Prove

Problem 801

Prove that $\frac{2 \cos (x)-1}{2 \cos ^{2}(x)-7 \cos (x)+3}=\frac{1}{\cos x-3}$ is an identity.

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Problem 802

If : $\boldsymbol{x}=\frac{1}{\sqrt{3}-\sqrt{2}} \quad, \quad y=\sqrt{3}-\sqrt{2}$ Prove that $X, y$ are conjugate, then find the value of : $(x+y)^{2}$

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Problem 803

Show that point $A(63,140)$ is a solution for the inequalities $5.2x - y \geq 180$ and $4.2x - y \leq 141$ .

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Problem 804

Verify if point A (63, 140) satisfies these inequalities: 5.2x - y ≥ 180 and 4.2x - y ≤ 141.

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Problem 805

Find the gradient of the output $f(\mathbf{x})$ with respect to $\mathbf{W}^{(2)}$ for the neural network defined as $f(\mathbf{x})=\sigma\left(\sigma\left(\mathbf{x} \cdot \mathbf{W}^{(1)}\right) \cdot \mathbf{W}^{(2)}\right)$ . Optional: Calculate the gradient with respect to $\mathbf{W}^{(1)}$ .

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Problem 806

Prove $P_{n}$ : $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}$ for all positive integers $n$ . What’s the first induction step?

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Problem 807

\text{Show that the series } \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}} \text{ converges.}

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Problem 808

3. Two triangles have two pairs of corresponding sides that are congruent. What else must be true for the triangles to be congruent by the HL Theorem? (A) The included angles must be right angles. B. They have one pair of congruent angles.
C Both riangles must be isosceles.
78. There are right angles adjacent to just one pair of congruen

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Problem 809

(b) $\lim _{x \rightarrow+\infty, x>0} \frac{1}{x}=0$ .

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Problem 810

$\sqrt{\circ} 6$ gokazamb $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$

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Problem 811

( 20 علمة) السؤال الثاني: 1- 1- برهن باسنتغام جدول الصو اب أن 2- برهن أن (20 علامة) السوال الثالث:

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Problem 812

( 20 علامة) $\sum_{r=1}^{n} 2 r-1=n^{2}$
السؤال الثالث:
برهن باستخدام الاستقراء الرياضي أن:

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Problem 813

ـ بين لماذا تكون المصفوفة القابلة للإنعكاس (Invertible) يكون لها محدد غير مفري.

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Problem 814

Q2) Let $f: X \rightarrow Y$ be a function and $B \subseteq Y$ , prove that $f^{-1}(Y-B)=X-f^{-1}(B) .$

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Problem 815

Prove that a positive integer $n$ is even if it can be expressed as $2m$ for some integer $m$ using induction.

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Problem 816

Д13.1. Докажите, что можно так занумеровать вершины связного неориентированного графа на $n$ вершинах числами от 1 до $n$ , что для каждого $1 \leqslant k \leqslant n$ связен подграф, индуцированный множеством вершин с номерами от 1 до $k$ . Множество $S$ вершин графа $G=(V, E)$ индуцирует подграф с множеством вершин $S$ , рёбрами которого являются все рёбра из $E$ с обоими концами в $S$ .

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Problem 817

Д13.4. Пусть $K$ - множество конечных подмножеств натуральных чисел, упорядоченных по включению (если $a, b \in K$ , то $a \leqslant b \Leftrightarrow a \subseteq b$ ); $M$ - множество положительных натуральных чисел, свободных от квадратов (которые не делятся на $p^{2}$ ни для какого простого $p$ ), упорядоченных по отношению делимости (если $a, b \in M$ , то $a \leqslant b \Leftrightarrow b$ делится без остатка на $a$ ). Докажите, что эти два порядка изоморфны.

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Problem 818

fonksiyon için $\delta$ , sadece $\epsilon$ sayısına bağlıdır yani $x=t$ den bağımsız olmaktadır. 3.11 Örnek $f:(0,1] \longrightarrow \mathbb{R}, f(x)=\frac{1}{x}$ fonksiyonunun sürekli ancak düzgün sürekli olmadığım gösteriniz.

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Problem 819

$\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ True False

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Problem 820

Prove the Linear Pairs Theorem: If $\angle ABC$ and $\angle CBD$ are a linear pair, then they are supplementary.

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Problem 821

Prove by induction that for positive integers $n$ :
$1 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}$ .

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Problem 822

यदि $a+b+c=0$ , तब $a^{3}+b^{3}+c^{3}-a b c$ का मान क्या है? (a) $3 a b c$ (b) $2 a b c$ (c) $4 a b c$ (d) $a b c$

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Problem 823

MATh140 Second Firstsem-2024-2025, (161745)(-4) mentie
Question 8 of 18 A matrix that is both sympetric and upper triangular must be a diagonal matrix. True Falle

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Problem 824

1. Prove that $2+3 \sqrt{3}$ is an irrational number. It is given that $\sqrt{ } 3$ is an irrational number.

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Problem 825

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

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Problem 826

b) $\sin 2 x=2 \sin x \cos x$ c) $\tan x=\frac{\sin x}{\cos x}$ d) all of these - The height of the tip of one blade of a wind turbine above the ground, $h(t)$ , can be modelled by $h(t)=18 \cos \left(\pi t+\frac{\pi}{4}\right)+2$ where $t$ is the time passed in seconds. Whic, time interval describes a period when the bl tip is at least 30 m above the ground? a) $5.24 \leq t \leq 7.33$ (c) $1.37 \leq t \leq 2$ . ) $0.42 \leq t \leq 1.08$ d) $0.08 \leq t \leq 1$ .
Iify $\cos \frac{\pi}{5} \cos \frac{\pi}{6}-\sin \frac{\pi}{5} \sin \frac{\pi}{6}$

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Problem 827

Question: Convergence in Probability Let $X_{1}, X_{2}, \ldots$ be a sequence of independent and identically distributed (i.i.d.) random variables, where each $X_{i}$ has the following probability distribution: $P\left(X_{i}=0\right)=\frac{1}{2}, \quad P\left(X_{i}=1\right)=\frac{1}{2} .$ 1
Define the sample mean $\bar{X}_{n}$ as: $\bar{X}_{n}=\frac{1}{n} \sum_{i=1}^{n} X_{i} .$
We want to analyze the behavior of $\bar{X}_{n}$ as $n \rightarrow \infty$ . (a) Show that $E\left[X_{i}\right]=\frac{1}{2}$ and $\operatorname{Var}\left(X_{i}\right)=\frac{1}{4}$ . (b) Using the weak law of large numbers (WLLN), show that $\bar{X}_{n} \xrightarrow{P} \frac{1}{2}$ as $n \rightarrow \infty$ . That is, prove that $\bar{X}_{n}$ converges to $\frac{1}{2}$ in probability. (c) For a sequence $Y_{1}, Y_{2}, \ldots$ of independent random variables where $P\left(Y_{i}=\right.$ 1) $=1-\frac{1}{i}$ and $P\left(Y_{i}=0\right)=\frac{1}{i}$ , determine whether $Y_{n}$ converges in probability to 1 as $n \rightarrow \infty$ . Justify your answer using the definition of convergence in probability.

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Problem 828

CCA2 > Chapter 2 > Lesson 2.1.2 > Problem 2-24
Consider the equations $y=3(x-1)^{2}-5$ and $y=3 x^{2}-6 x-2$ . a. Verify that they are equivalent by creating a table or graph for each equation. $\square$ $\checkmark$ Hint (a): Here are a couple of points on the table. Make sure you get these points and continue both of your tables for at leas \begin{tabular}{c|c|} \hline $x$ & $y$ \\ \hline-2 & 22 \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & -5 \\ \hline 2 & \\ \hline \end{tabular}

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Problem 829

$\sin 2x + \sin 2y = 2 \sin (x+y) \cos (x-y)$ Prove this using only the double angle formula, without using the sum-to-product identities.

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Problem 830

f) $\cos ^{4} \theta-\sin ^{4} \theta=1-2 \sin ^{2} \theta$

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Problem 831

Determine if triangle $B C D$ and triangle $E F G$ are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)
Answer
The triangles $\square$ similar.

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Problem 832

Given: $A B C D$ is a parallelogram. Diagonals $\overline{\mathrm{AC}}, \overline{\mathrm{BD}}$ intersect at E . Prove: $\overline{\mathrm{AE}} \cong \overline{\mathrm{CE}}$ and $\overline{\mathrm{BE}} \cong \overline{\mathrm{DE}}$ Statements
1. ABCD is a parallelogram
2. $\overline{\mathrm{AB}} \| \overline{\mathrm{CD}}$
3. $\angle \mathrm{BAE}$ and $\angle \mathrm{DCE}$ are alt. interior angles
Reasons
1. given
2. def. of parallelogram
3. def. of alt. interior angles
CorrectIAssemble the next statement. Intro

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Problem 833

Is it true or false that every irrational number is a natural number? Choose: True or False.

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Problem 834

Is the statement true or false? Justify: Does the augmented matrix $\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right]$ imply $b$ is in Span $\{a_{1}, a_{2}, a_{3}\}$ ?

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Problem 835

Is the statement true or false? The linear system from $\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right]$ has a solution if $b$ is in Span $\{a_{1}, a_{2}, a_{3}\}$ . Choose A, B, C, or D.

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Problem 836

Prove or disprove that there are no positive integers $a, b, c$ and integer $n>2$ such that $a^{n}+b^{n}=c^{n}$ .

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Problem 837

50. Given: $\overline{A C} \cong \overline{E C}, \overline{B C} \cong \overline{D C}$
Prove: $\triangle A B C \cong \triangle E D C$ \begin{tabular}{l|l} \multicolumn{1}{c|}{ Statements } & \multicolumn{1}{c}{ Reasons } \\ \hline 1. $\overline{A C} \cong \overline{E C}$ & 1. Given \\
2. $\overline{B C} \cong \overline{D C}$ & 2. GNen \\
3. $\angle A C B \cong \angle E C D$ & 3. \\ $4 . \triangle A B C \cong \triangle E D C$ & 4. \end{tabular}

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Problem 838

$m \angle x=0 u$
Prove: $m \angle 7=120^{\circ}$
Statements Reasons $m|\mid n$ $m \angle 2=60^{\circ}$ $\angle 8 \approx \angle 2$ $m \angle 8=m \angle 2$ 2 $m(8-60)$ Angle Congruence Postulate Substitution Property of Equality $m \angle 7+m \angle 8=180^{\circ}$ Linear Pair Postulate

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Problem 839

se the given information to prove that $\triangle P Q R \cong \triangle T S R$ .
Given: $\overline{Q R} \cong \overline{S R}$ Send To Proof $\angle P Q R \cong \angle T S R$ Send To Proof
Prove: $\triangle P Q R \cong \triangle T S R$ Send To Proof Statement Reason
1 $\square$ Reason?
Validate

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Problem 840

$13 \frac{d}{d x}[\csc x]=-\cot x \csc x$

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Problem 841

Name: $\qquad$ Section 2 - B. 2 4) [IC] (/2) Zhen claims that in exponential functions, they act like a parabola, so a vertical stretch by a factor of $a$ will result in the same graph as a horizontal compression by $a$ . Use the functions $f(x)=3(2)^{x}$ and $g(x)=(2)^{3 x}$ to either back up Zhen's claim or reject his claim. Use of the grid provided is optional.

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Problem 842

3. Provet $\frac{\sin 4 \pm-\sin 2 \pi}{\sin 2 x}=\frac{\cos 3 x}{\cos x}$

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Problem 843

Question 3 8 pts
Given: $\angle T V W \cong \angle V W X$ , and $\angle V W T \cong \angle X V W$ Prove: $\triangle T V W \cong \triangle X W V$
1. $\angle T V W \cong \angle V W X$
1. Given
2. $\angle V W T \cong \angle X V W$
2. Given
3. [Select]
3. [Select]
4. [Select]
4. [Select]

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Problem 844

Prove that the repeating decimal $0.999\ldots$ is equal to 1.

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Problem 845

Draw Venn diagrams for sets $A$ and $B$ to show that $A = (A \cap B) \cup (A \cap \bar{B})$ .

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Problem 846

Complete the proof that $\triangle F G H \cong$ $\triangle F H G$ .

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Problem 847

III. Given: $\overline{B E} \perp \overline{E D}$ and $\overline{E B} \perp \overline{B A}$ , and $C$ is the midpoint of $\overline{B E}$ Prove: $\triangle A B C \cong \triangle D E C$ \begin{tabular}{|l|l|l|} \hline & & \\ \hline Corresponding, Congruent Parts: & Corresponding, Congruent Parts: & Corresponding, Congruent Parts: \\ \hline Explanation: & & \\ \hline \end{tabular}

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Problem 848

false: (5 points) form of $y_{p}$ for $y^{\prime \prime \prime}-3 y^{\prime}+2 y=x e^{x}$ is $\left(\mathrm{Ax}^{3}+B x^{2}\right) e^{x}$ the roots of the indicial equation are $-0.3,1.7$ then the D.E. has two nearly independent solutions $W(f, g, h)=\sin t$ then the functions $f, g, h$ are linearly dependent er bound for the radius of convergence for the series 1 of $\left(1-\mathrm{x}^{3}\right) y^{\prime \prime}+4 x y^{\prime}+y=0 \quad, \mathrm{x}_{0}=3 \quad$ is 2 $=1$ is a R.S.P for $(x-1)^{2} y^{\prime \prime}+3 y^{\prime}+(x-1) y=0$

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Problem 849

ue or false: (5 points) - The form of $y_{p}$ for $y^{\prime \prime \prime}-3 y^{\prime}+2 y=x e^{x}$ is $\left(\mathrm{Ax}^{3}+B x^{2}\right) e^{x}$ - If the roots of the indicial equation are $-0.3,1.7$ then the D.E. has two linearly independent solutions - If $W(f, g, h)=\sin t$ then the functions $f, g, h$ are linearly dependent lower bound for the radius of convergence for the series lution of $\left(1-\mathrm{x}^{3}\right) y^{\prime \prime}+4 x y^{\prime}+y=0 \quad, \mathrm{x}_{0}=3 \quad$ is 2 - $\mathrm{x}=1$ is a R.S.P for $(x-1)^{2} y^{\prime \prime}+3 y^{\prime}+(x-1) y=0$

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Problem 850

Q1) True or false: (5 points)
1- The form of $y_{p}$ for $y^{\prime \prime \prime}-3 y^{\prime}+2 y=x e^{x}$ is $\left(\mathrm{Ax}^{3}+B x^{2}\right) e^{x}$ 2- If the roots of the indicial equation are $-0.3,1.7$ then the D.E. has two linearly independent solutions
3- If $\mathrm{W}(\mathrm{f}, \mathrm{g}, \mathrm{h})=\sin \mathrm{t}$ then the functions $\mathrm{f}, \mathrm{g}, \mathrm{h}$ are linearly dependent 4- The lower bound for the radius of convergence for the series solution of $\left(1-\mathrm{x}^{3}\right) y^{\prime \prime}+4 x y^{\prime}+y=0 \quad, \mathrm{x}_{0}=3 \quad$ is 2
5- $\quad \mathrm{x}=1$ is a R.S.P for $(x-1)^{2} y^{\prime \prime}+3 y^{\prime}+(x-1) y=0$

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Problem 851

II. Given the triangles below, determine what additional piece of information is needed to prove $\triangle A B C \cong \triangle C E D$ by AAS? state and mark on diagram

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Problem 852

Which of the following explains how $\triangle A E B$ could be proven similar to $\triangle D E C$ using the $A A$ similarity postulate? $\angle A E B \cong \angle C E D$ because vertical angles are congruent; reflect $\triangle C E D$ across segment $F G$ , then translate point $D$ to point $A$ to confirm $\angle E A B \cong \angle E D C$ . $\angle A E B \approx \angle C E D$ because vertical angles are congruent; rotate $\triangle C E D 180^{\circ}$ around point $E$ , then dilate $\triangle C E D$ to confirm $\overline{E B} \approx \overline{E C}$ . $\angle A E B \cong \angle D E C$ because vertical angles are congruent; rotate $\triangle C E D 180^{\circ}$ around point $E$ , then translate point $D$ to point $A$ to confirm $\angle E A B=\angle E D C$ . $\angle A E B \cong \angle D E C$ because vertical angles are congruent; reflect $\triangle C E D$ across segment $F G$ , then dilate $\triangle C E D$ to confirm $\overline{E B} \approx \overline{E D}$

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Problem 853

Q1) True or false: (5 points) 1- The form of $y_{p}$ for $y^{\prime \prime \prime}-3 y^{\prime}+2 y=x e^{\prime}$ is $\left(\mathrm{Ax}^{3}+B x^{2}\right) e^{x}$

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Problem 854

Let $(Y_1, Y_2)$ have joint density $f_{Y_1, Y_2}(y_1, y_2)$ . Define $U_1 = Y_1 + Y_2$ , $U_2 = Y_2$ .
a) Show $f_{U_1, U_2}(u_1, u_2) = f_{Y_1, Y_2}(u_1 - u_2, u_2)$ .
b) Find $f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1, Y_2}(u_1 - u_2, u_2) du_2$ .
c) If $Y_1$ and $Y_2$ are independent, show $f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1}(u_1 - u_2) f_{Y_2}(u_2) du_2$ .

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Problem 855

Prove the trigonometric equation: $\theta \sin \left(\frac{1}{n}\right) \frac{x}{\sqrt{1-x^{2}}} = \tan^{-1}(x)$ .

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Problem 856

Show that multiplication is commutative by proving $q \cdot c = c \cdot q$ for variables 'q' and 'c'.

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Problem 857

Verify that the equation is an identity. $\frac{\csc \alpha}{\cot \alpha}=\sec \alpha$
To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step. $\frac{\csc \alpha}{\cot \alpha}=\frac{\square}{\square}$
What transforniation is made in the numerator? $\square$ What transformation is made in the denominator? $\square$

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Problem 858

Détermine if the two functions are inverses of each othe $\begin{array}{l} f(x)=\sqrt[3]{3-x} \\ g(x)=x^{3}-3 \end{array}$ No because $f(g(x))=\sqrt[3]{-x^{3}+6}$ and $g(f(x))=-x$ No because $f(g(x))=-x$ and $g(f(x))=-x$ Yes because $f(g(x))=-x$ and $g(f(x))=-x$ Yes because $f(g(x))=\sqrt[3]{-x^{3}+6}$ and $g(f(x))=-x$

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Problem 859

Find a matrix for the linear transformation $T(x_{1}, x_{2}, x_{3})=\left(x_{1}-8 x_{2}+5 x_{3}, x_{2}-7 x_{3}\right)$ . $A=$

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Problem 860

Find a matrix for the linear transformation $T(x_{1}, x_{2}, x_{3})=\left(x_{1}-4 x_{2}+7 x_{3}, x_{2}-6 x_{3}\right)$ . $A=\square$

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Problem 861

Exercice $\mathrm{n}^{\circ} 9$ : Soit $a \in] 0,1$ . Montrer à l'aide des accroissements finis que pour tout $n \in \mathbb{N}^{*}$ , $\frac{a}{(n+1)^{1-a}} \leqslant(n+1)^{a}-n^{a} \leqslant \frac{a}{n^{1-a}} .$

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Problem 862

Question: Sufficient Estimator for Poisson Distribution
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a ${ }^{* *}$ Poisson distribution** with an unknown parameter $\lambda$ , where $\lambda>0$ . The probability mass function (PMF) of each $X_{i}$ is given by: $f(x ; \lambda)=\frac{\lambda^{x} e^{-\lambda}}{x!}, \quad x=0,1,2, \ldots$ (a) Write the likelihood function $L(\lambda)$ based on the random sample $X_{1}, X_{2}, \ldots, X_{n}$ . (b) Use the ${ }^{* *}$ Factorization Theorem** to show that the statistic $T=\sum_{i=1}^{n} X_{i}$ is a ${ }^{* }$ sufficient statistic ${ }^{ }$ for $\lambda$ . (c) Find the ${ }^{ }$ maximum likelihood estimator (MLE) ${ }^{ }$ of $\lambda$ . (d) Show that the statistic $T=\sum_{i=1}^{n} X_{i}$ is a ${ }^{ *}$ complete and sufficient** statistic for $\lambda$ . Justify your answer.

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Problem 863

Дана пирамида EABCD. Её основание - параллелограмм, диагонали которого пересекаются в точке $O$ . Определи, справедливо ли равенство: $1.2 \overrightarrow{O D}-\overrightarrow{A D}+\overrightarrow{A C}=\overrightarrow{B E}$ $\square$
2. $\overrightarrow{O D}+\overrightarrow{O E}-\overrightarrow{C E}+0,5 \overrightarrow{C A}=\overrightarrow{O B}$ . $\square$
3. $\overrightarrow{A E}-\overrightarrow{O E}+0,5 \overrightarrow{B D}=\overrightarrow{D A}$ . $\square$

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Problem 864

Soit $\left\{U_{n}\right\}$ et $\left(V_{n}\right)$ deux suites définies par: $U_{n}=\frac{2^{n}+4 n+3}{2}$ et $V_{n}=\frac{2^{n}-4 n+3}{2}$ On pose $T_{1}=U_{n}+V_{n}$ et $T_{2}=U_{n}-V_{n}$ 1) Montrer que $T_{1}$ est géométrique et que $T_{2}$ est arithmétique ? 2) En déduire $S_{1}$ et $S_{2}$ en fonction de $n$ tels que: $S_{1}=\sum_{K=0}^{n} \boldsymbol{U}_{K}$ et $S_{2}=\sum_{K=0}^{n} V_{K}$

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Problem 865

Q1: A) Prove that $\left[\frac{5}{3-w}-\frac{5}{3-w^{2}}\right]^{2}=-\frac{75}{169}$

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Problem 866

B) If $z=\cos 2 \theta+i \sin 2 \theta$ prove that $\frac{Z+1}{Z-1}=-i \cot \theta$

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1 ...6 7 8 9

Prove

Problem 801

Prove that 2cos⁡(x)−12cos⁡2(x)−7cos⁡(x)+3=1cos⁡x−3\frac{2 \cos (x)-1}{2 \cos ^{2}(x)-7 \cos (x)+3}=\frac{1}{\cos x-3}2cos2(x)−7cos(x)+32cos(x)−1​=cosx−31​ is an identity.

Problem 802

If : x=13−2,y=3−2\boldsymbol{x}=\frac{1}{\sqrt{3}-\sqrt{2}} \quad, \quad y=\sqrt{3}-\sqrt{2}x=3​−2​1​,y=3​−2​ Prove that X,yX, yX,y are conjugate, then find the value of : (x+y)2(x+y)^{2}(x+y)2

Problem 803

Show that point A(63,140)A(63,140)A(63,140) is a solution for the inequalities 5.2x−y≥1805.2x - y \geq 1805.2x−y≥180 and 4.2x−y≤1414.2x - y \leq 1414.2x−y≤141.

Problem 804

Verify if point A (63, 140) satisfies these inequalities: 5.2x - y ≥ 180 and 4.2x - y ≤ 141.

Problem 805

Problem 806

Prove PnP_{n}Pn​: ∑k=1n1k(k+1)=nn+1\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}∑k=1n​k(k+1)1​=n+1n​ for all positive integers nnn. What’s the first induction step?

Problem 807

\text{Show that the series } \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}} \text{ converges.}

Problem 808

Problem 809

(b) lim⁡x→+∞,x>01x=0\lim _{x \rightarrow+\infty, x>0} \frac{1}{x}=0limx→+∞,x>0​x1​=0.

Problem 810

∘6\sqrt{\circ} 6∘​6 gokazamb lim⁡x→∞(1+1x)x=e\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=ex→∞lim​(1+x1​)x=e

Problem 811

( 20 علمة) السؤال الثاني: 1- 1- برهن باسنتغام جدول الصو اب أن 2- برهن أن (20 علامة) السوال الثالث:

Problem 812

( 20 علامة) ∑r=1n2r−1=n2\sum_{r=1}^{n} 2 r-1=n^{2}r=1∑n​2r−1=n2السؤال الثالث:برهن باستخدام الاستقراء الرياضي أن:

Problem 813

ـ بين لماذا تكون المصفوفة القابلة للإنعكاس (Invertible) يكون لها محدد غير مفري.

Problem 814

Q2) Let f:X→Yf: X \rightarrow Yf:X→Y be a function and B⊆YB \subseteq YB⊆Y, prove that f−1(Y−B)=X−f−1(B).f^{-1}(Y-B)=X-f^{-1}(B) .f−1(Y−B)=X−f−1(B).

Problem 815

Prove that a positive integer nnn is even if it can be expressed as 2m2m2m for some integer mmm using induction.

Problem 816

Problem 817

Problem 818

fonksiyon için δ\deltaδ, sadece ϵ\epsilonϵ sayısına bağlıdır yani x=tx=tx=t den bağımsız olmaktadır. 3.11 Örnek f:(0,1]⟶R,f(x)=1xf:(0,1] \longrightarrow \mathbb{R}, f(x)=\frac{1}{x}f:(0,1]⟶R,f(x)=x1​ fonksiyonunun sürekli ancak düzgün sürekli olmadığım gösteriniz.

Problem 819

log⁡(xy)=log⁡xlog⁡y\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}log(yx​)=logylogx​ True False

Problem 820

Prove the Linear Pairs Theorem: If ∠ABC\angle ABC∠ABC and ∠CBD\angle CBD∠CBD are a linear pair, then they are supplementary.

Problem 821

Prove by induction that for positive integers nnn:1+2(12)+3(12)2+…+n(12)n−1=4−n+22n−11 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}1+2(21​)+3(21​)2+…+n(21​)n−1=4−2n−1n+2​.

Problem 822

यदि a+b+c=0a+b+c=0a+b+c=0, तब a3+b3+c3−abca^{3}+b^{3}+c^{3}-a b ca3+b3+c3−abc का मान क्या है? (a) 3abc3 a b c3abc (b) 2abc2 a b c2abc (c) 4abc4 a b c4abc (d) abca b cabc

Problem 823

MATh140 Second Firstsem-2024-2025, (161745)(-4) mentieQuestion 8 of 18 A matrix that is both sympetric and upper triangular must be a diagonal matrix. True Falle

Problem 824

1. Prove that 2+332+3 \sqrt{3}2+33​ is an irrational number. It is given that 3\sqrt{ } 3​3 is an irrational number.

Problem 825

1−2sin⁡2xsin⁡x+cos⁡x+2sin⁡x2cos⁡x2=cos⁡x\frac{1-2 \sin ^{2} x}{\sin x+\cos x}+2 \sin \frac{x}{2} \cos \frac{x}{2}=\cos xsinx+cosx1−2sin2x​+2sin2x​cos2x​=cosx

Problem 826

Problem 827

Problem 828

Problem 829

sin⁡2x+sin⁡2y=2sin⁡(x+y)cos⁡(x−y)\sin 2x + \sin 2y = 2 \sin (x+y) \cos (x-y)sin2x+sin2y=2sin(x+y)cos(x−y) Prove this using only the double angle formula, without using the sum-to-product identities.

Problem 830

f) cos⁡4θ−sin⁡4θ=1−2sin⁡2θ\cos ^{4} \theta-\sin ^{4} \theta=1-2 \sin ^{2} \thetacos4θ−sin4θ=1−2sin2θ

Problem 831

Determine if triangle BCDB C DBCD and triangle EFGE F GEFG are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)AnswerThe triangles □\square□ similar.

Problem 832

Problem 833

Is it true or false that every irrational number is a natural number? Choose: True or False.

Problem 834

Is the statement true or false? Justify: Does the augmented matrix [a1a2a3b]\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right][a1​​a2​​a3​​b​] imply bbb is in Span {a1,a2,a3}\{a_{1}, a_{2}, a_{3}\}{a1​,a2​,a3​}?

Problem 835

Is the statement true or false? The linear system from [a1a2a3b]\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right][a1​​a2​​a3​​b​] has a solution if bbb is in Span {a1,a2,a3}\{a_{1}, a_{2}, a_{3}\}{a1​,a2​,a3​}. Choose A, B, C, or D.

Problem 836

Prove or disprove that there are no positive integers a,b,ca, b, ca,b,c and integer n>2n>2n>2 such that an+bn=cna^{n}+b^{n}=c^{n}an+bn=cn.

Problem 837

Problem 838

Problem 839

Problem 840

13ddx[csc⁡x]=−cot⁡xcsc⁡x13 \frac{d}{d x}[\csc x]=-\cot x \csc x13dxd​[cscx]=−cotxcscx

Problem 841

Problem 842

3. Provet sin⁡4±−sin⁡2πsin⁡2x=cos⁡3xcos⁡x\frac{\sin 4 \pm-\sin 2 \pi}{\sin 2 x}=\frac{\cos 3 x}{\cos x}sin2xsin4±−sin2π​=cosxcos3x​

Problem 843

Problem 844

Prove that the repeating decimal 0.999…0.999\ldots0.999… is equal to 1.

Problem 845

Draw Venn diagrams for sets AAA and BBB to show that A=(A∩B)∪(A∩Bˉ)A = (A \cap B) \cup (A \cap \bar{B})A=(A∩B)∪(A∩Bˉ).

Problem 846

Complete the proof that △FGH≅\triangle F G H \cong△FGH≅ △FHG\triangle F H G△FHG.

Prove that $\frac{2 \cos (x)-1}{2 \cos ^{2}(x)-7 \cos (x)+3}=\frac{1}{\cos x-3}$ is an identity.

If : $\boldsymbol{x}=\frac{1}{\sqrt{3}-\sqrt{2}} \quad, \quad y=\sqrt{3}-\sqrt{2}$ Prove that $X, y$ are conjugate, then find the value of : $(x+y)^{2}$

Show that point $A(63,140)$ is a solution for the inequalities $5.2x - y \geq 180$ and $4.2x - y \leq 141$ .

Prove $P_{n}$ : $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}$ for all positive integers $n$ . What’s the first induction step?

(b) $\lim _{x \rightarrow+\infty, x>0} \frac{1}{x}=0$ .

$\sqrt{\circ} 6$ gokazamb $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$

( 20 علامة) $\sum_{r=1}^{n} 2 r-1=n^{2}$
السؤال الثالث:
برهن باستخدام الاستقراء الرياضي أن:

Q2) Let $f: X \rightarrow Y$ be a function and $B \subseteq Y$ , prove that $f^{-1}(Y-B)=X-f^{-1}(B) .$

Prove that a positive integer $n$ is even if it can be expressed as $2m$ for some integer $m$ using induction.

fonksiyon için $\delta$ , sadece $\epsilon$ sayısına bağlıdır yani $x=t$ den bağımsız olmaktadır. 3.11 Örnek $f:(0,1] \longrightarrow \mathbb{R}, f(x)=\frac{1}{x}$ fonksiyonunun sürekli ancak düzgün sürekli olmadığım gösteriniz.

$\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ True False

Prove the Linear Pairs Theorem: If $\angle ABC$ and $\angle CBD$ are a linear pair, then they are supplementary.

Prove by induction that for positive integers $n$ :
$1 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}$ .

यदि $a+b+c=0$ , तब $a^{3}+b^{3}+c^{3}-a b c$ का मान क्या है? (a) $3 a b c$ (b) $2 a b c$ (c) $4 a b c$ (d) $a b c$

MATh140 Second Firstsem-2024-2025, (161745)(-4) mentie
Question 8 of 18 A matrix that is both sympetric and upper triangular must be a diagonal matrix. True Falle

1. Prove that $2+3 \sqrt{3}$ is an irrational number. It is given that $\sqrt{ } 3$ is an irrational number.

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

$\sin 2x + \sin 2y = 2 \sin (x+y) \cos (x-y)$ Prove this using only the double angle formula, without using the sum-to-product identities.

f) $\cos ^{4} \theta-\sin ^{4} \theta=1-2 \sin ^{2} \theta$

Determine if triangle $B C D$ and triangle $E F G$ are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)
Answer
The triangles $\square$ similar.

Is the statement true or false? Justify: Does the augmented matrix $\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right]$ imply $b$ is in Span $\{a_{1}, a_{2}, a_{3}\}$ ?

Is the statement true or false? The linear system from $\left[\begin{array}{llll}a_{1} & a_{2} & a_{3} & b\end{array}\right]$ has a solution if $b$ is in Span $\{a_{1}, a_{2}, a_{3}\}$ . Choose A, B, C, or D.

Prove or disprove that there are no positive integers $a, b, c$ and integer $n>2$ such that $a^{n}+b^{n}=c^{n}$ .

$13 \frac{d}{d x}[\csc x]=-\cot x \csc x$

3. Provet $\frac{\sin 4 \pm-\sin 2 \pi}{\sin 2 x}=\frac{\cos 3 x}{\cos x}$

Prove that the repeating decimal $0.999\ldots$ is equal to 1.

Draw Venn diagrams for sets $A$ and $B$ to show that $A = (A \cap B) \cup (A \cap \bar{B})$ .

Complete the proof that $\triangle F G H \cong$ $\triangle F H G$ .

II. Given the triangles below, determine what additional piece of information is needed to prove $\triangle A B C \cong \triangle C E D$ by AAS? state and mark on diagram

Q1) True or false: (5 points) 1- The form of $y_{p}$ for $y^{\prime \prime \prime}-3 y^{\prime}+2 y=x e^{\prime}$ is $\left(\mathrm{Ax}^{3}+B x^{2}\right) e^{x}$

Prove the trigonometric equation: $\theta \sin \left(\frac{1}{n}\right) \frac{x}{\sqrt{1-x^{2}}} = \tan^{-1}(x)$ .

Show that multiplication is commutative by proving $q \cdot c = c \cdot q$ for variables 'q' and 'c'.

Find a matrix for the linear transformation $T(x_{1}, x_{2}, x_{3})=\left(x_{1}-8 x_{2}+5 x_{3}, x_{2}-7 x_{3}\right)$ . $A=$

Find a matrix for the linear transformation $T(x_{1}, x_{2}, x_{3})=\left(x_{1}-4 x_{2}+7 x_{3}, x_{2}-6 x_{3}\right)$ . $A=\square$

Q1: A) Prove that $\left[\frac{5}{3-w}-\frac{5}{3-w^{2}}\right]^{2}=-\frac{75}{169}$

B) If $z=\cos 2 \theta+i \sin 2 \theta$ prove that $\frac{Z+1}{Z-1}=-i \cot \theta$