it can be emptied by another tap in 10 hours. If both the taps are opened simultaneously, after how much time will the cistern get filled? | tiger_math |
Given a triangular hole created by a dropped cube, with side lengths of \(68\), \(75\), and \(77\), calculate the depth of the hole. | automath |
The Regex Rockets racing team is planning a multi-city tour for upcoming races. They have mapped out a route starting at their home base in Regexville, then proceeding to three race cities: Algorithmtown, Loopburg, and Functionford. First, they drive to Algorithmtown, which is 120 miles east and 150 miles north of Rege... | tiger_math |
Lou Senior took 3 cookies out of the cookie jar and ate them. Since he didn't get caught by his wife, he went back the next day and took another 3 cookies out of the jar. But after eating just one of the cookies, he felt guilty about it and put the other two cookies back. His son, Louie Junior saw that his Dad was e... | tiger_lab_math |
You have $45 to spend at the amusement park. You buy a ticket for the Ferris wheel for $5, a bag of popcorn for $4, and spend 1/4 of what's left on a game. How much money do you have left? | nvidia_math |
In the figure, $m\angle A = 28^{\circ}$, $m\angle B = 74^\circ$ and $m\angle C = 26^{\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x - y$?
[asy]
size(150);
draw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));
draw((0,5)--(2,2)--(5,5)--(12,-2)--(15,5),linewidth(.7));
l... | tiger_math |
What is the value of the integral
$$\int_{\mathbb{R}^3} \nabla^2\left(\frac{1}{|\mathbf{x}|}\right) \phi(\mathbf{x}) \, d\mathbf{x}$$
for a smooth test function $\phi$ that is compactly supported and does not contain the origin, and how does this relate to the behavior of $\nabla^2 (1/|\mathbf{x}|)$ as a distributi... | automath |
Determine the remainder when the sum of the digits of the number $12^{13}$ is divided by 7. | nvidia_math |
If nn is a positive integer and (n+1)(n+3)(n+1)(n+3) is odd, then (n+2)(n+4)(n+2)(n+4) must be a multiple of which one of the following?
Answer Choices: (A) 2 (B) 4 (C) 6 (D) 8 (E) 9 | tiger_lab_math |
Find the integer $m$, $1 \le m \le 10$, such that \[m \equiv 23456 \pmod{11}.\] | tiger_math |
Let \( X \) be a smooth projective variety, and let \( i: X \rightarrow X \times X \) be the diagonal morphism. If \( i_* \mathcal{O}_X \) and \( \mathcal{O}_\Delta \) are canonically isomorphic in the derived category \( D^b(X) \), what is the derived functor \( \mathbf{R} i_* \mathcal{O}_X \) in terms of the derived ... | automath |
A starfish informed me that his kelp forest home is $321_{5}$ years old. How many years is this in base ten? | tiger_math |
Solve the equation for \( x \):
\[
5(2x + 3) - 3(4 - x) = 2(x + 6)
\] | automath |
A pen factory manufactures 6 types of pens. They package 3 pens of each type in a pack. The factory has the capacity to produce enough pens to fill 8 packs per hour. How many pens does the factory produce in 6 hours? | nvidia_math |
In the magical land of Mathlandia, three types of mystical flowers bloom in a garden: Sunfloria that doubles every day, Moonblossom that triples every day, and Starpetal that quadruples every day. At sunrise on the first day, the garden starts with 2 Sunfloria, 1 Moonblossom, and 1 Starpetal. How many Starpetal flowers... | tiger_math |
Find the area of the quadrilateral formed by the points $(2,3),$ $(5,6),$ $(8,1),$ and $(4,2)$ by using the Shoelace formula. | nvidia_math |
When Ava started her savings account, her parents deposited $\$3000$, which was invested at an interest rate of $3\%$ per year, compounded yearly. How much money will Ava have when she withdraws the money at the age of $22$? Give your answer to the nearest hundredth of a dollar. | nvidia_math |
Does the series \(\sum\limits_{j=1}^\infty \frac{(1+\frac{1}{j})^{2j}}{e^j}\) converge or diverge? Determine the criteria you would use to conclude your answer, and explain your reasoning. | automath |
Consider a sorted \( n \times n \) matrix \( M \) where each row and column is sorted in increasing order. Given two integers \( k_{\min} \) and \( k_{\max} \), what is the maximum number of elements that can fall within the interval \([k_{\min}, k_{\max}]\) if \( k_{\min} \) is less than or equal to the smallest eleme... | automath |
Rationalize the denominator of $\frac{1-\sqrt{5}}{1+\sqrt{5}}$. Express your answer in the form $A + B\sqrt{C}$, where $A$, $B$, and $C$ are integers. What is the value of $A+B+C$? | tiger_math |
I run at a constant pace, and it takes me 20 minutes to run to the store from my house. If the store is 2.5 miles away, and my friend's house is 1.25 miles away from my house, how many minutes will it take me to run from my house to my friend's house? | tiger_math |
A chef is preparing garnishes for a special event and wants to add a few rosemary sprigs to each appetizer plate. However, the supply of rosemary is limited, so the chef decides to split some sprigs into smaller pieces to make the most of what's available. If 10 plates are garnished with a full rosemary sprig and 15 pl... | nvidia_math |
A certain lottery ticket allows you to choose a four-digit number (each digit can be from 0 to 9). If you choose a number that matches the winning number exactly, you win $1000. If your number differs from the winning number by exactly one digit, you win $10. If your number differs from the winning number by exactly tw... | nvidia_math |
In a competitive examination in State A, 6% candidates got selected from the total appeared candidates. State B had an equal number of candidates appeared and 7% candidates got selected with 81 more candidates got selected than State A. What was the number of candidates appeared from each state? | tiger_math |
Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary? Let's write a Python program. | tiger_lab_math |
Given a Cournot oligopoly consisting of three identical firms, each with a marginal cost of \(C = 10\) and facing a market demand represented by the equation \(P(Q) = 300 - 10Q\), where \(Q = q_1 + q_2 + q_3\) is the total quantity produced by the three firms, calculate the Cournot equilibrium quantities produced by ea... | automath |
A part-time employee whose hourly wage was increased by 10 percent decided to reduce the number of hours worked per week so that the employee's total weekly income would remain unchanged. By what percent should the number of hours worked be reduced?
Answer Choices: (A) 9.09 (B) 10 (C) 11 (D) 12 (E) 13 | tiger_lab_math |
Passage: Hoping to rebound from their loss to the Chargers, the Raiders flew to Arrowhead Stadium for a Week 2 AFC West duel with the Kansas City Chiefs. Oakland would trail in the first quarter as Chiefs kicker Ryan Succop got a 23-yard field goal. The Raiders would tie the game in the second quarter as kicker Seba... | tiger_lab_math |
What is the equation of the normal line to the parabola \( y = x^2 - 5x + 4 \) that is parallel to the line \( x - 3y = 5 \)? | automath |
The population of a town increased from 1,24,300 to 1,62,400 in a decade. The average percent increase of population per year is:
Answer Choices: (A) 4.37% (B) 4% (C) 2.3% (D) 3.75% (E) None | tiger_lab_math |
Betty is saving money for a new wallet which costs x. Betty has only half of the money she needs. Her parents decided to give her $15 for that purpose, and her grandparents twice as much as her parents. Betty needs 5 more money to buy the wallet. What is the value of unknown variable x? What is the value of unknown var... | tiger_math |
The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find
\[\frac{f(5) - f(1)}{f(2)}.\] | nvidia_math |
Jane bought 2 skirts for $13 each. She also bought 3 blouses for $6 each. She paid the cashier $100. How much change did she receive? | tiger_lab_math |
Sara got her first paycheck of two weeks of work. She had worked 40 hours a week at $11.50 per hour. The first thing she did was buy a new set of tires for her car for $410. How much money was she left with? | tiger_lab_math |
Find the area of a square whose side length is $\frac{2}{3}$. | tiger_math |
Given the equation $x - y = 10$, where $x$ and $y$ are non-negative integers, how many distinct ordered-pair solutions $(x, y)$ exist? | tiger_math |
find the area of the quadrilateral of one of its diagonals is 22 cm and its off sets 9 cm and 6 cm ? Let's write a program. | tiger_lab_math |
It will take 16 days for Mary to complete a certain task alone. She worked for 8 days before she was joined by her sister. both of them completed the remaining task in 2 and half days. if her sister had joined her when she started the task, how many days would it have taken?
Answer Choices: (A) 6 (B) 8 (C) 2 (D) 5 (E) ... | tiger_lab_math |
Mary is writing a story, and wants her 60 characters to be split according to their initials. Half of her characters have the initial A, and half of this amount have the initial C. Mary wants the rest of her characters to have the initials D and E, but she wants there to be twice as many characters with the initial D a... | tiger_lab_math |
Determine the equation of the line that is the perpendicular bisector of the line segment joining the points $(-2, 5)$ and $(4, -1)$. | tiger_math |
The average of 30 numbers is 48. If three numbers namely 45, 55 and 43 are discarded, the average of remaining numbers is?
Answer Choices: (A) 47 (B) 47.03 (C) 48.03 (D) 48 (E) 49 | tiger_lab_math |
If Nathan played amateur baseball for 3 hours every day for two weeks and Tobias played for 5 hours every day for one week, what is the total number of hours that Nathan and Tobias played together? | tiger_math |
What is the smallest integer \( k \) such that every set of integers \( A \) with positive upper density contains an arithmetic progression of length \( k \)? Provide an example to illustrate your answer. | automath |
Let $y$ be a positive real number such that $y + \frac{1}{y} = 39.$ Find
\[\sqrt[3]{y} + \frac{1}{\sqrt[3]{y}}.\] | nvidia_math |
Two Equal Amounts of Money are lent out at 6% and 5 % simple Interest respectively at the same time. The former is recovered two years earlier than the latter and the amount so recovered in each case is Rs.2800. Determine the amount that is lent out?
Answer Choices: (A) 3348 (B) 368 (C) 37077 (D) 1750 (E) 2701 | tiger_lab_math |
Given a power series \( f(x) = \sum_{n \geq 1} a_n \frac{x^n}{n!} \) with \( a_1 \neq 0 \), if \( g(x) \) is its series reversion such that \( f(g(x)) = g(f(x)) = x \), how do the coefficients \( b_n \) of the series reversion \( g(x) = \sum_{n \geq 1} b_n \frac{x^n}{n!} \) relate to the phylogenetic trees on \( \{1, \... | automath |
Triangle DEF is an isosceles triangle with a base angle of 45 degrees and O is the center of its circumscribed circle. If the area of the circle is $9\pi$ sq cm, what is the area, in square centimeters, of triangle DEF? Express your answer in simplest radical form.
Note: The area of a circle with radius $r$ is $\pi r^... | nvidia_math |
Seven distinct points are identified on the circumference of a circle. How many different quadrilaterals can be formed if each vertex must be one of these 7 points? | tiger_math |
If John initially receives $100 from his uncle and gives his sister Jenna 1/4 of that money, then goes on to spend $40 on groceries, how much money does John have left? | tiger_math |
a room is 7 meters 20 centimeters in length and 4 meters 32 centimeters in width . find the least number of square tiles of equal size required to cover the entire floor of the room . Let's program in Python in the response. | tiger_lab_math |
Let $p$ be a prime number such that $p^2 + 2p + 2$ is also prime. Find the remainder when $p^3$ is divided by $p^2 + 2p + 2$. | nvidia_math |
Consider the transformation of the coordinates given by the equations \(x'^{\mu} = ax^{\mu} + b\). If we let \(b=0\) and \(a=2\), express the transformation of an infinitesimal displacement \(v^{\mu}\) and find the transformed components of the vector if \(v^1 = 3\) and \(v^0 = 5\). What are the values of \(v'^0\) and ... | automath |
Find the minimum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x.$ | nvidia_math |
What is the value of $x$ in the expression $\frac{5x+2}{2} - \frac{3x-1}{2}$ if it is equal to $\frac{7}{2}$? | nvidia_math |
In triangle $ABC,$ $\sin A = \frac{3}{5}$ and $\cos B = \frac{5}{13}.$ Find $\cos C.$ | nvidia_math |
Emily visited a butterfly garden and spotted 23 monarch butterflies and 38 non-monarch butterflies. Out of those 38 non-monarch butterflies, a quarter of them had distinctive blue markings. Out of the 23 monarch butterflies, 17 of them did not have any blue markings. How many more non-monarch butterflies with blue mark... | nvidia_math |
What is the least common multiple of 6, 8, and 10? | nvidia_math |
Kristy, a sales representative earns a basic salary of $7.50 per hour plus a 16% commission on everything she sells. This month, she worked for a certain number of hours and sold $25000 worth of items. Her monthly budget for food, clothing, rent, transportation, bills and savings is 95% of her total monthly earnings an... | tiger_math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(x) + g(y) = g(xy)$ for all real numbers $x$ and $y.$ Suppose that $g(1) = 1.$ Let $a$ be the number of possible values of $g(0),$ and let $b$ be the product of all possible values of $g(0).$ Find $a \times b.$ | nvidia_math |
In triangle $ABC,$ $AB = 10,$ $BC = 15,$ and $AC = 20$ units. A point $P$ is chosen on segment $AB$ such that the area of triangle $APC$ is equal to the area of triangle $BPC.$ What is the length of segment $BP$? | nvidia_math |
Seven adults attended a concert with nine teenagers. Adult tickets cost $10 each, and the total bill was $133. How many dollars is one teenager's ticket? | nvidia_math |
Given two parents with real-valued positions \( p_1 = 2.55 \) and \( p_2 = 9.92 \) in a 1-dimensional genetic algorithm where the offspring are generated using the formulas \( c_1 = 0.5p_1 + 0.5p_2 \), \( c_2 = 1.5p_1 - 0.5p_2 \), and \( c_3 = -0.5p_1 + 1.5p_2 \), calculate the values of \( c_2 \) and \( c_3 \). After ... | automath |
What is the ratio of the diameter of the exoplanet GJ 1132b to that of Earth, given that GJ 1132b has a diameter of 9200 miles, and if Earth’s diameter is approximately 7926 miles? Express this ratio as a fraction in simplest form. | automath |
Emily purchased a cold drink before walking to the park. If the drink will lose its chill in 15 minutes, the park is 20 blocks away, and each block is 1/10th of a mile, how fast does Emily need to walk (in miles per hour) to get to the park before the drink loses its chill? | nvidia_math |
It takes 1.75 hours to clean a car and 0.25 hours to clean a motorcycle. What is the number of minutes it takes to clean 4 cars and 6 motorcycles? | nvidia_math |
Simplify $\displaystyle\frac{1-\sqrt{3}i}{2+\sqrt{2}i}$, where $i^2 = -1.$ | nvidia_math |
A sells a cricket bat to B at a profit of 20%. B sells it to C at a profit of 25%. The cost price of the cricket bat for A is $148. How much does C pay for the cricket bat? | tiger_math |
In the expansion of $(1+x)^{2014}$, what is the sum of the coefficients of the terms with even powers of $x$? | nvidia_math |
A right triangle has one leg measuring 15 cm. The angle opposite this leg measures $60^\circ$. Find the length of the hypotenuse in centimeters. | tiger_math |
Solve for $p$: $\frac 56 = \frac n{72} = \frac {m+n}{84}= \frac {p - m}{120}$. | nvidia_math |
For what values of $j$ does the equation $(2x+7)(x-5) = -43 + jx$ have exactly one real solution? Express your answer as a list of numbers, separated by commas. | tiger_lab_math |
A line is parameterized by a parameter $s$, so that the vector on the line at $s = 0$ is $\begin{pmatrix} -2 \\ 1 \end{pmatrix},$ and the vector on the line at $s = 4$ is $\begin{pmatrix} 6 \\ 5 \end{pmatrix}.$ Find the vector on the line at $s = -3$. | nvidia_math |
Train X crosses a stationary train Y in 60 seconds and a pole in 25 seconds with the same speed. The length of the train X is 300 m. What is the length of the stationary train Y ?
Answer Choices: (A) 360 m (B) 420 m (C) 460 m (D) 320 m (E) 340 m | tiger_lab_math |
Janice can type 6 sentences per minute. Today at work, Janice continued working on a paper she started typing yesterday. She typed for 20 minutes, took a break, and typed 15 minutes longer. She then had to erase 40 sentences she had typed incorrectly. After a meeting, she typed for 18 minutes more. In all, the paper ha... | tiger_math |
What number comes next in this Number series.
4 4 6 9 7 8 8 ?
*Hint: check calende
Answer Choices: (A) 1 (B) 3 (C) 6 (D) 7 (E) 8 | tiger_lab_math |
Given that ship A sails with a velocity of \( \vec{v_A} = (30i + 40j) \) km/h and ship B sails with a velocity of \( \vec{v_B} = (15i + 12j) \) km/h, and at one instant the position vector of ship B relative to ship A is \( \vec{r_{BA}} = (5i + 12j) \) km, determine the following:
1. Calculate the velocity of ship A r... | automath |
Emily purchases a vase and some decorative candles. The vase costs $45. The candles cost 75% as much as the vase. How much did she pay for everything? | nvidia_math |
The polynomial $P(x) = x^3 + ax^2 + bx + c$ is such that $P(1) = 1$, $P(2) = 2$, and $P(3) = 3$. Find the remainder when $P(x)$ is divided by $x - 4$. | nvidia_math |
Omi is twice as old as Kimiko. Arlette is 3/4 times as old as Kimiko. If Kimiko is 28 years old, calculate the average age of the three? | tiger_lab_math |
A box contains twelve marbles, some of which are blue and the rest of which are green. When two marbles are drawn at random at the same time, the probability that both marbles are blue is $\frac{1}{22}$. How many marbles in the box are blue? | nvidia_math |
Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that
\begin{align*}
a^2 + b^2 + c^2 + d^2 &= 3, \\
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 9.
\end{align*} | tiger_math |
How many digits does the smallest repeating block in the decimal expansion of $\frac{5}{7}$ contain? | tiger_math |
Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\$$20, what is the total amount of money shared? Let's program in Python in the response. | tiger_lab_math |
If 60% of the students at school are girls and the number of boys is 300, how many girls are at the school? | nvidia_math |
If Jack bought an ice cream cone that will melt in 10 minutes and the beach is 16 blocks away, with each block being 1/8th of a mile, what speed does Jack need to jog at (in miles per hour) in order to reach the beach before the ice cream melts? | tiger_math |
the product of two numbers is 120 and the sum of their squares is 289 . the sum of the number is : ? Please respond by writing a program in Python. | tiger_lab_math |
each month a retailer sells 100 identical items . on each item he makes a profit of $ 50 that constitutes 10 % of the item ' s price to the retailer . if the retailer contemplates giving a 5 % discount on the items he sells , what is the least number of items he will have to sell each month to justify the policy of the... | tiger_lab_math |
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