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What is the product of all integer solutions to $1<(x-2)^3<27$? | nvidia_math |
Express $0.725$ as a simplified fraction. | tiger_math |
M and N started a business by investing Rs.4000/- and Rs.5000/- respectively. Find the M’s share out of a total profit of Rs.1800:
Answer Choices: (A) Rs.500/- (B) Rs.550/- (C) Rs.600/- (D) Rs.650/- (E) Rs.800/- | tiger_lab_math |
Given that $x$ is a positive number and $x^2 = 729$, what is the value of $x$? | tiger_math |
A population of rabbits initially has 500 individuals and can sustain a maximum population of 10,000. The growth rate of the population is proportional to the product of the number of rabbits and the difference between 10,000 and the current population. Write a logistic growth model and find the population after 5 year... | tiger_lab_math |
Evaluate the sum \( 2^2 + 4^2 + 6^2 + \ldots + 100^2 \). | automath |
If Betty starts with 60 cookies and 10 brownies, and she eats 3 cookies and 1 brownie each day, how many more cookies than brownies does she have after a week of following this eating pattern? | tiger_math |
If \( V \) is a finite-dimensional vector space over the field \( \mathbb{F}_2 = \{0, 1\} \), what is the total number of elements in \( V \) if the dimension of \( V \) is \( k \)? Additionally, can you provide an example of a vector space over \( \mathbb{F}_2 \) that has 6 elements, or explain why such a space cannot... | automath |
Ava has to prepare packages to be shipped. She needs to send a gift to each of her two siblings and her best friend. She also has to mail her phone bill and credit card bill separately. She has three more book returns than bills to ship, and she has twice as many packages to send to her customers as book returns. How m... | nvidia_math |
Noemi lost $400 on roulette and $500 on blackjack. How much money did Noemi begin with if she still had $800 in her purse? Let's write a Python program. | tiger_lab_math |
1,000^2 + 1,001^2 +1,002^2 + 1,003^2 + 1,004^2 =
Answer Choices: (A) 6,030,053 (B) 6,030,054 (C) 6,030,055 (D) 6,020,030 (E) 6,030,057 | tiger_lab_math |
Emily purchases 25 cookies. Her friend Olivia pays for 8 of them. If each cookie costs $2, how much did Emily pay? | nvidia_math |
The positive number $b$ is chosen such that the terms $16, b, \frac{9}{4}$ are the first, second and third terms, respectively, of a geometric sequence. What is the value of $b$, if $b$ is positive? | nvidia_math |
If $x - y = 6$ and $x + y = 12$, what is the value of $y$? | nvidia_math |
if a lends rs . 1200 to b at 10 % per annum and b lends the same sum to c at 18 % per annum then the gain of b in a period of 2 years is ? Let's write a Python program. | tiger_lab_math |
A train 280 m long, running with a speed of 36 km / hr will pass a tree in
Answer Choices: (A) 12 sec (B) 15 sec (C) 16 sec (D) 28 sec (E) None | tiger_lab_math |
If \( \tau \) and \( \beta \) are constants, solve the equation
\[
\cos\left(\frac{\pi t}{\tau}\right) = \cos\left(\frac{\pi t}{\beta \tau}\right)
\]
for \( t \) in terms of \( \beta \) and \( \tau \). What are the possible values of \( t \)? | automath |
What is the reciprocal of the common fraction form of $.\overline{2}$? | nvidia_math |
Find the sum of the greatest common divisor and the least common multiple of $150$ and X if the product of GCD and LCM is 18000. | tiger_math |
What is 5/6 of 30? | tiger_math |
Madeline has 10 flowers. If 4 flowers are red, 2 flowers are white, and the rest are blue, what percentage of flowers are blue? | tiger_math |
Find the value of $\frac{4 \cdot 6! - 20\cdot 5!}{7!}$ | tiger_math |
Count the number of positive integers less than or equal to 20 that can be expressed as the difference of two primes. | nvidia_math |
Sophie's aunt gave her $260 to spend on clothes at the mall. She bought 2 shirts that cost $18.50 each and a pair of trousers that cost x. She then decides to purchase 4 more articles of clothing with her remaining budget. Each item would cost 40 if she split the cost of each item evenly. What is the value of unknown v... | tiger_math |
If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\sin \frac{P + Q}{2}}{\cos \frac{R}{2}} + \frac{\cos \frac{P + Q}{2}}{\sin \frac{R}{2}}.\] | nvidia_math |
Let X denote the number of tails George flips. Then X follows a binomial distribution:
$$X \sim Bin(7,\frac{3}{4})$$
Therefore, P(X = 2) = C(7,2)(\frac{3}{4})^{2}(\frac{1}{4})^{5} where C(7,2)=\frac{7!}{(7-2)!2!}
The final answer is 189/16384. | nvidia_math |
Maddox and Theo both bought 3 Polaroid Cameras, each sold at $20 per camera from Amazon, and decided to sell them on eBay. Maddox sold his cameras at $28 each, while Theo sold his cameras at $23 each. How much more profit did Maddox get from the sale of his cameras than Theo? | tiger_lab_math |
Alice is standing at point $A$ near Cape Town, South Africa, $34^\circ \text{ S}$ latitude and $18^\circ \text{ E}$ longitude. Bob is standing at point $B$ near Barrow, Alaska, USA, $71^\circ \text{ N}$ latitude and $156^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degr... | nvidia_math |
A man purchased 3 blankets @ Rs.100 each, 4 blankets @ Rs.150 each and two blankets at a certain rate which is now slipped off from his memory. But he remembers that the average price of the blankets was Rs.150. Find the unknown rate of two blankets?
Answer Choices: (A) A) 600 (B) B) 350 (C) C) 450 (D) D) 470 (E) E) 50... | tiger_lab_math |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | tiger_lab_math |
At a gathering, 33 guests arrived by car and 55 by bus. The average number of kilometers traveled by car is 120 km, and the average number of kilometers traveled by bus is 55 km. What is the average number of kilometers traveled by all guests? | nvidia_math |
Evaluate the integral $$\int \left( \frac{1}{2}x^{\frac{1}{3}} + 5x^{\frac{3}{4}} \right) \, dx.$$ | automath |
A geometric sequence has a first term of 4 and a common ratio of 2. What is the sum of the first 12 terms? | nvidia_math |
The sequence $2, 7, 12, a, b, 27$ is arithmetic. What is the value of $a + b$? Let's program in Python in the response. | tiger_lab_math |
Two pianos are tuned to a frequency of 523.25 Hz (which is the frequency of middle C) and are mistuned by ±1 Hz. What is the beat frequency produced by these two pianos, and what is the carrier frequency in radians per second? Express your answer for the beat frequency in Hz and the carrier frequency in rad/sec. | automath |
In a garden man has to pick some flowers ans has to pass thru 3 doors. Each door has a guard. Man has to give half of the flowers he had to the guard and in return guard returns the man him one flower back. how many flower should man pick so that at the end he is left with same number of flowers he has picked?
Answer C... | tiger_lab_math |
Find the coefficient of $x^2$ in the expansion of $-2(x^2 - 3) + 3(2x^2 + 4x - 1) - (x^2 - 5x + 2)$? | tiger_math |
What is the relationship between the number of vanishing moments \( M \) of a wavelet function \( \psi(t) \) and its ability to represent a signal \( s \) in the presence of Gaussian noise \( v \), when \( s \) has a sparse representation in a certain wavelet basis? Additionally, how does this relate to the effectivene... | automath |
What is the Cartesian equation of the polar graph defined by the equation \( r = 8 \cos(4\theta) \), and how many petals does this rose curve have? | automath |
Pauly is making ice cubes. He needs 10 pounds of cubes. He knows that 2 ounces of water make 1 cube and each cube weighs 1/16th of a pound. It takes him 1 hour to make 10 cubes. Every hour his ice maker run costs $1.50. Every ounce of water costs $0.10. How much will it cost to make all the ice? Let's write a Python pr... | tiger_lab_math |
Debby received twenty-one text messages before noon. After noon, the number of texts she received every hour doubled starting from 2 texts in the first hour. If this pattern continued until midnight, how many text messages did Debby receive in total? | tiger_math |
What is the relationship between the eigenvalues of the continuous-time state matrix \( A_c \) and the discrete-time state matrix \( A_d \) obtained through the bilinear transform, specifically how are they mapped using the formula \( \frac{1 + \lambda T/2}{1 - \lambda T/2} \)? Can you derive this mapping and provide a... | automath |
"statement 1": Ramesh can finish a work in 20 days and Sushil in 25 days, "statement 2" :Ramesh can finish a work in more than 10 days and Sushil in 25 days, "options: " Entailment or contradiction or neutral? Let's write a Python program. | tiger_lab_math |
There are 3 numbers. The third is 4 times the second and the second is two times the first. The second number of the 3 numbers is 90. What is the average of the 3 numbers? | tiger_math |
Determine the covariance matrix of the bivariate density function given by \( f(x, y) = A \exp\left[-\frac{1}{2}\left(x^2 + y^2 - 2pxy\right)\right] \), where \( A \) is the normalizing constant and \( p \) is the correlation between \( x \) and \( y \). Additionally, explain whether this density function represents a ... | automath |
Given a point \( p = (1, 2, 3) \) and a rotation about the axis \( a = (2, 3, 6) \) by an angle of \( 90^\circ \), determine the coordinates of the point \( p' \) after the rotation has been applied. Note that you need to normalize the axis \( a \) before performing the calculations. | automath |
Given the function \( s(t) = 4.5t^2 \), calculate the average rate of change of \( s \) over the intervals \([6, 12]\) and \([6, 9]\). Additionally, derive a formula for the average rate of change from \( t = 6 \) to \( t = 6 + h \), where \( h \) is a small positive number. What do the results indicate about the behav... | automath |
What are the spherical coordinate bounds for the triple integral
$$\int_{-a}^{a}\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\int_{0}^{\sqrt{a^2-x^2-y^2}}f(x,y,z)dzdydx$$
and how do they compare to the full sphere integration with respect to the given function? | automath |
Given the matrix \( A = D + aJ \), where \( D \) is a diagonal matrix with distinct diagonal entries and \( J \) is an all-ones matrix, derive the formula for the inverse of \( A \) and express it in terms of \( D^{-1} \) and scalar \( a \). What conditions must be satisfied for the inverse \( A^{-1} \) to exist? | automath |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 4,$ and
\[\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}.\]Compute $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}).$ | nvidia_math |
Let \(\gamma(t) = (t, t^2)\) be a parametrized curve in \(\mathbb{R}^2\). Calculate the curvature \(\kappa(t)\) of this curve at the point where \(t = 1\). | automath |
Consider the inductive type $\text{nat}$ defined as the smallest type containing the element $z$ and closed under the successor function $s$. If we define a recursive function $\text{rec}_{\text{nat}}(x.e_1;e_2)$ such that:
- $\text{rec}_{\text{nat}}(x,e_1;e_2)$ computes $e_1$ when $x$ is replaced by the introduction ... | automath |
Rodney has 35 dollars more than Ian. Ian has half as much money as Jessica has. If Jessica has 100 dollars, how much more money does Jessica have than Rodney? | tiger_lab_math |
Consider two positive even integers less than $15$ (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result? | nvidia_math |
A license plate in the country Kerrania consists of four digits followed by two letters. The letters A, B, and C are used only by government vehicles while the letters D through Z are used by non-government vehicles. Kerrania's intelligence agency has recently captured a message from the country Gonzalia indicating tha... | tiger_lab_math |
One side if a square is doubled. If the parameter is now 12x. What was the length of one of the sides before it was doubled?
Answer Choices: (A) 12x/4 (B) 12x/2 (C) 6x/2 (D) 12x/6 (E) 4x/1 | tiger_lab_math |
Find the polynomial of minimal degree, in $x,$ which has rational coefficients, leading coefficient $1$, and roots $2+\sqrt{5}$ and $3+\sqrt{7}.$ (Write the terms in decreasing order of degree.) | nvidia_math |
Let
\[f(x) = x^3 + 6x^2 + 16x + 28.\]The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b).$ Enter the ordered pair $(a,b).$ | nvidia_math |
In the coordinate plane, a reflection across a line with slope 3 followed by a translation of 5 units to the right will transform triangle ABC with vertices A(0,0), B(2,0), and C(1,1) to triangle A'B'C' with vertices A'(6,2), B'(8,-2), and C'(7,-1). Find the y-intercept of the line of reflection. | nvidia_math |
Michael has 2 cats, 3 dogs, 1 parrot, and 4 fish. He needs to pay a friend to watch them, who charges $13 a night per cat, $18 per dog, $10 per parrot, and $4 per fish. How much does Michael have to pay for a night of pet sitting? | tiger_math |
Patrick buys a season pass for each of his sons to visit the local amusement park as often as they want. The passes were $x each. The oldest son went to the park 35 times over the summer and the youngest went 15 times. The cost per trip to the park was 4. What is the value of unknown variable x? What is the value of ... | tiger_math |
Find the maximum and minimum values of the function \( f(x, y) = x^2 - 2xy + 2y^2 \) subject to the constraint \( x^2 + y^2 = 1 \). What are these extrema? | automath |
What is the minimum number of distinct vectors required to form a basis for the vector space \( V \) defined as the set of all polynomials of degree at most 3, denoted \( \{f | f(x) = ax^3 + bx^2 + cx + d, a,b,c,d \in \mathbb{R}\} \), and can you provide an example of such a basis? | automath |
If Katie's mother wants to provide 2 granola bars for each of the 30 kids, including Katie, after Katie's soccer game, and each box contains 12 bars, how many boxes should Katie's mother buy? | tiger_math |
One-third of a class of 39 students took part in a math competition. How many students did not participate in the competition? | tiger_lab_math |
Luna wants to send identical packets of cookies to her friends as gifts. She has 420 cookies in total and can pack them in bags of the same size. If she wants to make sure each bag contains no more than 12 cookies and she also wants to have at least 21 bags, what is the sum of all possible values for the number of cook... | nvidia_math |
Find the sum of the values of x for which |3x - 5| = 2. | nvidia_math |
In a magical world, a wizard collects enchanted stones in seven different colors corresponding to the days of the week. On Monday, he collects 3 blue stones, 5 green stones, and 2 red stones. On Tuesday, the total number of stones he collects is double the amount from Monday, but the stones are split evenly across blue... | tiger_math |
Suppose you have a fair six-sided dice. You roll the dice 100 times and record the number of times the dice landed on an odd number. Using Monte Carlo simulation, estimate the probability that the dice lands on an odd number. | tiger_lab_math |
What is the smallest Sophie Germain prime that is greater than 300? | tiger_lab_math |
A shop owner sells 50mtr of cloth and gains SP of 10 mtrs. Find the gain %?
Answer Choices: (A) 25% (B) 40% (C) 50% (D) 60% (E) 70% | tiger_lab_math |
Charles is moving from Springfield, which has 482,653 people, to Greenville, which has 119,666 fewer people. What is the total population of Springfield and Greenville? | tiger_lab_math |
Determine the remainder when the number $2^{17}+3^{23}+5^{19}+7^{13}$ is divided by 11. | nvidia_math |
The number $a-\sqrt{b}$ and its radical conjugate have a sum of $6$ and a product of $-3$. Find $a+b$. | tiger_math |
What is the maximum number of distinct denominators in a set of fractions, each of which has the form $\frac{p}{q}$, where $p$ and $q$ are positive integers less than 100, and the decimal representation of $\frac{p}{q}$ terminates? | nvidia_math |
The polynomial $f(x)=x^4+ax^3+bx^2+cx+d$ has real coefficients, and $f(2i)=f(2+i)=0$. What is $a+b+c+d$? | nvidia_math |
If Jason continues the babysitting pattern, but repays his debt by babysitting for 50 hours, how many dollars more would he earn than the amount he borrowed? | tiger_math |
A pineapple costs Rs 7 each and a watermelon costs Rs. 5 each. If I spend Rs 38 on total what is the number of pineapple I purchased?
Answer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) Data Inadequete | tiger_lab_math |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the maximum value of
\[\frac{abc}{(a^2 - 2a + 1)(b^2 - 2b + 1)(c^2 - 2c + 1)}.\] | nvidia_math |
How many operations at worst are needed to move one level up in the Calkin-Wilf tree from a rational number greater than 1, and what is the significance of this in terms of finding a secret rational number with a secret numerator and denominator? | automath |
Luna purchases 8 bouquets of flowers at $8 each and sells them for a total of $160. If she uses decorative ribbons that cost her $1.50 for each bouquet, how much profit does she make for all the bouquets she sells? | nvidia_math |
Is the duration of a fixed-rate bond a convex function of the interest rate, and if so, what is the behavior of its second derivative with respect to the interest rate in both positive and negative rate environments? | automath |
a farmer has an apple orchard consisting of fuji and gala apple trees . due to high winds this year 10 % of his trees cross pollinated . the number of his trees that are pure fuji plus the cross - pollinated ones totals 153 , while 3 / 4 of all his trees are pure fuji . how many of his trees are pure gala ? | tiger_math |
What is the greatest prime factor of the sum of all positive divisors of 100? | nvidia_math |
A bag contains 9 marbles: 2 are green, 4 are red, and 3 are blue. Laura chooses a marble at random, and without putting it back, chooses another one at random. What is the probability that the first marble she chooses is blue and the second marble she chooses is green? Write your answer as a fraction in simplest form. | automath |
What is the expanded form of the binomial \((a - b)^6\), including all coefficients and their correct signs? | automath |
Given the matrices
\[
A = \begin{pmatrix} \cos(a) & -\sin(a) & 0 \\ \sin(a) & \cos(a) & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]
and
\[
B = \begin{pmatrix} \cos(b) & 0 & -\sin(b) \\ 0 & 1 & 0 \\ \sin(b) & 0 & \cos(b) \end{pmatrix},
\]
both representing rotations in 3-dimensional space, find the axis of rotation and the an... | automath |
Mr. Smith takes his wife, his parents, and his 3 children to a seafood buffet. The price for the adult buffet is $30. The price for the children’s buffet is $15. Senior citizens get a 10% discount. How much does Mr. Smith spend on the buffet for his entire family? | tiger_lab_math |
Yanna bought ten shirts at $5 each and three pairs of sandals at $3 each. How much change did she get back if she gave a one hundred dollar bill? | nvidia_math |
Determine if the number \( 1000001 \) is prime. If it is not, provide its prime factorization. | automath |
The cost of fuming the engine of an army tank is proportional to the square of the speed and Rs 64 per hour for a speed of 16km/hr. Other costs amount to Rs 400 per hour. The tank has to make a journey of 400km at a constant speed.The most economical speed for this journey is :
Answer Choices: (A) 20 km/hr (B) 30 km/hr... | tiger_lab_math |
Casey is going to stay at a motel while she works her summer job as a ski instructor. The hotel charges $280/week or $1000/month. If Casey will be staying 3 months, each with exactly 4 weeks, how much money does she save by paying monthly? | tiger_math |
Three fair coins are flipped simultaneously. What is the probability that exactly two coins will land on heads? | tiger_math |
Consider the hypergraph \( H = \text{hyperGraph} \{ a*b*c, b*d, c*e, b*c*f \} \). Determine the number of vertices in \( H \) that are classified as leaves. | automath |
Let \(X\) be a random variable that follows a Beta distribution with parameters \(\nu = 3\) and \(\omega = 5\). What is the variance of \(X\)? | automath |
What is the value of Euler's Totient function \( \varphi(N) \) for the semiprime \( N = 131 \times 11953 \) using the formula \( \varphi(N) = N - 131 - 11953 + 1 \)? | automath |
What is the value of $\log_2 8$? | tiger_math |
Yoque borrowed money from her sister. She promised to pay it back in 11 months including an additional 10% of the money she borrowed. If she pays $15 per month, how much money did she borrow? | tiger_math |
Monica made cookies for herself and her family. Her father ate some cookies and her mother ate half as much as the father. Her brother ate 2 more than her mother. Monica has 8 cookies left from the 30 she made in total. How many cookies did her father eat? | tiger_math |
Given the vector field \( \mathbf{F_1}(x, y, z) = 0\mathbf{i} - 0.1667y\mathbf{j} + 0\mathbf{k} \) representing a gradient field pointing in the negative \( y \) direction, and the second gradient field \( \mathbf{F_2}(x, y, z) = -y\mathbf{i} + x\mathbf{j} + 0\mathbf{k} \) representing the equation of a line \( y = -x ... | automath |
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