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 |
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