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Meadowview is a large tract of undeveloped land. Black, the owner of Meadowview, prepared a development plan creating 200 house lots in Meadowview with the necessary streets and public areas. The plan was fully approved by all necessary governmental agencies and duly recorded. However, construction of the streets, util... | D |
Purvis purchased a used car from Daley, a used-car dealer. Knowing them to be false, Daley made the following statements to Purvis prior to the sale: Statement 1. This car has never been involved in an accident. Statement 2. This car gets 25 miles to the gallon on the open highway. Statement 3. This is as smooth-riding... | C |
Mrs. Ritter, a widow, recently purchased a new uncrated electric range for her kitchen from Local Retailer. The range has a wide oven with a large oven door. The crate in which Stove Company, the manufacturer, shipped the range carried a warning label that the stove would tip over with a weight of 25 pounds or more on ... | B |
Parents purchased a new mobile home from Seller. The mobile home was manufactured by Mobilco and had a ventilating system designed by Mobilco with both a heating unit and an air conditioner. Mobilco installed a furnace manufactured by Heatco and an air conditioning unit manufactured by Coolco. Each was controlled by an... | A |
When Esther, Gray's 21-year-old daughter, finished college, Gray handed her a signed memorandum stating that if she would go to law school for three academic years, he would pay her tuition, room, and board, and would "give her a $ 1,000$ bonus" for each "A" she got in law school. Esther's uncle, Miller, who was presen... | C |
Jackson and Brannick planned to break into a federal government office to steal food stamps. Jackson telephoned Crowley one night and asked whether Crowley wanted to buy some "hot" food stamps. Crowley, who understood that "hot" meant stolen, said, "Sure, bring them right over." Jackson and Brannick then successfully e... | C |
Park brought an action against Dan for injuries received in an automobile accident, alleging negligence in that Dan was speeding and inattentive. Park calls White to testify that Dan had a reputation in the community of being a reckless driver and was known as "Daredevil Dan." White's testimony is:
Options:
A. Admissi... | D |
Pemberton and three passengers, Able, Baker, and Charley, were injured when their car was struck by a truck owned by Mammoth Corporation and driven by Edwards. Helper, also a Mammoth employee, was riding in the truck. The issues in Pemberton $v$. Mammoth include the negligence of Edwards in driving too fast and in fail... | A |
Lord leased a warehouse building and the lot on which it stood to Taylor for a term of 10 years. The lease contained a clause prohibiting Taylor from subletting his interest. Can Taylor assign his interest under the lease?
Options:
A. Yes, because restraints on alienation of land are strictly construed.
B. Yes, becau... | A |
Testator devised his farm "to my son, Selden, for life, then to Selden's children and their heirs and assigns." Selden, a widower, had two unmarried adult children. In an appropriate action to construe the will, the court will determine that the remainder to the children is:
Options:
A. Indefeasibly vested.
B. Conting... | C |
Ohner holds title in fee simple to a tract of 1,500 acres. He desires to develop the entire tract as a golf course, country club, and residential subdivision. He contemplates forming a corporation to own and to operate the golf course and country club; the stock in the corporation will be distributed to the owners of l... | B |
Paulsen was eating in a restaurant when he began to choke on a piece of food that had lodged in his throat. Dow, a physician who was sitting at a nearby table, did not wish to become involved and did not render any assistance, although prompt medical attention would have been effective in removing the obstruction from ... | C |
Innes worked as a secretary in an office in a building occupied partly by her employer and partly by Glass, a retail store. The two areas were separated by walls and were in no way connected, except that the air conditioning unit served both areas and there was a common return-air duct. Glass began remodeling, and its ... | D |
When Denton heard that his neighbor, Prout, intended to sell his home to a minority purchaser, Denton told Prout that Prout and his wife and children would meet with "accidents" if he did so. Prout then called the prospective purchaser and told him that he was taking the house off the market. If Prout asserts a claim a... | A |
Dave is a six-year-old boy who has a welldeserved reputation for bullying younger and smaller children. His parents have encouraged him to be aggressive and tough. Dave, for no reason, knocked down, kicked, and severely injured Pete, a four-year-old. A claim has been asserted by Pete's parents for their medical and hos... | B |
In a suit attacking the validity of a deed executed 15 years ago, Plaintiff alleges mental incompetency of Joe, the grantor, and offers in evidence a properly authenticated affidavit of Harry, Joe's brother. The affidavit, which was executed shortly after the deed, stated that Harry had observed Joe closely over a peri... | B |
Oscar, the owner in fee simple, laid out a subdivision of 325 lots on 150 acres of land. He obtained governmental approval (as required by applicable ordinances) and, between 1968 and 1970 , he sold 140 of the lots, inserting in each of the 140 deeds the following provision: The Grantee, for himself and his heirs, assi... | D |
Dave is a six-year-old boy who has a welldeserved reputation for bullying younger and smaller children. His parents have encouraged him to be aggressive and tough. Dave, for no reason, knocked down, kicked, and severely injured Pete, a four-year-old. A claim has been asserted by Pete's parents for their medical and hos... | A |
Brown suffered from the delusion that he was a special agent of God. He frequently experienced hallucinations in the form of hearing divine commands. Brown believed God told him several times that the local Roman Catholic bishop was corrupting the diocese into heresy, and that the bishop should be "done away with." Bro... | B |
Dutton, disappointed by his eight-year-old son's failure to do well in school, began systematically depriving the child of food during summer vacation. Although his son became seriously ill from malnutrition, Dutton failed to call a doctor. He believed that as a parent he had the sole right to determine whether the chi... | B |
Tom had a heart ailment so serious that his doctors had concluded that only a heart transplant could save his life. They therefore arranged to have him flown to Big City to have the operation performed. Dan, Tom's nephew, who stood to inherit from him, poisoned him. The poison produced a reaction which required postpon... | A |
Parker sued Dodd over title to an island in a river. Daily variations in the water level were important. For many years Wells, a commercial fisherman, kept a daily log of the water level at his dock opposite the island in order to forecast fishing conditions. Parker employed Zee, an engineer, to prepare graphs from Wel... | D |
Paula sued for injuries she sustained in a fall in a hotel hallway connecting the lobby of the hotel with a restaurant located in the hotel building. The hallway floor was covered with vinyl tile. The defendants were Horne, owner of the hotel building, and Lee, lessee of the restaurant. The evidence was that the hallwa... | A |
Peters sued Davis for $ 100,000$ for injuries received in a traffic accident. Davis charges Peters with contributory negligence and alleges that Peters failed to have his lights on at a time when it was dark enough to require them. Davis offers to have Bystander testify that he was talking to Witness when he heard the ... | A |
Assume for the purposes of this question that you are counsel to the state legislative committee that is responsible for real estate laws in your state. The committee wants you to draft a statute, governing the recording of deeds, that fixes priorities of title, as reflected on the public record, as definitely as possi... | B |
Ogden was the fee simple owner of three adjoining vacant lots fronting on a common street in a primarily residential section of a city which had no zoning laws. The lots were identified as Lots 1, 2, and 3. Ogden conveyed Lot 1 to Akers and Lot 2 to Bell. Ogden retained Lot 3, which consisted of three acres of woodland... | D |
Philip was a 10-year-old boy. Macco was a company that sold new and used machinery. Macco stored discarded machinery, pending sale for scrap, on a large vacant area it owned. This area was unfenced and was one-quarter mile from the housing development where Philip lived. Macco knew that children frequently played in th... | D |
Compute$$ \int_{0}^{\pi} \frac{x \sin x}{1+\sin ^{2} x} d x . $$ | We use the example from the introduction for the particular function $f(x)=\frac{x}{1+x^{2}}$ to transform the integral into$$ \pi \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\sin ^{2} x} d x . $$This is the same as$$ \pi \int_{0}^{\frac{\pi}{2}}-\frac{d(\cos x)}{2-\cos ^{2} x}, $$which with the substitution $t=\cos x$ be... |
Compute up to two decimal places the number$$ \sqrt{1+2 \sqrt{1+2 \sqrt{1+\cdots+2 \sqrt{1+2 \sqrt{1969}}}}} $$where the expression contains 1969 square roots. | Let$$ x_{n}=\sqrt{1+2 \sqrt{1+2 \sqrt{1+\cdots+2 \sqrt{1+2 \sqrt{1969}}}}} $$with the expression containing $n$ square root signs. Note that$$ x_{1}-(1+\sqrt{2})=\sqrt{1969}-(1+\sqrt{2})<50 . $$Also, since $\sqrt{1+2(1+\sqrt{2})}=1+\sqrt{2}$, we have $$ \begin{aligned} x_{n+1}-(1+\sqrt{2})=& \sqrt{1+2 x_{n}}-\sqrt{1+2(... |
Let $M_{2 \times 2}$ be the vector space of all real $2 \times 2$ matrices. Let
$$
A=\left(\begin{array}{cc}
1 & 2 \\
-1 & 3
\end{array}\right) \quad B=\left(\begin{array}{cc}
2 & 1 \\
0 & 4
\end{array}\right)
$$
and define a linear transformation $L: M_{2 \times 2} \rightarrow M_{2 \times 2}$ by $L(X)=A X B$. Comput... | Identify $M_{2 \times 2}$ with $\mathbb{R}^{4}$ via
$$
\left(\begin{array}{ll}
a & b \\nc & d
\end{array}\right) \leftrightarrow\left(\begin{array}{l}
a \\nb \\nc \\nd
\end{array}\right)
$$
and decompose $L$ into the multiplication of two linear transformations,
$$
M_{2 \times 2} \simeq \mathbb{R}^{4} \stackrel{L_{... |
Define
$$
F(x)=\int_{\sin x}^{\cos x} e^{\left(t^{2}+x t\right)} d t .
$$
Compute $F^{\prime}(0)$.
| Let
$$
G(u, v, x)=\int_{v}^{u} e^{t^{2}+x t} d t .
$$
Then $F(x)=G(\cos x, \sin x, x)$, so
$$
\begin{aligned}
F^{\prime}(x) &=\frac{\partial G}{\partial u} \frac{\partial u}{\partial x}+\frac{\partial G}{\partial v} \frac{\partial v}{\partial x}+\frac{\partial G}{\partial x} \\n&=e^{u^{2}+x u}(-\sin x)-e^{\left(v^{... |
What is the probability that the sum of two randomly chosen numbers in the interval $[0,1]$ does not exceed 1 and their product does not exceed $\frac{2}{9}$? | Let $x$ and $y$ be the two numbers. The set of all possible outcomes is the unit square$$ D=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\} . $$The favorable cases consist of the region$$ D_{f}=\left\{(x, y) \in D \mid x+y \leq 1, x y \leq \frac{2}{9}\right\} . $$This is the set of points that lie below both the line $... |
Evaluate
$$
\int_{0}^{2 \pi} e^{\left(e^{i \theta}-i \theta\right)} d \theta .
$$
| By Cauchy's Integral Formula for derivatives, we have
therefore,
$$
\left.\frac{d}{d z} e^{z}\right|_{z=0}=\frac{1}{2 \pi i} \int_{|z|=1} \frac{e^{z}}{z^{2}} d z=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{e^{i \theta}-i \theta} d \theta
$$
$$
\int_{0}^{2 \pi} e^{e^{i \theta}-i \theta} d \theta=2 \pi .
$$
|
Compute the product$$ \left(1-\frac{4}{1}\right)\left(1-\frac{4}{9}\right)\left(1-\frac{4}{25}\right) \cdots $$ | For $N \geq 2$, define$$ a_{N}=\left(1-\frac{4}{1}\right)\left(1-\frac{4}{9}\right)\left(1-\frac{4}{25}\right) \cdots\left(1-\frac{4}{(2 N-1)^{2}}\right) . $$The problem asks us to find $\lim _{N \rightarrow \infty} a_{N}$. The defining product for $a_{N}$ telescopes as follows:$$ \begin{aligned} a_{N} &=\left[\left(1-... |
Suppose the coefficients of the power series
$$
\sum_{n=0}^{\infty} a_{n} z^{n}
$$
are given by the recurrence relation
$$
a_{0}=1, a_{1}=-1,3 a_{n}+4 a_{n-1}-a_{n-2}=0, n=2,3, \ldots .
$$
Find the radius of convergence $r$.
| From the recurrence relation, we see that the coefficients $a_{n}$ grow, at most, at an exponential rate, so the series has a positive radius of convergence. Let $f$ be the function it represents in its disc of convergence, and consider the polynomial $p(z)=3+4 z-z^{2}$. We have
$$
\begin{aligned}
p(z) f(z) &=\left(3... |
For integers $n \geq 2$ and $0 \leq k \leq n-2$, compute the determinant$$ \left|\begin{array}{ccccc} 1^{k} & 2^{k} & 3^{k} & \cdots & n^{k} \\ 2^{k} & 3^{k} & 4^{k} & \cdots & (n+1)^{k} \\ 3^{k} & 4^{k} & 5^{k} & \cdots & (n+2)^{k} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n^{k} & (n+1)^{k} & (n+2)^{k} & \cdots... | The polynomials $P_{j}(x)=(x+j)^{k}, j=0,1, \ldots, n-1$, lie in the $(k+1)$-dimensional real vector space of polynomials of degree at most $k$. Because $k+1<n$, they are linearly dependent. The columns consist of the evaluations of these polynomials at $1,2, \ldots, n$, so the columns are linearly dependent. It follow... |
Compute$$ \lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{4 n^{2}-1^{2}}}+\frac{1}{\sqrt{4 n^{2}-2^{2}}}+\cdots+\frac{1}{\sqrt{4 n^{2}-n^{2}}}\right] . $$ | We have$$ \begin{aligned} s_{n} &=\frac{1}{\sqrt{4 n^{2}-1^{2}}}+\frac{1}{\sqrt{4 n^{2}-2^{2}}}+\cdots+\frac{1}{\sqrt{4 n^{2}-n^{2}}} \\ &=\frac{1}{n}\left[\frac{1}{\sqrt{4-\left(\frac{1}{n}\right)^{2}}}+\frac{1}{\sqrt{4-\left(\frac{2}{n}\right)^{2}}}+\cdots+\frac{1}{\sqrt{4-\left(\frac{n}{n}\right)^{2}}}\right] . \end... |
Consider the sequences $\left(a_{n}\right)_{n}$ and $\left(b_{n}\right)_{n}$ defined by $a_{1}=3, b_{1}=100, a_{n+1}=3^{a_{n}}$, $b_{n+1}=100^{b_{n}}$. Find the smallest number $m$ for which $b_{m}>a_{100}$. | We need to determine $m$ such that $b_{m}>a_{n}>b_{m-1}$. It seems that the difficult part is to prove an inequality of the form $a_{n}>b_{m}$, which reduces to $3^{a_{n-1}}>100^{b_{m-1}}$, or $a_{n-1}>\left(\log _{3} 100\right) b_{m-1}$. Iterating, we obtain $3^{a_{n-2}}>\left(\log _{3} 100\right) 100^{b_{m-2}}$, that... |
Let the sequence $a_{0}, a_{1}, \ldots$ be defined by the equation
$$
1-x^{2}+x^{4}-x^{6}+\cdots=\sum_{n=0}^{\infty} a_{n}(x-3)^{n} \quad(0<x<1) .
$$
Find
$$
\limsup _{n \rightarrow \infty}\left(\left|a_{n}\right|^{\frac{1}{n}}\right) .
$$
| As
$$
1-x^{2}+x^{4}-x^{6}+\cdots=\frac{1}{1+x^{2}}
$$
which has singularities at $\pm i$, the radius of convergence of
$$
\sum_{n=0}^{\infty} a_{n}(x-3)^{n}
$$
is the distance from $3$ to $\pm i, and |3 \mp i|=\sqrt{10}$. We then have
$$
\limsup _{n \rightarrow \infty}\left(\left|a_{n}\right|^{\frac{1}{n}}\right)... |
Find the radius of convergence $R$ of the Taylor series about $z=1$ of the function $f(z)=1 /\left(1+z^{2}+z^{4}+z^{6}+z^{8}+z^{10}\right)$.
| The rational function
$$
f(z)=\frac{1-z^{2}}{1-z^{12}}
$$
has poles at all nonreal twelfth roots of unity (the singularities at $z^{2}=1$ are removable). Thus, the radius of convergence is the distance from 1 to the nearest singularity:
$$
R=|\exp (\pi i / 6)-1|=\sqrt{(\cos (\pi / 6)-1)^{2}+\sin ^{2}(\pi / 6)}=\sqr... |
Find the number of permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ of the numbers $1,2,3,4,5,6$ that can be transformed into $1,2,3,4,5,6$ through exactly four transpositions (and not fewer). | The condition from the statement implies that any such permutation has exactly two disjoint cycles, say $\left(a_{i_{1}}, \ldots, a_{i_{r}}\right)$ and $\left(a_{i_{r+1}}, \ldots, a_{i_{6}}\right)$. This follows from the fact that in order to transform a cycle of length $r$ into the identity $r-1$, transpositions are n... |
Given the fact that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$, evaluate the integral
$$
I=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(x^{2}+(y-x)^{2}+y^{2}\right)} d x d y .
$$ | We write
$$
\begin{aligned}
I &=\int_{-\infty}^{\infty} e^{-3 y^{2} / 2}\left(\int_{-\infty}^{\infty} e^{-2 x^{2}+2 x y-y^{2} / 2} d x\right) d y \\n&=\int_{-\infty}^{\infty} e^{-3 y^{2} / 2}\left(\int_{-\infty}^{\infty} e^{-2\left(x-\frac{y}{2}\right)^{2}} d x\right) d y
\end{aligned}
$$
making the substitution $t=... |
The zeros of the polynomial $P(x)=x^{3}-10 x+11$ are $u$, $v$, and $w$. Determine the value of $\arctan u+\arctan v+\arctan w$. | First solution: Let $\alpha=\arctan u, \beta=\arctan v$, and $\arctan w$. We are required to determine the sum $\alpha+\beta+\gamma$. The addition formula for the tangent of three angles, $$ \tan (\alpha+\beta+\gamma)=\frac{\tan \alpha+\tan \beta+\tan \gamma-\tan \alpha \tan \beta \tan \gamma}{1-(\tan \alpha \tan \beta... |
Find the maximum of $x_{1}^{3}+\ldots+x_{10}^{3}$ for $x_{1}, \ldots, x_{10} \in[-1,2]$ such that $$ x_{1}+\ldots+x_{10}=10.$$ | Look at the behavior of expression $x_{1}^{3}+x_{2}^{3}$ when we move the variables $x_{1} \leq x_{2}$ together, that is, replace them with $x_{1}+\varepsilon$ and $x_{2}-\varepsilon, 0<\varepsilon \leq \frac{x_{2}-x_{1}}{2}$, and when we move them apart, that is, replace them with $x_{1}-\varepsilon$ and $x_{2}+\varep... |
Two airplanes are supposed to park at the same gate of a concourse. The arrival times of the airplanes are independent and randomly distributed throughout the 24 hours of the day. What is the probability that both can park at the gate, provided that the first to arrive will stay for a period of two hours, while the sec... | The set of possible events is modeled by the square $[0, 24] \times[0,24]$. It is, however, better to identify the 0th and the 24th hours, thus obtaining a square with opposite sides identified, an object that in mathematics is called a torus (which is, in fact, the Cartesian product of two circles. The favorable regio... |
Find a limit $$ \lim _{n \rightarrow \infty}\left(\int_{0}^{1} e^{x^{2} / n} d x\right)^{n}. $$ | Fix an arbitrary $\varepsilon>0$. Since the function $y=e^{t}$ is convex, for $t \in[0, \varepsilon]$ its graph lies above the tangent $y=1+t$ but under the secant $y=1+a_{\varepsilon} t$, where $a_{\varepsilon}=\frac{1}{\varepsilon}\left(e^{\varepsilon}-1\right)$. Hence for $\frac{1}{n} \leq \varepsilon$ it holds $$ ... |
Let $x_{0}=1$ and
$$
x_{n+1}=\frac{3+2 x_{n}}{3+x_{n}}, \quad n \geqslant 0 .
$$
Find the limit $x_{\infty}=\lim _{n \rightarrow \infty} x_{n}$.
| Obviously, $x_{n} \geqslant 1$ for all $n$; so, if the limit exists, it is $\geqslant 1$, and we can pass to the limit in the recurrence relation to get
$$
x_{\infty}=\frac{3+2 x_{\infty}}{3+x_{\infty}} \text {; }
$$
in other words, $x_{\infty}^{2}+x_{\infty}-3=0$. So $x_{\infty}$ is the positive solution of this qu... |
Consider a sequence $x_{n}=x_{n-1}-x_{n-1}^{2}, n \geq 2, x_{1} \in(0,1)$. Calculate $$ \lim _{n \rightarrow \infty} \frac{n^{2} x_{n}-n}{\ln n} . $$ | By the monotone convergence theorem, it is easy to show that $x_{n} \rightarrow 0$, as $n \rightarrow \infty$. Next, by the Stolz-Cesaro theorem it holds $$ \begin{aligned} n x_{n}=\frac{n}{\frac{1}{x_{n}}} \sim \frac{1}{\frac{1}{x_{n}}-\frac{1}{x_{n-1}}}=\frac{x_{n-1} x_{n}}{x_{n-1}-x_{n}}=\\ \quad=\frac{x_{n-1}\left... |
Let $\varphi(x, y)$ be a function with continuous second order partial derivatives such that
1. $\varphi_{x x}+\varphi_{y y}+\varphi_{x}=0$ in the punctured plane $\mathbb{R}^{2} \backslash\{0\}$,
2. $r \varphi_{x} \rightarrow \frac{x}{2 \pi r}$ and $r \varphi_{y} \rightarrow \frac{y}{2 \pi r}$ as $r=\sqrt{x^{2}+y^{2... | Consider the annular region $\mathcal{A}$ between the circles of radius $r$ and $R$, then by Green's Theorem
$$
\begin{aligned}
&\int_{R} e^{x}\left(-\varphi_{y} d x+\varphi_{x} d y\right)-\int_{r} e^{y}\left(-\varphi_{y} d x+\varphi_{x} d y\right)= \\n&=\int_{\partial \mathcal{A}} e^{x}\left(-\varphi_{x} d x+\varphi_... |
Compute$$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\frac{k}{n^{2}}\right)^{\frac{k}{n^{2}}+1} . $$ | We use the fact that$$ \lim _{x \rightarrow 0^{+}} x^{x}=1 . $$As a consequence, we have$$ \lim _{x \rightarrow 0^{+}} \frac{x^{x+1}}{x}=1 . $$For our problem, let $\epsilon>0$ be a fixed small positive number. There exists $n(\epsilon)$ such that for any integer $n \geq n(\epsilon)$,$$ 1-\epsilon<\frac{\left(\frac{k}{... |
A sequence $\left\{x_{n}, n \geq 1\right\}$ satisfies $x_{n+1}=x_{n}+e^{-x_{n}}, n \geq 1$, and $x_{1}=1$. Find $\lim _{n \rightarrow \infty} \frac{x_{n}}{\ln n}$. | The sequence $\left\{x_{n}\right\}$ is increasing, hence it has a finite or infinite limit. If $\lim _{n \rightarrow \infty} x_{n}=$ $x<+\infty$, then $x=x+e^{-x}$, a contradiction. Therefore, $\lim _{n \rightarrow \infty} x_{n}=+\infty$. We use twice the Stolz-Cesaro theorem and obtain $$ \lim _{n \rightarrow \infty}... |
Find the limit $$ \lim _{N \rightarrow \infty} \sqrt{N}\left(1-\max _{1 \leq n \leq N}\{\sqrt{n}\}\right), $$ where $\{x\}$ denotes the fractional part of $x$. | Estimate $\max _{1 \leq n \leq N}\{\sqrt{n}\}$. For $k^{2} \leq n<(k+1)^{2}$ it holds $$ \{\sqrt{n}\}=\sqrt{n}-k \leq \sqrt{(k+1)^{2}-1}-k, $$ where equality is attained for $n=(k+1)^{2}-1$. If $k<l$ then $$ \begin{aligned} 1 &-\left\{\sqrt{(k+1)^{2}-1}\right\}=k+1-\sqrt{(k+1)^{2}-1}=\\ &=\frac{1}{k+1+\sqrt{(k+1)^{2... |
Compute
$$
L=\lim _{n \rightarrow \infty}\left(\frac{n^{n}}{n !}\right)^{1 / n} .
$$
| Let $p_{1}=1, p_{2}=(2 / 1)^{2}, p_{3}=(3 / 2)^{3}, \ldots, p_{n}=(n /(n-1))^{n}$. Then
$$
\frac{p_{1} p_{2} \cdots p_{n}}{n}=\frac{n^{n}}{n !},
$$
and since $p_{n} \rightarrow e$, we have $\lim \left(n^{n} / n !\right)^{1 / n}=e$ as well (using the fact that $\left.\lim n^{1 / n}=1\right)$.
|
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function of period $2 \pi$ such that $f(x)=x^{3}$ for $-\pi \leqslant x<\pi$.
The Fourier series for $f$ has the form $\sum_{1}^{\infty} b_{n} \sin n x$. Find
$$
\sum_{n=1}^{\infty} b_{n}^{2}.
$$
| Using Parseval's Identity
$$
\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty}\left(a_{n}^{2}+b_{n}^{2}\right)=\frac{1}{\pi} \int_{-\pi}^{\pi} f^{2}(x) d x
$$
and the fact that all $a_{n}=0$,
$$
\sum_{n=1}^{\infty} b^{2}=\frac{1}{\pi} \int_{-\pi}^{\pi} x^{6} d x=\frac{2}{7} \pi^{6} .
$$
|
For a real $2 \times 2$ matrix
$$
X=\left(\begin{array}{ll}
x & y \\
z & t
\end{array}\right),
$$
let $\|X\|=x^{2}+y^{2}+z^{2}+t^{2}$, and define a metric by $d(X, Y)=\|X-Y\|$. Let $\Sigma=\{X \mid \operatorname{det}(X)=0\}$. Let
$$
A=\left(\begin{array}{ll}
1 & 0 \\
0 & 2
\end{array}\right) \text {. }
$$
Find ... | For $X \in \Sigma$, we have
$$
\begin{aligned}
\|A-X\|^{2} &=(1-x)^{2}+y^{2}+z^{2}+(2-t)^{2} \\n&=y^{2}+z^{2}+1-2 x+x^{2}+(2-t)^{2} \\n& \geqslant \pm 2 y z+1-2 x+2|x|(2-t) \\n&=4|x|-2 x+2(\pm y z-|x| t)+1
\end{aligned}
$$
We can choose the sign, so $\pm y z-|x| t=0$ because $\operatorname{det} X=0$. As $4|x|-2 x \ge... |
Find the probability that in the process of repeatedly flipping an unbiased coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. | We call a successful string a sequence of $H$ 's and $T$ 's in which $H H H H H$ appears before $T T$ does. Each successful string must belong to one of the following three types:(i) those that begin with $T$, followed by a successful string that begins with $H$;(ii) those that begin with $H, H H, H H H$, or $H H H H$,... |
Compute the integral $\iint_{D} x d x d y$, where$$ D=\left\{(x, y) \in \mathbb{R}^{2} \mid x \geq 0,1 \leq x y \leq 2,1 \leq \frac{y}{x} \leq 2\right\} . $$ | The domain is bounded by the hyperbolas $x y=1, x y=2$ and the lines $y=x$ and $y=2 x$. This domain can mapped into a rectangle by the transformation$$ T: \quad u=x y, \quad v=\frac{y}{x} . $$Thus it is natural to consider the change of coordinates$$ T^{-1}: \quad x=\sqrt{\frac{u}{v}}, \quad y=\sqrt{u v} . $$The domain... |
A husband and wife agree to meet at a street corner between 4 and 5 o'clock to go shopping together. The one who arrives first will await the other for 15 minutes, and then leave. What is the probability that the two meet within the given time interval, assuming that they can arrive at any time with the same probabilit... | Denote by $x$, respectively, $y$, the fraction of the hour when the husband, respectively, wife, arrive. The configuration space is the square $$ D=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\} . $$In order for the two people to meet, their arrival time must lie inside the region$$ D_{f}=\left\{(x, y)|| x-y \mid \leq... |
Evaluate the product$$ \left(1-\cot 1^{\circ}\right)\left(1-\cot 2^{\circ}\right) \cdots\left(1-\cot 44^{\circ}\right) \text {. } $$ | We have$$ \begin{aligned} \left(1-\cot 1^{\circ}\right)\left(1-\cot 2^{\circ}\right) \cdots\left(1-\cot 44^{\circ}\right) \\ &=\left(1-\frac{\cos 1^{\circ}}{\sin 1^{\circ}}\right)\left(1-\frac{\cos 2^{\circ}}{\sin 2^{\circ}}\right) \cdots\left(1-\frac{\cos 44^{\circ}}{\sin 44^{\circ}}\right) \\ &=\frac{\left(\sin 1^{\c... |
Find the number of roots of
$$
z^{7}-4 z^{3}-11=0
$$
which lie between the two circles $|z|=1$ and $|z|=2$.
| Let $p(z)=z^{7}-4 z^{3}-11$. For $z$ in the unit circle, we have
$$
|p(z)-11|=\left|z^{7}-4 z^{3}\right| \leqslant 5<11
$$
so, by Rouch\u00e9's Theorem, the given polynomial has no zeros in the unit disc. For $|z|=2$,
$$
\left|p(z)-z^{7}\right|=\left|4 z^{3}+11\right| \leqslant 43<128=\left|z^{7}\right|
$$
so ther... |
Let $\mathcal{S}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\}$ denote the unit sphere in $\mathbb{R}^{3}$. Evaluate the surface integral over $\mathcal{S}$ :
$$
\iint_{\mathcal{S}}\left(x^{2}+y+z\right) d A .
$$
| Using the change of variables
$$
\begin{cases}x=\sin \varphi \cos \theta & 0<\theta<2 \pi \\ y=\sin \varphi \sin \theta & 0<\varphi<\pi \\ z=\cos \varphi & \end{cases}
$$
we have
$$
d A=\sin \varphi d \theta d \varphi
$$
and
$$
\iint_{\mathcal{S}}\left(x^{2}+y+z\right) d A=\int_{0}^{\pi} \int_{0}^{2 \pi}\left(\si... |
Find the sum of the series $$ \sum_{n=0}^{\infty} \frac{1}{n ! 2^{n}} \cos \frac{\pi n-1}{2} . $$ | For all $x \in \mathbb{R}$, it holds $$ \begin{aligned} &\sum_{n=0}^{\infty} \frac{\cos \left(\frac{\pi n}{2}-x\right)}{n !} x^{n}=\cos x \sum_{n=0}^{\infty} \frac{\cos \frac{\pi n}{2}}{n !} x^{n}+\sin x \sum_{n=0}^{\infty} \frac{\sin \frac{\pi n}{2}}{n !} x^{n}= \\ &=\cos x \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k) !... |
What is the probability that a uniformly random permutation of the first $n$ positive integers has the numbers 1 and 2 within the same cycle? | The total number of permutations is of course $n$ !. We will count instead the number of permutations for which 1 and 2 lie in different cycles.If the cycle that contains 1 has length $k$, we can choose the other $k-1$ elements in $\left(\begin{array}{c}n-2 \\ k-1\end{array}\right)$ ways from the set $\{3,4, \ldots, n\... |
Find$$ \int_{0}^{1} \frac{\ln (1+x)}{1+x^{2}} d x .$$ | With the substitution $\arctan x=t$ the integral takes the form$$ I=\int_{0}^{\frac{\pi}{4}} \ln (1+\tan t) d t . $$This we already computed in the previous problem. ("Happiness is longing for repetition," says M. Kundera.) So the answer to the problem is $\frac{\pi}{8} \ln 2$. |
Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference of $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with dia... | The pair $(p, q)$ is chosen randomly from the three-dimensional domain $C \times$ int $C$, which has a total volume of $2 \pi^{2}$ (here int $C$ denotes the interior of $C$ ). For a fixed $p$, the locus of points $q$ for which $R$ does not have points outside of $C$ is the rectangle whose diagonal is the diameter throu... |
Let $M$ be a subset of $\{1,2,3, \ldots, 15\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$. | Note that the product of the three elements in each of the sets $\{1,4,9\},\{2,6,12\}$, $\{3,5,15\}$, and $\{7,8,14\}$ is a square. Hence none of these sets is a subset of $M$. Because they are disjoint, it follows that $M$ has at most $15-4=11$ elements.Since 10 is not an element of the aforementioned sets, if $10 \no... |
Compute the limit
$$
\lim _{z \rightarrow 0}\left((\tan z)^{-2}-z^{-2}\right)
$$
where $z$ is a complex variable.
| As
$$
(\tan z)^{-2}-z^{-2}=\frac{z^{2}-(\tan z)^{2}}{z^{2}(\tan z)^{2}}
$$
the Maclaurin expansion of the numerator has no terms of degree up to 3 , whereas the expansion of the denominator starts with $z^{4}$, therefore, the limit is finite.
As
$$
\tan z=z+\frac{1}{3} z^{3}+o\left(z^{4}\right) \quad(z \rightarrow... |
Let $f(x)=\frac{1}{4}+x-x^{2}$. For any real number $x$, define a sequence $\left(x_{n}\right)$ by $x_{0}=x$ and $x_{n+1}=f\left(x_{n}\right)$. If the sequence converges, let $x_{\infty}$ denote the limit. For $x=0$, find $x_{\infty}=\lambda$.
| We have
$$
f(x)=\frac{1}{2}-\left(x-\frac{1}{2}\right)^{2}
$$
so $x_{n}$ is bounded by $1 / 2$ and, by the Induction Principle, nondecreasing. Let $\lambda$ be its limit. Then
$$
\lambda=\frac{1}{2}-\left(\lambda-\frac{1}{2}\right)^{2}
$$
and, as the sequence takes only positive values,
$$
\lambda=\frac{1}{2} \tex... |
The clock-face is a disk of radius 1 . The hour-hand is a disk of radius $1 / 2$ is internally tangent to the circle of the clock-face, and the minute-hand is a line segment of length 1 . Find the area of the figure formed by all intersections of the hands in 12 hours (i.e., in one full turn of the hour-hand). | Let the minute hand makes an angle $\varphi \in[0 ; 2 \pi)$ with the vertical position. At that moment, the hour hand makes one of the angles $$ \psi_{k}=\frac{\varphi+2 \pi k}{12}(\bmod 2 \pi), k \in \mathbb{Z}, $$ with the vertical position. The hands intersect along a segment of length $$ \max \left(0, \cos \left... |
Evaluate
$$
\iint_{\mathcal{A}} e^{-x^{2}-y^{2}} d x d y,
$$
where $\mathcal{A}=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leqslant 1\right\}$.
| Using polar coordinates, we have
$$
\begin{aligned}
\iint_{\mathcal{A}} e^{-x^{2}-y^{2}} d x d y &=\int_{0}^{2 \pi} \int_{0}^{1} \rho e^{-\rho^{2}} d \rho d \theta \\n&=-\frac{1}{2} \int_{0}^{2 \pi} \int_{0}^{1}-2 \rho e^{-\rho^{2}} d \rho d \theta \\n&=-\frac{1}{2} \int_{0}^{2 \pi}\left(e^{-1}-1\right) \\n&=\pi\left... |
Let $S_{9}$ denote the group of permutations of nine objects and let $A_{9}$ be the subgroup consisting of all even permutations. Denote by $1 \in S_{9}$ the identity permutation. Determine the minimum of all positive integers $m$ such that every $\sigma \in A_{9}$ satisfies $\sigma^{m}=1$.
| The order of a $k$-cycle is $k$, so the smallest $m$ which simultaneously annihilates all 9-cycles, 8-cycles, 7-cycles, and 5-cycles is $2^{3} \cdot 3^{2} \cdot 5 \cdot 7= 2520$. Any $n$-cycle, $n \leqslant 9$, raised to this power is annihilated, so $n=2520$.
To compute $n$ for $A 9$, note that an 8-cycle is an odd ... |
A bag contains 1993 red balls and 1993 black balls. We remove two uniformly random balls at a time repeatedly and
(i) discard them if they are of the same color,
(ii) discard the black ball and return to the bag the red ball if they are of different colors.
The process ends when there are less than two balls in the ... | First, observe that since at least one ball is removed during each stage, the process will eventually terminate, leaving no ball or one ball in the bag. Because red balls are removed 2 at a time and since we start with an odd number of red balls, the number of red balls in the bag at any time is odd. Hence the process ... |
Let $T_{1}, T_{2}, T_{3}$ be points on a parabola, and $t_{1}, t_{2}, t_{3}$ the tangents to the parabola at these points. Compute the ratio of the area of triangle $T_{1} T_{2} T_{3}$ to the area of the triangle determined by the tangents. | Choose a Cartesian system of coordinates such that the equation of the parabola is $y^{2}=4 p x$. The coordinates of the three points are $T_{i}\left(4 p \alpha_{i}^{2}, 4 p \alpha_{i}\right)$, for appropriately chosen $\alpha_{i}, i=1,2,3$. Recall that the equation of the tangent to the parabola at a point $\left(x_{0... |
For each continuous function $f:[0,1] \rightarrow \mathbb{R}$, we define $I(f)=\int_{0}^{1} x^{2} f(x) d x$ and $J(f)=\int_{0}^{1} x(f(x))^{2} d x$. Find the maximum value of $I(f)-J(f)$ over all such functions $f$. | We change this into a minimum problem, and then relate the latter to an inequality of the form $x \geq 0$. Completing the square, we see that$$ \left.x(f(x))^{2}-x^{2} f(x)=\sqrt{x} f(x)\right)^{2}-2 \sqrt{x} f(x) \frac{x^{\frac{3}{2}}}{2}=\left(\sqrt{x} f(x)-\frac{x^{\frac{3}{2}}}{2}\right)^{2}-\frac{x^{3}}{4} $$Hence... |
Compute the integral$$ \iint_{D} \frac{d x d y}{\left(x^{2}+y^{2}\right)^{2}}, $$where $D$ is the domain bounded by the circles$$ \begin{array}{ll} x^{2}+y^{2}-2 x=0, & x^{2}+y^{2}-4 x=0, \\ x^{2}+y^{2}-2 y=0, & x^{2}+y^{2}-6 y=0 . \end{array} $$ | The domain $D$ is depicted in Figure 71 . We transform it into the rectangle $D_{1}=$ $\left[\frac{1}{4}, \frac{1}{2}\right] \times\left[\frac{1}{6}, \frac{1}{2}\right]$ by the change of coordinates$$ x=\frac{u}{u^{2}+v^{2}}, \quad y=\frac{v}{u^{2}+v^{2}} . $$The Jacobian is Figure 71$$ J=-\frac{1}{\left(u^{2}+v^{2}\ri... |
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Find the probability that $\sqrt{2+\sqrt{3}} \leq|v+w|$. | Because the 1997 roots of the equation are symmetrically distributed in the complex plane, there is no loss of generality to assume that $v=1$. We are required to find the probability that$$ |1+w|^{2}=|(1+\cos \theta)+i \sin \theta|^{2}=2+2 \cos \theta \geq 2+\sqrt{3} . $$This is equivalent to $\cos \theta \geq \frac{1... |
The numbers $1,2,3,4,5,6,7$, and 8 are written on the faces of a regular octahedron so that each face contains a different number. Find the probability that no two consecutive numbers are written on faces that share an edge, where 8 and 1 are considered consecutive. | Consider the dual cube to the octahedron. The vertices $A, B, C, D, E, F, G$, $H$ of this cube are the centers of the faces of the octahedron (here $A B C D$ is a face of the cube and $(A, G),(B, H),(C, E),(D, F)$ are pairs of diagonally opposite vertices). Each assignment of the numbers $1,2,3,4,5,6,7$, and 8 to the f... |
Let $V$ be the vector space of all real $3 \times 3$ matrices and let $A$ be the diagonal matrix
$$
\left(\begin{array}{lll}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 1
\end{array}\right) .
$$
Calculate the determinant of the linear transformation $T$ on $V$ defined by $T(X)=\frac{1}{2}(A X+X A)$.
| Let $X=\left(x_{i j}\right)$ be any element of $M_{3}(\mathbb{R})$. A calculation gives
$$
T(X)=\left(\begin{array}{ccc}
x_{11} & 3 x_{12} / 2 & x_{13} \\n3 x_{21} / 2 & 2 x_{22} & 3 x_{23} / 2 \\nx_{31} & 3 x_{32} / 2 & x_{33}
\end{array}\right) .
$$
It follows that the basis matrices $M_{i j}$ are eigenvectors of ... |
An old woman went to the market and a horse stepped on her basket and smashed her eggs. The rider offered to pay for the eggs and asked her how many there were. She did not remember the exact number, but when she had taken them two at a time there was one egg left, and the same happened when she took three, four, five,... | We are to find the smallest positive solution to the system of congruences$$ \begin{aligned} &x \equiv 1(\bmod 60), \\ &x \equiv 0(\bmod 7) . \end{aligned} $$The general solution is $7 b_{1}+420 t$, where $b_{1}$ is the inverse of 7 modulo 60 and $t$ is an integer. Since $b_{1}$ is a solution to the Diophantine equatio... |
Consider a spinless particle represented by the wave function
$$
\psi=K(x+y+2 z) e^{-\alpha r}
$$
where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, and $K$ and $\alpha$ are real constants.
If the $z$-component of angular momentum, $L_{z}$, were measured, what is the probability that the result would be $L_{z}=+\hbar$ ?
You may f... | The wave function may be rewritten in spherical coordinates as
$$
\psi=K r(\cos \phi \sin \theta+\sin \phi \sin \theta+2 \cos \theta) e^{-\alpha r}
$$
its angular part being
$$
\psi(\theta, \phi)=K^{\prime}(\cos \phi \sin \theta+\sin \phi \sin \theta+2 \cos \theta),
$$
where $K^{\prime}$ is the normalization consta... |
A student near a railroad track hears a train's whistle when the train is coming directly toward him and then when it is going directly away. The two observed frequencies are $250$ and $200 \mathrm{~Hz}$. Assume the speed of sound in air to be $360 \mathrm{~m} / \mathrm{s}$. What is the train's speed? | Let $\nu_{0}, \nu_{1}, \nu_{2}$ be respectively the frequency of the whistle emitted by the train, and the frequencies heard by the student when the train is coming and when it is moving away. The Doppler effect has it that
$$
\begin{aligned}
&\nu_{1}=\left(\frac{c}{c-v}\right) \nu_{0}, \\
&\nu_{2}=\left(\frac{c}{c+v}... |
An opaque sheet has a hole in it of $0.5 \mathrm{~mm}$ radius. If plane waves of light $(\lambda=5000 \AA)$ fall on the sheet, find the maximum distance from this sheet at which a screen must be placed so that the light will be focused to a bright spot. | The maximum distance $r$ of the screen from the hole for which the light will be focused to a bright spot is that for which the area of the hole corresponds to the first Fresnel zone only, and is given by
$$
\rho_{1}^{2}=\lambda r,
$$
where $\rho$ is the radius of the hole. Hence
$$
r=\frac{\rho_{1}^{2}}{\lambda}=\f... |
Assume a visible photon of $3 \mathrm{eV}$ energy is absorbed in one of the cones (light sensors) in your eye and stimulates an action potential that produces a $0.07$ volt potential on an optic nerve of $10^{-9} \mathrm{~F}$ capacitance.
Calculate the charge needed. | $Q=V C=0.07 \times 10^{-9}=7 \times 10^{-11}$ Coulomb. |
Imagine the universe to be a spherical cavity, with a radius of $10^{28} \mathrm{~cm}$ and impenetrable walls.
If the temperature were $0 \mathrm{~K}$, and the universe contained $10^{80}$ electrons in a Fermi distribution, calculate the Fermi momentum of the electrons. | The number of photons in the angular frequency range from $\omega$ to $\omega+d \omega$ is
$$
d N=\frac{V}{\pi^{2} c^{3}} \frac{\omega^{2} d \omega}{e^{\beta \hbar \omega}-1}, \quad \beta=\frac{1}{k T} .
$$
The total number of photons is
$$
\begin{aligned}
N &=\frac{V}{\pi^{2} c^{3}} \int_{0}^{\infty} \frac{\omega^{... |
In a simplified model of a relativistic nucleus-nucleus collision, a nucleus of rest mass $m_{1}$ and speed $\beta_{1}$ collides head-on with a target nucleus of mass $m_{2}$ at rest. The composite system recoils at speed $\beta_{0}$ and with center of mass energy $\varepsilon_{0}$. Assume no new particles are created.... | As implied by the question the velocity of light is taken to be one for convenience.
For a system, $E^{2}-p^{2}$ is invariant under Lorentz transformation. In the laboratory frame $\Sigma$, writing $\gamma_{1}=\frac{1}{\sqrt{1-\beta_{1}^{2}}}$,
$$
E^{2}-p^{2}=\left(m_{1} \gamma_{1}+m_{2}\right)^{2}-\left(m_{1} \gamma... |
In a region of empty space, the magnetic field (in Gaussian units) is described by
$$
\mathrm{B}=B_{0} e^{a x} \hat{\mathbf{e}}_{z} \sin w
$$
where $w=k y-\omega t$.
Find the speed of propagation $v$ of this field. | Express $B$ as $\operatorname{Im}\left(B_{0} e^{a x} e^{i w}\right) \hat{\mathbf{e}}_{z}$.
Using Maxwell's equation
$$
\nabla \times \mathbf{B}=\frac{1}{c} \frac{\partial \mathrm{E}}{\partial t}
$$
and the definition $k=\frac{\omega}{c}$ for empty space, we obtain
$$
\mathbf{E}=\frac{i c}{\omega} \nabla \times \mat... |
A meteorite of mass $1.6 \times 10^{3} \mathrm{~kg}$ moves about the earth in a circular orbit at an altitude of $4.2 \times 10^{6} \mathrm{~m}$ above the surface. It suddenly makes a head-on collision with another meteorite that is much lighter, and loses $2.0 \%$ of its kinetic energy without changing its direction o... | The laws of conservation of mechanical energy and conservation of angular momentum apply to the motion of the heavy meteorite after its collision.
For the initial circular motion, $E<0$, so after the collision we still have $E<0$. After it loses $2.0 \%$ of its kinetic energy, the heavy meteorite will move in an ellipt... |
X-rays are reflected from a crystal by Bragg reflection. If the density of the crystal which is of an accurately known structure is measured with an rms error of 3 parts in $10^{4}$, and if the angle the incident and reflected rays make with the crystal plane is $6^{\circ}$ and is measured with an rms error of $3.4$ mi... | For simplicity consider a crystal whose primitive cell is simple cubic with edge $d$ (to be multiplied by a factor of about one for the primitive cells of other crystal structures). For first order reflection, $n=1$ and Bragg's law gives
$$
2 d \sin \theta=\lambda .
$$
Differentiating, we have
$$
\left|\frac{\Delta ... |
Find the angular separation in seconds of arc of the closest two stars resolvable by the following reflecting telescope: $8 \mathrm{~cm}$ objective, $1.5$ meter focal length, $80$X eyepiece. Assume a wavelength of $6000 \AA$. ($1 \AA=10^{-8}$ cm). | The angular resolving power of the telescope is
$$
\Delta \theta_{1}=1.22 \frac{\lambda}{D}=1.22 \times \frac{6000 \times 10^{-8}}{8} \approx 2^{\prime \prime} .
$$
The resolving power of the human eye is about 1 minute of arc. Using an eyepiece of magnification $80 \mathrm{X}$, the eye's resolving power is $\Delta \... |
A self-luminous object of height $h$ is $40 \mathrm{~cm}$ to the left of a converging lens with a focal length of $10 \mathrm{~cm}$. A second converging lens with a focal length of $20 \mathrm{~cm}$ is $30 \mathrm{~cm}$ to the right of the first lens.
Calculate the ratio of the height of the final image to the height ... | From $\frac{1}{f_{1}}=\frac{1}{u_{1}}+\frac{1}{v_{1}}$, where $f_{1}=10 \mathrm{~cm}, u_{1}=40 \mathrm{~cm}$, we obtain $v_{1}=13 \frac{1}{3} \mathrm{~cm}$.
From $\frac{1}{f_{2}}=\frac{1}{u_{2}}+\frac{1}{v_{2}}$, where $f_{2}=20 \mathrm{~cm}, u_{2}=\left(30-\frac{10}{3}\right) \mathrm{cm}$, we obtain $v_{2}=-100 \math... |
A currently important technique for precisely measuring the lifetimes of states of multiply ionized atoms consists of exciting a beam of the desired ions with a laser tuned to a resonance wavelength of the ion under study, and measuring the emission intensity from the ion beam as a function of down- stream distance. Wh... | Take two coordinate frames $\Sigma, \Sigma^{\prime} . \Sigma$ is the laboratory frame in which the laser source $L$ is at rest and the ions move with velocity $\beta c$ along the $x$-axis as shown in Fig. 1.46. $\Sigma^{\prime}$ is the rest frame of an ion in the beam and moves with velocity $\beta c$ relative to $\Sig... |
A one-dimensional square well of infinite depth and $1 \AA$ width contains 3 electrons. The potential well is described by $V=0$ for $0 \leq x \leq 1 \AA$ and $V=+\infty$ for $x<0$ and $x>1 \AA$. For a temperature of $T=0 \mathrm{~K}$, the average energy of the 3 electrons is $E=12.4 \mathrm{eV}$ in the approximation t... | For a one-dimensional potential well the energy levels are given by
$$
E_{n}=E_{1} n^{2},
$$
where $E_{1}$ is the ground state energy and $n=1,2, \ldots$. Pauli's exclusion principle and the lowest energy principle require that two of the three electrons are in the energy level $E_{1}$ and the third one is in the ene... |
Evaluate
$$
I=\int_{0}^{\infty} \frac{d x}{4+x^{4}} .
$$ | As
$$
y=1+2 x+3 x^{2}+4 x^{3}+\ldots+n x^{n-1}+\ldots
$$
we have
$$
x y=x+2 x^{2}+3 x^{3}+4 x^{4}+\ldots+n x^{n}+\ldots
$$
and thus
$$
y-x y=1+x+x^{2}+x^{3}+\ldots+x^{n-1}+\ldots=\frac{1}{1-x}
$$
since $|x|<1$. Hence
$$
y=\frac{1}{(1-x)^{2}} .
$$
The mean value of $x$ is
$$
\bar{x}=\frac{\int_{0}^{\infty} x f(... |
The deuteron is a bound state of a proton and a neutron of total angular momentum $J=1$. It is known to be principally an $S(L=0)$ state with a small admixture of a $D(L=2)$ state.
Calculate the magnetic moment of the pure $D$ state $n-p$ system with $J=1$. Assume that the $n$ and $p$ spins are to be coupled to make t... | The parities of the $S$ and $D$ states are positive, while the parity of the $P$ state is negative. Because of the conservation of parity in strong interaction, a quantum state that is initially an $S$ state cannot have a $P$ state component at any later moment.
The possible spin values for a system composed of a prot... |
At room temperature, $k_{\mathrm{B}} T / e=26 \mathrm{mV}$. A sample of cadmium sulfide displays a mobile carrier density of $10^{16} \mathrm{~cm}^{-3}$ and a mobility coefficient $\mu=10^{2} \mathrm{~cm}^{2} /$ volt sec.
The carriers are continuously trapped into immobile sites and then being thermally reionized into... | The electrical conductivity is given by $\sigma=n e \mu$. With $n=10^{22} \mathrm{~m}^{-3}$, $e=1.6 \times 10^{-19} \mathrm{C}, \mu=10^{-2} \mathrm{~m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}$, we have for the material $\sigma=$ $16 \Omega^{-1} \mathrm{~m}^{-1}$.The law of equipartition of energy
$$
\frac{1}{2} m \bar{v... |
A closely wound search coil has an area of $4 \mathrm{~cm}^{2}, 160$ turns and a resistance of $50 \Omega$. It is connected to a ballistic galvanometer whose resistance is $30 \Omega$. When the coil rotates quickly from a position parallel to a uniform magnetic field to one perpendicular, the galvanometer indicates a c... | Suppose the coil rotates from a position parallel to the uniform magnetic field to one perpendicular in time $\Delta t$. Since $\Delta t$ is very short, we have
$$
\varepsilon=\frac{\Delta \phi}{\Delta t}=i(R+r)
$$
As $q=i \Delta t$, the increase of the magnetic flux is
$$
\Delta \phi=q(R+r)=B A N,
$$
since the coi... |
A solenoid has an air core of length $0.5 \mathrm{~m}$, cross section $1 \mathrm{~cm}^{2}$, and 1000 turns. A secondary winding wrapped around the center of the solenoid has 100 turns. A constant current of $1 \mathrm{~A}$ flows in the secondary winding and the solenoid is connected to a load of $10^{3}$ ohms. The cons... | Let the current in the winding of the solenoid be $i$. The magnetic induction inside the solenoid is then $B=\mu_{0} n i$ with direction along the axis, $n$ being the number of turns per unit length of the winding.
The total magnetic flux linkage is
$$
\psi=N \phi=N B S=N^{2} \mu_{0} S i / l .
$$
Hence the self-indu... |
In a region of empty space, the magnetic field (in Gaussian units) is described by
$$
\mathrm{B}=B_{0} e^{a x} \hat{\mathbf{e}}_{z} \sin w
$$
where $w=k y-\omega t$.
Find the speed of propagation $v$ of this field. | Express $B$ as $\operatorname{Im}\left(B_{0} e^{a x} e^{i w}\right) \hat{\mathbf{e}}_{z}$.
Using Maxwell's equation
$$
\nabla \times \mathbf{B}=\frac{1}{c} \frac{\partial \mathrm{E}}{\partial t}
$$
and the definition $k=\frac{\omega}{c}$ for empty space, we obtain
$$
\mathbf{E}=\frac{i c}{\omega} \nabla \times \mat... |
The electric field of an electromagnetic wave in vacuum is given by
$$
\begin{gathered}
E_{x}=0, \\
E_{y}=30 \cos \left(2 \pi \times 10^{8} t-\frac{2 \pi}{3} x\right), \\
E_{z}=0,
\end{gathered}
$$
where $E$ is in volts/meter, $t$ in seconds, and $x$ in meters. Determine the wavelength $\lambda$. | $$
k=\frac{2 \pi}{3} \mathrm{~m}^{-1}, \quad \omega=2 \pi \times 10^{8} \mathrm{~s}^{-1} .
$$
$f=\frac{\omega}{2 \pi}=10^{8} \mathrm{~Hz}$.$\lambda=\frac{2 \pi}{k}=3 \mathrm{~m}$. |
A common lecture demonstration is as follows: hold or clamp a onemeter long thin aluminium bar at the center, strike one end longitudinally (i.e. parallel to the axis of the bar) with a hammer, and the result is a sound wave of frequency $2500 \mathrm{~Hz}$.
From this experiment, calculate the speed of sound in alumin... | The point where the bar is struck is an antinode and the point where it is held a node. With the bar held at the center and its one end struck, the wavelength $\lambda$ is related to its length $L$ by $\lambda=2 L$. Hence the speed of sound propagation in the aluminium bar is
$$
v_{\mathrm{Al}}=\nu \lambda=2 \nu L=2 \... |
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