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What is the best SciPy method to find the root of a nonlinear function with an analytically known sparse Jacobian, considering the function can be reformulated as a least squares minimization problem? Additionally, how can I incorporate the sparsity of the Jacobian when using this method? | automath |
Quadrilateral $ABCD$ with vertices of $A(0,2)$, $B(1,4)$, $C(7,1)$, and $D(3,0)$ is rotated by $90^\circ$ clockwise around the origin to quadrilateral $A'B'C'D'$. This quadrilateral is then translated 2 units to the left and 3 units upwards to quadrilateral $A''B''C''D''$. What are the coordinates of point $B''$? | nvidia_math |
If eight friends plan to attend a concert, but two of them are only 3/7 sure they'll stay for the entire event (the other six are certain they'll stay the whole time), what is the probability that at least 7 people stayed for the entire concert? | nvidia_math |
Bob is building a vegetable garden. It is 220 feet long by 120 feet wide. Half of the garden is going to be tilled land. A third of the rest of it will have trellises for vining plants, and the rest will be devoted to raised beds. How big will the raised bed section of the garden be? | tiger_math |
Here is a precise numerical problem for the student to solve:
The probability of rain on any randomly selected day in a certain city is 0.3. What is the probability that it will not rain for the next 3 consecutive days in this city? | tiger_lab_math |
If Conor can chop 12 eggplants, 9 carrots, and 8 potatoes in a single day, and he works 4 times a week, what is the total number of vegetables that he can chop? | tiger_math |
Peter and Andrew both enjoy running in the morning. Peter runs 3 miles more than Andrew's 2 miles. After 5 days, what is the total number of miles that they have run together? | tiger_math |
In a 1000 m race, A beats B by 100 meters or 25 seconds. Find the speed of B?
Answer Choices: (A) 8 (B) 9 (C) 7 (D) 5 (E) 4 | tiger_lab_math |
Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$. What is $ab$? Let's write a program. | tiger_lab_math |
What is the minimum number of edges in a non-planar graph with exactly 6 vertices, and which well-known non-planar graph does this correspond to?
(A) 6 edges,
(B) 8 edges,
(C) 9 edges,
(D) 10 edges.
Provide the correct answer with justification. | automath |
A manufacturing company produces two products, Product A and Product B. The cost of producing one unit of Product A is $10, and the cost of producing one unit of Product B is $15. The company only has a budget of $5000 for production. Product A sells for $25 per unit, and Product B sells for $30 per unit. What is the o... | tiger_lab_math |
A shopkeeper sells three items P, Q and R and incurs a loss of 21, 11 and 10respectively. The overall loss percentage on selling P and Q items is 14.33 and that of Qand R items is 10.4. Find the overall loss percentage on selling the three items?
Answer Choices: (A) 15% (B) 12.16% (C) 13.4% (D) 20.2% (E) None of these | tiger_lab_math |
What is the maximum distance from the origin to the normal line of the ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)? | automath |
Herbert is 10 years younger than Kris. If Kris is 24 years old now, how old will Herbert be next year? Let's write a Python program to solve it. | tiger_lab_math |
If $\log_a(4b+12) = \dfrac{5}{3}$, find $b$ in terms of $a$. | tiger_math |
Zilla spent 7% of her monthly earnings on rent, half of it on her other monthly expenses, and put the rest in her savings. If she spent $133 on her rent, how much does she deposit into her savings account in a month? Let's program in Python in the response. | tiger_lab_math |
Evaluate $$\int_{0}^{1} x(2-x)\sin(\pi x) dx$$ | nvidia_math |
How many distinct four-digit positive integers have exactly two even digits and two odd digits, with the condition that the even digits must be adjacent to each other? | nvidia_math |
Find the largest value of $x$ that satisfies the equation $\frac{1}{x-2} + \frac{1}{x-4} = \frac{3}{x-3}$. | tiger_math |
A robot is programmed to move around a rectangular room that is divided into squares. The robot can move one square at a time, either horizontally or vertically. It starts at the bottom left corner of the room.
The robot's movements are specified by a sequence of commands, which are represented as a string of the let... | nvidia_math |
In his room, Dumbo has three trousers. One of them is black, the second is blue, and the third is brown. In his room, he also has four shirts. One of them is black and the other 3 are white. He opens his room in the dark and picks out one shirt and one trouser pair without examining the colour. What is the likelihood t... | tiger_lab_math |
Given the constraints \( ||x||^2 + ||y||^2 = 1 \) and \( e^T x = 0 \), \( e^T y = 0 \), where \( e \) is a vector of ones, what can be said about the optimal solution of the embedding problem in terms of the relationship between \( x \) and \( y \) if the objective function is related to minimizing distances in the con... | automath |
The volumes of two cones are in the ratio 1 : 10 and the radii of the cones are in the ratio of 1 : 2. What is the length of the wire?
Answer Choices: (A) 2 : 5 (B) 2 : 6 (C) 2 : 0 (D) 2 : 5 (E) 2 : 1 | tiger_lab_math |
A pharmaceutical company produces a medication in 50-milliliter bottles, of which 25 milliliters is the actual medication and the rest is a preservative. If the company produces a new 75-milliliter bottle with the same ratio of medication to preservative, how many milliliters of medication will the new bottle contain? | nvidia_math |
Let \( U \) be an open subset of \( \mathbb{R}^n \) and consider the differentiability of the function \( f: U \to \mathbb{R} \) at the point \( x \in U \). If it is given that \( f \) satisfies the condition
\[
f(x + h) = f(x) + \alpha_1 h_1 + \cdots + \alpha_n h_n + r(h) \cdot h
\]
for some constants \( \alpha_1, ... | automath |
Shannon and Maxine work in the same building and leave work at the same time. Shannon lives due north of work and Maxine lives due south. The distance between Maxine's house and Shannon's house is 40 miles. If they both drive home at the rate 2R miles per hour, Maxine arrives home 40 minutes after Shannon. If Maxine ri... | tiger_lab_math |
Gwendolyn can read 200 sentences of a book in 1 hour. She just brought a book from the library, having 20 paragraphs per page. What's the total time Gwendolyn will take to read the book if each paragraph has x sentences and the book has 50 pages?
If we know the answer to the above question is 50, what is the value of u... | tiger_math |
Given a watercraft crossing a stream of width \( W \) with a speed \( V_B \) relative to the water and a stream flowing at speed \( V_W \), if the watercraft always points directly towards a pier located on the opposite bank, derive the ordinary differential equation for the path of the watercraft in the form of \( \fr... | automath |
What is the probability that exactly \( X \) unique items appear in all 5 random samplings when drawing 100 items from a population of 10,000 items? | automath |
John is a hack author. He writes a book every 2 months. He has been writing for 20 years and has earned an average of $30,000 per book. How much money has he made writing? | nvidia_math |
During the outbreak of the coronavirus, a company had to ramp up its toilet paper production three times more to cover the increased demand. If the company was producing 7000 toilet paper per day, calculate its total toilet paper production during March of 2020 after increasing its production. | tiger_lab_math |
The integer $m$ is between $30$ and $80$ and is a multiple of $6$. When $m$ is divided by $8,$ the remainder is $2$. Similarly, when $m$ is divided by $5,$ the remainder is $2$. What is the value of $m$? | tiger_math |
Suppose $\cos x X and $\cos(x+z)= \frac{1}{2}$. What is the smallest possible positive value of $z,$ in radians?
If we know the answer to the above question is \frac{\pi}{6}, what is the value of unknown variable X? | tiger_math |
What is the radius of the circle inscribed in triangle $ABC$ if $AB = 22, AC=12,$ and $BC=14$? Express your answer in simplest radical form. Let's write a Python program. | tiger_lab_math |
Scott runs 3 miles every Monday through Wednesday. Then he runs twice as far he ran on Monday every Thursday and Friday. How many miles will he run in a month with 4 weeks? | tiger_lab_math |
In a scenario where Anna takes two balls from the first box containing 15 white and 12 black balls, and transfers them to a second box that initially has 14 white and 18 black balls, what is the probability that she transferred one black ball and one white ball to the second box, given that she later drew a black ball ... | automath |
"statement 1": Randy has 78.0 blocks and he uses 19.0 blocks to build a tower, "statement 2" :63.0 blocks are left, "options: " Entailment or contradiction? Please write a program to solve it | tiger_lab_math |
George has 7 colors available to paint his room. He wants to paint the ceiling, walls, and floor in different colors, but he has certain restrictions:
* The ceiling cannot be painted in blue or green.
* The walls cannot be painted in red or yellow.
* The floor must be painted in a color that is not used on the ceiling... | nvidia_math |
Given an undirected graph \( G \) with vertex set \( \{(i,j) \mid 1 \leq i \leq 12, 1 \leq j \leq 12\} \) where there is an edge between vertices \( (a,b) \) and \( (c,d) \) if \( |a-c| \leq 1 \) and \( |b-d| \leq 1 \), how many edges are there in this graph? | automath |
Some insects called aphids attack a large farm. In response, the farmer releases ladybugs onto the fields. There are some ladybugs with spots and 54912 ladybugs without spots. There are in all 67082 ladybugs. How many ladybugs with spots are there? | tiger_math |
In an election, candidate A got 80% of the total valid votes. If 15% of the total votes were declared invalid and the total numbers of votes is 560000, find the number of valid vote polled in favor of candidate?
Answer Choices: (A) 355600 (B) 380800 (C) 356500 (D) 356800 (E) 357000 Please respond by writing a program i... | tiger_lab_math |
Alex's grandmother gave him $1.20 and his brother gave him 60 cents for helping with the yard work. He went to the store and spent 80 cents on a small toy for his friend's birthday. His aunt gave him another $1.00 as a surprise gift. How many cents does he have now? | nvidia_math |
What is the area of a square field whose diagonal of length 20 m?
Answer Choices: (A) 238 (B) 379 (C) 200 (D) 328 (E) 397 Let's write a Python program to solve it. | tiger_lab_math |
If \( x = 3^n \) where \( n \) is a positive integer, what is the probability that \( x \) has 3 as the unit's digit? | automath |
A rectangular solid has dimensions 3, 4, and 12. If this rectangular solid is inscribed in a sphere such that all its vertices touch the sphere, what is the diameter of the sphere?
(A) 13
(B) 15
(C) 18
(D) 19
(E) 20 | automath |
Alex spends time watching 3 forty-five minute documentaries and 2 fifteen-minute comedy sketches. How long does he spend watching TV? | nvidia_math |
What is the dividend if the divisor is 800, the quotient is 594, and the remainder is -968? | tiger_math |
Solve the equation 2sin(x) + sqrt(3) = 3cos(x) for x, where 0<= x <= 360 degrees. | tiger_lab_math |
14 less than x times the number of koi fish in the pond is 64. You will find 39 koi fish in the pond. What is the value of unknown variable x? | tiger_math |
Given a set of data points \((x_i, y_i)\) from a simple linear regression model where the responses are modeled as \(Y_i = \beta_0 + \beta_1 x_i + \epsilon_i\) with \(\epsilon_i \sim N(0, \sigma^2)\), if we compute the coefficient of determination \(R^2\) and find that \(R^2 = 0.85\), what is the proportion of variabil... | automath |
Given that \( X \), \( Y \), and \( Z \) are positive integers, determine the conditions under which the product \((X-Y)(Y-Z)(X-Z)\) is greater than zero.
Is either of the following statements sufficient to conclude this?
1. \( X^2 + YZ = XY + XZ \)
2. \( XY - Y^2 = XZ - YZ \)
Provide a clear explanation for your ... | automath |
Consider the system of ordinary differential equations defined by
$$\dot x = ax + o(|x| + |y|),$$
$$\dot y = -by + o(|x| + |y|)$$
where \( a, b > 0 \). If a Chetaev function for this system is given by \( v(x, y) = x^2 - c^2y^2 \) for any \( c \neq 0 \), what can be concluded about the stability of the fixed point... | automath |
The roots $x_1$ and $x_2$ of the equation $$x^2+px+q=0$$ are such that $x_1+x_2=4$ and $x_1^2-x_2^2=16$. Determine $p$ and $q$. | nvidia_math |
The vertices of a regular octagon are connected to form triangles. The least possible perimeter of one of these triangles is found to be 6 inches. What is the side length of the octagon? | nvidia_math |
"statement 1": What is the height difference between the two if the Empire
State Building is 356 m tall and the Petronas Towers is 370 m tall ?, "statement 2" :What is the height difference between the two if the Empire State Building is less than 356 m tall and the Petronas Towers is 370 m tall ?, "options: " Entailm... | tiger_lab_math |
Compute
\[\begin{vmatrix} 7 & 3 \\ -1 & 2 \end{vmatrix}.\] | tiger_math |
The matrix \[\begin{pmatrix} 2 & a \\ b & -1 \end{pmatrix}\] is equal to its own transpose. Find the ordered pair $(a, b)$. | nvidia_math |
What is the least number of colors needed to color the vertices of the graph shown, such that no two adjacent vertices are the same color?
[asy]
draw((-8,-2)--(8,-2));
draw((-8,2)--(8,2));
draw((-8,6)--(8,6));
draw((-8,10)--(8,10));
draw((-8,14)--(8,14));
path a=(-8,14)--(-6.5,12)--(-10,8)--(-6.5,4)--(-10,0)--(-8,-2);... | nvidia_math |
How many ways are there to partition a set of 8 objects into 4 non-empty subsets using Stirling numbers of the second kind? | tiger_lab_math |
A submarine is traveling at a speed of 8 m/s due north relative to the water. After 2 minutes, it collides with a sunken pirate ship located 600 m from its starting point at an angle of 25° east of north. What is the speed and direction of the underwater current that caused the submarine to drift off its intended cours... | automath |
Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is?
Answer Choices: (A) 10.9 sec (B) 14.8 sec (C) 10.6 sec (D) 10.8 sec (E) 20.8 sec | tiger_lab_math |
A certain smoothie A is three-fifths blueberries and 40% raspberries by weight. Another smoothie B is three-quarters banana and one-quarter raspberries by weight. If the mixed smoothie of A and B is 30% raspberries by weight, how much weight of this mixture belongs to smoothie A?
Answer Choices: (A) 10% (B) 33.33% (C) ... | tiger_lab_math |
It starts raining at 7:00 and pours heavily until its stops at 17:00 on a particular day. On the second day, the rain takes 2 more hours than it took on the first day to stop. On the third day, the rain pours for twice the amount of time it took on the second day. Calculate the total time it was raining in the three d... | tiger_math |
Given the mass-luminosity relation for main sequence stars, \( L \propto M^{3.5} \), where \( L \) is luminosity and \( M \) is the mass of the star, if both luminosity and mass are measured in solar units (i.e., the luminosity and mass of the Sun are set to 1), what is the constant of proportionality in this relation? | automath |
If ( 1 / 2 ) ^ 22 times ( 1 / 81 ) ^ a certain variable equals 1 / 18 ^ 22, and the variable equals 11, what is the variable? | tiger_math |
Tom has a red marble, a green marble, a blue marble, and three identical yellow marbles. If Tom randomly chooses two marbles without replacement, what is the probability that he chooses two yellow marbles? | tiger_math |
In a nearby garden, a single sunflower seed sprouted and grew into a mature plant, producing a large flower. After two months, the flower transformed into 75 seeds, which were dispersed by the wind. However, only a third of these seeds landed in fertile soil and began to grow, mirroring the development of the initial s... | nvidia_math |
Find the maximum value of
\[\frac{\sqrt{x + 1}}{x + 2}\]
for real numbers $x > 1$. | nvidia_math |
Can you construct an explicit bijection \( f: (0,1) \to (0,1) \setminus \mathbb{Q} \) and demonstrate that it is indeed a bijection? | automath |
Consider the RLC (resistor, capacitor, and inductor) circuit shown below, where R = 100Ω, L = 0.2H, C = 10μF, and the source voltage is 110 V AC at 60 Hz.
Find the solution to the differential equation for the voltage across the capacitor:
$$
\frac{d^2V_c}{dt^2} + \fr... | tiger_lab_math |
Two functions $f(x)$ and $g(x)$ both start at 20 when $x = 0$ and have constant rates of change of absolute value 5, with $f(x)$ increasing and $g(x)$ decreasing. However, $f(x)$ and $g(x)$ are not linear. Instead, they are both quadratic functions with leading coefficients of 1 and -1, respectively. What is the absolu... | nvidia_math |
A rectangle has a length that is 5 times its width, $w$. The rectangle is divided into two parts by a diagonal line, forming two triangles. Write an expression for the sum of the areas of the two triangles as a function of the width $w$. | nvidia_math |
Kaitlyn has a cake that is cut into 24 pieces. After she and her family finish eating, there are 6 pieces left. If each of them ate 3 pieces of cake, how many people ate the cake? | nvidia_math |
A periodic function f(t) with a period of T = 2π has the following Fourier series:
f(t) = 2 + 3cos(t) - 4sin(2t) - 2cos(3t) + 5sin(4t) + ...
What is the value of the coefficient a5 in the Fourier series of f(t)? | tiger_lab_math |
Consider the geometric sequence $3$, $\dfrac{9}{2}$, $\dfrac{27}{4}$, $\dfrac{81}{8}$, $\ldots$. Find the eighth term of the sequence. Express your answer as a common fraction. | tiger_lab_math |
Mia has a triangular-shaped garden and wants to place a large planter at each corner. Each planter will have a large peace lily that costs $20.00 per plant, 3 lavender plants that cost $5.00 per plant, and 3 begonias that cost $2.50 per plant. How much will it cost to fill all the pots? | nvidia_math |
Alicia has 20 gumballs. Pedro has that many gumballs plus an additional number of gumballs equal to 150% of the number Alicia has. They put their gumballs in a bowl, and later Pedro takes out 55% of the gumballs to eat. How many gumballs are remaining in the bowl? | tiger_math |
In an election between two candidates, the first candidate got 70% of the votes and the second candidate got 240 votes. What was the total number of votes?
Answer Choices: (A) 800 (B) 900 (C) 1000 (D) 1100 (E) 1200 | tiger_lab_math |
Find the smallest positive integer $x$ such that the product of $3^{2017}$, $5^{2019}$, and $x$ is a perfect square. | nvidia_math |
The bacteria in a lab dish double in number every four hours. If 500 bacteria cells are in the dish now, in how many hours will there be exactly 32,000 bacteria? | tiger_math |
If $$1+22+333+4444+55555+666666+7777777+88888888$$is congruent to $n$ modulo 9, where $0\le n<9$, then what is the value of $n$? Let's program in Python in the response. | tiger_lab_math |
If Tommy ordered a total weight of 10 kilograms of fruits, including 3 kilograms of apples, 1 kilogram of oranges, and 3 kilograms of grapes, how many kilograms of strawberries did he order? | tiger_math |
The side lengths of a triangle are 10 cm, 24 cm and 26 cm. How many square centimeters are in the area of the triangle? | tiger_math |
In how many ways can you select 3 different flavors from a set of 5 distinct frozen yogurt flavors (banana, chocolate, lemon, strawberry, vanilla) if each selection allows for repetitions (i.e., you can choose the same flavor more than once)? | automath |
Given the origin of a ray \( R_0 = (0, 0, 0) \) and a direction vector \( R_d = (d_X, d_Y, d_Z) \), if you want to find the point on the ray at a distance \( t \) from the origin, can you express the coordinates \( (x, y, z) \) of that point in terms of \( t \) and the components of \( R_d \)? Specifically, what are th... | automath |
A compound has a certain number of Barium and 1 Oxygen atom. The molecular weight of the compound is 153. How many Barium atoms are in the compound? | tiger_math |
A dealer sold 200 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were Hondas, how many Hondas did she sell?
\begin{tabular}{ |c | c|}
\hline \textbf{Type of Car} & \textbf{$\%$ of Total Cars Sold} \\ \hline
Audi & $15\%$ \\ \hline
Toyota & $22\%$ \\ \hline
Acura... | nvidia_math |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | nvidia_math |
"statement 1": He gave 19 toys to his brother Gourav , while Gourav playing all but 6 got destroyed, "statement 2" :He gave less than 19 toys to his brother Gourav , while Gourav playing all but 6 got destroyed, "options: " Entailment or contradiction or neutral? Let's write a program. | tiger_lab_math |
If an infinite square well is centered at the origin with walls located at \(x = -L\) and \(x = L\), what are the eigen-energies of the system, and how do the corresponding eigen-functions compare to those from a well that spans from \(x = 0\) to \(x = 2L\)? Specifically, provide the expression for the eigen-energies i... | automath |
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color.... | nvidia_math |
A man sells a horse for Rs.800 and loses something, if he had sold it for Rs.980, his gain would have been double the former loss. Find the cost price of the horse?
Answer Choices: (A) 337 (B) 278 (C) 288 (D) 860 (E) 721 | tiger_lab_math |
Triangle STV has sides ST = TV = 10, and SV = 12. What is the area?
Answer Choices: (A) 36 (B) 40 (C) 44 (D) 48 (E) 52 | tiger_lab_math |
A taxi company charges a flat fee of $2.00 plus Y dollars for each mile traveled. If a 7-mile taxi ride costs $4.50, determine the value of Y. | tiger_math |
If √10 = 3.16, find the value of If √5/2
Answer Choices: (A) 1.3 (B) 1.58 (C) 2.03 (D) 2.15 (E) 3.15 | tiger_lab_math |
Mrs. Fredrickson has 80 chickens where 1/4 are roosters and the rest are hens. Only three-fourths of those hens lay eggs. How many chickens does Mr. Fredrickson have that do not lay eggs? Let's write a Python program. | tiger_lab_math |
What is the value of $n$ in the equation $n + (n + 1) + (n + 2) = 9$? | nvidia_math |
During the hot dog eating contest, the 1st competitor ate 12 hot dogs in 3 minutes. The 2nd competitor ate twice that amount and the 3rd competitor ate 25% less than the 2nd competitor. How many hotdogs did the 3rd competitor eat? | tiger_math |
What mass of sodium carbonate (Na₂CO₃) is required to prepare 100 mL of a 0.15 molal solution? | automath |
Let \( A \) be a finite abelian semigroup with \( n \) generators. If the group \( K(A) \) is defined as \( F(A) / E(A) \), and \( E(A) \) is generated by the relations \( a + a' - (a \oplus a') \) for \( a, a' \in A \), how many elements does the group \( K(A) \) contain? Provide a justification for your answer in ter... | automath |
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