instance_id stringlengths 6 10 | context stringlengths 31 1.8k | correct_response stringclasses 2
values | response_a stringlengths 578 8.09k | response_b stringlengths 457 7.72k | difficulty float64 5.5 10 |
|---|---|---|---|---|---|
74047_6.5 | Let $\omega = \cos\left(\frac{2\pi}{10}\right) + i\sin\left(\frac{2\pi}{10}\right)$, which is a primitive $10$th root of unity. Determine the relationship between the fields $K = \mathbb{Q}(\omega^2)$ and $L = \mathbb{Q}(\omega)$, where $[\cdot, \cdot]$ denotes the degree of the field extension. Choose the correct opti... | A | We are given:
- $\omega = \cos\left(\frac{2\pi}{10}\right) + i\sin\left(\frac{2\pi}{10}\right)$, a **primitive 10th root of unity**.
- $K = \mathbb{Q}(\omega^2)$
- $L = \mathbb{Q}(\omega)$
- We are to determine the relationship between the fields $K$ and $L$, and choose the correct option from A to D.
---
### Step 1... | We are given:
- $\omega = \cos\left(\frac{2\pi}{10}\right) + i\sin\left(\frac{2\pi}{10}\right)$, a **primitive 10th root of unity**.
- $K = \mathbb{Q}(\omega^2)$
- $L = \mathbb{Q}(\omega)$
We are to determine the relationship between $K$ and $L$, and choose the correct option among:
A. $[L : \mathbb{Q}] = 10$
B. $... | 6.5 |
65443_6.5 | Calculate the flux integral of the vector field \( \vec{G} = (y, x, 2+z) \) through the surface defined by \( z=4-x^2-y^2 \) for \( z \geq 0 \). | A | We are asked to compute the **flux integral** of the vector field
$$
\vec{G} = (y, x, 2+z)
$$
through the surface defined by
$$
z = 4 - x^2 - y^2, \quad \text{for } z \geq 0.
$$
---
### Step 1: Understand the Surface
The surface is a **paraboloid** opening downward, given by
$$
z = 4 - x^2 - y^2, \quad z \ge... | We are asked to compute the **flux integral** of the vector field
$$
\vec{G} = (y, x, 2+z)
$$
through the surface defined by
$$
z = 4 - x^2 - y^2, \quad \text{for } z \geq 0.
$$
---
### **Step 1: Understand the Surface**
The surface is a **paraboloid** opening downward, given by
$$
z = 4 - x^2 - y^2.
$$
Th... | 6.5 |
39830_6.0 | Evaluate the contour integral \( \oint_{|z|=1} \frac{z^2+1}{(2z-i)^2} \, dz \) using the residue theorem. | B | We are asked to evaluate the contour integral:
$$
\oint_{|z|=1} \frac{z^2+1}{(2z - i)^2} \, dz
$$
using the **Residue Theorem**.
---
### Step 1: Identify the singularities
The integrand is:
$$
f(z) = \frac{z^2 + 1}{(2z - i)^2}
$$
This function has a **pole of order 2** at the point where the denominator is zero:... | We are asked to evaluate the contour integral:
$$
\oint_{|z|=1} \frac{z^2+1}{(2z - i)^2} \, dz
$$
using the **Residue Theorem**.
---
### Step 1: Identify the singularities
The integrand is:
$$
f(z) = \frac{z^2 + 1}{(2z - i)^2}
$$
The denominator is $(2z - i)^2$, which has a **double pole** at:
$$
2z - i = 0 \Ri... | 6 |
45574_8.0 | Evaluate the integral \( \int_{-\infty}^{+\infty}\int_{0}^{+\infty}xe^{-x^{2}e^{y}} \Gamma \left(0,x^2\right)\, dydx. \) | B | We are asked to evaluate the double integral:
$$
\int_{-\infty}^{+\infty}\int_{0}^{+\infty} xe^{-x^{2}e^{y}} \Gamma \left(0,x^2\right)\, dydx.
$$
---
### Step 1: Understand the components
Let’s recall the **lower incomplete gamma function**:
$$
\Gamma(s, x) = \int_{x}^{\infty} t^{s-1} e^{-t} dt.
$$
So the **lower... | We are asked to evaluate the double integral:
$$
\int_{-\infty}^{+\infty}\int_{0}^{+\infty} xe^{-x^{2}e^{y}} \Gamma \left(0,x^2\right)\, dydx.
$$
---
### Step 1: Understand the integrand
Let’s denote the integrand as:
$$
f(x, y) = x e^{-x^2 e^y} \Gamma(0, x^2)
$$
Recall the definition of the **lower incomplete ga... | 8 |
36156_7.0 | Identify a non-abelian group \( G \) such that its abelianization \( G/[G, G] \) has a proper, non-trivial homomorphic image. | B | We are asked to **identify a non-abelian group $ G $** such that its **abelianization** $ G/[G, G] $ has a **proper, non-trivial homomorphic image**.
---
### Step 1: Understand the Abelianization
The **abelianization** of a group $ G $ is the **largest abelian quotient** of $ G $, defined as:
$$
G_{\text{ab}} = G/[... | We are asked to **identify a non-abelian group $ G $** such that its **abelianization $ G/[G, G] $** has a **proper, non-trivial homomorphic image**.
---
### Step 1: Understand the Abelianization
The **abelianization** of a group $ G $, denoted $ G/[G, G] $, is the largest **abelian quotient** of $ G $. That is, it ... | 7 |
55284_8.5 | Determine whether an injective bounded linear functional on a Banach space attains its norm on the closed unit ball. | B | We are asked to determine whether an **injective bounded linear functional** on a **Banach space** **attains its norm on the closed unit ball**.
---
### Step 1: Understand the Problem
Let $ X $ be a **Banach space**, and let $ f \in X^* $ be a **bounded linear functional** (i.e., $ f \in X^* $, the dual space of $ X... | We are asked to determine whether an **injective bounded linear functional** on a **Banach space** **attains its norm on the closed unit ball**.
---
### Step 1: Understanding the Problem
Let $ X $ be a **Banach space**, and let $ f \in X^* $ be a **bounded linear functional** (i.e., $ f \in X^* $, the dual space of ... | 8.5 |
21339_5.5 | Determine whether the limit \( \lim_{x\to a^+}\frac{\cos{x}\ln(x-a)}{\ln(e^x-e^a)} \) exists. | A | We are given the limit:
$$
\lim_{x \to a^+} \frac{\cos{x} \ln(x - a)}{\ln(e^x - e^a)}
$$
We are to determine **whether this limit exists**.
---
### Step 1: Understand the behavior of the denominator and numerator as $ x \to a^+ $
Let’s analyze the **denominator**:
$$
\ln(e^x - e^a)
$$
As $ x \to a^+ $, $ e^x \t... | We are asked to evaluate the limit:
$$
\lim_{x \to a^+} \frac{\cos{x} \ln(x - a)}{\ln(e^x - e^a)}
$$
---
### Step 1: Analyze the behavior as $ x \to a^+ $
As $ x \to a^+ $, the term $ x - a \to 0^+ $, so $ \ln(x - a) \to -\infty $.
Also, the denominator is $ \ln(e^x - e^a) $. Let's examine the behavior of $ e^x - ... | 5.5 |
22782_8.5 | Determine whether every $\mathbb{R}$-algebra homomorphism $h: \mathbb{R}[V] \to \mathbb{R}$ is of the form $ev_x$ for some point $x \in V$, where $ev_x(f) = f(x)$ for $f \in \mathbb{R}[V]$. Provide a justification for your answer. | A | We are asked to determine whether **every** $\mathbb{R}$-algebra homomorphism
$$
h: \mathbb{R}[V] \to \mathbb{R}
$$
is of the form $ev_x$, where $ev_x(f) = f(x)$ for some point $x \in V$, and $f \in \mathbb{R}[V]$.
---
### Step 1: Understand the Setting
Let $V \subseteq \mathbb{R}^n$ be a **real algebraic variety*... | We are asked to determine whether **every** $\mathbb{R}$-algebra homomorphism $ h : \mathbb{R}[V] \to \mathbb{R} $ is of the form $ \operatorname{ev}_x $ for some point $ x \in V $, where
$$
\operatorname{ev}_x(f) = f(x) \quad \text{for } f \in \mathbb{R}[V].
$$
---
### Step 1: Understand the Setup
Let $ V \subsete... | 8.5 |
63293_5.5 | Evaluate the sum \( \sum_{k=1}^n \frac{a_k-b_k}{k} \) where \( a_k = \frac{1}{\binom{n}{k}} \) and \( b_k = 2^{k-n} \). | B | We are given the sum:
$$
\sum_{k=1}^n \frac{a_k - b_k}{k}
$$
with the definitions:
- $ a_k = \frac{1}{\binom{n}{k}} $
- $ b_k = 2^{k-n} $
---
### Step 1: Write out the expression
$$
\sum_{k=1}^n \frac{a_k - b_k}{k} = \sum_{k=1}^n \left( \frac{1}{k \binom{n}{k}} - \frac{2^{k-n}}{k} \right)
$$
---
### Step 2: Ana... | We are given the sum:
$$
\sum_{k=1}^n \frac{a_k - b_k}{k}
$$
where:
- $ a_k = \frac{1}{\binom{n}{k}} $
- $ b_k = 2^{k - n} $
So the sum becomes:
$$
\sum_{k=1}^n \frac{1}{k} \left( \frac{1}{\binom{n}{k}} - 2^{k - n} \right)
$$
We'll evaluate this step by step.
---
### Step 1: Understand the components
We have t... | 5.5 |
44887_5.5 | Find the point on the surface defined by the function \( f(x,y) = 4x^2+7y^2+5xy+14 \) where the vector \( 6\hat{i} + 69\hat{j} + 3\hat{k} \) is normal to the tangent plane. | A | We are given a function:
$$
f(x, y) = 4x^2 + 7y^2 + 5xy + 14
$$
and a vector:
$$
\vec{v} = 6\hat{i} + 69\hat{j} + 3\hat{k}
$$
We are to find the point $(x, y)$ on the surface such that the vector $\vec{v}$ is **normal to the tangent plane** at that point.
---
### Step 1: Understand the geometry
The **normal vect... | We are given a function:
$$
f(x, y) = 4x^2 + 7y^2 + 5xy + 14
$$
and a vector:
$$
\vec{v} = 6\hat{i} + 69\hat{j} + 3\hat{k}
$$
We are to find the point $(x, y)$ on the surface where the **tangent plane** to the surface has a **normal vector** equal to $\vec{v}$.
---
### Step 1: Understand the surface and its norma... | 5.5 |
44570_8.0 | Determine whether it is possible to construct a subfield of \( \mathbb{C} \) that is isomorphic to a proper subfield of \( \mathbb{R} \). | B | We are asked to determine whether it is **possible** to construct a **subfield of $ \mathbb{C} $** that is **isomorphic to a proper subfield of $ \mathbb{R} $**.
---
### Step 1: Understand the question
We are looking for a subfield $ K \subseteq \mathbb{C} $ such that:
- $ K $ is a **subfield** of $ \mathbb{C} $
- ... | We are asked to determine whether it is **possible to construct a subfield of $ \mathbb{C} $ that is isomorphic to a proper subfield of $ \mathbb{R} $**.
---
### Step 1: Understand the question
We are to determine if there exists a **subfield** $ K \subset \mathbb{C} $ such that:
$$
K \cong L \subset \mathbb{R}
$$
... | 8 |
82875_8.0 | If an eigenvalue of a matrix $M \in SL(n, \mathbb{Z})$ lies on the unit circle, is its argument necessarily a rational multiple of $\pi$? Answer yes or no and justify your answer. | A | **Answer: No**
---
### Justification:
Let $ M \in SL(n, \mathbb{Z}) $, meaning $ M $ is an $ n \times n $ integer matrix with determinant 1.
Suppose $ \lambda $ is an eigenvalue of $ M $ lying on the unit circle. Then $ |\lambda| = 1 $, so $ \lambda $ is a complex number of modulus 1.
We are asked whether the **ar... | **Answer:** No.
---
### **Justification:**
Let $ M \in SL(n, \mathbb{Z}) $, i.e., $ M $ is an integer matrix with determinant 1. Suppose that an **eigenvalue** $ \lambda $ of $ M $ lies on the **unit circle**, meaning $ |\lambda| = 1 $.
We are asked whether the **argument** (i.e., the angle) of $ \lambda $ must be ... | 8 |
40191_6.5 | Let $p$ be a prime number and consider a group $G$ with order $p^4$ such that the center $Z(G)$ has order $p^2$. Determine the number of conjugacy classes of $G$. | A | We are given:
- $ G $ is a group of order $ p^4 $, where $ p $ is a prime.
- The center $ Z(G) $ has order $ p^2 $.
We are to **determine the number of conjugacy classes** of $ G $.
---
### Step 1: Use the class equation
The **class equation** for a group $ G $ is:
$$
|G| = |Z(G)| + \sum_{x \in G \setminus Z(G)} ... | We are given:
- $ G $ is a group of order $ p^4 $, where $ p $ is a prime.
- The center $ Z(G) $ has order $ p^2 $.
We are to determine the **number of conjugacy classes** of $ G $.
---
### Step 1: Use the Class Equation
The **class equation** for a finite group $ G $ is:
$$
|G| = |Z(G)| + \sum_{i} [G : C_G(g_i)]... | 6.5 |
13167_6.5 | Evaluate the infinite series \( \sum_{n=1}^{\infty}\frac{1+2+\cdots +n}{n!} \). | B | We are asked to evaluate the infinite series:
$$
\sum_{n=1}^{\infty} \frac{1 + 2 + \cdots + n}{n!}
$$
---
### Step 1: Use the formula for the sum of the first $ n $ natural numbers
The sum $ 1 + 2 + \cdots + n $ is a well-known arithmetic series, and its closed form is:
$$
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
$$
... | We are asked to evaluate the infinite series:
$$
\sum_{n=1}^{\infty} \frac{1 + 2 + \cdots + n}{n!}
$$
---
### Step 1: Use the formula for the sum of the first $ n $ natural numbers
Recall that:
$$
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
$$
So the series becomes:
$$
\sum_{n=1}^{\infty} \frac{\frac{n(n+1)}{2}}{n!} =... | 6.5 |
24991_8.0 | Let \((W_t)_t\) be a Brownian motion with respect to \((F_t)_t\). Compute the expectation \(E[(W_t^2 - t)(W_s^2 - s)]\) for \(0 < s < t\). | B | We are given a Brownian motion $(W_t)_t$ with respect to a filtration $(\mathcal{F}_t)_t$, and we are asked to compute the expectation:
$$
\mathbb{E}[(W_t^2 - t)(W_s^2 - s)] \quad \text{for } 0 < s < t.
$$
---
### Step 1: Use the definition of Brownian motion
We know the following properties of a standard Brownian ... | We are given a Brownian motion $(W_t)_t$ with respect to a filtration $(\mathcal{F}_t)_t$, and we are asked to compute the expectation:
$$
\mathbb{E}[(W_t^2 - t)(W_s^2 - s)], \quad \text{for } 0 < s < t.
$$
---
### Step 1: Expand the product
We expand the expression:
$$
(W_t^2 - t)(W_s^2 - s) = W_t^2 W_s^2 - t W_s... | 8 |
83021_6.0 | Evaluate the sum for $x \neq n \cdot 2 \pi$:
\[ 1+ \sum_{k=1}^{\infty}(e^{ikx}+e^{-ikx}) \] | A | We are asked to evaluate the infinite sum:
$$
1 + \sum_{k=1}^{\infty}(e^{ikx} + e^{-ikx})
$$
for $ x \neq n \cdot 2\pi $, where $ n \in \mathbb{Z} $.
---
### Step 1: Simplify the summand
Note that:
$$
e^{ikx} + e^{-ikx} = 2\cos(kx)
$$
So the sum becomes:
$$
1 + \sum_{k=1}^{\infty} 2\cos(kx)
$$
---
### Step 2:... | We are given the sum:
$$
1 + \sum_{k=1}^{\infty}(e^{ikx} + e^{-ikx})
$$
and we are to evaluate it for $ x \neq n \cdot 2\pi $, where $ n \in \mathbb{Z} $.
---
### Step 1: Simplify the expression
We can write:
$$
1 + \sum_{k=1}^{\infty}(e^{ikx} + e^{-ikx}) = 1 + \sum_{k=1}^{\infty} 2\cos(kx)
$$
Because:
$$
e^{ik... | 6 |
71854_7.5 | Calculate the residue of the function \( f(z) = \frac{1}{z^2 \sin(\pi z)} \) at the point \( z = 0 \) without using the Laurent series expansion. Use the formula for residues at a pole of order \( n \):
\[ \mathrm{Res}[f,0] = \lim_{z \to 0} \frac{1}{(n-1)!} \frac{\mathrm{d}^{n-1}}{\mathrm{d}z^{n-1}} \left[ (z-z_0)^n f... | A | We are asked to compute the **residue** of the function
$$
f(z) = \frac{1}{z^2 \sin(\pi z)}
$$
at $ z = 0 $, **without using Laurent series expansion**, using the **residue formula for a pole of order $ n $**:
$$
\mathrm{Res}[f, 0] = \lim_{z \to 0} \frac{1}{(n-1)!} \frac{\mathrm{d}^{n-1}}{\mathrm{d}z^{n-1}} \left[ (... | We are given the function:
$$
f(z) = \frac{1}{z^2 \sin(\pi z)}
$$
and we are asked to compute the **residue** of $ f(z) $ at $ z = 0 $ **without using the Laurent series expansion**, using the formula for residues at a pole of order $ n $:
$$
\mathrm{Res}[f, 0] = \lim_{z \to 0} \frac{1}{(n-1)!} \frac{d^{n-1}}{dz^{n-... | 7.5 |
69341_8.5 | Calculate \( \mathbb{E}[(\int_{0}^{\infty}f(t)dW_t)^2] \) where \( f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t), t \ge 0 \), and \( (W_t)_{t\ge0} \) is a Brownian motion. | B | We are given a stochastic integral:
$$
\mathbb{E}\left[\left(\int_{0}^{\infty} f(t) \, dW_t\right)^2\right]
$$
where the function $ f(t) $ is defined as:
$$
f(t) = (W_2 - W_1) \cdot 1_{[2,3)}(t) + (W_3 - W_1) \cdot 1_{[3,5)}(t)
$$
and $ (W_t)_{t \ge 0} $ is a standard Brownian motion.
---
### Step 1: Understand t... | We are given the function:
$$
f(t) = (W_2 - W_1) \cdot 1_{[2,3)}(t) + (W_3 - W_1) \cdot 1_{[3,5)}(t), \quad t \ge 0
$$
and asked to compute the expectation:
$$
\mathbb{E}\left[\left(\int_0^\infty f(t)\,dW_t\right)^2\right]
$$
where $ (W_t)_{t \ge 0} $ is a **Brownian motion**.
---
### Step 1: Understand the struc... | 8.5 |
34512_7.5 | Calculate the value of the integral \( \int_{X\times Y}\chi_\Delta d(\lambda\times \nu) \), where \( (X,\mathcal{M},\lambda) \) is a measure space with Lebesgue measure \( \lambda \), \( (Y,\mathcal{N},\nu) \) is a measure space with an arbitrary measure \( \nu \), and \( \Delta=\{(x,y)|x=y,x\in X,y\in Y\} \). | A | We are given the integral:
$$
\int_{X \times Y} \chi_\Delta \, d(\lambda \times \nu)
$$
Where:
- $ (X, \mathcal{M}, \lambda) $ is a measure space with Lebesgue measure $ \lambda $,
- $ (Y, \mathcal{N}, \nu) $ is a measure space with an arbitrary (possibly non-Lebesgue) measure $ \nu $,
- $ \Delta = \{(x, y) \in X \t... | We are given the integral:
$$
\int_{X \times Y} \chi_\Delta \, d(\lambda \times \nu)
$$
Where:
- $ (X, \mathcal{M}, \lambda) $ is a measure space with Lebesgue measure $ \lambda $,
- $ (Y, \mathcal{N}, \nu) $ is a measure space with an arbitrary measure $ \nu $,
- $ \Delta = \{(x, y) \in X \times Y \mid x = y\} $ is... | 7.5 |
14235_7.5 | Evaluate the triple integral \( \int \int \int z \, dx \, dy \, dz \) over the region \( M \), where \( M = \{ (x, y, z) \mid x \ge 0, y \ge 0, z \ge \sqrt{x^2 + y^2}, x^2 + y^2 + z^2 \le 2x \} \). | B | We are asked to evaluate the **triple integral**:
$$
\iiint_M z \, dx \, dy \, dz
$$
over the region:
$$
M = \left\{ (x, y, z) \mid x \ge 0, y \ge 0, z \ge \sqrt{x^2 + y^2}, x^2 + y^2 + z^2 \le 2x \right\}
$$
---
### **Step 1: Understand the Region**
Let’s analyze the region $ M $:
- $ x \ge 0, y \ge 0 $: the re... | We are asked to evaluate the **triple integral**:
$$
\iiint_M z \, dx \, dy \, dz
$$
over the region $ M $ defined by:
$$
M = \left\{ (x, y, z) \mid x \ge 0, \, y \ge 0, \, z \ge \sqrt{x^2 + y^2}, \, x^2 + y^2 + z^2 \le 2x \right\}
$$
---
### Step 1: Understand the Region
Let’s analyze the constraints:
1. $ x \g... | 7.5 |
57892_8.0 | Find the normal curvature in the direction of an arbitrary unit tangent vector at the origin $p = (0, 0, 0)$ for the helicoid $S$ defined by the parametric equation $\sigma(u, v) = (u \cos v, u \sin v, v)$. Provide your answer as a single value. | B | We are given a **helicoid** defined by the parametric equation:
$$
\sigma(u, v) = (u \cos v, u \sin v, v)
$$
We are to find the **normal curvature** in the direction of an **arbitrary unit tangent vector** at the origin $ p = (0, 0, 0) $.
---
### Step 1: Understand the geometry
The **helicoid** is a surface of rev... | We are given the **helicoid** surface $ S $ with parametric representation:
$$
\sigma(u, v) = (u \cos v, u \sin v, v)
$$
We are to find the **normal curvature** in the **direction of an arbitrary unit tangent vector** at the origin $ p = (0, 0, 0) $.
---
### Step 1: Understand the Surface
This is a **helicoid**, a... | 8 |
34752_8.5 | Is it always possible to find a non-irreducible polynomial in $K[x]$ that vanishes at two distinct algebraic elements in an algebraic field extension $E/K$, given a field isomorphism between their subfields that fixes $K$? | A | We are given:
- A field extension $ E/K $, where $ E $ is an algebraic field extension of $ K $.
- A field isomorphism $ \sigma: L \to M $, where $ L $ and $ M $ are subfields of $ E $, and $ \sigma $ fixes $ K $ (i.e., $ \sigma|_K = \text{id}_K $).
- The question is: **Is it always possible to find a non-irreducible ... | We are given:
- A field extension $ E/K $, where $ E $ is an algebraic field extension of $ K $.
- A field isomorphism $ \sigma: K(\alpha) \to K(\beta) $ that fixes $ K $, where $ \alpha, \beta \in E $ are algebraic over $ K $.
- The question is: **Is it always possible to find a non-irreducible polynomial in $ K[x] $... | 8.5 |
9335_8.5 | Determine the fundamental group of a Klein bottle with two points removed. | A | We are asked to determine the **fundamental group** of the **Klein bottle with two points removed**.
---
### Step 1: Understand the Klein Bottle
The **Klein bottle** is a non-orientable surface. It can be constructed by taking a square and identifying the edges as follows:
- Top and bottom edges are identified with... | We are asked to compute the **fundamental group** of the **Klein bottle with two points removed**.
---
### Step 1: Understand the Klein Bottle
The **Klein bottle** is a non-orientable surface with no boundary. It can be constructed by identifying the edges of a square as follows:
- Identify the top and bottom edges... | 8.5 |
72090_8.5 | Determine whether there exists a sextic polynomial that is irreducible over the field of rational numbers \(\mathbb{Q}\) and has a Galois group isomorphic to the symmetric group \(S_5\). | A | We are asked to determine whether there exists a **sextic polynomial** (degree 6) that is **irreducible over** $\mathbb{Q}$ and has a **Galois group isomorphic to** $S_5$.
---
### Step 1: Understand the problem
- A **sextic polynomial** is a polynomial of degree 6.
- The polynomial is **irreducible over** $\mathbb{Q... | We are asked to determine whether there exists a **sextic polynomial** (degree 6) that is **irreducible over $\mathbb{Q}$** and has **Galois group isomorphic to the symmetric group $S_5$**.
---
### Step 1: Understand the Problem
- A **sextic polynomial** is a polynomial of degree 6.
- It is **irreducible over $\math... | 8.5 |
18390_8.0 | Determine whether the function \( h(x,y) = \frac{\arccos[(\cos x)(\cos y) + k(\sin x)(\sin y)]^2}{(\sin^2 x) - 2k(\sin x)(\sin y) + (\sin^2 y)} \) has a limit of 1 as \((x,y)\) approaches \((0,0)\), where \(-1 \leq k \leq 1\). | A | We are given the function:
$$
h(x,y) = \frac{\arccos\left[(\cos x)(\cos y) + k(\sin x)(\sin y)\right]^2}{(\sin^2 x) - 2k(\sin x)(\sin y) + (\sin^2 y)}
$$
and we are to determine whether the limit of $ h(x,y) $ as $ (x,y) \to (0,0) $ is equal to 1, for $ -1 \leq k \leq 1 $.
---
### Step 1: Analyze the numerator
Let... | We are given the function:
$$
h(x,y) = \frac{\arccos\left[(\cos x)(\cos y) + k(\sin x)(\sin y)\right]^2}{(\sin^2 x) - 2k(\sin x)(\sin y) + (\sin^2 y)}
$$
and we are to determine whether:
$$
\lim_{(x,y) \to (0,0)} h(x,y) = 1
$$
for $ -1 \leq k \leq 1 $.
---
### Step 1: Analyze the numerator
Let’s define:
$$
A(x,... | 8 |
15365_6.5 | Given that \( f(x) = x^2 f(2x) \) and the values \( f(4) = -2 \), \( f'(4) = 1 \), and \( f''(4) = -1 \), find \( f''(2) \). | A | We are given the functional equation:
$$
f(x) = x^2 f(2x)
$$
and the values:
- $ f(4) = -2 $
- $ f'(4) = 1 $
- $ f''(4) = -1 $
We are to find $ f''(2) $.
---
### Step 1: Use the functional equation to express $ f(x) $ in terms of $ f(2x) $
Given:
$$
f(x) = x^2 f(2x)
$$
Let’s plug in $ x = 2 $:
$$
f(2) = 2^2 f(... | We are given the functional equation:
$$
f(x) = x^2 f(2x)
$$
and the values:
$$
f(4) = -2, \quad f'(4) = 1, \quad f''(4) = -1
$$
We are asked to find $ f''(2) $.
---
### Step 1: Use the functional equation to relate $ f(2) $ to $ f(4) $
From the functional equation:
$$
f(x) = x^2 f(2x)
$$
Let $ x = 2 $:
$$
f(... | 6.5 |
79743_6.0 | Let $T: V \rightarrow V$ be a linear operator with distinct eigenvalues $\mu_1, \ldots, \mu_t$. Define $W_j = \ker(T - \mu_j \cdot \operatorname{id})$ as the eigenspaces corresponding to these eigenvalues. Consider the subspace $E = \{ S \in \operatorname{Hom}(V,V) \mid S(W_j) \subseteq W_j \text{ for all } 1 \leq j \l... | B | We are given a linear operator $ T: V \to V $ with **distinct eigenvalues** $ \mu_1, \mu_2, \ldots, \mu_t $, and the corresponding **eigenspaces** $ W_j = \ker(T - \mu_j \cdot \operatorname{id}) $. We are also given the subspace:
$$
E = \left\{ S \in \operatorname{Hom}(V,V) \mid S(W_j) \subseteq W_j \text{ for all } 1... | We are given a linear operator $ T: V \to V $ with **distinct eigenvalues** $ \mu_1, \mu_2, \ldots, \mu_t $, and the corresponding **eigenspaces**:
$$
W_j = \ker(T - \mu_j \cdot \operatorname{id}), \quad \text{for } j = 1, 2, \ldots, t.
$$
We are to consider the subspace:
$$
E = \left\{ S \in \operatorname{Hom}(V,V)... | 6 |
45254_6.5 | Consider an open and connected subset of the plane that intersects the unit circle. Does this subset necessarily contain points within a small distance of the circle's center? | B | We are given:
- A **connected** and **open** subset $ U \subseteq \mathbb{R}^2 $.
- $ U $ **intersects** the **unit circle** $ \mathbb{S}^1 = \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} $.
- The question is: **Does $ U $ necessarily contain points within a small distance of the circle's center (i.e., the origin)?... | We are given:
- A **connected** and **open** subset $ U \subset \mathbb{R}^2 $,
- $ U $ **intersects** the **unit circle** $ S^1 = \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} $,
- The question is: **Does $ U $ necessarily contain points within a small distance of the circle's center (i.e., the origin)?**
---
##... | 6.5 |
52468_5.5 | Given real numbers $a, b, c$ such that $|x| \leq 1$ and $|ax^2 + bx + c| \leq 1$, find the maximum value of $|2ax + b|$. | A | We are given:
- Real numbers $ a, b, c $,
- A constraint: $ |x| \leq 1 $,
- Another constraint: $ |ax^2 + bx + c| \leq 1 $ for all $ x $ with $ |x| \leq 1 $,
- We are to **find the maximum value of** $ |2ax + b| $, **for $ x \in [-1, 1] $**.
---
### Step 1: Understand the function
Let us define the quadratic functi... | We are given:
- Real numbers $ a, b, c $,
- $ |x| \leq 1 $,
- $ |ax^2 + bx + c| \leq 1 $ for all $ x $ with $ |x| \leq 1 $,
- We are to find the **maximum value** of $ |2ax + b| $.
---
### Step 1: Understand the Function
Let $ f(x) = ax^2 + bx + c $.
We are told that $ |f(x)| \leq 1 $ for all $ x \in [-1, 1] $.
W... | 5.5 |
16543_8.0 | Is it necessary for \(M\) to be a finitely generated projective \(\mathbb{Z}\)-module for the functor composition \(\text{Hom}(-,N)\otimes_{\mathbb{Z}}M\) to be representable? | A | We are asked whether it is **necessary** for $ M $ to be a **finitely generated projective** $ \mathbb{Z} $-module for the functor
$$
\text{Hom}(-, N) \otimes_{\mathbb{Z}} M
$$
to be **representable**.
---
### Step 1: Understanding the Functor
We are given a fixed $ \mathbb{Z} $-module $ N $, and we are considerin... | We are given a question about the representability of the functor:
$$
\text{Hom}(-,N)\otimes_{\mathbb{Z}}M
$$
and asked whether it is **necessary** for $ M $ to be a **finitely generated projective** $ \mathbb{Z} $-module for this functor to be representable.
---
### Step 1: Understand the Functor
Let’s recall tha... | 8 |
27461_5.5 | Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)} \] | A | We are asked to evaluate the limit:
$$
\lim_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)}
$$
---
### Step 1: Use polar coordinates
Let us switch to polar coordinates, where:
$$
x = r\cos\theta, \quad y = r\sin\theta, \quad x^2 + y^2 = r^2
$$
As $(x, y) \to (0, 0)$, we have $r \to 0$, and $\theta$ is arb... | We are asked to evaluate the following limit:
$$
\lim_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)}
$$
---
### Step 1: Use Polar Coordinates
Let’s switch to **polar coordinates** to simplify the expression. Recall:
$$
x = r\cos\theta, \quad y = r\sin\theta, \quad x^2 + y^2 = r^2
$$
So as $(x, y) \to (0,... | 5.5 |
48334_8.0 | Consider polynomials $P_1(x), P_2(x) \in \mathbb{Z}[x]$, and integers $n, a$. Suppose that the polynomial $x^n-a$ is irreducible over $\Bbb Z[i][x]$. Determine the number of ordered pairs $(P_1(x), P_2(x))$ that satisfy the equation $(P_1(x))^2+(P_2(x))^2=(x^n-a)^2$, excluding trivial pairs like $(x^n-a,0)$. How many s... | B | We are given that the polynomial $ x^n - a $ is **irreducible over $ \mathbb{Z}[i][x] $** and we are to find the number of **ordered pairs** $ (P_1(x), P_2(x)) \in \mathbb{Z}[i][x] $ such that:
$$
(P_1(x))^2 + (P_2(x))^2 = (x^n - a)^2,
$$
excluding **trivial pairs** like $ (x^n - a, 0) $.
---
### Step 1: Understand... | We are given:
- $ P_1(x), P_2(x) \in \mathbb{Z}[x] $
- $ n, a \in \mathbb{Z} $
- $ x^n - a $ is **irreducible over $ \mathbb{Z}[i][x] $**
- We are to find the number of **ordered pairs** $ (P_1(x), P_2(x)) $ such that:
$$
(P_1(x))^2 + (P_2(x))^2 = (x^n - a)^2
$$
excluding the **trivial pairs** like $ (x^n - a, 0) $.... | 8 |
71232_6.0 | Calculate the value of \( \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} 4^{-(i+j+k)} \) where \( i, j, \) and \( k \) are distinct positive integers. | B | We are given the triple sum:
$$
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} 4^{-(i+j+k)}
$$
with the **condition** that $ i, j, k $ are **distinct** positive integers.
---
### Step 1: Understand the full sum (without the distinct condition)
If there were **no restriction** on $ i, j, k $, the tripl... | We are asked to evaluate the following triple sum:
$$
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} 4^{-(i+j+k)}
$$
with the **condition** that $ i, j, k $ are **distinct positive integers**.
---
### Step 1: Understand the structure of the sum
We are summing over all **positive integers** $ i, j, k $... | 6 |
37680_7.5 | Evaluate the integral \( \int_{0}^{\infty}\frac{\sin(3x)\sin(x)}{x^2}\,\mathrm{d}x \) using integration by parts. | A | We are asked to evaluate the integral:
$$
\int_{0}^{\infty}\frac{\sin(3x)\sin(x)}{x^2}\,\mathrm{d}x
$$
### Step 1: Use a Trigonometric Identity
We can use the identity:
$$
\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]
$$
Apply this to the integrand:
$$
\sin(3x)\sin(x) = \frac{1}{2}[\cos(2x) - \cos(4x)]
$... | We are asked to evaluate the integral:
$$
\int_{0}^{\infty} \frac{\sin(3x)\sin(x)}{x^2} \, dx
$$
using **integration by parts**.
---
### Step 1: Use a trigonometric identity
We can use the identity:
$$
\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]
$$
Let $ A = 3x $, $ B = x $, so:
$$
\sin(3x)\sin(x) =... | 7.5 |
29512_5.5 | Evaluate the limit: \[ \lim_{ x \to 0^- }\frac{2^{\frac{1}{x}}+2^{\frac{-1}{x}}}{3^{\frac{1}{x}}+3^{\frac{-1}{x}}} \] without using L'Hôpital's rule. | B | We are asked to evaluate the following limit:
$$
\lim_{x \to 0^-} \frac{2^{\frac{1}{x}} + 2^{-\frac{1}{x}}}{3^{\frac{1}{x}} + 3^{-\frac{1}{x}}}
$$
---
### Step 1: Understand the behavior of the function as $ x \to 0^- $
As $ x \to 0^- $, the quantity $ \frac{1}{x} \to -\infty $, and so:
- $ 2^{\frac{1}{x}} = 2^{-\... | We are asked to evaluate the limit:
$$
\lim_{x \to 0^-} \frac{2^{\frac{1}{x}} + 2^{-\frac{1}{x}}}{3^{\frac{1}{x}} + 3^{-\frac{1}{x}}}
$$
---
### Step 1: Understand the behavior as $ x \to 0^- $
As $ x \to 0^- $, the term $ \frac{1}{x} \to -\infty $, and $ -\frac{1}{x} \to +\infty $.
So, we analyze the behavior of ... | 5.5 |
77793_6.0 | Evaluate the line integral \( \int_{C_2} \frac{x\vec{e_1}+y\vec{e_2}}{x^2+y^2} d\vec{r} \) where \( C_2 \) is a square with vertices \((1,1), (-1,1), (-1,-1), (1,-1)\). | A | We are asked to evaluate the **line integral**:
$$
\int_{C_2} \frac{x\vec{e_1} + y\vec{e_2}}{x^2 + y^2} \, d\vec{r}
$$
where $ C_2 $ is the **square** with vertices:
$$
(1,1),\ (-1,1),\ (-1,-1),\ (1,-1)
$$
---
### Step 1: Understand the vector field
The vector field is:
$$
\vec{F}(x, y) = \frac{x\vec{e_1} + y\ve... | We are asked to evaluate the line integral:
$$
\int_{C_2} \frac{x\vec{e_1} + y\vec{e_2}}{x^2 + y^2} \, d\vec{r}
$$
where $ C_2 $ is the square with vertices at $ (1,1), (-1,1), (-1,-1), (1,-1) $, i.e., the square centered at the origin with side length 2.
---
### Step 1: Understand the vector field
Let’s denote th... | 6 |
44108_7.0 | Let $f: [0, +\infty) \to \mathbb{R}$ be a continuous function. If the integral $\int_{0}^{+\infty}f(x)dx$ converges, does it imply that the series $\sum_{n\geq1}f(n + x)$ converges uniformly for $x \in [0, 1]$? Answer yes or no and justify your answer. | A | We are given:
- A function $ f: [0, +\infty) \to \mathbb{R} $ is **continuous**.
- The **integral** $ \int_0^{+\infty} f(x)\,dx $ **converges**.
- We are to determine whether the **series** $ \sum_{n=1}^{\infty} f(n + x) $ **converges uniformly** for $ x \in [0, 1] $.
---
### Step 1: Understanding the Problem
We ar... | We are given:
- A function $ f: [0, +\infty) \to \mathbb{R} $ is **continuous**.
- The **improper integral** $ \int_0^{+\infty} f(x)\,dx $ **converges**.
- We are asked whether the **series** $ \sum_{n=1}^\infty f(n + x) $ **converges uniformly** for $ x \in [0, 1] $.
---
### Step 1: Understand the Series
We are lo... | 7 |
9617_6.0 | Evaluate the sum \( \sum_{n=1}^{\infty}\frac{1}{n!(n+2)} \). | B | We are asked to evaluate the infinite sum:
$$
\sum_{n=1}^{\infty} \frac{1}{n!(n+2)}
$$
---
### Step 1: Understanding the Series
We are summing the terms:
$$
\frac{1}{n!(n+2)}
$$
This is a series involving factorials and a linear term in the denominator.
---
### Step 2: Consider a General Strategy
Let’s conside... | We are asked to evaluate the infinite sum:
$$
\sum_{n=1}^{\infty} \frac{1}{n!(n+2)}
$$
---
### Step 1: Rewrite the term
Let’s write the general term as:
$$
\frac{1}{n!(n+2)}
$$
We want to simplify or find a closed-form expression for the sum:
$$
\sum_{n=1}^{\infty} \frac{1}{n!(n+2)}
$$
---
### Step 2: Use part... | 6 |
32443_6.0 | Evaluate the limit: \[ \lim_{ x \to -\infty }\frac{\sqrt[n]{x+1}+\sqrt[n]{x+2}-2\sqrt[n]{x-3}}{\sqrt[n]{x-1}+\sqrt[n]{x-2}-2\sqrt[n]{x+3}} \] given that \( n \in \mathbb{N}, n>2, \text{ and } n \text{ is odd}."} | A | We are asked to evaluate the limit:
$$
\lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}}
$$
where $ n \in \mathbb{N}, n > 2 $, and $ n $ is **odd**.
---
### Step 1: Understand the behavior of the nth root for large negative $ x $
Let’s denote... | We are given the limit:
$$
\lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}}
$$
where $ n \in \mathbb{N}, n > 2 $, and $ n $ is **odd**.
---
### Step 1: Understanding the behavior of the nth root as $ x \to -\infty $
Since $ n $ is **odd**, t... | 6 |
39427_6.0 | Calculate the limit: $$\Large \lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt $$ | A | We are given the limit:
$$
\lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt
$$
---
### Step 1: Understand the structure
We have a limit involving an integral of the **inverse sine function**, divided by $\sin^2 x$, as $x \to 0^+$.
Let’s denote:
$$
I(x) = \int_{\frac{x}{2}}^x \sin^{-1} t \, dt
$$
... | We are asked to evaluate the following limit:
$$
\lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt
$$
---
### Step 1: Understand the structure of the expression
We are taking the limit of a ratio:
- Numerator: $\int_{x/2}^x \sin^{-1} t \, dt$
- Denominator: $\sin^2 x$
We will use **L’Hospital’s Ru... | 6 |
17710_6.0 | Determine the smallest integer $N$ such that the remainder $R_N$ of the series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} \) after summing the first $N$ terms is less than 0.1. That is, find $N$ such that:
\[
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1.
\] | A | We are given the infinite series:
$$
\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}
$$
and we are asked to find the **smallest integer** $ N $ such that the **remainder** after summing the first $ N $ terms is less than 0.1:
$$
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1
$$
---
### Step 1: Use the **Integral ... | We are given the infinite series:
$$
\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}
$$
and we are to find the **smallest integer** $ N $ such that the **remainder** after the first $ N $ terms is less than $ 0.1 $, i.e.,
$$
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1
$$
---
### Step 1: Use an Integral Test Es... | 6 |
71170_8.0 | Determine whether it is possible to create a topology on a finite set that is normal but not regular. | A | To determine whether it is **possible to create a topology on a finite set that is normal but not regular**, we need to understand the definitions and properties of **normal** and **regular** topologies, and how they behave on **finite sets**.
---
### Definitions:
- A topological space is **regular** if for every po... | To determine whether it is possible to create a **topology on a finite set** that is **normal but not regular**, we need to recall the definitions of **normal** and **regular** topological spaces, and analyze their properties in the context of **finite sets**.
---
### Definitions:
- A topological space is **regular*... | 8 |
40532_6.0 | Evaluate the surface integral \( \iint_S (y^2z^2 \textbf{i} \, +z^2x^2\textbf{j}+z^2y^2\textbf{k})\cdot\textbf{n} \,\mathrm{d}S \) where \( S \) is the part of the sphere \( x^2+y^2+z^2=1 \) that lies above the \( xy \) plane. | B | We are given the surface integral:
$$
\iint_S (y^2z^2 \textbf{i} + z^2x^2 \textbf{j} + z^2y^2 \textbf{k}) \cdot \textbf{n} \, dS
$$
where $ S $ is the part of the sphere $ x^2 + y^2 + z^2 = 1 $ that lies **above the $ xy $-plane**, i.e., $ z \geq 0 $.
---
### Step 1: Understand the Surface
The surface $ S $ is the... | We are asked to evaluate the surface integral:
$$
\iint_S (y^2z^2 \, \mathbf{i} + z^2x^2 \, \mathbf{j} + z^2y^2 \, \mathbf{k}) \cdot \mathbf{n} \, dS
$$
where $ S $ is the **upper hemisphere** of the sphere $ x^2 + y^2 + z^2 = 1 $, i.e., the part of the sphere with $ z \geq 0 $.
---
### Step 1: Understand the vecto... | 6 |
4008_6.0 | When \((x^2 + x + 1)^{54}\) is expanded, how many coefficients of the resulting polynomial are divisible by 3? | A | We are given the expression $(x^2 + x + 1)^{54}$, and we are to find how many **coefficients** in the expanded polynomial are **divisible by 3**.
---
### Step 1: Understand the expression
We are looking at the expansion of:
$$
(x^2 + x + 1)^{54}
$$
This is a polynomial of degree $2 \cdot 54 = 108$, so the expanded ... | We are given the expression:
$$
(x^2 + x + 1)^{54}
$$
We are to find **how many coefficients** in the expanded form of this polynomial are **divisible by 3**.
---
### Step 1: Understand the structure of the expression
We are raising a trinomial $x^2 + x + 1$ to the 54th power. The expanded form will be a polynomia... | 6 |
28144_5.5 | Evaluate the complex integral \( \int\limits_{\Gamma} |z|^2 \, dz \), where \( \Gamma \) is the boundary of a square with vertices at \( 0, 1, 1+i, i \), traversed anti-clockwise starting at \( 0 \). | A | We are asked to evaluate the complex integral:
$$
\int\limits_{\Gamma} |z|^2 \, dz
$$
where $ \Gamma $ is the **boundary of a square** with vertices at $ 0, 1, 1+i, i $, traversed **anti-clockwise** starting at $ 0 $.
---
### Step 1: Understand the Path
The square has vertices:
- $ A = 0 $
- $ B = 1 $
- $ C = 1+i... | We are asked to evaluate the complex integral:
$$
\int\limits_{\Gamma} |z|^2 \, dz
$$
where $ \Gamma $ is the **boundary of a square** with vertices at $ 0, 1, 1+i, i $, traversed **anti-clockwise** starting at $ 0 $.
---
### Step 1: Understand the path $ \Gamma $
The square has vertices at:
- $ 0 $
- $ 1 $
- $ 1+... | 5.5 |
70961_7.5 | Calculate the surface area of the intersection between two cylinders given by the equations $x^2+y^2=a^2$ and $x^2+z^2=a^2$. | B | We are asked to find the **surface area of the intersection** of two cylinders:
$$
x^2 + y^2 = a^2 \quad \text{(cylinder along the } z\text{-axis)}
$$
$$
x^2 + z^2 = a^2 \quad \text{(cylinder along the } y\text{-axis)}
$$
This is a classic problem in geometry, and the intersection of these two cylinders is known as a... | We are given two cylinders:
1. $ x^2 + y^2 = a^2 $ (a cylinder extending along the $ z $-axis)
2. $ x^2 + z^2 = a^2 $ (a cylinder extending along the $ y $-axis)
We are to **find the surface area of the intersection** of these two cylinders.
---
### Step 1: Understand the Geometry
The intersection of the two cylin... | 7.5 |
41150_6.5 | Evaluate the limit: $$\lim_{x \to 0+} \frac{x-\sqrt{x^2+1}\cdot \ln \big(\sqrt{x^2+1} + x \big)}{2x^2\sqrt{x^2+1}\cdot \ln\big(\sqrt{x^2+1} + x \big)}$$ | A | We are asked to evaluate the limit:
$$
\lim_{x \to 0^+} \frac{x - \sqrt{x^2+1} \cdot \ln\left(\sqrt{x^2+1} + x\right)}{2x^2\sqrt{x^2+1} \cdot \ln\left(\sqrt{x^2+1} + x\right)}
$$
---
### Step 1: Simplify the logarithmic expression
Let’s define:
$$
L(x) = \ln\left(\sqrt{x^2 + 1} + x\right)
$$
Note that:
$$
\sqrt{... | We are asked to evaluate the limit:
$$
\lim_{x \to 0+} \frac{x - \sqrt{x^2 + 1} \cdot \ln\left(\sqrt{x^2 + 1} + x\right)}{2x^2\sqrt{x^2 + 1} \cdot \ln\left(\sqrt{x^2 + 1} + x\right)}
$$
---
### Step 1: Simplify the expression
Let’s denote:
$$
L = \lim_{x \to 0+} \frac{x - \sqrt{x^2 + 1} \cdot \ln\left(\sqrt{x^2 + ... | 6.5 |
22448_7.0 | Evaluate the integral \( \int_{C(0,1)} z^3\exp\left(\frac{1}{z^2}\right)dz \), where \( C(0,1) \) is the contour of the unit circle centered at the origin. | A | We are asked to evaluate the integral:
$$
\int_{C(0,1)} z^3\exp\left(\frac{1}{z^2}\right)dz
$$
where $ C(0,1) $ is the **unit circle centered at the origin**, traversed in the **positive (counterclockwise) direction**.
---
### Step 1: Understand the Contour and the Function
- The contour $ C(0,1) $ is a **closed, ... | We are asked to evaluate the complex integral:
$$
\int_{C(0,1)} z^3 \exp\left(\frac{1}{z^2}\right) \, dz
$$
where $ C(0,1) $ is the **unit circle centered at the origin**, traversed in the **positive (counterclockwise) direction**.
---
### Step 1: Understand the Contour
- The contour $ C(0,1) $ is the **unit circl... | 7 |
72571_7.5 | Evaluate the limit \( \lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x}+n^2x}dx \). | B | We are asked to evaluate the limit:
$$
\lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x} + n^2 x} \, dx
$$
---
### Step 1: Understand the behavior of the integrand
Let’s consider the integrand:
$$
f_n(x) = \frac{2n}{e^{2x} + n^2 x}
$$
We want to understand how this behaves as $ n \to \infty $, and then integrate... | We are asked to evaluate the limit:
$$
\lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x} + n^2x} \, dx
$$
---
### Step 1: Understand the integrand
We are integrating a function of the form:
$$
f_n(x) = \frac{2n}{e^{2x} + n^2x}
$$
We are interested in the behavior of this integral as $ n \to \infty $.
---
### S... | 7.5 |
75851_6.5 | Let $X$ be a topological space with a subspace $Y$. Suppose there is a strong deformation retraction onto $Y$, given by a continuous map $F: X \times [0,1] \to X$ such that for all $x \in X$, $F(x,0) = x$, $F(x,1) \in Y$, and for all $y \in Y$ and $t \in [0,1]$, $F(y,t) = y$. Determine if it is true that for any $x \in... | A | We are given a **strong deformation retraction** from a topological space $ X $ onto its subspace $ Y $, defined by a continuous map:
$$
F: X \times [0,1] \to X
$$
with the following properties:
1. $ F(x, 0) = x $ for all $ x \in X $ (i.e., $ F $ fixes $ X $ at time 0),
2. $ F(x, 1) \in Y $ for all $ x \in X $ (i.e.... | We are given a **strong deformation retraction** $ F: X \times [0,1] \to X $ from a topological space $ X $ onto a subspace $ Y \subseteq X $, with the following properties:
1. $ F(x, 0) = x $ for all $ x \in X $,
2. $ F(x, 1) \in Y $ for all $ x \in X $,
3. $ F(y, t) = y $ for all $ y \in Y $ and $ t \in [0,1] $.
We... | 6.5 |
62366_7.0 | Given three points A, B, and C, where AB forms one side of a triangle and C is the center of a logarithmic spiral, determine the ratio of the lengths of the other two sides of the triangle if the angle opposite AB is \( \theta \) and the spiral has a constant \( k \). | B | We are given:
- Points $ A $, $ B $, and $ C $
- $ AB $ is one side of a triangle
- $ C $ is the **center** of a **logarithmic spiral**
- The angle opposite $ AB $ is $ \theta $
- The spiral has a **constant** $ k $
- We are to find the **ratio** of the lengths of the **other two sides** of the triangle.
---
### Ste... | We are given a triangle $ \triangle ABC $, where:
- $ AB $ is one side of the triangle,
- $ C $ is the center of a **logarithmic spiral**,
- The angle opposite side $ AB $ is $ \theta $,
- The spiral has a **constant** $ k $,
- We are to find the **ratio** of the lengths of the other two sides of the triangle (i.e., t... | 7 |
55132_8.0 | Determine if there exists a bound on the exponent $a$ such that $x^a = x^b$ holds for all $x$ in any ring $R$, independent of the ring $R$. | A | We are asked to determine whether there exists a **bound on the exponent $ a $** such that the identity
$$
x^a = x^b
$$
holds for **all** $ x \in R $, **independent of the ring $ R $**. That is, we want to know whether there exists a fixed integer $ a $ (depending only on $ b $) such that the equation $ x^a = x^b $ i... | We are asked to determine whether there exists a **bound on the exponent $ a $** such that the equation
$$
x^a = x^b
$$
holds for **all** $ x \in R $, **independent of the ring $ R $**.
---
### Step 1: Understanding the Problem
We are given a ring $ R $, and we are to find whether there exists some **fixed integer... | 8 |
62595_7.5 | Find the minimum value of the expression \(|5^{4m+3} - n^2|\) for positive integers \(n\) and \(m\). | B | We are asked to find the **minimum value** of the expression:
$$
|5^{4m+3} - n^2|
$$
for **positive integers** $ m $ and $ n $.
---
### Step 1: Understand the form of the expression
Let’s denote:
$$
x = 5^{4m+3}
$$
So we are trying to find the **minimum value of**:
$$
|x - n^2| = |5^{4m+3} - n^2|
$$
We want to... | We are given the expression:
$$
|5^{4m+3} - n^2|
$$
and we are to **find the minimum value** of this expression for **positive integers** $ m $ and $ n $.
---
### Step 1: Understand the structure of the expression
We are looking for the **minimum absolute difference** between $ 5^{4m+3} $ and a perfect square $ n^... | 7.5 |
56188_8.0 | Let $k, r, n$ be positive integers such that $k+1$ and $r-1$ are relatively prime. Let $N = nk$ and $M = n(n-1)nk/2$. Is it true that the dimension of the vector space $V^{(k,r)}_{N,M}$ is 1 under these conditions? | A | We are given the following:
- $ k, r, n $ are **positive integers**.
- $ k+1 $ and $ r-1 $ are **relatively prime**.
- Define:
$$
N = nk, \quad M = \frac{n(n-1)nk}{2}
$$
- We are asked whether the **dimension of the vector space** $ V^{(k,r)}_{N,M} $ is **1** under these conditions.
---
### Step 1: Understand ... | We are given:
- $ k, r, n $ are **positive integers**.
- $ k+1 $ and $ r-1 $ are **relatively prime**.
- $ N = nk $
- $ M = \frac{n(n-1)nk}{2} $
We are asked whether the **dimension of the vector space** $ V^{(k,r)}_{N,M} $ is **1** under these conditions.
---
### Step 1: Understanding the Notation
We are told tha... | 8 |
16559_7.0 | Let \( A \) be an \( n \times n \) matrix with all eigenvalues equal to 1 or 0. Does there exist a matrix \( B = XAX^{-1} \) for some invertible matrix \( X \) such that all elements of \( B \) are either 1 or 0? | A | We are given that $ A $ is an $ n \times n $ matrix with all **eigenvalues equal to 0 or 1**, and we are asked whether there exists an **invertible matrix** $ X $ such that the matrix $ B = X A X^{-1} $ has all entries **either 0 or 1**.
---
### Step 1: Understand the structure of $ A $
Since all eigenvalues of $ A ... | We are given:
- $ A $ is an $ n \times n $ matrix.
- All eigenvalues of $ A $ are either 0 or 1.
- We are to determine whether there exists an invertible matrix $ X $ such that the matrix $ B = X A X^{-1} $ has all entries either 0 or 1.
---
### Step 1: Understand the structure of $ A $
Since all eigenvalues of $ A... | 7 |
61376_7.5 | Compute the limit \(\lim_{x \to {0_+}} {\ln (x \ln a) \ln \left({{\ln (ax)}\over{\ln({x\over a})}}\right)}\) where \(a > 1\). | A | We are asked to compute the following limit:
$$
\lim_{x \to 0_+} \ln(x \ln a) \cdot \ln\left(\frac{\ln(ax)}{\ln\left(\frac{x}{a}\right)}\right)
$$
where $ a > 1 $.
---
### Step 1: Simplify the inner logarithm
We are dealing with:
$$
\ln\left(\frac{\ln(ax)}{\ln\left(\frac{x}{a}\right)}\right)
$$
Let’s simplify th... | We are asked to compute the limit:
$$
\lim_{x \to 0_+} \ln(x \ln a) \cdot \ln\left(\frac{\ln(ax)}{\ln\left(\frac{x}{a}\right)}\right)
$$
where $ a > 1 $.
---
### Step 1: Analyze the inner logarithmic expression
We look at:
$$
\frac{\ln(ax)}{\ln\left(\frac{x}{a}\right)}
$$
Let’s simplify the logarithms:
- $ \ln(... | 7.5 |
73003_6.5 | Let $(G, \cdot)$ be a centerless group with $n$ elements, where $n$ is an integer not divisible by 4. Determine if there exist distinct elements $a, b \in G$ such that $C(a) = C(b)$. Answer with 'True' if such elements exist, otherwise 'False'. | A | We are given:
- $ G $ is a **centerless group**, i.e., $ Z(G) = \{e\} $, the trivial group.
- $ |G| = n $, and $ n $ is **not divisible by 4**.
- We are to determine whether **there exist distinct elements** $ a, b \in G $ such that $ C(a) = C(b) $, where $ C(x) $ denotes the **centralizer** of $ x $ in $ G $, i.e., t... | We are given:
- A group $ (G, \cdot) $ is **centerless**, i.e., its **center** $ Z(G) = \{e\} $, where $ e $ is the identity element.
- The group has $ n $ elements, and $ n $ is **not divisible by 4**.
- We are to determine whether **there exist distinct elements** $ a, b \in G $ such that their **centralizers** are ... | 6.5 |
33017_5.5 | Find the Laplace Transform of the function \( \delta(t-2\pi)\cos(t) \), where \( \delta(t) \) is the Dirac Delta Function. | B | We are asked to find the **Laplace transform** of the function:
$$
f(t) = \delta(t - 2\pi)\cos(t)
$$
---
### Step 1: Use the **time-shift property** of the Laplace transform
The **time-shift property** of the Laplace transform states that:
$$
\mathcal{L}\{\delta(t - a)g(t)\} = e^{-as}G(s)
$$
where $ G(s) = \mathc... | We are asked to find the **Laplace transform** of the function:
$$
f(t) = \delta(t - 2\pi)\cos(t)
$$
---
### Step 1: Recall the Laplace transform of a **shifted Dirac delta function**
The Laplace transform of a **shifted Dirac delta function** is given by the formula:
$$
\mathcal{L}\left\{ \delta(t - a) \right\} =... | 5.5 |
66573_8.0 | Let $p\colon X\to Y$ be a covering map where $Y$ is path-connected, locally path-connected, and semi-locally simply connected. Suppose $\pi_1(Y,p(x))$ acts transitively on $p^{-1}(p(x))$ for some $x\in X$. Is it true that $\pi_1(Y,p(x'))$ acts transitively on $p^{-1}(p(x'))$ for all $x'\in X$? | B | We are given:
- $ p \colon X \to Y $ is a **covering map**.
- $ Y $ is **path-connected**, **locally path-connected**, and **semi-locally simply connected**.
- $ \pi_1(Y, p(x)) $ acts **transitively** on $ p^{-1}(p(x)) $ for some $ x \in X $.
- The question is: **Does $ \pi_1(Y, p(x')) $ act transitively on $ p^{-1}(p... | We are given:
- $ p: X \to Y $ is a **covering map**.
- $ Y $ is **path-connected**, **locally path-connected**, and **semi-locally simply connected**.
- $ \pi_1(Y, p(x)) $ **acts transitively** on $ p^{-1}(p(x)) $ for **some** $ x \in X $.
We are to determine whether this implies that $ \pi_1(Y, p(x')) $ acts transi... | 8 |
79130_6.5 | In the field \(\mathbb{F}_2\) of order 2, consider the ideal \(I\) in \(\mathbb{F}_2[x]\) generated by the polynomials \(x^5 + x^2 + x + 1\) and \(x^7 + x^5 + x^3 + x\). How many polynomials of degree 3 are elements of \(I\)? | B | We are given the field $\mathbb{F}_2 = \{0, 1\}$, and the ideal $I$ in $\mathbb{F}_2[x]$ generated by the polynomials:
- $f(x) = x^5 + x^2 + x + 1$
- $g(x) = x^7 + x^5 + x^3 + x$
We are to find **how many polynomials of degree 3** are in the ideal $I$.
---
### Step 1: Understand the structure of the ideal
In a pol... | We are given the ideal $ I $ in the polynomial ring $ \mathbb{F}_2[x] $, generated by two polynomials:
$$
f(x) = x^5 + x^2 + x + 1, \quad g(x) = x^7 + x^5 + x^3 + x
$$
We are to find **how many polynomials of degree 3** are in the ideal $ I $.
---
### Step 1: Understand the Structure of the Ideal
In $ \mathbb{F}_2... | 6.5 |
67847_8.0 | Let $F$ be a field, and $V$ be an $F$-vector space. Define the ring $R = F \oplus V$ with multiplication given by $xy = 0$ for $x, y \in V$. Determine if the Krull dimension of $R$ is zero. | B | We are given a ring $ R = F \oplus V $, where $ F $ is a field and $ V $ is an $ F $-vector space, with the multiplication defined by:
$$
xy = 0 \quad \text{for all } x, y \in V.
$$
Our goal is to determine whether the **Krull dimension** of $ R $ is **zero**.
---
### Step 1: Understand the ring structure
We are g... | We are given a ring $ R = F \oplus V $, where $ F $ is a field and $ V $ is an $ F $-vector space. The multiplication is defined such that for any $ x, y \in V $, we have $ xy = 0 $. We are to determine whether the **Krull dimension** of $ R $ is zero.
---
### Step 1: Understand the ring structure
Let’s write down t... | 8 |
34451_5.5 | Calculate the scalar surface integral of the function \( f(x, y, z) = yz \) over the half cylinder \( S \), defined by the equation \( x^2 + z^2 = a^2 \) with \( z \leq 0 \) and \( 0 \leq y \leq h \). Verify if the result is \( -a^2h^2 \). | B | We are asked to compute the **scalar surface integral** of the function
$$
f(x, y, z) = yz
$$
over the surface $ S $, which is a **half-cylinder** defined by:
$$
x^2 + z^2 = a^2, \quad z \leq 0, \quad 0 \leq y \leq h
$$
This is a **cylindrical surface** oriented along the **$ y $-axis**, with **radius $ a $**, an... | We are given the scalar surface integral:
$$
\iint_S f(x, y, z) \, dS
$$
with:
- $ f(x, y, z) = yz $
- $ S $ is the **half cylinder** defined by:
- $ x^2 + z^2 = a^2 $ (a cylinder of radius $ a $)
- $ z \leq 0 $ (only the lower half of the cylinder)
- $ 0 \leq y \leq h $
We are to **verify if the result is $ ... | 5.5 |
85400_5.5 | Given the equation \( \frac{x}{1+x-2x^{2}} = a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n} + \cdots \) for any \( x \in \left( -\frac{1}{2}, 1 \right) \), find the value of \( a_{3} + a_{4} \). | B | We are given the equation:
$$
\frac{x}{1 + x - 2x^2} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots
$$
and we are to find the value of $ a_3 + a_4 $, where the right-hand side is a power series expansion valid for $ x \in \left( -\frac{1}{2}, 1 \right) $.
---
### Step 1: Rewrite the denominator
We start by f... | We are given the equation:
$$
\frac{x}{1 + x - 2x^2} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots
$$
We are to find the value of $ a_3 + a_4 $, assuming the right-hand side is the power series expansion of the left-hand side in the interval $ x \in \left( -\frac{1}{2}, 1 \right) $.
---
### Step 1: Understan... | 5.5 |
9607_6.5 | Determine the values of \( n \in (1, \infty) \) for which the inequality \( x^n \leq n^x \) holds for all \( x \in (1, \infty) \). | A | We are given the inequality:
$$
x^n \leq n^x \quad \text{for all } x \in (1, \infty)
$$
We are to find all $ n \in (1, \infty) $ for which this inequality holds **for all** $ x > 1 $.
---
### Step 1: Take logarithms
To simplify the inequality, take the natural logarithm of both sides:
$$
\ln(x^n) \leq \ln(n^x)
$$... | We are given the inequality:
$$
x^n \leq n^x \quad \text{for all } x \in (1, \infty)
$$
and we are to determine the values of $ n \in (1, \infty) $ for which this inequality holds **for all** $ x > 1 $.
---
### Step 1: Take logarithms
To simplify the inequality, take the natural logarithm of both sides (since the ... | 6.5 |
29812_8.0 | Is an anti-unitary operator $J$ on a complex Hilbert space $\mathscr{H}$, with $J^2 = I$, always diagonalizable by an orthonormal basis? | A | We are asked whether an **anti-unitary operator** $ J $ on a complex Hilbert space $ \mathscr{H} $, satisfying $ J^2 = I $, is always **diagonalizable by an orthonormal basis**.
---
### Step 1: Understand the terms
- An **anti-unitary operator** $ J $ on a complex Hilbert space $ \mathscr{H} $ is a linear operator (... | We are asked whether an **anti-unitary operator** $ J $ on a complex Hilbert space $ \mathscr{H} $, satisfying $ J^2 = I $, is always **diagonalizable by an orthonormal basis**.
---
### Step 1: Understanding the Terms
- **Anti-unitary operator**: An operator $ J $ on a complex Hilbert space $ \mathscr{H} $ is called... | 8 |
78129_6.0 | Given four points $x_1, x_2, x_3, x_4$ on the unit circle $\mathbb{S}^1$ in the plane $\mathbb{R}^2$ such that $\sum_{i=1}^4 x_i = 0$, does this imply that the points form a rectangle or two pairs of antipodal points? Answer "yes" or "no." | B | We are given four points $ x_1, x_2, x_3, x_4 $ on the **unit circle** $ \mathbb{S}^1 \subset \mathbb{R}^2 $, and the condition:
$$
x_1 + x_2 + x_3 + x_4 = 0
$$
We are asked whether this implies that the points form a **rectangle** or **two pairs of antipodal points**. The answer is:
$$
\boxed{\text{no}}
$$
---
##... | We are given four points $ x_1, x_2, x_3, x_4 $ on the unit circle $ \mathbb{S}^1 \subset \mathbb{R}^2 $, and it is given that:
$$
x_1 + x_2 + x_3 + x_4 = 0
$$
We are to determine whether this implies that the points form a **rectangle** or **two pairs of antipodal points**.
---
### Step 1: Understand the Setup
- ... | 6 |
35736_7.0 | Evaluate the sum: $$\sum_{j=0}^\infty \left( \sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k \right)$$ | A | We are asked to evaluate the following infinite double sum:
$$
\sum_{j=0}^\infty \left( \sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k \right)
$$
---
### Step 1: Understand the inner sum
Let’s denote the inner sum as:
$$
S_j = \sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k
$$
We can interpre... | We are asked to evaluate the infinite double sum:
$$
\sum_{j=0}^\infty \left( \sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k \right)
$$
---
### Step 1: Understand the inner sum
Let’s look at the inner sum:
$$
\sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k
$$
This is a well-known sum. Recall ... | 7 |
27965_9.0 | Does there exist a selective ultrafilter \( \mathcal{U} \) and a bijection \( \varphi: \omega \to \omega \) such that \( \varphi(\mathcal{F_1}) \subset \mathcal{U} \) in ZFC and CH? | B | We are asked:
> Does there exist a **selective ultrafilter** $ \mathcal{U} $ and a **bijection** $ \varphi: \omega \to \omega $ such that $ \varphi(\mathcal{F}_1) \subset \mathcal{U} $ in **ZFC** and **CH**?
Let’s break this down carefully.
---
### Definitions
- **Ultrafilter** $ \mathcal{U} \subseteq \mathcal{P}(... | We are asked whether **there exists a selective ultrafilter** $ \mathcal{U} $ and a **bijection** $ \varphi: \omega \to \omega $ such that:
$$
\varphi(\mathcal{F}_1) \subset \mathcal{U}
$$
in **ZFC** and **CH** (the **Continuum Hypothesis**).
---
### Step 1: Understand the Notation and Definitions
- $ \omega $: th... | 9 |
42171_6.0 | Let $h$ and $k$ be functions of $u$ and $v$ such that $h(2, 1) = 1$ and $k(2, 1) = 3$. At the point $(2, 1)$, the differentials are given by $dh = 3du - 2dv$ and $dk = -du + 4dv$. The function $w$ is defined implicitly by the equation $w^3h - wk + u + v = 0$, with the condition $w(2, 1) = 2$. Find the value of $\frac{\... | B | We are given that the function $ w $ is defined **implicitly** by the equation:
$$
w^3 h - w k + u + v = 0
$$
We are to find $ \frac{\partial w}{\partial v} $ at the point $ (u, v) = (2, 1) $, with the following known values:
- $ h(2, 1) = 1 $
- $ k(2, 1) = 3 $
- $ w(2, 1) = 2 $
- Differentials:
- $ dh = 3\,du - 2... | We are given an implicit function:
$$
w^3 h - w k + u + v = 0
$$
and we are to find the partial derivative $\frac{\partial w}{\partial v}$ at the point $(u, v) = (2, 1)$, with the condition that $w(2, 1) = 2$, and the values of the functions $h$ and $k$ at that point:
$$
h(2, 1) = 1, \quad k(2, 1) = 3
$$
Also, the ... | 6 |
39655_7.0 | Determine the range of values of \( p \) for which the improper integral \( \int_{0}^{1} \frac{\ln x}{x^p(1-x)^{1-p}} \, dx \) converges. | A | We are asked to determine the **range of values of $ p $** for which the improper integral
$$
\int_{0}^{1} \frac{\ln x}{x^p(1-x)^{1-p}} \, dx
$$
**converges**.
---
### Step 1: Understand the integrand
Let’s rewrite the integrand:
$$
\frac{\ln x}{x^p(1-x)^{1-p}} = \ln x \cdot x^{-p}(1-x)^{p-1}
$$
So the integrand... | We are asked to determine the **range of values of $ p $** for which the **improper integral**
$$
\int_{0}^{1} \frac{\ln x}{x^p(1-x)^{1-p}} \, dx
$$
**converges**.
---
### Step 1: Understand the integrand
The integrand is:
$$
f(x) = \frac{\ln x}{x^p(1 - x)^{1 - p}}
$$
This function is defined on $ (0, 1) $, but ... | 7 |
90587_6.5 | Determine the minimum number of tokens that must be placed in the cells of a $99 \times 99$ grid such that every $4 \times 4$ subgrid contains at least eight tokens. | B | We are given a $99 \times 99$ grid, and we need to place the **minimum number of tokens** such that **every $4 \times 4$ subgrid** contains **at least 8 tokens**.
---
### Step 1: Understand the Problem
We are to find the **minimum number of tokens** to place in the $99 \times 99$ grid such that **every $4 \times 4$ ... | We are given a $99 \times 99$ grid, and we are to **determine the minimum number of tokens** that must be placed in the grid such that **every $4 \times 4$ subgrid contains at least eight tokens**.
---
### Step 1: Understand the problem
We are to place tokens in the grid so that **every $4 \times 4$ subgrid** has **... | 6.5 |
34572_8.0 | In a Noetherian ring $R$, if a prime ideal $p$ contains only zero divisors, does it necessarily correspond to an associated prime? In other words, is there an element $x \in R$ such that $p = \text{Ann}(x)$? | B | We are given a **Noetherian ring** $ R $, and a **prime ideal** $ \mathfrak{p} \subseteq R $ such that **every element of $ \mathfrak{p} $ is a zero divisor**. The question is whether this necessarily implies that $ \mathfrak{p} $ is an **associated prime** of $ R $, i.e., whether there exists some $ x \in R $ such tha... | We are given:
- $ R $ is a **Noetherian ring**.
- $ p $ is a **prime ideal** of $ R $.
- Every element of $ p $ is a **zero divisor**.
- The question is: **Does this imply that $ p $ is an associated prime of $ R $?**
That is, does there exist an element $ x \in R $ such that $ p = \text{Ann}(x) $?
---
### Definiti... | 8 |
85154_7.5 | Consider the sequence of polynomials \( P_{n}(x) \) defined recursively by \( P_{0}(x) = 1 \) and \( P_{n+1}(x) = x^{17(n+1)} - P_{n}(x) \) for all integers \( n \geq 0 \). Determine all real roots of \( P_{n}(x) \) for each \( n \geq 0 \). | B | We are given a recursive sequence of polynomials:
$$
P_0(x) = 1
$$
$$
P_{n+1}(x) = x^{17(n+1)} - P_n(x)
$$
We are to determine **all real roots** of $ P_n(x) $ for each $ n \geq 0 $.
---
### Step 1: Compute the first few polynomials
Let’s compute the first few $ P_n(x) $ to look for a pattern.
- $ P_0(x) = 1 $
- ... | We are given a sequence of polynomials defined recursively:
- $ P_0(x) = 1 $
- $ P_{n+1}(x) = x^{17(n+1)} - P_n(x) $ for all $ n \geq 0 $
We are to **determine all real roots of $ P_n(x) $ for each $ n \geq 0 $**.
---
### Step 1: Understand the recursive structure
Let’s compute the first few polynomials to see a p... | 7.5 |
30601_6.0 | Find \( \lim\limits_{n\to\infty}\left(\frac{a_1}{a_0S_1}+\frac{a_2}{S_1S_2}+...+\frac{a_n}{S_{n-1}S_n}\right) \) where \( a_n=2015^n \) and \( S_n=\sum\limits_{k=0}^{n}a_k \). Express \( S_n \) as the geometric sum \( S_n=\frac{2015^{n+1}-1}{2014} \). | B | We are given the expression:
$$
\lim_{n \to \infty} \left( \frac{a_1}{a_0 S_1} + \frac{a_2}{S_1 S_2} + \cdots + \frac{a_n}{S_{n-1} S_n} \right)
$$
where:
- $ a_n = 2015^n $
- $ S_n = \sum_{k=0}^n a_k = \frac{2015^{n+1} - 1}{2014} $
---
### Step 1: Understand the general term
Let’s look at the general term of the s... | We are given the expression:
$$
\lim_{n\to\infty} \left( \frac{a_1}{a_0 S_1} + \frac{a_2}{S_1 S_2} + \cdots + \frac{a_n}{S_{n-1} S_n} \right)
$$
where:
- $ a_n = 2015^n $
- $ S_n = \sum_{k=0}^n a_k = \frac{2015^{n+1} - 1}{2014} $
---
### Step 1: Understand the general term
We are summing terms of the form:
$$
\f... | 6 |
1067_6.5 | Let $A$ be an $n \times n$ real symmetric non-singular matrix. Suppose there exists $x \in \mathbb{R}^n$ such that $x'Ax < 0$. Determine which of the following statements is true:
1. $\det(A) < 0$
2. $B = -A$ is positive definite.
3. $\exists y \in \mathbb{R}^n: y'A^{-1}y < 0$
4. $\forall y \in \mathbb{R}^n: y'A^{-1}y... | A | We are given:
- $ A $ is an **$ n \times n $ real symmetric non-singular matrix**.
- There exists $ x \in \mathbb{R}^n $ such that $ x'Ax < 0 $.
- We are to determine which of the four statements is **true**.
---
### Step 1: Understand the given condition
We are told that:
$$
x'Ax < 0 \quad \text{for some } x \in ... | We are given:
- $ A $ is an **$ n \times n $ real symmetric non-singular matrix**.
- There exists $ x \in \mathbb{R}^n $ such that $ x'Ax < 0 $.
We are to determine which of the following statements is **true**:
1. $ \det(A) < 0 $
2. $ B = -A $ is positive definite.
3. $ \exists y \in \mathbb{R}^n: y'A^{-1}y < 0 $
4... | 6.5 |
53291_7.0 | Compute the surface integral \( \iint_{S_1} \mathbf{F} \cdot \hat{\mathbf{n}} \ dS \) for the vector field \( \mathbf{F}(x,y,z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}} \) over the sphere \( S_1 \) given by \( x^2 + (y-2)^2 + z^2 = 9 \), oriented outwards. | B | We are asked to compute the **surface integral** of the vector field
$$
\mathbf{F}(x, y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2 + y^2 + z^2]^{3/2}}
$$
over the **sphere**
$$
S_1: x^2 + (y - 2)^2 + z^2 = 9,
$$
oriented **outward**.
---
### Step 1: Understand the Vector Field
The vector field is:
$$
\... | We are asked to compute the **surface integral** of a vector field over a sphere:
$$
\iint_{S_1} \mathbf{F} \cdot \hat{\mathbf{n}} \, dS
$$
where the vector field is:
$$
\mathbf{F}(x, y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2 + y^2 + z^2]^{3/2}}
$$
and the surface $ S_1 $ is the sphere:
$$
x^2 + (y - 2)^2... | 7 |
11919_8.5 | Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n$-dimensional manifold $X$. If this mapping is diffeomorphic onto its image, does it follow that it is also biholomorphic onto its image? | B | We are given a holomorphic mapping:
$$
\varphi : U \rightarrow X
$$
where $ U \subseteq \mathbb{C} $ is an open set, and $ X $ is a complex $ n $-dimensional manifold. The mapping $ \varphi $ is **diffeomorphic onto its image**, and we are asked whether it follows that $ \varphi $ is **biholomorphic onto its image**.... | We are given a holomorphic mapping:
$$
\varphi : U \rightarrow X
$$
where:
- $ U \subseteq \mathbb{C} $ is an open set,
- $ X $ is a complex $ n $-dimensional manifold,
- $ \varphi $ is **holomorphic**,
- $ \varphi $ is **diffeomorphic** onto its image.
We are asked: **Does it follow that $ \varphi $ is biholomorph... | 8.5 |
23177_8.0 | Let $X$ be an open, simply connected subset of $\mathbb{R}^2$ containing the point $0$. For every pair of points $p, q \in X \setminus \{0\}$, is there an open, simply connected subset $A_{pq} \subseteq X \setminus \{0\}$ such that $p, q \in A_{pq}$? | B | We are given:
- $ X \subseteq \mathbb{R}^2 $ is **open** and **simply connected**.
- $ 0 \in X $, and $ X \setminus \{0\} $ is the domain of interest.
- For **every pair** of points $ p, q \in X \setminus \{0\} $, we are to determine whether there exists an **open, simply connected subset** $ A_{pq} \subseteq X \setmi... | We are given:
- $ X $ is an **open**, **simply connected** subset of $ \mathbb{R}^2 $, containing the point $ 0 $.
- For **every pair** of points $ p, q \in X \setminus \{0\} $, we are asked whether there exists an **open**, **simply connected** subset $ A_{pq} \subseteq X \setminus \{0\} $ such that $ p, q \in A_{pq}... | 8 |
20556_5.5 | Solve the differential equation \( \frac{dy}{dx} = \frac{6}{x+y} \) with the initial condition \( y(0) = 0 \). Find the value of \( y \) when \( x+y = 6 \). | B | We are given the differential equation:
$$
\frac{dy}{dx} = \frac{6}{x + y}
$$
with the initial condition:
$$
y(0) = 0
$$
We are to **find the value of $ y $ when $ x + y = 6 $**.
---
### Step 1: Substitution
Let’s make a substitution to simplify the equation. Let:
$$
u = x + y
$$
Then:
$$
\frac{du}{dx} = \fra... | We are given the differential equation:
$$
\frac{dy}{dx} = \frac{6}{x + y}
$$
with the initial condition:
$$
y(0) = 0
$$
We are asked to **find the value of $ y $ when $ x + y = 6 $**.
---
### Step 1: Substitution
Let’s define a new variable:
$$
u = x + y
$$
Then:
$$
\frac{du}{dx} = \frac{d}{dx}(x + y) = 1 + ... | 5.5 |
49654_8.5 | In Rudin's proof that a subalgebra $\mathscr{A}$ of $C(X,\mathbb{R})$ on a compact space $X$ is dense in $C(X,\mathbb{R})$ under the uniform topology if it separates points and vanishes nowhere, identify the step that relies on the Hausdorffness condition of $X$. The proof consists of the following steps: (1) $|f|\in \... | A | To answer this question, we need to understand **Rudin's proof** (from *Principles of Mathematical Analysis*, or *Real and Complex Analysis*) of the **Stone–Weierstrass theorem**. The theorem states:
> Let $ X $ be a compact Hausdorff space, and let $ \mathscr{A} $ be a subalgebra of $ C(X, \mathbb{R}) $ that separate... | To identify which step in Rudin's proof relies on the **Hausdorffness** of the compact space $ X $, we need to analyze the logical structure of the proof and determine which part **requires** the space $ X $ to be **Hausdorff**.
---
### **Summary of Rudin's Proof Outline**
The key steps in Rudin's proof that a subal... | 8.5 |
46006_6.5 | Evaluate the integral \( \displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx \). | A | We are asked to evaluate the integral:
$$
\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx
$$
---
### Step 1: Analyze the integrand
The integrand is:
$$
\frac{x^2 + 2x}{\ln(x+1)}
$$
Let’s note that the logarithm $\ln(x+1)$ is defined for $x+1 > 0$, i.e., $x > -1$. Since the integral is from $-1$ to $0$, the logarithm appro... | We are asked to evaluate the integral:
$$
\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx
$$
---
### Step 1: Substitution
Let’s make the substitution:
$$
u = x + 1 \Rightarrow x = u - 1
$$
Then:
- When $ x = -1 $, $ u = 0 $
- When $ x = 0 $, $ u = 1 $
Also, $ dx = du $
Now, substitute into the integral:
$$
\int_{-1}^{... | 6.5 |
39039_7.0 | Determine the number of distinct conjugacy classes in a finite group whose order is a multiple of 8, considering only the case where the order is even. | A | We are asked to **determine the number of distinct conjugacy classes** in a finite group whose order is a **multiple of 8**, and specifically consider the case where the order is **even**.
---
### Step 1: Understanding the Problem
Let $ G $ be a finite group with order $ |G| = n $, where $ n $ is **even** and a **mu... | We are asked to **determine the number of distinct conjugacy classes** in a finite group $ G $ whose **order is a multiple of 8**, and we are to consider only the case where the order is **even**.
---
### Step 1: Understanding the Problem
We are given:
- $ |G| $ is a **multiple of 8**, i.e., $ |G| = 8k $ for some i... | 7 |
79941_6.0 | Consider a sequence of independent random variables \(X_{n1}, X_{n2}, \ldots, X_{nn}\), all sharing the same distribution given by:
\[ P(X_{nk} = 0) = 1 - \frac{2}{n} - \frac{4}{n^3} \]
\[ P(X_{nk} = 1) = \frac{2}{n} \]
\[ P(X_{nk} = 2) = \frac{4}{n^3} \]
Let \(S_n = \sum^n_{k=1} X_{nk}\). Determine the limit in proba... | B | We are given a sequence of independent random variables $ X_{nk} $, for $ k = 1, 2, \ldots, n $, and each $ X_{nk} $ has the same distribution:
$$
P(X_{nk} = 0) = 1 - \frac{2}{n} - \frac{4}{n^3}, \quad
P(X_{nk} = 1) = \frac{2}{n}, \quad
P(X_{nk} = 2) = \frac{4}{n^3}
$$
We are also given that:
$$
S_n = \sum_{k=1}^n X... | We are given a sequence of independent and identically distributed (i.i.d.) random variables:
$$
P(X_{nk} = 0) = 1 - \frac{2}{n} - \frac{4}{n^3}, \quad
P(X_{nk} = 1) = \frac{2}{n}, \quad
P(X_{nk} = 2) = \frac{4}{n^3}
$$
Let us define the sum:
$$
S_n = \sum_{k=1}^n X_{nk}
$$
We are asked to find the **limit in proba... | 6 |
21054_6.0 | Evaluate the double integral \( \iint_D x^6y^6 \, dx \, dy \) where the region \( D \) is defined by \( D = \{ (x,y) : x^{2} \le y \le x^{1/8} \} \). | B | We are asked to evaluate the double integral:
$$
\iint_D x^6 y^6 \, dx \, dy
$$
where the region $ D $ is defined by:
$$
D = \{ (x,y) : x^2 \le y \le x^{1/8} \}
$$
---
### Step 1: Understand the region of integration
We are given bounds on $ y $ in terms of $ x $:
$$
x^2 \le y \le x^{1/8}
$$
We need to determin... | We are asked to evaluate the double integral:
$$
\iint_D x^6 y^6 \, dx \, dy
$$
where the region $ D $ is defined by:
$$
D = \{ (x, y) : x^2 \le y \le x^{1/8} \}
$$
---
### Step 1: Understand the region $ D $
We are given the bounds for $ y $ in terms of $ x $:
$$
x^2 \le y \le x^{1/8}
$$
This defines a region ... | 6 |
92677_6.0 | Inside the tetrahedron \(ABCD\), there is a point \(O\) such that line segments \(AO, BO, CO, DO\) intersect the faces \(BCD, ACD, ABD, ABC\) at points \(A_1, B_1, C_1, D_1\) respectively, with \(\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k\). Determine all possible values of \(k\). | A | We are given a tetrahedron $ABCD$ with a point $O$ inside it such that the line segments $AO, BO, CO, DO$ intersect the opposite faces at points $A_1, B_1, C_1, D_1$ respectively, and for each of these, the ratio of the segments is the same:
$$
\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k
... | We are given a tetrahedron $ABCD$ with a point $O$ inside it such that the segments $AO, BO, CO, DO$ intersect the opposite faces at points $A_1, B_1, C_1, D_1$, respectively. It is given that:
$$
\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k
$$
We are to **determine all possible values of... | 6 |
67721_7.5 | Consider the sequence of numbers: $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{64}$. A transformation is applied to any two terms $a$ and $b$ in the sequence, replacing them with $a+b+ab$. Is it possible to reach a state where all terms in the sequence are identical after a certain number of steps? If so, how many dis... | B | We are given a sequence of numbers:
$$
1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{64}
$$
and a transformation: for any two terms $ a $ and $ b $, replace them with $ a + b + ab $.
We are asked:
1. **Is it possible** to reach a state where **all terms are identical** after a certain number of steps?
2. **If so**, ... | We are given a sequence of numbers:
$$
1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{64}
$$
This is the sequence of reciprocals of the first 64 positive integers:
$$
\left\{ \frac{1}{n} \mid n = 1, 2, 3, \dots, 64 \right\}
$$
We are allowed to perform a transformation on any two terms $ a $ and $ b $, replacing them... | 7.5 |
70816_6.0 | Evaluate the integral \( \int_0^{2\pi}\frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, dx \) for any \(N\in\mathbb{N}\). | B | We are asked to evaluate the integral:
$$
\int_0^{2\pi}\frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, dx
$$
for any $ N \in \mathbb{N} $.
---
### Step 1: Simplify the Expression
Let’s denote the integral as:
$$
I = \int_0^{2\pi} \frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, ... | We are asked to evaluate the integral:
$$
\int_0^{2\pi} \frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, dx
$$
for any natural number $ N \in \mathbb{N} $.
---
### Step 1: Simplify the expression
Let’s write the integral in a more compact form:
$$
\frac{1}{2\pi(N+1)} \int_0^{2\pi} \frac{\sin^2\le... | 6 |
73106_6.5 | Determine whether all $k$-hyperperfect numbers for $k > 1$ are odd. Answer true or false. | A | We are asked to determine whether **all $ k $-hyperperfect numbers for $ k > 1 $** are **odd**. The answer should be **true** or **false**.
---
### Step 1: Understand what a $ k $-hyperperfect number is
A **$ k $-hyperperfect number** is a positive integer $ n $ such that:
$$
n = 1 + k(p - 1)
$$
where $ p $ is a p... | We are asked to determine whether **all $ k $-hyperperfect numbers for $ k > 1 $ are odd**. The answer should be **true or false**, and we must provide the final answer in **LaTeX \boxed{}** format.
---
### Step 1: Understand what a $ k $-hyperperfect number is
A **$ k $-hyperperfect number** is a positive integer $... | 6.5 |
76340_8.0 | Evaluate the limit: $$\lim_{n \to \infty} \left(\frac{\sum_{k=1}^n k^k}{\left(\frac{n+1}{e}\right)^{n+1}\cdot \sqrt{2\pi (n+1)}}\right)^{\frac{1}{n+1}}$$ | B | We are asked to evaluate the limit:
$$
\lim_{n \to \infty} \left(\frac{\sum_{k=1}^n k^k}{\left(\frac{n+1}{e}\right)^{n+1}\cdot \sqrt{2\pi (n+1)}}\right)^{\frac{1}{n+1}}
$$
---
### Step 1: Understand the structure of the expression
Let’s denote:
- $ A_n = \sum_{k=1}^n k^k $
- $ B_n = \left(\frac{n+1}{e}\right)^{n+1... | We are asked to evaluate the following limit:
$$
\lim_{n \to \infty} \left(\frac{\sum_{k=1}^n k^k}{\left(\frac{n+1}{e}\right)^{n+1} \cdot \sqrt{2\pi (n+1)}}\right)^{\frac{1}{n+1}}
$$
---
### Step 1: Understand the structure of the expression
Let’s denote the expression inside the limit as:
$$
A_n = \left(\frac{\su... | 8 |
87058_6.5 | 2021 people are seated around a circular table. In one move, you may swap the positions of two adjacent people. What is the minimum number of moves required to shift each person 1000 positions to the left from their original position? | A | We are given:
- There are **2021 people** seated around a **circular table**.
- We can perform a **swap of two adjacent people** in one move.
- We want to **shift each person 1000 positions to the left** (i.e., each person moves **1000 positions counterclockwise** around the table).
- We are to find the **minimum numb... | We are given a circular table with **2021 people**, and we are to **shift each person 1000 positions to the left** (i.e., each person moves to the position that is 1000 positions counterclockwise around the circle). We are allowed to **swap two adjacent people** in one move, and we want to find the **minimum number of ... | 6.5 |
91773_7.5 | Find the maximum value of the expression \(A = \left(3(x_{1} + \ldots + x_{n}) - 5(y_{1} + \ldots + y_{n})\right) \cdot \left(5(x_{1} + \ldots + x_{n}) + 3(y_{1} + \ldots + y_{n})\right)\) given that \(x_{1}^{2} + \ldots + x_{n}^{2} + y_{1}^{2} + \ldots + y_{n}^{2} \leq 1\). | A | We are given the expression:
$$
A = \left(3(x_{1} + \ldots + x_{n}) - 5(y_{1} + \ldots + y_{n})\right) \cdot \left(5(x_{1} + \ldots + x_{n}) + 3(y_{1} + \ldots + y_{n})\right)
$$
and the constraint:
$$
x_{1}^{2} + \ldots + x_{n}^{2} + y_{1}^{2} + \ldots + y_{n}^{2} \leq 1
$$
---
### Step 1: Simplify the Expression... | We are given the expression:
$$
A = \left(3(x_{1} + \ldots + x_{n}) - 5(y_{1} + \ldots + y_{n})\right) \cdot \left(5(x_{1} + \ldots + x_{n}) + 3(y_{1} + \ldots + y_{n})\right)
$$
and the constraint:
$$
x_{1}^{2} + \ldots + x_{n}^{2} + y_{1}^{2} + \ldots + y_{n}^{2} \leq 1
$$
---
### Step 1: Simplify the Expression... | 7.5 |
50958_8.5 | Does there exist a sequence of radii \(r_i\) such that the open balls \(B_{r_i}(q_i)\) cover \(\mathbb{R}\) for a given enumeration \(\{q_i\}_{i\in I}\) of the rationals, even if \(\liminf r_i = 0\)? | A | We are given a sequence of rational numbers $\{q_i\}_{i \in \mathbb{N}}$, which is an enumeration of $\mathbb{Q}$, and we are to determine whether there exists a sequence of radii $r_i > 0$ such that the open balls $B_{r_i}(q_i) = \{x \in \mathbb{R} : |x - q_i| < r_i\}$ cover $\mathbb{R}$, even if $\liminf r_i = 0$.
-... | We are asked whether there exists a sequence of radii $ r_i $ such that the open balls $ B_{r_i}(q_i) $ cover $ \mathbb{R} $, where $ \{q_i\}_{i \in I} $ is a given enumeration of the rationals $ \mathbb{Q} $, and even if $ \liminf r_i = 0 $.
---
### Step 1: Understanding the Problem
We are given:
- A countable den... | 8.5 |
48110_6.5 | Let $X = Y = \Bbb{N}$, where $\Bbb{N}$ is the set of natural numbers. Define $f: X \to Y$ by $f(x) = 2x$ and define $g: Y \to X$ by $g(y) = 3y$. Let $W = \bigcup_{n=0}^\infty (g(fg))^n(Y - D)$, where $D = f(X)$. $W$ is a subset of $X$. Find an element of $X$ that is not in $W$. | B | We are given:
- $ X = Y = \mathbb{N} $, the set of natural numbers.
- $ f: X \to Y $ is defined by $ f(x) = 2x $.
- $ g: Y \to X $ is defined by $ g(y) = 3y $.
- $ D = f(X) = \{2x \mid x \in \mathbb{N}\} = \{2, 4, 6, 8, \dots\} $, the set of even natural numbers.
- $ W = \bigcup_{n=0}^\infty (g \circ f \circ g)^n(Y \s... | We are given:
- $ X = Y = \mathbb{N} $, the set of natural numbers.
- $ f: X \to Y $ is defined by $ f(x) = 2x $.
- $ g: Y \to X $ is defined by $ g(y) = 3y $.
- $ D = f(X) $, so $ D = \{2x \mid x \in \mathbb{N}\} = \{2, 4, 6, 8, \dots\} $.
- $ W = \bigcup_{n=0}^\infty (g \circ f \circ g)^n(Y \setminus D) $, and we ar... | 6.5 |
61858_8.5 | In a Cartesian closed category, consider a pullback square with a monic arrow. Does the uniqueness of a certain arrow imply the equality of two given arrows? | B | We are given a **pullback square** in a **Cartesian closed category**, and one of the arrows in the square is **monic**. The question is:
> Does the **uniqueness** of a certain arrow imply the **equality** of two given arrows?
Let’s break this down step by step.
---
### 1. **Recall the Setup: Pullback Square**
A *... | We are given a **pullback square** in a **Cartesian closed category**, and one of the arrows in the square is **monic** (i.e., a monomorphism). The question is whether the **uniqueness** of a certain arrow implies the **equality** of two given arrows.
---
### Step 1: Understand the Pullback Square
In a pullback squa... | 8.5 |
64702_6.5 | Calculate the surface area of the portion of the cone \(z^2 = x^2 + y^2\) that is contained within the sphere \(x^2 + y^2 + z^2 = 8\). | B | We are given:
- A **cone**:
$$
z^2 = x^2 + y^2
$$
- A **sphere**:
$$
x^2 + y^2 + z^2 = 8
$$
We are to **find the surface area of the portion of the cone** that lies **inside the sphere**.
---
### Step 1: Understand the Geometry
The cone equation $ z^2 = x^2 + y^2 $ represents a double-napped cone s... | We are given:
- A **cone**: $ z^2 = x^2 + y^2 $
- A **sphere**: $ x^2 + y^2 + z^2 = 8 $
We are to find the **surface area** of the **portion of the cone** that lies **inside** the sphere.
---
### Step 1: Understand the Geometry
The cone $ z^2 = x^2 + y^2 $ is symmetric about the $ z $-axis. In cylindrical coordina... | 6.5 |
82926_6.5 | Evaluate the integral \( \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\delta (x+y) \delta (x-y) e^{-x^2-y^2}dxdy \) using the properties of the Dirac delta function. | B | We are asked to evaluate the double integral:
$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(x+y) \delta(x-y) e^{-x^2 - y^2} \, dx \, dy
$$
This is a double integral involving the **Dirac delta function**, and the integrand is the product of two delta functions and a Gaussian function.
---
### Step 1: Un... | We are asked to evaluate the double integral:
$$
\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\delta (x+y) \delta (x-y) e^{-x^2-y^2} \, dx \, dy
$$
---
### Step 1: Understand the Dirac delta functions
We have two delta functions:
- $ \delta(x + y) $: This is zero unless $ x + y = 0 $, i.e., $ y = -x $
- $ \... | 6.5 |
28818_7.5 | Does there exist a sequence \((x_n)\) belonging to \(\ell_1 \cap \ell_2\) which converges in one space but not the other? | B | We are asked whether there exists a sequence $(x_n) \in \ell_1 \cap \ell_2$ that **converges in one of the spaces** but **not in the other**.
---
### Definitions:
- A sequence $(x_n)$ is in $\ell_1$ if:
$$
\sum_{n=1}^\infty |x_n| < \infty
$$
- A sequence $(x_n)$ is in $\ell_2$ if:
$$
\sum_{n=1}^\infty |x_... | We are asked whether there exists a sequence $(x_n) \in \ell_1 \cap \ell_2$ that **converges in one space but not the other**.
---
### Definitions:
- A sequence $(x_n)$ is in $\ell_1$ if:
$$
\sum_{n=1}^\infty |x_n| < \infty
$$
- A sequence $(x_n)$ is in $\ell_2$ if:
$$
\sum_{n=1}^\infty |x_n|^2 < \infty
... | 7.5 |
75965_7.5 | Determine if the density of the set of numbers expressible as \(a^2 + b^2 + c^2 - abc\) for positive integers \(a, b, c\) is exactly 50%. | A | We are asked to determine whether the **density** of the set of numbers expressible as:
$$
a^2 + b^2 + c^2 - abc
$$
for positive integers $ a, b, c $ is **exactly 50%**. That is, we want to know whether **half** of the positive integers can be written in this form.
---
### Step 1: Understanding the Expression
We a... | We are given the set of numbers expressible as:
$$
S = \{ a^2 + b^2 + c^2 - abc \mid a, b, c \in \mathbb{Z}^+ \}
$$
We are to **determine whether the density of this set is exactly 50%**.
---
### Step 1: Understand the Expression
Let’s define the function:
$$
f(a, b, c) = a^2 + b^2 + c^2 - abc
$$
We are interest... | 7.5 |
76863_8.0 | For an affine variety $V$ in $
abla^{n}$ with a coordinate ring that is a unique factorization domain (UFD), determine if it is true that any closed subvariety of $V$ with codimension 1 can be defined by a single equation. | B | We are given an **affine variety** $ V \subset \mathbb{A}^n $ (affine $ n $-space over an algebraically closed field, say $ \mathbb{C} $) whose **coordinate ring** is a **unique factorization domain (UFD)**. The question is:
> **Is it true that any closed subvariety of $ V $ with codimension 1 can be defined by a sing... | We are given an **affine variety** $ V \subset \mathbb{A}^n $ (affine $ n $-space over an algebraically closed field, say $ \mathbb{C} $) whose **coordinate ring** is a **unique factorization domain (UFD)**. We are to determine whether **any closed subvariety of $ V $ of codimension 1** can be defined by a single equat... | 8 |
66877_5.5 | Evaluate the integral: $$\int_{1}^{2} \left( \sum_{k=0}^{\infty} \frac{k}{(2x)^{k+1}} \right)dx$$ | A | We are asked to evaluate the integral:
$$
\int_{1}^{2} \left( \sum_{k=0}^{\infty} \frac{k}{(2x)^{k+1}} \right)dx
$$
---
### Step 1: Interchange the sum and the integral
We will first try to interchange the sum and the integral. Since the sum is a power series, we need to check whether it converges uniformly on the ... | We are given the integral:
$$
\int_{1}^{2} \left( \sum_{k=0}^{\infty} \frac{k}{(2x)^{k+1}} \right)dx
$$
---
### Step 1: Interchange Sum and Integral (if valid)
We can consider interchanging the sum and the integral, provided the convergence is uniform. Let's proceed with that and verify later if it's valid.
$$
\in... | 5.5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.