inclusionAI/Ring-lite-rl-data huggingface dataset:
Use Green's theorem to calculate the line integral \( \oint\limits_{C2} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy \), where \( C1: x^{2}+y^2 = 1 \) and \( C2: r = \frac{2}{\sqrt{2-\cos(\theta)}} \). The region \( R \) is the area between \( C1 \) and \( C2 \), and both curves are oriented positively. Given that \( \oint\limits_{C1} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy = 2\pi \) and \( \iint\limits_R \frac{1}{(x^{2}+y^2)^2} \, dx \, dy = \frac{\pi}{2} \), find the value of the line integral over \( C2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find a possible expression for the differentiable function $g(x)$ given the integral relationship: \[ \int g(x)e^{\frac{x}{4}}dx = 4g(x)e^{\frac{x}{4}} - \int 8x^2e^{\frac{x}{4}}dx, \] where integration by parts is applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}x^3}
Evaluate the sum \( \sum_{k=0}^{4}\csc^2\left(\frac{\pi}{6}+\frac{k\pi}{5}\right) \) given that \( \csc^2(5 \times \frac{\pi}{6}) = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Find the limit \( \lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Construct a field extension $E$ of the field $\mathbb{F}_{11}$ such that the degree of the extension, $[E:\mathbb{F}_{11}]$, is equal to $3$. Provide the irreducible polynomial used for this extension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 + x + 4}
Evaluate the expectation $E[AX^TBA]$ in terms of $E[X]$, given that $A \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{n \times p}$, $\|A\| \leq 1$, and $E[X] \geq 0$. Assume $B$ is a known matrix of appropriate dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A (E[X])^\top B A}
Evaluate the double integral \( \int \int_D xydA \) where \( D \) is the union of the regions defined by the following bounds: \( \int_{-1}^{1}\int _{0}^{\sqrt{1-x^2}}xydydx + \int_{-1}^{1}\int _{-x/3-1/\sqrt{3}}^{\sqrt{x^2-1}}xydydx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$x = \frac{2}{3}$}
Find a polynomial in two variables with integer coefficients whose range is the positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$p(x,y)=x^2+(xy-1)^2$.}
Given the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}_2[x] \) defined by \( T(1,0) = 1+x \) and \( T(1,1) = 1-x \), find \( T^{-1}(2x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, -1)}
Determine the set of real numbers \(\alpha\) for which the function \(f_\alpha(x) = \sum_{n=1}^\infty n^\alpha x e^{-nx}\) is continuous on \([0, \infty)\) and integrable with respect to the Lebesgue measure on \([0, \infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
Evaluate the integral \( \oint_C \frac{zf'(z)}{f(z)} \,ds \) using the argument principle, where \( f(z) = z^4 - 2z^3 + z^2 - 12z + 20 \) and \( C \) is the circle \( |z| = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\pi}
Evaluate the integral \( \int_{\lvert z \rvert = \frac{1}{2}} \frac{dz}{z \sin\left( \frac{1}{z} \right)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose that vectors $x_1, x_2, \dots, x_n$ have the following property: for each $i$, the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1, x_2, \dots, x_n$ are not parallel, what is the value of the sum $x_1 + x_2 + \dots + x_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called balanced if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of nonempty balanced sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3751}
Find the direct limit of the sequence of groups \( \frac{1}{n}\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}}
Calculate the limit \( \lim_{n \to \infty} \int f_n \, d\mu \) where \( S = \mathbb{N} \), \( A = P(\mathbb{N}) \), \( \mu = h \, dv \) with \( v \) being the counting measure and \( h(k) = \frac{1}{4^k} \), and \( f_n(k) = \left(\frac{k}{n} - 2\right)^k \cdot \mathbf{1}_{(1,2,3,...,n)}(k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Find the equation of a polynomial of degree 6 whose roots are the products of the roots of the polynomial \(g(x) = x^3 - 2x^2 + 3x - 4\) taken three at a time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 4)^6}
Find the greatest integer $k$ such that among 2016 customers who visited a store, we can select $k$ customers where either all $k$ were in the store at the same time, or no two of them were in the store at the same time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given a complex number ω such that ω^5 = 1 and ω ≠ 1, evaluate the expression:

\[ \frac{\omega}{1+\omega^2} + \frac{\omega^2}{1+\omega^4} + \frac{\omega^3}{1+\omega} + \frac{\omega^4}{1+\omega^3} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the range of the function, $f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{\pi}{2}, \frac{\pi}{2}]}
Determine the least number of terms needed to approximate the integral \( \int_0^{1/2} \frac{x - \arctan(x)}{x^2} \, dx \) using the series expansion \( \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \) for all \( x \in [-1,1] \), such that the error is less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The inclination angle of the line $x- \sqrt {3}y+3=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{6}}
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n=0,1,2$}
In measuring the sides of a rectangle, one side is taken 15% in excess, and the other 11% in deficit. Find the error percent in the area calculated from these measurements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.35\%}
Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
The price of a math textbook in the school bookshop is $45. If those sold in bookshops outside the school cost 20% less, Peter is considering buying from other bookshops. Additionally, there is a science textbook sold at the school bookshop for $60, which costs 25% less in external bookshops, and a literature textbook priced at $35 in the school bookshop, which costs 15% less in outside bookshops. The school bookshop has a tax rate of 7%, while alternative bookshops have a tax rate of 6%. If Peter needs to pay a shipping cost of $10 for the books purchased from the alternative bookshop, how much can Peter save in total (including tax and shipping) by purchasing these three textbooks from alternative bookshops instead of the school's?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$22.40}
Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. She continues rolling the dice and recording the product until one of the products appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given that the three vertices of $\triangle ABC$ are all on the hyperbola $\frac{x^{2}}{2} - \frac{y^{2}}{4} = 1$, the slopes of the three sides $AB$, $BC$, $AC$ all exist and are non-zero, and their sum is $-1$; furthermore, the midpoints of $AB$, $BC$, $AC$ are $M$, $N$, $P$ respectively, and $O$ is the coordinate origin. The slopes of the lines $OM$, $ON$, $OP$ are $k_{1}$, $k_{2}$, $k_{3}$ respectively, and are all non-zero. Find $\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Using arithmetic operation signs, write the largest natural number using two twos.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given the joint moment generating function of random variables X and Y:
\[ M_{X,Y}(a,b) = \frac{4}{5}\left[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}\right] \]
find the expected value of the product E(XY).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{20}}
When Erick went to the market to sell his fruits, he noticed that the price of lemons had risen by 50% and the price of grapes had increased by 25%. In anticipation of these changes, he brought different quantities of each fruit to the market. He had 80 lemons, 140 grapes, 60 oranges, and 100 apples in his basket. He originally planned to sell the lemons at $8 each, grapes at $7 each, oranges at $5 each, and apples at $4 each. However, since the price of oranges increased by 10% and apples by 20%, he needed to update his selling prices.

How much money did he collect from selling the fruits at the new prices if he sold all the fruits in his basket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2995}
James decides to sell 80% of his toys. He bought them for $20 each and sells them for $30 each. He had a certain number of toys and made $800 more compared to before he bought them. How many toys did he have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Determine the minimal value of the maximum order (max(a, b, c)) among the subgroups $P_{23}$, $P_{13}$, and $P_{12}$, generated by permutations $p_1$, $p_2$, and $p_3$ in the symmetric group $S_n$ for $n > 7$. What is this minimal maximum order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) of the vector field \( \mathbf{F}(x,y,z) = -x^2\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) along the boundary \( C \) of the surface \( S \) defined by \( f(x,y) = 4x - 8y + 30 \), within the rectangle \( R = \{(x,y): -2 < x < 3, 0 < y < 2\} \). The boundary \( C \) is oriented counterclockwise when viewed from above. Use Stokes' Theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Evaluate the limit: \[ \lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for $x + y + z$. Provide your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
A computer store offers employees different discount tiers off the retail price based on the length of their employment and additional conditions such as bulk purchases or store credit usage. The store has three computer models with different retail prices: Model A costs $500, Model B costs $650, and Model C costs $800. These retail prices are determined after applying varying markup percentages for each model (12% for Model A, 15% for Model B, and 18% for Model C). Employees who have worked for less than 1 year receive a 10% discount, employees who have worked for 1-2 years receive a 15% discount, and employees who have worked for 2 years or more receive a 20% discount. In addition, employees get an extra 5% discount if they purchase 5 or more computers, and employees can use their $100 store credit towards the purchase.

An employee who has worked for 1.5 years decides to purchase computers in bulk. If the employee purchases 5 Model A computers, 3 Model B computers, and 2 Model C computers, how much would they save on the total cost with their employee discount, bulk purchase discount, and store credit usage, compared to the final retail prices of all the computers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1310}
Let $f : \mathbb{N} \to \mathbb{R}$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \left(\frac{n}{p}\right) - f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the quarter circle defined by the equation $x^2 + y^2 = 16$, consider the quarter arc $AD$. Points $B$ and $C$ are arbitrary points on the arc $AD$, with $C$ located between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $\odot$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ and $c$, $(a + b) \odot c = (a \odot c) + (b \odot c)$ and $a \odot (b + c) = (a \odot b) \odot c.$ Given that $5 \odot 5 = 160$, find the value of $7 \odot 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{896}
从 $1 \sim 9$ 这九个正整数中任选三个,则这三个数两两互素的概率是 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Determine the values of \(x\) for which the series \(\sum_{n=1}^\infty \frac{n! \> x^n}{(x+1)(2x+1) \cdots (nx+1)}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1)}
4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\qquad$ in total.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{797}
Provide an example of a group \( G \) that contains an element \( a \) of order 3 and an element \( b \) of order 4, such that the order of the product \( ab \) is less than 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_4}
Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$       
, and the determinant  $D_3=$         
|1 1 1|
|1 3 3|
|1 3 5|         
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Dave had some apps on his phone. After deleting 18 of them, he installed "x" number of new apps and then had 5 apps left. How many apps did he have on his phone at the start?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let  $\triangle ABC$  be a triangle with circumcenter  $O$  satisfying  $AB=13$ ,  $BC = 15$ , and  $AC = 14$ .  Suppose there is a point  $P$  such that  $PB \perp BC$  and  $PA \perp AB$ .  Let  $X$  be a point on  $AC$  such that  $BX \perp OP$ . What is the ratio  $AX/XC$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{169}{225}}
Given $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$ in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is $8102$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1991}
Four copies of the triangle shown are joined together, without gaps or overlaps, to make a parallelogram. The triangle has edge lengths of 13 cm, 13 cm, and 12 cm. Find the largest possible perimeter of the parallelogram. Express your answer in centimeters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Determine the signature of the matrix \( \begin{pmatrix} -3 & 0 & -1 \\ 0 & -3 & 0 \\ -1 & 0 & -1 \end{pmatrix} \). You may use methods such as Gaussian elimination to find the answer without computing eigenvalues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 3, 0)}
Let $T = \{9^k : k ~ {is ~an~ integer}, 0 \le k \le 4000\}$ . Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{184}
Calculate the expected present value of a continuous payment stream of £190 per annum for 20 years, beginning at time 0. For the first 12 years, the force of interest is 5%. After 12 years, the interest rate changes to 3% with probability 0.25, remains at 5% with probability 0.5, and increases to 7% with probability 0.25. What is the expected present value of this payment stream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2403.38}
Evaluate the second mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & \text{if }(x,y) \neq (0,0) \\ 0 & \text{otherwise} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the Cauchy principal value of the integral: $$ \int_0^{\infty}\frac{\tan x}{x}\mathrm dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{203}}
已知x、y是正实数，且满足$x y + x + y = 7 1 , x ^ { 2 } y + x y ^ { 2 } = 8 8 0$，则$x ^ { 2 } + y ^ { 2 } =$ ·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{146}
How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16}
Solve the recurrence relation \( T(n) = 2T(\sqrt{n}) + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Determine the values of \( p \in \mathbb{R} \) for which the integral \( \int_0^\infty \frac{\sin x}{x^p} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Find the minimum value of \( \int_{0}^{1} (f'(x))^2 \, dx \) given that \( f \in C^{(1)} [0,1] \), \( f(1) = b \), and \( \int_{0}^{1} f(x) \, dx = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3b^2}
若$\sin x+\sqrt{3}\cos x+1\le m$在$0\le x\le \frac{\pi }{2}$上有解，则实数$\textit{m}$的最小值为____________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum area of $\triangle PAB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + \sqrt{2}}
Evaluate the integer part of the sum \( \left[\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}}\right] \) using the concept of definite integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Each of the integers $1,2, \ldots, 729$ is written in its base-3 representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122 \ldots \ldots$ How many times in this string does the substring 012 appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148}
A cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29.6300\%}
Determine the dimension of the vector space $\mathfrak{m}/\mathfrak{m}^2$ for the curve $\mathcal{C}=\{ (x,y): x^3+y^3-1=0 \} \subset \mathbb{C}^2$, where $\mathfrak{m} = \langle x-a,y-b\rangle/\langle x^3 + y^3 -1\rangle$ and $p=(a,b) \in \mathcal{C}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the determinant of the matrix \(A= \begin{bmatrix} 1 & x & x^2 & x^3\\ x & x^2 & x^3 & 1 \\ x^2 & x^3 & 1 & x \\ x^3 & 1 & x&x^2 \end{bmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^4 - 1)^3}
Determine the cardinality of the set \( S = \{ h: D \rightarrow D \mid h \text{ is analytic in } D \text{ and } h(z)^2 = \overline{h(z)} \text{ for all } z \in D \} \), where \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The base of a prism is a square with a side length of \( a \). One of the lateral faces is also a square, and another is a rhombus with an angle of \( 60^{\circ} \). Determine the total surface area of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 (4 + \sqrt{3})}
Evaluate the limit \( \lim_{Z \to Zi} (Z^2 - \bar{Z}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 + i}
Evaluate the limit: $$\lim\limits_{x\to 0^{+}}\lim\limits_{n\to \infty}\frac{d^n}{dx^n}\left(\frac{e^{\sqrt{x}}+e^{-\sqrt{x}}}{x^n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A man engaged a servant on the condition that he would pay him Rs. 900 and a uniform after one year of service. The servant served only for 9 months and received a uniform and a certain amount of money. The price of the uniform is Rs. 100. How much money did the servant receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{775}