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Evaluate the integral
\[
\int (3t^2-4)^{\frac{5}{2}} \, dt.
\]
Additionally, clarify whether
\[
(3t^2 - 4)^{\frac{5}{2}}
\]
is equivalent to
\[
\sqrt{(3t^2 - 4)^5}
\]
or
\[
\left(\sqrt{3t^2 - 4}\right)^5.
\]
Find the final result of the integral and verify both interpretations. | automath |
Jar X contains exactly 5 green marbles and 10 yellow marbles. Marco then removes the same number of green marbles as yellow marbles from Jar X and places them in an empty Jar Y. Jar X now has $\frac{3}{4}$ of its original number of marbles. If Marco were now to randomly choose a marble from Jar X and a marble from J... | nvidia_math |
A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space? | nvidia_math |
Let \( f(x, y, z) = x^2 + y^2 + z^2 \) and consider the path \( P \) in Euclidean space defined by the following segments: from \((0, 0, 0)\) to \((1, 0, 0)\), then from \((1, 0, 0)\) to \((1, 2, 0)\), and finally from \((1, 2, 0)\) to \((1, 2, 3)\). Using the Fundamental Theorem for line integrals, calculate the integ... | automath |
Let $\mathbf{Q}$ be the matrix for projecting onto the vector $\begin{pmatrix} 2 \\ 5 \end{pmatrix}.$ Find $\mathbf{Q}^{-1}.$
If the inverse does not exist, then enter the zero matrix. | nvidia_math |
Two trains each 250 m in length are running on the same parallel lines in opposite directions with the speed of 80 kmph and 70 kmph respectively. In what time will they cross each other completely?
Answer Choices: (A) 32 sec (B) 17 sec (C) 12 sec (D) 46 sec (E) 15 sec | tiger_lab_math |
The angles of quadrilateral $ABCD$ are in the ratio $2:3:4:5$. What is the degree measure of $\angle B$, rounded to the nearest whole number? | tiger_math |
Given that $b$ is a multiple of $162$, find the greatest common divisor of $4b^3+b^2+7b+31$ and $b$. | nvidia_math |
If a rectangle is 7 inches wide and 4 times as long as it is wide, what is the area of the rectangle? | tiger_math |
Look at this series: 21, 9, 21, 11, 21, 13, 21, ... What number should come next?
Answer Choices: (A) 14 (B) 15 (C) 21 (D) 23 (E) None of these | tiger_lab_math |
A group of people pays $720 for admission tickets to an amusement park. The price of an adult ticket is x, and a child ticket is $8. There are 25 more adults than children. There are 15 children in the group. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
Every hour Joanne has to collect the coins out of the fountain inside the mall. During the first hour, she collected 15 coins. For the next two hours, she collected 35 coins from the fountain. In the fourth hour, she collected 50 coins from the fountain but she gave 15 of them to her coworker so she could buy a soda. H... | tiger_math |
Find the product of the solutions for $x$ in the equation $16x^2 - 60x + 36 = 0$? | tiger_math |
A rectangle has an area of 500 cm², and the length is twice the width. What are the dimensions of the rectangle? | automath |
a man can do a piece of work in 5 days , but with the help of his son , he can finish it in 4 days . in what time can the son do it alone ? Let's write a Python program to solve it. | tiger_lab_math |
What is the value of the definite integral \( \int_0^{\frac{\pi}{2}} \frac{dx}{\sec(2x)} \)? | automath |
Given three pairwise coprime integers \(a\), \(b\), and \(c\) representing the dimensions of a box \(a \times b \times c\), determine the number of unit cubes that will be cut into two pieces by a single plane slice that intersects the box. For the following input values for \(a\), \(b\), and \(c\):
1. \( \{1, 2, 3\} ... | automath |
Our club has 20 members, 10 boys and 10 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender? Assume no one can hold both offices. | tiger_math |
What is the expected number of times I can run three independent machines, each failing with a probability of \( \frac{1}{3} \), until all three machines fail simultaneously? | automath |
The teacher gave candy to Yoongi, Jimin, and Taehyung. If Yoongi got more candies than Taehyung and Taehyung got more candies than Jimin, who got the fewest candies? | tiger_math |
In a debate competition, Sophia scored a total of 80 points and Olivia scored 20 more points than Sophia. In the second round, Sophia scored three times as many points as Olivia in the first round and Olivia scored 5 less points than Sophia. How many more points in total did Olivia score than Sophia? | nvidia_math |
The value of $a$ is one-third the value of $b$, and the value of $b$ is one-fourth the value of $c$. If $c$ is 72, what is the value of $a$? | tiger_math |
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 |
A whole pizza was cut into 8 slices. Angeli and Marlon ate 3/2 slices each. How many slices of pizza are left? | tiger_math |
If $m+\frac{1}{m}=8$, then what is the value of $m^{2}+\frac{1}{m^{2}}+4$? | tiger_math |
Given that the temperature decreases at a rate of 6 ∘C per kilometer of elevation, how much will the temperature drop if a climber ascends 1.5 kilometers? | automath |
Consider the piecewise function defined as follows:
\[
f(x) =
\begin{cases}
3 - x^2 & \text{if } -1 \leq x < 2 \\
2x - 4 & \text{if } 2 \leq x \leq 4
\end{cases}
\]
(a) Is the function \( f(x) \) continuous at \( x = 2 \)? Justify your answer with limits.
(b) Is the function \( f(x) \) differentiable at \( x = 2... | automath |
Divide the number 81 into 4 positive integers such that each integer is one more than a multiple of 5. What is the largest possible remainder when 81 is divided by the product of these four numbers? | nvidia_math |
A company has observed that the amount of money spent on advertising and the number of customers acquired are directly proportional. In the first quarter, they spent $1000 on advertising and acquired 200 customers. To reach a total of 500 customers by the end of the second quarter, how much should they spend on adverti... | nvidia_math |
Determine the range of the function f(x) = x/(x+1) on the interval (-∞, 0]. | nvidia_math |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | tiger_math |
If $5,000,000 is the initial amount placed in an account that collects 7% annual interest, which of the following compounding rates would produce the largest total amount after two years?
Answer Choices: (A) compounding annually (B) compounding quarterly (C) compounding monthly (D) compounding daily (E) All four of the... | tiger_lab_math |
What is the value of $x^2y^3z$ when $x = \frac13$, $y = \frac23$, and $z = -9$? | tiger_math |
A magician has a top hat with 20 red marbles and a top hat with 30 blue marbles. If he takes away 3 red marbles and four times as many blue marbles as red marbles (without looking), how many marbles in total does he have left? Let's write a program. | tiger_lab_math |
A garrison of 2000 men has provisions for 54 days. At the end of 15 days, a reinforcement arrives, and it is now found that the provisions will last only for 30 days more. What is the reinforcement?
Answer Choices: (A) 1800 men (B) 3900 men (C) 600 men (D) 3800 men (E) 900 men | tiger_lab_math |
Consider the series of cubes of positive integers:
$1^3+2^3+3^3+\ldots+k^3 = \left(\frac{k(k+1)}{2}\right)^2$.
Find the smallest positive integer $k$ such that $1^3+2^3+3^3+\ldots+k^3$ is a multiple of $243$. | nvidia_math |
the average of 7 consecutive numbers a , b , c , d , e , f and g is 9 . what percent of g is a ? Let's write a Python program. | tiger_lab_math |
In propositional logic, when analyzing the implication \( A \rightarrow B \), how many distinct truth assignments exist that result in the implication being false, and what are those assignments? | automath |
Lily has 5 times as many seashells as Alex, and Alex has 3 times as many seashells as Ben. If Ben has 24 seashells, how many do Lily and Alex have combined? | nvidia_math |
Lily has $3$ gallons of milk and gives $\frac{12}{5}$ gallons to James. What fraction of a gallon does she have left? Please respond by writing a program in Python. | tiger_lab_math |
Find the mass percentage of C in CuCO3 | tiger_math |
Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$? | nvidia_math |
Alice has a deck of 20 cards, each card having either a number (from 1 to 5) or a letter (from A to E). She draws 5 cards at random without replacement. What is the probability that she gets exactly two cards with numbers and three cards with letters, given that the cards are arranged in a specific order (e.g., number-... | nvidia_math |
Find the largest integer value of $n$ such that $n^2-7n+12$ is negative. | tiger_math |
In a rhombus, a circle is inscribed such that the diagonals of the rhombus are the diameters of the circle. What is the ratio of the area of the rhombus to the area of the circle? Express your answer as a common fraction in terms of π. | nvidia_math |
A plane is observed from a point on the ground. Initially, the angle of elevation to the plane is \(45^\circ\). After 20 seconds, the angle of elevation decreases to \(30^\circ\). If the plane is flying at a constant speed of \(360\) km/h, what is the altitude of the plane above the ground in meters? | automath |
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set
$\{-7,-5,-3,-2,2,4,6,13\}.$
Find the minimum possible value of
$(a+b+c+d)^2+(e+f+g+h)^2.$ | nvidia_math |
A man is 25 years older than his son. In two years, his age will be twice the age of his son. The present age of the son is?
Answer Choices: (A) 11 (B) 23 (C) 27 (D) 22 (E) 91 | tiger_lab_math |
Factor the following expression: $55z^{17}+121z^{34}$. | nvidia_math |
Find the cross product of $\begin{pmatrix} 5 \\ 2 \\ -6 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}.$ | nvidia_math |
Reuben opens a sandwich shop selling his namesake sandwich and pastrami sandwiches. Pastrami cost $2 more than the Reuben. He sells 10 Reubens and 5 Pastrami sandwiches and earns $55. How much does a pastrami sandwich cost? | nvidia_math |
A company produces two products: product A and product B. Each unit of product A requires 3 hours of machine time and 2 hours of labor to produce, and it sells for $50 per unit. Each unit of product B requires 2 hours of machine time and 1 hour of labor to produce, and it sells for $35 per unit. The company has 120 hou... | tiger_lab_math |
a and b can do a work in 4 hours and 12 hours respectively . a starts the work at 6 am and they work alternately for one hour each . when will the work be completed ? Let's write a program. | tiger_lab_math |
What is the least number which when divided by 6, 9, 12 and 18 leaves remainder 4 in each care?
Answer Choices: (A) 30 (B) 40 (C) 36 (D) 56 (E) 66 | tiger_lab_math |
What is the value of the binomial coefficient \({5 \choose 3}\)? | automath |
In a small town, a mini-golf course has 4 different themed holes: Jungle, Castle, Pirate, and Space. If a player wants to plan their game so that they play each themed hole exactly once but in different orders over multiple games, how many different play order combinations can they achieve? | tiger_math |
Consider the integral
$$I = \int_0^{\frac{\pi}{2}} \frac{dx}{a\sin^2 x + b\sin x \cos x + c\cos^2 x},$$
where \(a\), \(b\), and \(c\) are constants such that \(a \neq 0\).
Determine the value of \(I\) in terms of \(a\), \(b\), and \(c\) for the following scenarios:
1. When \(b^2 - 4ac > 0\).
2. When \(b^2 - 4ac =... | automath |
Find the number of three-digit positive integers that have exactly two digits in common with the number 647. | nvidia_math |
What is the smallest possible degree of a polynomial $P(x)$ such that $P(x^2)$ can be factored into a product of 3 non-constant polynomials with integer coefficients, one of which is $x^4+2x^2+1$? | nvidia_math |
Sure, here's a problem for a math student to solve:
Consider the heat equation, u_t = k u_xx, where u(x,t) is the temperature at position x and time t, k is a constant, and subscripts denote differentiation. Suppose the temperature is initially given by u(x,0) = 2x - x^2, for 0 <= x <= 2. What is the temperature distr... | tiger_lab_math |
The average of three consecutive terms in a geometric sequence is 12. If the middle term is $x$, what is the value of $x$? | nvidia_math |
Solve the following system of linear equations using matrix methods:
```
2x + 3y - z = 4
4x - 2y + 3z = -7
3x + 4y - 5z = 1
``` | tiger_lab_math |
The value of the expression \[\frac{(2^{1004}+5^{1005})^2-(2^{1004}-5^{1005})^2}{2^{1004}\cdot5^{1005}}\]is $k$ for some positive integer $k$. What is $k$? | tiger_math |
Danny bought 2Q steaks for W dollars. Jerome buys R steaks for a 50% discount, how much will the steaks cost him in cents?
Answer Choices: (A) 50RW/Q. (B) 50QR/W. (C) 25RQ/W. (D) 25RW/Q. (E) RW/ (4Q) . | tiger_lab_math |
Is the product of compact sets compact in the box topology? If it is not, provide a specific counterexample and explain why it demonstrates that the property does not hold. | automath |
What is the supremum of the set of real numbers \( \{ x \in \mathbb{R} : x < 5 \} \) when considering the extended real number line \( \overline{\mathbb{R}} \)? | automath |
A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | nvidia_math |
A certain company has 12 machines that produce identical cylindrical tubes with different dimensions. The tubes are meant to contain 385 grams of a substance, which exactly fills the volume of each tube. What is the greatest possible internal diameter (in cm) of the tubes if the company's specifications require a heigh... | nvidia_math |
In a complete locally convex topological vector space \( V \), consider a sequence of non-zero vectors \( (v_k)_{k=1}^\infty \). Under what conditions can we guarantee the existence of a sequence of positive real numbers \( (\alpha_k)_{k=1}^\infty \) such that the series \( \sum_{k=1}^\infty \alpha_k v_k \) converges t... | automath |
Evaluate the triple integral
\[
\int_0^2 \int_0^{\infty} \int_1^{\infty} x e^{y - 2z} \, dz \, dy \, dx.
\]
What is the value of this integral? | automath |
A manufacturing company produces 5000 units of a product per day. The probability of a certain type of defect occurring in any given unit is 0.03. What is the probability that at least 150 units produced on a given day will be defective? | tiger_lab_math |
When expanding the expression $(ax^b+cx^d+e)(fx^{g}-hx^i+jx^k)-(mx^n+p)^q$, what is the highest possible degree term in the resulting polynomial, given that all variables and constants are non-zero? | tiger_math |
In a small village, there are seven households. Each household decides to plant trees in their backyard. Each household can choose between planting one, two, or three apple trees. What is the probability that exactly five of the seven households decide to plant three apple trees? | tiger_math |
If Adam has $15$ of a certain type of rare coin and he knows that $5$ of these coins are worth $12 in total, how many dollars is his entire collection worth? | tiger_math |
2, -6, 18, -54, 162, (...)
Answer Choices: (A) -486 (B) 486 (C) -428 (D) 422 (E) 423 | tiger_lab_math |
Emberly takes her mornings walks every day. If each walk takes her 1 hour covering 4 miles, and she didn't walk for 4 days in March, calculate the total number of miles she's walked. | tiger_lab_math |
A Hénon map is defined by the equations:
$x_{n+1} = 1 - 1.4x_n^2 + 0.3y_n$
$y_{n+1} = 0.6x_n$
Given the initial conditions $x_0 = 0$ and $y_0 = 0$, what is the value of $x_{10}$? | tiger_lab_math |
What is $\frac{0.\overline{72}}{0.\overline{27}}$? Express your answer as a common fraction in lowest terms. | tiger_math |
In a special year-long math competition, three students, Alice, Bob, and Charlie, correctly answer problems every week and accumulate points. Alice scores 2 points for every problem she solves, Bob scores 3 points per problem, and Charlie scores 5 points per problem. At the end of the year, Alice solved 50 problems, Bo... | tiger_math |
Given the correlation coefficients r = 0.85, r = -0.50, r = 0.15, and r = 0.70, identify which of these values indicates a strong positive linear relationship and provide the corresponding strength of correlation based on the general interpretation rules for Pearson's correlation coefficient. | automath |
Consider a complex polynomial of the lowest degree whose zeros are exactly the vertices of a regular heptagon on the complex plane, centered at the origin. How many complex coefficients of this polynomial are integers? | nvidia_math |
Two rival colleges are planning their annual sports tournaments, and each athlete from both colleges requires 8 bottles of water every day to stay hydrated. However, the athletes from the second college are given 1 bottle less than the athletes from the first college. If the first college has 3000 athletes participatin... | nvidia_math |
Let \[ f(x) =
\begin{cases}
-x^2 & \text{if } x \geq 0,\\
x+8& \text{if } x <0.
\end{cases}
\]Compute $f(f(f(f(f(1))))).$ Please respond by writing a program in Python. | tiger_lab_math |
Determine the smallest positive integer $n$ that satisfies the congruence $3n \equiv 1356 \pmod{22}$. | tiger_math |
In a family, 15 people eat only vegetarian, 8 people eat only non-veg, and some people eat both veg and non-veg. There are 26 people who eat veg in the family. How many people eat both veg and non-veg? | tiger_math |
Olga has three stripes on the side of each of her tennis shoes. Rick has one less stripe per shoe than does Olga. But Hortense has double the number of stripes on her tennis shoes as does Olga. In total, what is the combined number of stripes on all of their pairs of tennis shoes? | tiger_lab_math |
m = { - 6 , - 5 , - 4 , - 3 , - 2 , - 1 } t = { - 5 , - 4 , - 3 , - 2 , - 1 , 0 , 1 , 2 } if an integer is to be randomly selected from set m above and an integer is to be randomly selected from set t above , what is the probability that the product of the two integers will be negative ? | tiger_math |
Alex, Ben, and Chris want to buy erasers from an office supply store. The price of one eraser is 15 cents. Alex wants 4 erasers, Ben wants 6 erasers, and Chris wants 3 erasers. How much (in dollars) will the students spend altogether at the office supply store? | nvidia_math |
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$. What is the $x$-coordinate of the vertex of the parabola? | tiger_math |
Find the area of a hyperbolic triangle with sides of length 3, 4, and 5 units in hyperbolic geometry. | tiger_lab_math |
The length of the bridge, which a train 110 metres long and travelling at 45 km/hr can cross in 30 seconds, is?
Answer Choices: (A) 255 (B) 265 (C) 245 (D) 277 (E) 211 Let's write a Python program to solve it. | tiger_lab_math |
Suppose I have a bag with 12 slips of paper in it. Some of the slips have a 2 on them, and the rest have a 7 on them. If the expected value of the number shown on a slip randomly drawn from the bag is $3.25$, then how many slips have a 2? | nvidia_math |
Karen drove from 9:40 a.m. until 1:20 p.m. of the same day. During the first two hours, she drove 75 miles. If her average speed for the entire trip was 45 miles per hour, how many miles did she drive after the first two hours? | tiger_math |
Daniel works for a sports stadium filling water bottles for athletes. He filled 254 water bottles this season. The football team had 11 players that wanted 6 bottles each. The soccer team had him fill 53 bottles. The lacrosse team needed 12 more bottles than the football team. Each team coach during the season also wan... | tiger_lab_math |
What is the value of $Var(X - 2Y + 8)$ given that $Var(X) = 1$, $Var(Y) = 2$, and the correlation $\rho(X,Y) = \frac{1}{6}$? | automath |
Ava walked 3 blocks from her school to the park. She rode her bike 5 blocks to the bookstore. Later, she went back to school the same way. How many blocks did Ava travel in all? | nvidia_math |
What is the derivative of y = 3sin(2x) at x = π/4? | tiger_lab_math |
Simplify $\sin^2 \frac{\pi}{12} + \sin^2 \frac{5\pi}{12}$ | nvidia_math |
Let \( f: [-1, 1] \to [0, 1] \) be a continuous function such that \( x^2 + (f(x))^2 \le 1 \) for all \( x \in [-1, 1] \) and \( \int_0^1 f(x) \, dx = \frac{\pi}{4} \). If we define \( f(x) = \sqrt{1-x^2} \) for all \( x \in [0, 1] \), what is the value of the integral
\[
\int_{-1/2}^{1/\sqrt{2}} \frac{f(x)}{1-x^2} \... | automath |
In a small village, the town council needs to create a new decorating committee for the upcoming festival. The council has a pool of 10 residents from which to choose members for the committee. If the committee requires exactly 4 residents, how many different ways can the members be selected? | tiger_math |
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