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Problem 401

10. $\sin ^{4} x-\cos ^{4} x=1-2 \cos ^{2} x$

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Problem 402

1. Let $f(x)=x^{2} \cos \frac{1}{x}, \quad x \neq 0$ (a) Use a graphing calculator to sketch the graph of $y=f(x)$ . (b) Show that $-x^{2} \leq x^{2} \cos \frac{1}{x} \leq x^{2}$ holds for $x \neq 0$ . (c) Use your result in (b) and the sandwich theorem to show that $\lim _{x \rightarrow 0} x^{2} \cos \frac{1}{x}=0$

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Problem 403

The radian is an alternative unit to the degree for angle measurement. True False

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Problem 404

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

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Problem 405

All integers can be expressed as fractions, confirming they are rational numbers.

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Problem 406

Prove that for $\triangle ABC$ with external angle $\angle ACD$ , $m \angle ACD = m \angle B + m \angle A$ .

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Problem 407

Explain why $\overline{B D} \cong \overline{B D}$ in the proof of $\angle A \simeq \angle C$ in $\triangle A B C$ .

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Problem 408

Which statement proves $\triangle D E F \cong \triangle A B C$ ? a. $A B=D E$ , $B C=E F$ b. $\angle D \cong \angle A$ , $\angle B \cong \angle E$ , $\angle C \cong \angle F$ c. Rigid motions map $A$ to $D$ , $A B$ to $D E$ , $\angle B$ to $\angle E$ d. Rigid motions map $A B$ to $D E$ , $B C$ to $E F$ , $A C$ to $D F$ .

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Problem 409

Prove that if $m \angle \mathrm{A}+m \angle \mathrm{B}=m \angle \mathrm{B}+m \angle \mathrm{C}$ , then $m \angle \mathrm{C}=m \angle \mathrm{A}$ . Write a paragraph proof.

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Problem 410

7. Let $E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}$ . Show that $\iiint_{E} e^{z} d V=e-\frac{5}{2}$

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Problem 411

Question 10 (1 point) The radian measure of an angle is defined as the length of the arc that subtends the angle divided by the radius of the circle. True False

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Problem 412

Problème \# 5 : Si $a(a-7 b)=-b^{2}$ , Prouve que $\log \left(\frac{a+b}{3}\right)=\frac{\log a+\log b}{2}$

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Problem 413

5. Given the expression $2 \cot (x)-2 \cot (x) \cos ^{2}(x)$ , a. Use technology to graph the expression [3 marks] b. Determine an equivalent trigonometric expression [2 marks] c. Then prove that your expression is equal to the given expression. [ 3 marks]

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Problem 414

True or False: 10 and $10 z$ are like terms. True False

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Problem 415

Problem Solving 11) A sector of a circle of radius 28 cm has perimeter $P \mathrm{~cm}$ and area $A \mathrm{~cm}^{2}$ . Given that $A=4 P$ , find the value of $P$ . 12) The percentage error for $\sin \theta$ for a given positive value of $\theta$ is $1 \%$ . Show that $100 \theta=101 \sin \theta$ .
Answers

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Problem 416

Submit Question
Question 2 $0 / 1$ pt 5 99 Details
ANOVA is a statistical procedure that compares two or more treatment conditions for differences in variance. True False
Question Help: Written Example Post to forum Submit Question

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Problem 417

S
8. Given: $\overline{\mathrm{DE}} \cong \overline{\mathrm{HG}}, \mathrm{DE} / / \mathrm{GH}$
Prove: $\overline{\mathrm{DF}} \cong \overline{\mathrm{HF}}$ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular}

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Problem 418

2. Flächeninhalt
Die Funktion $f(x)=(x-1) \cdot e^{x}(s$ . Bild oben) beschreibt den Verlauf eines Flusses, der von zwei Straßen überbrückt wird, die längs der Koordinatenachsen laufen. (1 LE = 1 km ) Die beiden Straßen und der Fluss schließen im 4. Quadranten ein Grundstück A ein, welches für $80 €$ pro $\mathrm{m}^{2}$ zum Kauf angeboten wird. a) Zeigen Sie, dass $F(x)=(x-2) \cdot e^{x}$ eine Stammfunktion von $f$ ist.

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Problem 419

Unit 4: Trigonometric Functions Activity 8: Trigonometric Identities
Assignment Prove each of the following trigonometric identities.
1. $\sin x \sin 2 x+\cos x \cos 2 x=\cos x$
2. $\cot x=\sin x \sin \left(\frac{\pi}{2}-x\right)+\cos ^{2} x \cot x$
3. $2 \csc 2 x=\sec x \csc x$

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Problem 420

(i) [3pts] $x^{-1} y^{-1} x y \in H$ for all $x, y \in G$ then $I I$ is normal subgroup of G

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Problem 421

Given: $C$ is the midpoint of $A D$ and $C$ is the midpoint of $E B$ . Prove: $\angle A \cong \angle D$ \begin{tabular}{l|l} STATEMENTS & REASONS \\ \hline & \\ & \end{tabular}

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Problem 422

In triangles $P Q R$ and $U V W$ , angles $Q$ and $V$ each have measure $75^{\circ}, P Q=9$ , and $U V=27$ . Which additional piece of information is sufficient to prove that triangle $P Q R$ is similar to triangle $U V W$ ?

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Problem 423

c approach. $(\sin x+\cos x)^{2}=1+2 \sin x \cos x$

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Problem 424

Richiesta 3 (5 punti) Dimostrare la seguente formula per calcolare l'area di un rombo: $A=l^{2} \sin (\alpha)$ dove $l$ è la misura di un suo lato e $\alpha$ è l'ampiezza di un suo angolo interno. Suggerimento: utilizzare la seguente formula di duplicazione: $\sin (2 \cdot \theta)=2 \cdot \sin (\theta) \cdot \cos (\theta)$

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Problem 425

oom DeltaMath All Bookma:
Math tnment \#3 at 10:00 PM ruence, Flowchart Proof (Level (Level 1) (Level 2) (Level 1) uence Criteria
Given: $\overline{A B} \cong \overline{A C}$ and $\angle B A D \cong \angle C A D$ . Prove: $\angle D B C \cong \angle D C B$ . Note: quadrilateral properties are not permitted in this proof.

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Problem 426

The result of dividing an integer by a decimal is not always an integer. Show examples like $3 \div 1.5$ and $1.8 \div 0.3$ .

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Problem 427

Prove that $\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}$ by finding $\delta$ in terms of $\varepsilon$ .

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Problem 428

Prove that $\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}$ by finding $\delta$ for any $\varepsilon>0$ .

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Problem 429

Are triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

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Problem 430

Dilation is a non-isometric transformation. True or False?

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Problem 431

If $\overline{P Q} \cong \overline{Q R}$ with $P Q=3x-8$ , $Q R=2x$ , and $R S=1.5x+4$ , is $P S=24$ true or false?

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Problem 432

Is it true or false that every real number is a noninteger rational number? Choose: True or False.

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Problem 433

In a G.P., if the $p^{\text{th}}$ , $q^{\text{th}}$ , and $r^{\text{th}}$ terms are $a, b, c$ , prove that: $a^{q-r} \cdot b^{r-p} \cdot c^{p-q}=1$ .

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Problem 434

Prove that if $a, b, c$ are in A.P. and $x, y, z$ are in G.P., then $x^{b-c} \cdot y^{c-a} \cdot z^{a-b}=1$ .

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Problem 435

Prove: $\triangle A E C \cong \triangle D F B$ .
Step Statement $\overline{A E} \| \overline{F D}$ 1 $\overline{B F} \| \overline{E C}$ $\overline{A C} \cong \overline{B D}$
Reason
Given try Type of Statement

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Problem 436

rcice 2 : Dans cet exercice toutes les récurrences devront être faites sans considérer qu'elles sont évidentes ; Soit $\left(u_{n}\right)_{n \geq 0}$ la suite de nombres réels définie par $\left.\left.u_{0} \in\right] 1,2\right]$ et par la relation de récurrence $u_{n+1}=\frac{\left(u_{n}\right)^{2}}{4}+\frac{3}{4}$ Exercice 5: Soit $\left(u_{n}\right)_{n}$
1. Montrer que: $\forall n \in \mathbb{N}, u_{n}>1$ .
2. Montrer que: $\forall n \in \mathbb{N}, u_{n} \leq 2$ .
3. Montrer que la suite est monotone. En déduire que la suite est convergente.
4. Déterminer la limite de la suite $\left(u_{n}\right)_{n \geq 0}$ .
Exercice 3 : Soient $u_{0}, a$ et $b$ trois réels. On considère la suite $\left(u_{n}\right)_{n \geq 0}$ de nombres réels définie par $u_{0}$ et la relation de récurrence: $\begin{array}{l} u_{n+1}=a u_{n}+b \\ u_{n} \end{array}$
1. Comment appelle-t-on la suite $\left(u_{n}\right)_{n \geq 0}$ lorsque $a=1$ ? Lorsque que $b=0$ et $a \neq 1$ ?
2. Exprimer $u_{n}$ dans les deux cas particulier de la question 1 .
3. Dans le cas général, calculer $u_{1}, u_{2}$ et $u_{3}$ en fonction de $u_{0}, a$ et $b$ .
4. Démontrer par récurrence que le terme général de la suite est donné par: $u_{n}=a^{n} u_{0}+b \sum_{k=1}^{n} a^{n-k}, n \in \mathbb{N}^{*}$

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Problem 437

1. Let the function $f(z)=u(x, y)+i v(x, y)$ and it satisfies the Cauchy-Riemann conditions: $\begin{array}{l} \frac{\partial u(x, y)}{\partial x}=\frac{\partial v(x, y)}{\partial y} \\ \frac{\partial u(x, y)}{\partial y}=-\frac{\partial v(x, y)}{\partial x} \end{array}$ then $f(z)$ is said to be analytical and $v(x, y)$ is said to be harmonic conjugate of $u(x, y)$ . It is said to be harmonic if $\begin{array}{l} \frac{\partial^{2} u(x, y)}{\partial x^{2}}+\frac{\partial^{2} u(x, y)}{\partial y^{2}}=0 \\ \frac{\partial^{2} v(x, y)}{\partial x^{2}}+\frac{\partial^{2} v(x, y)}{\partial y^{2}}=0 \end{array}$
Show that the following $u(x, y)$ are harmonic and find its harmonic conjugate (a) $u(x, y)=2 x(1-y)$ (b) $u(x, y)=\sinh (x) \sin (y)$

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Problem 438

2. Let $z=r e^{i \Theta}$ , then $\log (z)=\ln (r)+i(\Theta+2 n \pi)$ . The principle value of $\log (z)$ is obtained by setting n to zero and is written as $\log (z)=\ln (r)+i \Theta$ Find the following: (a) $\log (\sqrt{i})$ (b) Show that for any two nonzero complex numbers $z_{1}$ and $z_{2}$ , $\begin{array}{r} \log \left(z_{1} z_{2}\right)=\log \left(z_{1}\right)+\log \left(z_{2}\right)+2 N \pi i \\ N=-1,0,1 \end{array}$

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Problem 439

3. Prove the following identities: (a) $\sinh (z+\pi)=-\sinh (z)$ (b) $\cosh (z)=\cosh (x) \cos (y)+i \sinh (x) \sin (y)$

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Problem 440

The resistance of the LDR decreases as the incident light increases. True False

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Problem 441

In einem Waldstuick wird der derzeitige Halzbestand auf 4000 Festmeter geschatat. Der Holzzuwachs beträgt voraussichtlich in den nächsten 50 Jahren Jahrlich etwa 2\% des Bestands. Modellieren Sie die zeitliche Lntvicklung des Holzbestands als Exponentialfunktion der Form $\mathrm{H}: \mathrm{t} \mapsto \mathrm{b} \cdot \mathrm{a}^{t}(\mathrm{H}(\mathrm{t})$ in Festmetern, $t$ in Jahreri). Zeigen Sie, dass die Funktion $\mathrm{Hzu} \mathrm{t} \mapsto \mathrm{b} \cdot \mathrm{e}^{\mathrm{kt}}$ mit $\mathrm{k}=\ln (\mathrm{a}) \mathrm{urn}-$ geformt werden kann. Weiche der beiden Darstellungstormen ist zur Bestimmung der Wachstumsgeschwindigkeit

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Problem 442

(3) لاى ثلات مجمو عات C,B,A اثبت صحة الثو انين التنالية : (i) $A \Delta(B \Delta C)=(A \Delta B) \Delta C$ (ii) $A \cap(B \cup C)^{c}=(A-B) \cup(A-C)$ (iii) $(A \cap B) \cup\left(A^{c} \cap C\right) \cup(B \cap C)=(A \cap B) \cup\left(A^{c} \cap C\right)$

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Problem 443

(3) لأى ثلاث مجموعات C, B , A أثبت صحة القو انين التالية : (i) $A \Delta(B \Delta C)=(A \Delta B) \Delta C$ (ii) $A \cap(B \cup C)^{c}=(A-B) \cup(A-C)$ (iii) $(A \cap B) \cup\left(A^{c} \cap C\right) \cup(B \cap C)=(A \cap B) \cup\left(A^{c} \cap C\right)$

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Problem 444

14. For any integer $a$ , show the following: (a) $\operatorname{gcd}(2 a+1,9 a+4)=1$ . (b) $\operatorname{gcd}(5 a+2,7 a+3)=1$ . (c) If $a$ is odd, then $\operatorname{gcd}(3 a, 3 a+2)=1$ .

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Problem 445

3) لأى ثلاث مجمو عات C, B , A أثبت صحة القو انين التالية : (i) $A \Delta(B \Delta C)=(A \Delta B) \Delta C$ (ii) $A \cap(B \cup C)^{c}=(A-B) \cup(A-C)$ (iii) $(A \cap B) \cup\left(A^{c} \cap C\right) \cup(B \cap C)=(A \cap B) \cup\left(A^{c} \cap C\right)$

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Problem 446

(i) $A \Delta(B \Delta C)=(A \Delta B) \Delta C$ (ii) $A \cap(B \cup C)^{c}=(A-B) \cup(A-C)$

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Problem 447

Given that; Torque $=\mathrm{F} * \mathrm{r} \& \mathrm{a}=\mathrm{r} $ alpha (angular acceleration) Prove T=InertiaAlpha (angular acceleration)

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Problem 448

6) Given that $\sin \frac{\pi}{6}=\frac{1}{2}$ , use an equivalent trigonometric expression to show that $\cos \frac{\pi}{3}=\frac{1}{2}$

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Problem 449

Given: $\overline{A B} \cong \overline{B C}, \angle A \cong \angle C$ and $\overline{B D}$ bisects $\overline{A C}$ . Prove: $\triangle A B D \cong \triangle C B D$ .

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Problem 450

Example 4: Prove $\cos x=\frac{1}{\cos x}-\sin x \tan x$
LS RS

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Problem 451

Create a new conditional statement using the Law of Syllogism from these true statements:
1. If a figure is a rhombus, then it is a parallelogram.
2. If a figure is a parallelogram, then it has two pairs of parallel sides.

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Problem 452

Identify the logic law shown: If $x>12$ , then $x+9>20$ . Given $x=14$ , confirm $x+9>20$ .

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Problem 453

Which coordinates for points $A^{\prime}$ and $B^{\prime}$ show that lines $AB$ and $A^{\prime}B^{\prime}$ are perpendicular?
1. $A^{\prime}:(p, m)$ and $B^{\prime}:(z, w)$
2. $A^{\prime}:(p, m)$ and $B^{\prime}:(z,-w)$
3. $A^{\prime}:(p,-m)$ and $B^{\prime}:(z, w)$
4. $A^{\prime}:(p,-m)$ and $B^{\prime}:(z,-w)$

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Problem 454

Demostrar que la serie $\sum_{n=1}^{\infty}(\operatorname{sen} n x) / n^{2}$ converge para todo $x$ , y que $f(x)$ es continua en $[0, \pi]$ . Luego, probar que $\int_{0}^{\pi} f(x) d x=2 \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{3}}$ .

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Problem 455

Prove the identity $(\sec \theta-\cos \theta)(\csc \theta-\sin \theta)=\frac{\tan \theta}{1+\tan ^{2} \theta}$

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Problem 456

Здесь всюоду $A, B, \ldots$ , это какие-то непустые подмножества на прямой $\mathbb{R}$ . (1) Используя лишь определение компактности доказите, что (a) прямая $\mathbb{R}$ не компактна,

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Problem 457

множество А компактно, где $A=\left\{\frac{2}{3}\right\} \cup\left\{\frac{2 n^{2}-3 n+1}{3 n^{2}-n+10}, n=1,2,3, \ldots\right\}$

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Problem 458

(2) Докажите, что переселение любого семейства компактных подмнозеств компактно.

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Problem 459

Докажите, что множество $[0,1) \cap \mathbb{Q}$ не компактно.

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Problem 460

2. If $x=-3, y=4$ and $z=-5$ , then verify each of the following: (a) $|x+z| \leq|x|+|z|$ (b) ly- (c) $|x y z|=|x| \cdot|y| \cdot|z|$ (d) $\left|\frac{\mathrm{x}}{\mathrm{z}}\right|=$

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Problem 461

63 Une expérience aléatoire est représentée par l'arbre pondéré ci-dessous. - Justifier que $P(S)=0,63$ .

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Problem 462

6. Prove the Trig Identities. a. $\cos \theta \times \tan \theta=\sin \theta$

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Problem 463

Prove that $\frac{\cos ^{2} \phi}{1-\sin \phi}=1+\sin \phi$ , where $\sin \phi \neq 1$ , by expressing $\cos ^{2} \phi$ in terms of $\sin \phi$ .

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Problem 464

Dimostrare che $\forall n \geq 0,11$ divide $9^{n+1}+2^{6 n+1}$

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Problem 465

$\sin x \cos x \tan x=1-\cos ^{2} x$

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Problem 466

A square matrix $A$ is idempotent if $A^{2}=A$ . Let $V$ be the vector space of all $2 \times 2$ matrices with real entries. Let $H$ be the set of all $2 \times 2$ idempotent matrices with real entries. Is $H$ a subspace of the vector space $V$ ?
1. Does $H$ contain the zero vector of $V$ ? choose
2. Is $H$ closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in $H$ whose sum is not in $H$ , using a comma separated list and syntax such as $[[1,2],[3,4]],[[5,6],[7,8]]$ for the answer $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\left[\begin{array}{ll}5 & 6 \\ 7 & 8\end{array}\right]$ . (Hint: to show that $H$ is not closed under addition, it is sufficient to find two idempotent matrices $A$ and $B$ such that $(A+B)^{2} \neq(A+B)$ .) $\square$
3. Is $H$ closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in $\mathbb{R}$ and a matrix in $H$ whose product is not in $H$ , using a comma separated list and syntax such as $2,[[3,4],[5,6]]$ for the answer $2,\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]$ . (Hint: to show that $H$ is not closed under scalar multiplication, it is sufficient to find a real number $r$ and an idempotent matrix $A$ such that $(r A)^{2} \neq(r A)$ .) $\square$
4. Is $H$ a subspace of the vector space $V$ ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose

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Problem 467

Show how to use rigid motions (translations, reflections, rotations) to prove $\triangle ABC \cong \triangle XYZ$ given $\angle A \cong \angle X$ and $\angle B \cong \angle Y$ .

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Problem 468

Check if the point $(1,2)$ is on the line defined by the equation $y=3x-2$ .

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Problem 469

Find the limit: $271 \lim _{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$ . What is the result?

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Problem 470

Verify the product law for differentiation, $(\mathbf{A B})^{\prime}=\mathbf{A}^{\prime} \mathbf{B}+\mathbf{A} \mathbf{B}^{\prime}$ where $\mathbf{A}(t)=\left[\begin{array}{rr}3 t & 3 t-1 \\ t^{3} & \frac{1}{t}\end{array}\right]$ and $\mathbf{B}(t)=\left[\begin{array}{rr}1-t & 1+t \\ 2 t^{2} & 3 t^{3}\end{array}\right]$ .
To calculate $(\mathbf{A B})^{\prime}$ , first calculate $\mathbf{A B}$ . $A B=$ $\square$ Now take the derivative of $A B$ to find $(A B)^{\prime}$ . $(A B)^{\prime}=$ $\square$ To calculate $\mathbf{A}^{\prime} \mathbf{B}+\mathbf{A B ^ { \prime }}$ , first calculate $\mathbf{A}^{\prime}$ . $A^{\prime}=$ $\square$ Now find $\mathbf{A}^{\prime} \mathbf{B}$ . $A^{\prime} B=$ $\square$ Now find $\mathbf{B}^{\prime}$ . $\mathbf{B}^{\prime}=\square$ $\square$ Now calculate $\mathbf{A B}^{\prime}$ . $\mathbf{A B}^{\prime}=\square$ $\square$ Finally, find $\mathbf{A}^{\prime} \mathbf{B}+\mathbf{A B} \mathbf{B}^{\prime}$ . $A^{\prime} B+A B^{\prime}=$ $\square$

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Problem 471

The figure shows the healthy weight region for various heights for people ages 35 and older. If $x$ represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the system of linear inequalities below. Using this information, show that point A is a solution of the system of inequalities that describes healthy weight for this age group. $\left\{\begin{array}{l} 5.2 x-y \geq 178 \\ 4.1 x-y \leq 142 \end{array}\right.$
Healthy Weight Region for Men and Women, Ages 35 and Older
Substitute the x - and y -coordinates of point A for x and y in the first inequality. $\square$ $\geq 178$ (Type an integer or a decimal.) View an example Get more help - Final check

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Problem 472

Subspaces
1. Show that the sets consisting of vectors of the following form are subspaces of $\mathbf{R}^{2}$ by showing that they are closed under addition and under scalar multiplication. (a) $(a, 3 a)$

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Problem 473

2. Show that the sets consisting of vectors of the following form are subspaces of $\mathbf{R}^{3}$ or $\mathbf{R}^{4}$ . (c) $(a, 2 a,-a)$

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Problem 474

3. Determine whether the sets defined by the following vectors are subspaces of $\mathbf{R}^{3}$ . (a) $(a, b, 2 a+3 b)$

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Problem 475

8. Consider the following homogeneous system of linear equations in four variables. For convenience, the general solution is given. Show that the set of solutions forms a subspace of $\mathbf{R}^{4}$ . $\begin{aligned} x_{1}+x_{2}-3 x_{3}+5 x_{4} & =0 \\ x_{2}-x_{3}+3 x_{4} & =0 \\ x_{1}+2 x_{2}-4 x_{3}+8 x_{4} & =0 \end{aligned}$
General solution is $(2 r-2 s, r-3 s, r, s)$ .

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Problem 476

10. (a) Show that the vectors $(1,0,0),(0,1,0),(0,0,1)$ span $\mathbf{R}^{3}$ and that they are also linearly independent. (b) Show that the vectors $(1,0,0),(0,1,0),(0,0,1)$ , $(0,1,1)$ span $\mathbf{R}^{3}$ . Demonstrate that it is not an efficient spanning set by showing that an arbitrary vector in $\mathbf{R}^{3}$ can be expressed in more than one way as a linear combination of these vectors. We can think of $(0,1,1)$ as being a redundant vector. (c) Show that $\{(1,0,0),(0,1,0),(0,0,1),(0,1,1)\}$ is linearly dependent and is thus not a basis for $\mathbf{R}^{2}$ . A basis consists of a set of vectors, all of which are needed.

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Problem 477

Hilfsmittelteil (erlaubte Hilfsmittel: graphikfähiger Taschenrechner, Formelsammlung) Aufgabe 4: (37 Punkte) Die Abbildung zeigt den Würfel ABCDEFGH mit $\mathrm{G}(5|5| 5)$ und $\mathrm{H}(0|5| 5)$ in einem kartesischen Koordinatensystem. Die Punkte I(5|0|1), J(2|5|0), $\mathrm{K}(0|5| 2)$ und $L(1|0| 5)$ liegen jeweils auf einer Kante des Würfels.
8 多 (2P) - $A$ - e) Zeigen Sie, dass das Viereck IJKL ein Trapez ist, in dem zwei Seiten gleich lang sind. Weisen Sie nach, dass die Seite $\overline{\mathrm{L}}$ des Trapezes doppelt so lang ist wie die Seite JK. (7P) f) Berechnen Sie die Größe eines Innenwinkels des Trapezes IJKL. (6P) (4P)
Der Punkt P (4|0|2) liegt auf der Strecke $\overline{\mathrm{IL}}$ . Die Strecke $\overline{\mathrm{JP}}$ steht dabei senkrecht zur Strecke $\overline{\mathrm{IL}}$ . g) Berechnen Sie den Flächeninhalt des Trapezes IJKL. (5P) h) Gegeben ist die Ebene $S: \vec{x}=v \cdot\left(\begin{array}{c}-1 \\ -5 \\ 5\end{array}\right)+w \cdot\left(\begin{array}{c}-5 \\ 5 \\ 1\end{array}\right)$ mit $v, w \in \mathbb{R}$ .
Der Punkt K liegt in einer Ebene T, die parallel zu S ist. Untersuchen Sie, ob auch der Punkt L in T liegt. (5P)

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Problem 478

Exercice 4 : Soit $P \in \mathbb{K}[X]$ non constant. On pose $Q=P(X+1)-P(X)$ . Montrer que $\operatorname{dcg}(Q)=\operatorname{dog}(P)-1$ . Indication : commencer par le cas où $P=X^{n}, n \geq 1$ .

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Problem 479

2. Prove: $1+\tan ^{2} \theta=\frac{1}{\cos ^{2} \theta}$ (4 marks)

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Problem 480

1. $2 \cos x \cos y=\cos (x+y)+\cos (x-y)$

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Problem 481

The functions $g$ and $t$ are defined for $x \in \mathbb{R}$ as follows: $\begin{array}{l} g: x \rightarrow 4x - 5 \\ t: x \rightarrow x^2 - 5x + 1 \end{array}$ (a) Find $t(6)$
(b) Show that $t(g(x)) = 16x^2 - 60x + 51$

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Problem 482

b Arầtati că numărul $\overline{a 5}^{3}-25$ se divide cu 100 . 20 a Efectuati: $1(2 k)^{2}$ ; in $(2 k+1)^{2}$ . b Deduceți că un număr natural care dă restul 3 la impărțirea cu 4 nu este pătrat perfect. c Arătați că numărul 111151 nu este pătrat perfect. $i i n(3 k+2)^{2}$

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Problem 483

$\overleftrightarrow{T W}$ bisects $\angle U W Y$ and $\angle X \cong \angle V$ . Complete the proof that $\triangle T V W \cong \triangle T X W$ . \begin{tabular}{|c|c|c|c|c|} \hline & Statement & & Reason & \\ \hline 1 & $\overleftrightarrow{T W}$ bisects $\angle U W Y$ & & Glven & \\ \hline 2 & $\angle X \cong \angle V$ & & Given & \\ \hline 3 & $\angle X W Y \cong \angle U W V$ & & Vertical Angle Theorem & \\ \hline 4 & $\angle T W Y \cong \angle T W U$ & & Definition of angle bisector & \\ \hline 5 & $m \angle T W X=m \angle T W Y+m \angle X W Y$ & & Additive Property of Angle Measure & \\ \hline 6 & $m \angle T W V=m \angle T W U+m \angle U W V$ & & | & - \\ \hline 7 & $m \angle T W X=m \angle T W U+m \angle U W V$ & + & Substitution & \\ \hline 8 & $m \angle T W V=m \angle T W X$ & & Transitive Property of Equality & \\ \hline 9 & $\overline{T W} \cong \overline{T W}$ & & Reflexive Property of Congruence & \\ \hline 10 & $\triangle T V W \cong \triangle T X W$ & & & . \\ \hline \end{tabular}

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Problem 484

Cheating Challenge in Sims 4 - Youlude Complete the proof that $\triangle P T V \cong \triangle R U S$ . \begin{tabular}{|c|c|c|c|c|} \hline & Statement & \multicolumn{3}{|l|}{Reason} \\ \hline 1 & $\angle P T V \cong \angle R U S$ & Given & & \\ \hline 2 & $\overline{S T} \cong \overline{U V}$ & Given & & \\ \hline 3 & $\angle P \cong \angle R$ & Given & & \\ \hline 4 & $T V=U V+T U$ & & & - \\ \hline 5 & $S U=S T+T U$ & Additive & & \\ \hline 6 & $T V=S T+T U$ & & & \\ \hline 7 & $S U=T V$ & Transitiv & & \\ \hline 8 & $\triangle P T V \cong \triangle R U S$ & & - & \\ \hline \end{tabular}

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Problem 485

Complete the proof that $\triangle E F G \cong \triangle E G F$ . \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & $\angle F \cong \angle G$ & Given \\ 2 & $\overline{F G} \cong \overline{F G}$ & Reflexive Property of Congruence \\ 3 & $\triangle E F G \cong \triangle E G F$ & \\ \hline \end{tabular}

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Problem 486

True or False? The number of electrons equals the atomic mass?

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Problem 487

True or False? When an atom loses electrons, its overall charge is $+$ (positive).

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Problem 488

Is the statement "A trillion is one followed by 12 zeros" true or false? If false, correct it. Options: A. 9 zeros, B. 3 zeros, C. True, D. 6 zeros.

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Problem 489

Find the equivalent equation for $2^{3 x}=10$ . Options include $\log _{2} 10=3 x$ , $\log 2=3 x$ , etc.

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Problem 490

Find a counterexample to show that the sum of two five-digit numbers can be a five-digit number.

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Problem 491

11. True or false? Explain. a. If a counting number is divisible by 9 , it must be divisible by 3 . b. If a counting number is divisible by 3 and 11 , it must be divisible by 33 .

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Problem 492

33. Fill in the blank. The sum of three consecutive counting numbers always has a divisor (other than 1 ) of $\qquad$ . Prove.

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Problem 493

11. True or false? Explain. a. If a counting number is divisible by 6 and 8 , it must be divisible by 48 . b. If a counting number is divisible by 4 , it must be divisible by 8 .

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Problem 494

The number of bacteria $N$ , in a culture is modeled by the exponential growth model, $N(t)=300 e^{0.025 t}$ , where $t$ represents time in hours. The growth rate of the population of this bacterium is represented by $2.5 \%$ per hour. True False

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Problem 495

Les questions sont indépendantes . 1 Montrer que : $\forall z \in \mathbb{C},\left|e^{z}\right| \leq e^{|z|}$ , étudier le cas d'égalité. 2 Si $Z$ est un complexe non nul , montrer que les images des solutions complexes de l'équation $e^{z}=Z$ sont des points alignés.
3 Montrer que si $a, b, c$ sont des complexes de module 1 alors $|a b+b c+c a|=|a+b+c|$ . 4 Soit $(n, m) \in \mathbb{N}^{* 2}$ . Montrer que $\mathbb{U}_{m} \subset \mathbb{U}_{n} \Leftrightarrow m$ divise $n$ 5 Si $n$ est impaire, montrer que $\mathbb{U}_{n}=\mathbb{V}_{n}$ où $\mathbb{V}_{m}=\left\{z^{2} / z \in \mathbb{U}_{n}\right\}$ $6 a, b$ sont des complexes distincts de module 1 et $z \in \mathbb{C}$ . On pose $u=\frac{z+a b \bar{z}-a-b}{b-a}$ . Montrer que $u^{2}$ est un réel négatif . 7 Résoudre le système $\left\{\begin{array}{l}|z-1|=|z-2| \\ \operatorname{Arg}(z+i) \equiv \operatorname{Arg}(z-1)[2 \pi]\end{array}\right.$ . Interpréter la solution géométriquement.

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Problem 496

(II) A lever such as that shown in Fig. 7-20 can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, $F_{\mathrm{O}}$ , to input force, $F_{\mathrm{I}}$ , is related to the lengths $\ell_{\mathrm{I}}$ and $\ell_{\mathrm{O}}$ from the pivot by $F_{\mathrm{O}} / F_{\mathrm{I}}=\ell_{\mathrm{I}} / \ell_{\mathrm{O}}$ . Ignore friction and the mass of the lever, and assume the work output equals the work input. (a)
FIGURE 7-20 A lever. Problem 11. (b)

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Problem 497

a) $\alpha+\frac{4}{\alpha} \geq 4$
ק) $\left(\alpha+\frac{4}{\alpha}\right) \cdot\left(\beta+\frac{4}{\beta}\right) \geq 16$

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Problem 498

8. Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differenchable in $(a, b)$ . Show that if $\lim _{x \rightarrow a} f^{\prime}(x)=A$ , then $f^{\prime}(a)$ exists and equals $A$ . [Hint: Use the definition of $f^{\prime}(a)$ and the Mean Value Theorem.]
9. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x):=2 x^{4}+x^{4} \sin (1 / x)$ for $x \neq 0$ and $f(0):=0$ . Show that $f$ has an absolute minimum at $x=0$ , but that its derivative has both positive and negative values in every neighborhood of 0 .

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Problem 499

Q3: $\operatorname{For} z \in \mathbb{C}$ , show that: (a) $\sin \bar{z}=\overline{\sin z}$ ; (b) $\cosh \bar{z}=\overline{\cosh z}$ .

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Problem 500

The ratios of corresponding sides in the two triangles are equal.
What other information is needed to prove that $\triangle F G E$ $\sim \triangle \mathrm{IJH}$ by the SAS similarity theorem? $\angle F \cong \angle J$ $\angle I \cong \angle F$ $\angle E \cong \angle H$ $\angle G \cong \angle I$

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Prove

Problem 401

10. sin⁡4x−cos⁡4x=1−2cos⁡2x\sin ^{4} x-\cos ^{4} x=1-2 \cos ^{2} xsin4x−cos4x=1−2cos2x

Problem 402

Problem 403

The radian is an alternative unit to the degree for angle measurement. True False

Problem 404

Show that if y=5+xxy=\frac{5+\sqrt{x}}{\sqrt{x}}y=x​5+x​​, then 6dydx+4xd2ydx2=06 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=06dxdy​+4xdx2d2y​=0.

Problem 405

All integers can be expressed as fractions, confirming they are rational numbers.

Problem 406

Prove that for △ABC\triangle ABC△ABC with external angle ∠ACD\angle ACD∠ACD, m∠ACD=m∠B+m∠Am \angle ACD = m \angle B + m \angle Am∠ACD=m∠B+m∠A.

Problem 407

Explain why BD‾≅BD‾\overline{B D} \cong \overline{B D}BD≅BD in the proof of ∠A≃∠C\angle A \simeq \angle C∠A≃∠C in △ABC\triangle A B C△ABC.

Problem 408

Problem 409

Prove that if m∠A+m∠B=m∠B+m∠Cm \angle \mathrm{A}+m \angle \mathrm{B}=m \angle \mathrm{B}+m \angle \mathrm{C}m∠A+m∠B=m∠B+m∠C, then m∠C=m∠Am \angle \mathrm{C}=m \angle \mathrm{A}m∠C=m∠A. Write a paragraph proof.

Problem 410

7. Let E={(x,y,z)∣x+y+z≤1,x≥0,y≥0,z≥0}E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}E={(x,y,z)∣x+y+z≤1,x≥0,y≥0,z≥0}. Show that ∭EezdV=e−52\iiint_{E} e^{z} d V=e-\frac{5}{2}∭E​ezdV=e−25​

Problem 411

Question 10 (1 point) The radian measure of an angle is defined as the length of the arc that subtends the angle divided by the radius of the circle. True False

Problem 412

Problème \# 5 : Si a(a−7b)=−b2a(a-7 b)=-b^{2}a(a−7b)=−b2, Prouve que log⁡(a+b3)=log⁡a+log⁡b2\log \left(\frac{a+b}{3}\right)=\frac{\log a+\log b}{2}log(3a+b​)=2loga+logb​

Problem 413

Problem 414

True or False: 10 and 10z10 z10z are like terms. True False

Problem 415

Problem 416

Submit QuestionQuestion 2 0/10 / 10/1 pt 5 99 DetailsANOVA is a statistical procedure that compares two or more treatment conditions for differences in variance. True FalseQuestion Help: Written Example Post to forum Submit Question

Problem 417

Problem 418

Problem 419

Problem 420

(i) [3pts] x−1y−1xy∈Hx^{-1} y^{-1} x y \in Hx−1y−1xy∈H for all x,y∈Gx, y \in Gx,y∈G then III III is normal subgroup of G

Problem 421

Given: CCC is the midpoint of ADA DAD and CCC is the midpoint of EBE BEB. Prove: ∠A≅∠D\angle A \cong \angle D∠A≅∠D \begin{tabular}{l|l} STATEMENTS & REASONS \\ \hline & \\ & \end{tabular}

Problem 422

In triangles PQRP Q RPQR and UVWU V WUVW, angles QQQ and VVV each have measure 75∘,PQ=975^{\circ}, P Q=975∘,PQ=9, and UV=27U V=27UV=27. Which additional piece of information is sufficient to prove that triangle PQRP Q RPQR is similar to triangle UVWU V WUVW ?

Problem 423

c approach. (sin⁡x+cos⁡x)2=1+2sin⁡xcos⁡x(\sin x+\cos x)^{2}=1+2 \sin x \cos x(sinx+cosx)2=1+2sinxcosx

Problem 424

Problem 425

Problem 426

The result of dividing an integer by a decimal is not always an integer. Show examples like 3÷1.53 \div 1.53÷1.5 and 1.8÷0.31.8 \div 0.31.8÷0.3.

Problem 427

Prove that lim⁡x→8(17x+6)=507\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}limx→8​(71​x+6)=750​ by finding δ\deltaδ in terms of ε\varepsilonε.

Problem 428

Prove that lim⁡x→6(17x−9)=−577\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}limx→6​(71​x−9)=−757​ by finding δ\deltaδ for any ε>0\varepsilon>0ε>0.

Problem 429

Are triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF similar given ∠A=30∘\angle A=30^{\circ}∠A=30∘, ∠D=30∘\angle D=30^{\circ}∠D=30∘, ∠F=38∘\angle F=38^{\circ}∠F=38∘? Justify.

Problem 430

Dilation is a non-isometric transformation. True or False?

Problem 431

If PQ‾≅QR‾\overline{P Q} \cong \overline{Q R}PQ​≅QR​ with PQ=3x−8P Q=3x-8PQ=3x−8, QR=2xQ R=2xQR=2x, and RS=1.5x+4R S=1.5x+4RS=1.5x+4, is PS=24P S=24PS=24 true or false?

Problem 432

Is it true or false that every real number is a noninteger rational number? Choose: True or False.

Problem 433

In a G.P., if the pthp^{\text{th}}pth, qthq^{\text{th}}qth, and rthr^{\text{th}}rth terms are a,b,ca, b, ca,b,c, prove that: aq−r⋅br−p⋅cp−q=1a^{q-r} \cdot b^{r-p} \cdot c^{p-q}=1aq−r⋅br−p⋅cp−q=1.

Problem 434

Prove that if a,b,ca, b, ca,b,c are in A.P. and x,y,zx, y, zx,y,z are in G.P., then xb−c⋅yc−a⋅za−b=1x^{b-c} \cdot y^{c-a} \cdot z^{a-b}=1xb−c⋅yc−a⋅za−b=1.

Problem 435

Prove: △AEC≅△DFB\triangle A E C \cong \triangle D F B△AEC≅△DFB.Step Statement AE‾∥FD‾\overline{A E} \| \overline{F D}AE∥FD 1 BF‾∥EC‾\overline{B F} \| \overline{E C}BF∥EC AC‾≅BD‾\overline{A C} \cong \overline{B D}AC≅BDReasonGiven try Type of Statement

Problem 436

Problem 437

Problem 438

Problem 439

3. Prove the following identities: (a) sinh⁡(z+π)=−sinh⁡(z)\sinh (z+\pi)=-\sinh (z)sinh(z+π)=−sinh(z) (b) cosh⁡(z)=cosh⁡(x)cos⁡(y)+isinh⁡(x)sin⁡(y)\cosh (z)=\cosh (x) \cos (y)+i \sinh (x) \sin (y)cosh(z)=cosh(x)cos(y)+isinh(x)sin(y)

Problem 440

The resistance of the LDR decreases as the incident light increases. True False

Problem 441

Problem 442

Problem 443

Problem 444

14. For any integer aaa, show the following: (a) gcd⁡(2a+1,9a+4)=1\operatorname{gcd}(2 a+1,9 a+4)=1gcd(2a+1,9a+4)=1. (b) gcd⁡(5a+2,7a+3)=1\operatorname{gcd}(5 a+2,7 a+3)=1gcd(5a+2,7a+3)=1. (c) If aaa is odd, then gcd⁡(3a,3a+2)=1\operatorname{gcd}(3 a, 3 a+2)=1gcd(3a,3a+2)=1.

Problem 445

Problem 446

(i) AΔ(BΔC)=(AΔB)ΔCA \Delta(B \Delta C)=(A \Delta B) \Delta CAΔ(BΔC)=(AΔB)ΔC (ii) A∩(B∪C)c=(A−B)∪(A−C)A \cap(B \cup C)^{c}=(A-B) \cup(A-C)A∩(B∪C)c=(A−B)∪(A−C)

Problem 447

Given that; Torque =F∗r&a=r∗=\mathrm{F} * \mathrm{r} \& \mathrm{a}=\mathrm{r} *=F∗r&a=r∗ alpha (angular acceleration) Prove T=Inertia*Alpha (angular acceleration)

Problem 448

10. $\sin ^{4} x-\cos ^{4} x=1-2 \cos ^{2} x$

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

Prove that for $\triangle ABC$ with external angle $\angle ACD$ , $m \angle ACD = m \angle B + m \angle A$ .

Explain why $\overline{B D} \cong \overline{B D}$ in the proof of $\angle A \simeq \angle C$ in $\triangle A B C$ .

Prove that if $m \angle \mathrm{A}+m \angle \mathrm{B}=m \angle \mathrm{B}+m \angle \mathrm{C}$ , then $m \angle \mathrm{C}=m \angle \mathrm{A}$ . Write a paragraph proof.

7. Let $E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}$ . Show that $\iiint_{E} e^{z} d V=e-\frac{5}{2}$

Problème \# 5 : Si $a(a-7 b)=-b^{2}$ , Prouve que $\log \left(\frac{a+b}{3}\right)=\frac{\log a+\log b}{2}$

True or False: 10 and $10 z$ are like terms. True False

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Question 2 $0 / 1$ pt 5 99 Details
ANOVA is a statistical procedure that compares two or more treatment conditions for differences in variance. True False
Question Help: Written Example Post to forum Submit Question

(i) [3pts] $x^{-1} y^{-1} x y \in H$ for all $x, y \in G$ then $I I$ is normal subgroup of G

Given: $C$ is the midpoint of $A D$ and $C$ is the midpoint of $E B$ . Prove: $\angle A \cong \angle D$ \begin{tabular}{l|l} STATEMENTS & REASONS \\ \hline & \\ & \end{tabular}

In triangles $P Q R$ and $U V W$ , angles $Q$ and $V$ each have measure $75^{\circ}, P Q=9$ , and $U V=27$ . Which additional piece of information is sufficient to prove that triangle $P Q R$ is similar to triangle $U V W$ ?

c approach. $(\sin x+\cos x)^{2}=1+2 \sin x \cos x$

The result of dividing an integer by a decimal is not always an integer. Show examples like $3 \div 1.5$ and $1.8 \div 0.3$ .

Prove that $\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}$ by finding $\delta$ in terms of $\varepsilon$ .

Prove that $\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}$ by finding $\delta$ for any $\varepsilon>0$ .

Are triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

If $\overline{P Q} \cong \overline{Q R}$ with $P Q=3x-8$ , $Q R=2x$ , and $R S=1.5x+4$ , is $P S=24$ true or false?

In a G.P., if the $p^{\text{th}}$ , $q^{\text{th}}$ , and $r^{\text{th}}$ terms are $a, b, c$ , prove that: $a^{q-r} \cdot b^{r-p} \cdot c^{p-q}=1$ .

Prove that if $a, b, c$ are in A.P. and $x, y, z$ are in G.P., then $x^{b-c} \cdot y^{c-a} \cdot z^{a-b}=1$ .

Prove: $\triangle A E C \cong \triangle D F B$ .
Step Statement $\overline{A E} \| \overline{F D}$ 1 $\overline{B F} \| \overline{E C}$ $\overline{A C} \cong \overline{B D}$
Reason
Given try Type of Statement

3. Prove the following identities: (a) $\sinh (z+\pi)=-\sinh (z)$ (b) $\cosh (z)=\cosh (x) \cos (y)+i \sinh (x) \sin (y)$

14. For any integer $a$ , show the following: (a) $\operatorname{gcd}(2 a+1,9 a+4)=1$ . (b) $\operatorname{gcd}(5 a+2,7 a+3)=1$ . (c) If $a$ is odd, then $\operatorname{gcd}(3 a, 3 a+2)=1$ .

(i) $A \Delta(B \Delta C)=(A \Delta B) \Delta C$ (ii) $A \cap(B \cup C)^{c}=(A-B) \cup(A-C)$

Given that; Torque $=\mathrm{F} * \mathrm{r} \& \mathrm{a}=\mathrm{r} $ alpha (angular acceleration) Prove T=InertiaAlpha (angular acceleration)

6) Given that $\sin \frac{\pi}{6}=\frac{1}{2}$ , use an equivalent trigonometric expression to show that $\cos \frac{\pi}{3}=\frac{1}{2}$

Given: $\overline{A B} \cong \overline{B C}, \angle A \cong \angle C$ and $\overline{B D}$ bisects $\overline{A C}$ . Prove: $\triangle A B D \cong \triangle C B D$ .

Example 4: Prove $\cos x=\frac{1}{\cos x}-\sin x \tan x$
LS RS

Create a new conditional statement using the Law of Syllogism from these true statements:
1. If a figure is a rhombus, then it is a parallelogram.
2. If a figure is a parallelogram, then it has two pairs of parallel sides.

Identify the logic law shown: If $x>12$ , then $x+9>20$ . Given $x=14$ , confirm $x+9>20$ .

Prove the identity $(\sec \theta-\cos \theta)(\csc \theta-\sin \theta)=\frac{\tan \theta}{1+\tan ^{2} \theta}$

множество А компактно, где $A=\left\{\frac{2}{3}\right\} \cup\left\{\frac{2 n^{2}-3 n+1}{3 n^{2}-n+10}, n=1,2,3, \ldots\right\}$

Докажите, что множество $[0,1) \cap \mathbb{Q}$ не компактно.

63 Une expérience aléatoire est représentée par l'arbre pondéré ci-dessous. - Justifier que $P(S)=0,63$ .

6. Prove the Trig Identities. a. $\cos \theta \times \tan \theta=\sin \theta$

Prove that $\frac{\cos ^{2} \phi}{1-\sin \phi}=1+\sin \phi$ , where $\sin \phi \neq 1$ , by expressing $\cos ^{2} \phi$ in terms of $\sin \phi$ .

Dimostrare che $\forall n \geq 0,11$ divide $9^{n+1}+2^{6 n+1}$

$\sin x \cos x \tan x=1-\cos ^{2} x$

Show how to use rigid motions (translations, reflections, rotations) to prove $\triangle ABC \cong \triangle XYZ$ given $\angle A \cong \angle X$ and $\angle B \cong \angle Y$ .

Check if the point $(1,2)$ is on the line defined by the equation $y=3x-2$ .

Find the limit: $271 \lim _{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$ . What is the result?

Subspaces
1. Show that the sets consisting of vectors of the following form are subspaces of $\mathbf{R}^{2}$ by showing that they are closed under addition and under scalar multiplication. (a) $(a, 3 a)$

2. Show that the sets consisting of vectors of the following form are subspaces of $\mathbf{R}^{3}$ or $\mathbf{R}^{4}$ . (c) $(a, 2 a,-a)$

3. Determine whether the sets defined by the following vectors are subspaces of $\mathbf{R}^{3}$ . (a) $(a, b, 2 a+3 b)$