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Consider the matrices \( A = \begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \) and \( B = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{pmatrix} \).
Find a diagonal matrix \( M \) consisting of entries \( 1 \) and \( -1 \) s... | automath |
In a lot of $20$ items, exactly $2$ are defective. The lot is divided randomly into two sublots of $10$ items each. What is the probability that both defective items are in the same sublot? | nvidia_math |
A lawn has equally distributed grass. It takes 24 days to 70 cows and 60 days to 30 cows to eat the whole of grass. How many cows will be needed to eat the grass in 96 days.
Answer Choices: (A) 20 (B) 30 (C) 40 (D) 50 (E) 60 | tiger_lab_math |
Given Banach spaces \( X = Y = \ell^2 \) and an embedding \( \Phi : \ell^2 \rightarrow \ell^2 \) defined by \( \Phi((x_n)_{n=1}^\infty) = (x_n/n)_{n=1}^\infty \), showing that the induced map \( \Phi' : \ell^2' \rightarrow \ell^2' \) is not surjective, find a specific functional in \( \ell^2' \) that cannot be represen... | automath |
Carla bought x bags of mini peanut butter cups on clearance. Each bag was $6.00 but was 75% off. She spent 3 on 2 bags of candy. What is the value of unknown variable x? | tiger_math |
What are the complex zeros of the polynomial \( P(x) = x^4 - 16 \)? | automath |
What is the value of the following sum:
\[
S = \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} ?
\]
Provide a numerical approximation of \( S \) to five decimal places. | automath |
In a basket of flowers, there are 5 roses, three times as many carnations, and some daisies. In total, there are 27 flowers in the basket. How many daisies are in the basket? | nvidia_math |
Find the smallest positive integer $n$ such that $n^4$ is divisible by 12960000. | nvidia_math |
The average weight of a group of boys is some kg. After a boy of weight 40 kg joins the group, the average weight of the group goes up by 1 kg. There were originally 5 boys in the group. What was the average weight of the group before the new boy joined? | tiger_math |
Given the sample list of values {7, 8, 0, 9, 2, 10, 4, 1, 6, 0, 8, 7, 6, 0, 0} and the corresponding frequency data {{7, 2}, {8, 2}, {0, 4}, {9, 1}, {2, 1}, {10, 1}, {4, 1}, {1, 1}, {6, 2}}, calculate the mean of the list using both the raw data and the frequency data. Are the two means equal, and if so, what is their ... | automath |
Find the maximum value of
\[x^2 + \frac{1}{x^2} + x + \frac{1}{x}\]for $x > 0.$ | nvidia_math |
A right circular cylinder has a height of 15 cm and a base radius of 5 cm. What is the length, in cm, of the diagonal connecting point A and point B? Express your answer in simplest radical form.
(Note: Point A is on the top circle of the cylinder and point B is on the bottom circle of the cylinder.) | nvidia_math |
A regular hexagon is inscribed in a circle. A diagonal of the hexagon is a line segment connecting two non-adjacent vertices. How many diagonals of the hexagon are also diameters of the circle? | nvidia_math |
Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost? | tiger_lab_math |
Tabitha has 25 dollars. She gives her mom 8 dollars and invests half what is left in a money market. She spends some money on 5 items that costs 50 cents each. How much money does Tabitha have left? | tiger_lab_math |
A certain number of jars are filled with a combination of red, blue, and green marbles. Each jar contains exactly three marbles, and the following conditions are known:
- 1/3 of the jars contain two red marbles,
- 1/4 of the jars contain at least one blue marble,
- 1/5 of the jars contain only green marbles.
What is ... | nvidia_math |
Find the second derivative of the function f(x) = 3x^4 + 2x^3 - 5x^2 + 7x - 4. | tiger_lab_math |
Triangle $PQR$ has sides $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ of length $a$, $b$, and $c$, respectively. Let $K$ be the circle inscribed in $\triangle PQR$ and let $E$ be the intersection of $K$ and the angle bisector of $\angle PQR$ that is not on the same side of $\overline{QR}$ as $P$. The length of... | nvidia_math |
In a gradient defined by the luminance formula \( Y = 0.299r + 0.587g + 0.114b \), where \( r \), \( g \), and \( b \) are the red, green, and blue color components, respectively, what is the maximum possible value of \( Y \) when the color components are represented as integers in the range [0, 255]? | automath |
If the complement of a certain angle is four times the measure of that certain angle, then what is the measure of that certain angle?
Answer Choices: (A) 45° (B) 30° (C) 22.5° (D) 18° (E) 15° Let's write a Python program. | tiger_lab_math |
Leo and Agrey went fishing. Agrey caught 20 more fish than Leo. If Leo caught x fish, The total number of fish they caught together is 100. What is the value of unknown variable x? | tiger_math |
In a dairy farm, 50 cows eat 50 bags of husk in 50 days. In how many days one cow will eat one bag of husk ?
Answer Choices: (A) 34 days (B) 40 days (C) 46 days (D) 50 days (E) None of these | tiger_lab_math |
Tim decided to build his savings by depositing money into a savings account. On the first day, he deposited $100. Each subsequent day, he doubled the amount he had deposited the previous day. For example, if he deposited $100 on one day, he would deposit $200 the next day, then $400, and so on. If he continued this pat... | tiger_math |
Evaluate the integral \( \int_0^1 x^3 \sqrt{1 - x^2} \, dx \) using the substitution \( u = 1 - x^2 \). What is the exact value of this integral? | automath |
Andy completes a piece of work in 10 days, Raju completes the same work in 40 days. If both of them work together, then the number of days required to complete the work is
Answer Choices: (A) 6 days (B) 8 days (C) 9 days (D) 10 days (E) 11 days | tiger_lab_math |
Megan's glass, shaped like a cylinder, has a height of 15 inches and a radius of 3 inches. What is the volume of the glass in cubic inches, and how much milk (in cubic inches) can it hold? | automath |
Ashley purchases a salad bowl for dinner at $8.00 and adds a protein pack for an extra $2.00. If Ashley buys this same meal every evening for 7 days, how much does she spend on dinner? | nvidia_math |
Problem:
Compute the remainder when $(2^3\times3\times5^3\times7)^2$ is divided by $11$. | nvidia_math |
in a class of students , 1 / 2 of the number of girls is equal to 1 / 5 of the total number of students . what is the ratio of boys to girls in the class ? Please write a program to solve it | tiger_lab_math |
Can you explain why the running time of the naive CLIQUE algorithm, which is given as \( \Omega(k^2 \binom{|V|}{k}) \), is considered polynomial time when \( k \) is a constant? Specifically, I would like to understand how \( \binom{|V|}{k} \) is polynomial in terms of \( |V| \) when \( k \) remains constant. For insta... | automath |
Sally wants to bake cookies for her friends. She has a recipe that requires 3 cups of flour and makes 24 cookies. Sally wants to bake enough cookies to give 5 cookies to each of her 8 friends. How many cups of flour does she need? | tiger_math |
Marcus had some pebbles. He skipped half of them across the lake, but Freddy gave him another 30 pebbles. Marcus has 39 pebbles now. How many pebbles did Marcus have initially? | tiger_math |
If the probability of Keith winning a chess game against Sooeng is 0.4, what is the probability that Keith will not win 3 games in a row? | automath |
Emily fine-tunes her bicycle, which boosts its speed by 25%. She then installs lighter wheels, reducing the bike's weight by 10%, and this increases the speed by an additional 5 mph. If her bicycle's original speed was 80 mph, what is the new speed? | nvidia_math |
A rabbit community consists of 100 bunnies, with three different groups that come out of their burrows at varying frequencies:
- Group A consists of 40 bunnies, and each bunny comes out 3 times per minute.
- Group B consists of 30 bunnies, and each bunny comes out 5 times every 2 minutes.
- Group C consists of 30 bunn... | tiger_math |
Samantha's family is comparing internet plans offered by two different service providers, FastLink and SpeedNet. At FastLink, the plan costs $60 per month for the first 100 GB of data and $10 for each additional 50 GB. At SpeedNet, the plan costs $55 per month for the first 100 GB of data and $8 for each additional 50 ... | nvidia_math |
A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars? | nvidia_math |
If Addison sold 181 raffle tickets on Friday and twice as many on Saturday, and then sold 78 raffle tickets on Sunday, how many more raffle tickets were sold on Saturday than on Sunday? | tiger_math |
Consider the partition \(\lambda = (3, 1, 1)\) of the integer 5. Calculate the value of the monomial symmetric polynomial \(m_{\lambda}(r_1, r_2, r_3)\) in terms of the variables \(r_1, r_2, r_3\). What is the resulting expression after symmetrizing the corresponding monomial? | automath |
Let \(\epsilon = \emptyset\). Determine the values of \(\bigcup \epsilon\) and \(\bigcap \epsilon\). Are these values well-defined? If so, state what they are. | automath |
Caroline has 40 pairs of socks. She loses 4 pairs of socks at the laundromat. Of the remaining pairs of socks, she donates two-thirds to the thrift store. Then she purchases 10 new pairs of socks, and she receives x new pairs of socks as a gift from her dad. Caroline has 25 pairs of socks in total. What is the value of... | tiger_math |
Given the initial positions and velocities of flights 12 and 33, determine the time (in minutes) at which the two planes' paths intersect, if at all, assuming the altitude remains constant at 30,000 feet. Provide your answer with a verification of whether a collision occurs within that timeframe. | automath |
If Carson needs to mow 40 lines to cover the entire yard, and it takes him 2 minutes to mow one line, and he also needs to plant 8 rows of flowers, with each row containing 7 flowers, and it takes him half a minute to plant each flower, how many minutes will Carson spend on gardening in total? | tiger_math |
Here's a numerical problem for a math student to solve related to survival analysis:
A hospital conducted a study to determine the survival time (in weeks) of patients with a certain illness. They followed 10 patients and recorded their survival times in weeks: 12, 27, 15, 23, 31, 10, 8, 20, 25, 18. Using survival ana... | tiger_lab_math |
162 students represent x percent of the boys at a school. The boys at the school make up a certain percentage of the total school population of x students, which is 180. What percentage of the total school population do the boys make up? | tiger_math |
What is the residue of the function \( f(z) = \frac{z}{1 + \sin(z)} \) at the point \( z = -\frac{\pi}{2} \)? | automath |
The probability of rain for each of the next five days is $\frac{2}{3}$. What is the probability that it will rain every day during the next five days? Express your answer as a common fraction. | nvidia_math |
In a directed graph with 6 nodes, where the edges have weights represented by distances, how many distinct shortest paths exist between two nodes that are x blocks horizontally and y blocks vertically away from each other, assuming the shortest path allows for revisiting nodes and reusing edges? | automath |
What is the volume of the solid obtained by rotating the region between the curves \( y = 2x \) and \( y = x^2 \) about the x-axis? | automath |
There are 100 jellybeans in a glass jar. Mrs. Copper’s kindergarten class normally has 24 kids, but 2 children called in sick and stayed home that day. The remaining children who attended school eat 3 jellybeans each. How many jellybeans are still left in the jar? Please write a program to solve it | tiger_lab_math |
Consider two square matrices, $\mathbf{P}$ and $\mathbf{Q}$, such that $\mathbf{P} \mathbf{Q} = \mathbf{P} - \mathbf{Q}$.
Given that $\mathbf{P} - \mathbf{Q} = \begin{pmatrix} 1 & -2 \\ 3 & 4 \end{pmatrix}$, find $\mathbf{Q} \mathbf{P}$. | nvidia_math |
Find $AX$ in the diagram if $CX$ bisects $\angle ACB$.
[asy]
import markers;
real t=.56;
pair A=(0,0);
pair B=(3,2);
pair C=(.5,1.5);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SE);
//markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1... | nvidia_math |
Given that the ratio between x and y is 7 / 5, find the ratio between x^3 and y^2, where x is raised to the power of 3, and y is squared. | tiger_math |
The graph of the function $g(x)$ is a circle centered at the origin with radius $5$. Find the sum of the $x$-coordinates of all points where $g(x) = x^2-1$. | nvidia_math |
Find the positive value of $t$ for which $|{3+ti}| = 4\sqrt{2}$. | nvidia_math |
A work can be finished in 16 days by 30 women. The same work can be finished in 10 days by sixteen men. The ratio between the capacity of a man and a woman is
Answer Choices: (A) 3:1 (B) 4:3 (C) 2:3 (D) 2:1 (E) 2:4 | tiger_lab_math |
Let's say that $a + \sqrt{b}$ is a root of the equation $$x^2-2x-1=0.$$ The other root of this equation is $1 + \sqrt{2}$.
Determine $\sqrt{a}+b$. | nvidia_math |
If Mary takes 40 ounces of raw spinach and cooks it until it reduces to 20% of its initial volume, and then mixes it with 6 ounces of cream cheese and 4 ounces of eggs, what is the total volume of the spinach quiche? | tiger_math |
Let sets \( A \) and \( B \) be defined as \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{3, 4, 5, 6, 7\} \). Define a function \( m: A \cup B \to \{0, 1\} \) by \( m(x) = 1 \) if \( x \) is an even number, and \( m(x) = 0 \) if \( x \) is an odd number.
Using set-builder notation, define the set \( K \) such that \( K \) ... | automath |
Alex drives her motorcycle 72 miles to the east and 48 miles to the south. If Alex maintains a constant speed of 30 miles per hour throughout the entire trip, how many hours did Alex drive? | nvidia_math |
If Jean is wanted for 3 counts of arson, 2 burglary charges, and six times as many petty larceny charges as burglary charges, with each arson count carrying a 36-month sentence, each burglary charge carrying an 18-month sentence, and each petty larceny charge being 1/3rd as long as a burglary charge, how many months in... | tiger_math |
The town’s annual budget totals $x million. If half of the budget goes towards policing and $12 million goes towards education. How much money is left for managing public spaces?
If we know the answer to the above question is 4, what is the value of unknown variable x? | tiger_math |
Evaluate $\dfrac{\sqrt{y-12}}{\sqrt{y+12}}$ if $12 \star y = 5$. | tiger_math |
Let $f(x)$ be a function that satisfies the inequality $f(x) \le x+1$ for all $x \in [0, 5]$. What is the smallest possible value of $\frac{f(x)}{x}$ for $2 \le x \le 3$? | nvidia_math |
A developer plans to subdivide a piece of land with lake frontage into lots. If \( \frac{1}{9} \) of the lots will each have 80 feet of lake frontage and there are 40 lots each with 100 feet of lake frontage, what is the total length of the lake frontage \( x \) in feet?
(A) 400
(B) 3,200
(C) 3,700
(D) 4,400
... | automath |
Given the second-order transfer function model estimated from a dataset where the initial response is provided as `ic = initialCondition with properties: A: [2x2 double], X0: [2x1 double], C: [-1.6158 5.1969], Ts: 0`, if the initial state vector `X0` for the estimation data is `[-0.5053; -1.2941]` and for the validatio... | automath |
Given a string of length \(N = 8\) generated from the alphabet \(\{A, B, C\}\), what is the probability that exactly \(k = 5\) characters in the string are the same, taking into account that the other characters may be distributed among the remaining letters? | automath |
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$? | nvidia_math |
What is the probability that a needle of length 1, dropped at random onto a plane with parallel lines spaced 1 unit apart, will intersect one of the lines? | automath |
What is the product of the reciprocals of the non-negative integer exponents in the prime factorization of 6! (6 factorial)? | nvidia_math |
Alex was 7 years old. His brother Ryan was 3 years older than Alex. Their oldest brother Jack was 3 times as old as Ryan. Tom was 5 years older than Jack. How old was Tom? | nvidia_math |
Let a smooth manifold M be given with a metric g, defined as:
g = dx^2 + 2dy^2 + 2dz^2
Find the volume form of this metric over the region R where:
-1 <= x <= 1
-2 <= y <= 2
-3 <= z <= 3 | tiger_lab_math |
The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(1,3)$ to $(5,7)$. What is the value of b? | tiger_math |
Without any stoppage, a person travels a certain distance at an average speed of 42 km/h, and with stoppages he covers the same distance at an average speed of 28 km/h. How many minutes per hour does he stop?
Answer Choices: (A) 14 minutes (B) 15 minutes (C) 28 minutes (D) 38 minutes (E) None of these | tiger_lab_math |
Use the Trapezoidal Rule with to approximate $\int_0^{\pi} sin^2(x)dx$. | tiger_lab_math |
Let $Q(z)$ be a cubic polynomial with complex coefficients, one of whose roots is $\sqrt{2}+i$. When the three roots of $Q(z)$ are plotted in the complex plane, they form a triangle whose centroid is the origin. Given that the sum of the pairwise products of the roots is $-13$, find the modulus of the constant term o... | nvidia_math |
Given that the expected survival time in the absence of a contaminant is modeled with a normal distribution that has 10% of its mass below \( \log(10) \) and 10% above \( \log(40) \), calculate the standard deviation \( \sigma \) of this distribution if the mean is set to \( \mu = \frac{\log(10) + \log(40)}{2} \). Use ... | automath |
Find the inflection points and the intervals of concavity for the function \( f(x) = x^3 - 3x + 2 \). | automath |
Calculate the magnetic field inside a solenoid that has 100 turns, carries a current of 10 A, and is 2 meters long. Use the formula for the magnetic field in a solenoid:
\[ B = \frac{\mu_0 \cdot I \cdot N}{L} \]
where \( \mu_0 = 1.25664 \times 10^{-6} \, \text{T m/A} \). What is the value of the magnetic field in Te... | automath |
The time taken by a man to row his boat upstream is some multiple of the time taken by him to row the same distance downstream. The speed of the boat in still water is 51 kmph, and the speed of the stream is 17 kmph. What is the ratio of the time taken to row upstream to the time taken to row downstream? | tiger_math |
a and b together have some money. If 1/3 of a's amount is equal to 1/4 of b's amount, and b has $484, how much money do a and b have together? | tiger_math |
Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with... | nvidia_math |
Consider the functions defined by the following tables:
A.
| x | f(x) |
| --- | --- |
| -1 | 3 |
| 1 | 4 |
| 3 | 1 |
| 5 | 2 |
B.
| x | g(x) |
| --- | --- |
| -2 | 3 |
| 0 | 0 |
| 1 | 1 |
| 3 | 2 |
| 4 | 3 |
C.
| x | h(x) |
| --- | --- |
| -3 | 1 |
| 0 | 2 |
| 1 | 3 |
| 2 | 4 |
| 4 | 5 |
D.
| x | i(x) |
| --- ... | nvidia_math |
The owner of a furniture shop charges his customer 20% more than the cost price. If a customer paid Rs. 8400 for a computer table, then what was the cost price of the computer table?
Answer Choices: (A) 7000 (B) 2699 (C) 2670 (D) 6725 (E) 2601 | tiger_lab_math |
How many different combinations of large boxes (which hold 8 items each) and small boxes (which hold 3 items each) can the firm use to package exactly 84 items while using no more than 18 total boxes? | automath |
Mr Cruz went to his doctor to seek advice on the best way to gain weight. His doctor told him to include more proteins in his meals and live a generally healthy lifestyle. After a month of following his doctor's advice, Mr Cruz had a weight gain of 20 pounds. He gained 30 more pounds in the second month after more heal... | tiger_math |
How many natural numbers that are less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 & 8?
Answer Choices: (A) 5000 (B) 4096 (C) 6560 (D) 4095 (E) 8000 | tiger_lab_math |
Maya loves to read books. Last week she read 5 books. Each book had x pages of text. This week she read twice as much. How many pages did Maya read in total?
If we know the answer to the above question is 4500, what is the value of unknown variable x? | tiger_math |
How many whole numbers are between $\sqrt[3]{10}$ and $\sqrt[3]{200}$? | nvidia_math |
Evaluate the limit:
\[
\lim_{x \rightarrow -\infty} \frac{3x-7}{4x+\sqrt[3]{x^3+5x}}.
\]
What is the value of this limit? | automath |
Suraj has a certain average of runs for 16 innings. In the 17th innings he scores 92 runs thereby increasing his average by 4 runs. What is his average after the 17th innings?
Answer Choices: (A) 48 (B) 64 (C) 28 (D) 72 (E) 27 | tiger_lab_math |
one fourth of a solution that was 8 % sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight . the second solution was what percent sugar by weight ? Please write a program to solve it | tiger_lab_math |
Let \( k \) be a field and consider the subset \( S = \{(x, y) \in \mathbb{A}^2_k : x^2 + y^2 = 1\} \) in the affine space \( \mathbb{A}^2_k \). What is the vanishing ideal \( \mathcal{I}(S) \) of the set \( S \) in the polynomial ring \( k[X_1, X_2] \)? Provide a generator for this ideal. | automath |
At a bicycle shop, there are bicycles and tricycles. Each bicycle has 2 wheels and each tricycle has 3 wheels. The shop has 24 tricycles on display. In total, all the vehicles have 120 wheels. How many bicycles are in the shop? | nvidia_math |
If an angle measures 120 degrees, what is its equivalent measurement in radians? Provide your answer in terms of \(\pi\). | automath |
The snack machine at Richmond High School sells candy bars for $2 each and chips for $.50 each. How much money, in dollars, will 5 students need in total if each of them gets 1 candy bar and 2 bags of chips? Please respond by writing a program in Python. | tiger_lab_math |
Greg and Katie went to Mrs. Scarlet's gold store to have their gold chests opened. They realized Greg had four times less gold than Katie when Mrs. Scarlet opened the chests. If the total amount of gold is 100, how much gold does Greg have? | nvidia_math |
On his farm, Mr. Valentino has 200 chickens, twice as many ducks as chickens and three times as many turkeys as ducks. How many birds does Mr. Valentino have on the farm in total? | tiger_lab_math |
After spending $17.80 at the toy store to purchase 4 toy cars at $0.95 each and a race track for $6.00, how much money does Edward have remaining to buy additional toys? | tiger_math |
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