text:
# Showing annotations¶

Annotations are shapes that can be manually added to the viewer in order to highlight specific objects of interest, either for research purposes or when making a tour/presentation. It’s possible to generate several annotations at once, and you can choose from circles, polygons, and lines.

To add an annotation to the viewer, use either add_circle(), add_polygon(), or add_line() depending on the shape you want to add. These methods will then return an object that can be used to further set the visual properties of the annotation. See the sections below for more information about each specific shape. Note that coordinates must be provided as SkyCoord objects.

## Circles¶

Circles are the simplest kind of annotation and are defined using a center value and a radius. The radius can be assigned in pixels or degrees/arcseconds as preferred:

>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> crc1 = wwt.add_circle(SkyCoord(188, -57, unit=u.deg),
...                       radius=10 * u.degree,
...                       fill=True, fill_color='red')

Note that the properties of the circle can either be passed to add_circle() or can be set/changed on the returned object afterwards:

>>> crc1.fill_color = '#008CA8'

See Circle for a full list of available properties.

Note that if you don’t provide center coordinates, the circle will automatically be generated in the center of your view:

>>> wwt.center_on_coordinates(SkyCoord(190, -55, unit=u.deg))
>>> crc2 = wwt.add_circle(radius=10 * u.pixel, opacity=.4,
fill=True, fill_color='#C4D600')

Once an annotation is no longer needed, it can be removed via its pywwt.Circle.remove() method. The main WorldWide Telescope object (wwt in this case) also has a dedicated method for erasing every existing annotation from view called clear_annotations().

## Lines¶

Lines are added, modified, and removed in a similar way to circles, except that they are defined by a set of positions on the sky rather than a center and a radius. The following example shows how to create a line of custom width that traces the Big Dipper, fetching stellar coordinates through a SkyCoord method and joining them with the concatenate() function from Astropy:

>>> from astropy import units as u
>>> from astropy.coordinates import concatenate, SkyCoord
>>> bd = concatenate((SkyCoord.from_name('Alkaid'),  # stars in Big Dipper
...                   SkyCoord.from_name('Mizar'),
...                   SkyCoord.from_name('Alioth'),
...                   SkyCoord.from_name('Megrez'),
...                   SkyCoord.from_name('Phecda'),
...                   SkyCoord.from_name('Merak'),
...                   SkyCoord.from_name('Dubhe'))
>>> wwt.center_on_coordinates(SkyCoord.from_name('Megrez'))
>>> line = wwt.add_line(bd, width=3 * u.pixel)

This code block results in the following picture.

See Line for a full list of available properties.

The constellation is not completely connected, but as shown below, points can also be added after the line has been initialized using add_point():

>>> line.add_point(SkyCoord.from_name('Megrez'))

The line color can also be changed using either a color name, color hex string, or a tuple of (red, green, blue) values (each in the range [0:1]), e.g.:

>>> line.color = 'salmon'          # html colors
>>> line.color = 'g'               # matplotlib default colors
>>> line.color = '#C4D600'         # hex strings
>>> line.color = (.7, .1, ,.3, .5) # tuples with (or without) opacity

The above example results in the following line annotation:

## Polygons¶

Polygons are made in the same way as lines, but using the add_polygon() method:

>>> polygon = wwt.add_polygon(bd)

See Polygon for a full list of available properties. The main difference compared to lines is that the viewer will automatically connect the last point added to the first in order to form a closed shape. This is still the case if new points are added after the shape is initialized. The closed nature of the shape leads to the availability of a fill color (which can be toggled on and off). Using polygons allows arbitrarily complex shapes to be shown on the sky:


text:
## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  609.10034
Autor:  Erdös, Paul; Pomerance, C.; Sárközy, A.
Title:  On locally repeated values of certain arithmetic functions. II. (In English)
Source:  Acta Math. Hung. 49, 251-259 (1987).
Review:  [For part I, cf. J. Number Theory 21, 319-332 (1985; Zbl 574.10012).]
It seems reasonable, in view of the Erdös-Kac theorem, to conjecture that the number of n \leq x with \nu(n) = \nu(n+1) is of exact order x(log log x)^-. The principal result of the paper is that there is a constant c such that |\nu(n)-\nu(n+1)| \leq c has at least order x(log log x)^- solutions n \leq x. It is stated that c = 3 is admissible. The proof uses the Erdös-Kac theorem to produce many pairs a,b with \nu(a) = \nu(b). One then takes a0,b0 to solve ab0-ba0 = 1, and uses the small sieve to find integers m for which \nu(am+a0), \nu(bm+b0) are both small. One may then take n = b(am+a0). The reviewer [Mathematika 31, 141-149 (1984; Zbl 529.10040)] showed by a different approach that the related equation d(n) = d(n+1) (where d(n) is the divisor function) has infinitely many solutions. It seems likely that one could handle \nu(n) = \nu(n+1) the same way, but as yet there seems no hope of "hybridizing" the two methods so as to solve the original conjecture.
In the final section of the paper, the number of solutions n \leq x of the equation \phi(n) = \phi(n+1), (where \phi(n) is Euler's function) is considered, and shown to be at most x/\exp((log x)1/3). The proof is closely related to C. Pomerance's treatment [J. Reine Angew. Math. 325, 183-188 (1981; Zbl 448.10007)] of amicable numbers.
Reviewer:  D.R.Heath-Brown
Classif.:  * 11N05 Distribution of primes
11N35 Sieves
11A25 Arithmetic functions, etc.
11A25 Arithmetic functions, etc.
Keywords:  sieve method; Erdös-Kac theorem; Euler's function
Citations:  Zbl 542.10037; Zbl 574.10012; Zbl 529.10040; Zbl 448.10007

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag

format: plain


text:
# [Tex/LaTex] Preventing line break in figure caption

captionscitingfloatsline-breakingline-spacing

I have the following figure with the corresponding caption:

As you can see the caption inappropriately breaks the line, which I want to avoid. Optimally without simply reducing the font size of the caption text.

Code as follows:

\documentclass[12pt, a4paper] {article}
\usepackage[skip=10pt, labelfont=bf, labelsep=space]{caption}
\usepackage{tikz}
\usepackage[numbers,sort,authoryear]{natbib}

\begin{document}

\newcommand{\mytab}[1]{%
\begin{tabular}{@{}c@{}}
#1
\end{tabular}
}
\begin{figure} [h!]
\label{fig: timeline}
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (11,0);
\foreach \x in {0.8,4,5.5,7,10.2}
\draw(\x cm,3pt) -- (\x cm, -3pt);
\draw (0.8,0) node[below=3pt] {$T_0$};
\draw (4,0) node[below=3pt] {$T_1$};
\draw (5.5,0) node[below=3pt] {$0$};
\draw (7,0) node[below=3pt] {$T_2$};
\draw (10.2,0) node[below=3pt] {$T_3$};
\draw (2.35,0) node[above=12pt, align=center] {
$\left(\mytab{estimation \\ window}\right]$};
\draw (5.5,0) node[above=12pt, align=center]{
$\left(\mytab{event \\ window}\right]$};
\draw(8.65,0) node[above=12pt, align=center]{
$\left(\mytab{post-event \\ window}\right]$};
\end{tikzpicture}
\end{center}
\caption{Time Line for an Event Study (\cite{campbell1996}, p. 157})
\end{figure}

\bibliographystyle{agsm}
\bibliography{./references}
\end{document}

• Since you're (i) using the agsm bibliography style, which is distributed with the harvard citation management package and (ii) loading the natbib package rather than the harvard package, you should load the har2nat package as well. As its name indicates, it "translates" various macros that are defined by the harvard package (and which are used in agsm.bst) into equivalent natbib macros.

• You're currently loading the natbib package with the options numbers, sort, and authoryear. You should drop the first two options: The agsm style is meant for authoryear-style citation callouts -- pointless to try to use it with numeric-style citation callouts; the sort option only makes sense if numbers is the callout style.

• Now to the main point of your query: You ought to write the \caption command as

\caption{Time Line for an Event Study \citep[p.~157]{campbell1996}}

Note the use of the ~ ("tie") character, which acts like an unbreakable space character.

• Two minor points: (i) The \label instruction should come after the \caption instruction -- especially if you want to be able to cross-reference the figure elsewhere in the document. (ii) Use of \begin{center}...\end{center} adds a lot of (vertical) whitespace; use the \centering directive instead.

\documentclass[12pt, a4paper]{article}
\usepackage{filecontents}
\begin{filecontents}{references.bib}
@book{campbell1996,
author = "John Y. Campbell and Andrew W. Lo and A. Craig McKinlay",
title  = "The Econometrics of Financial Markets",
year   = 1996,
publisher = "Princeton University Press",
}
\end{filecontents}
\usepackage[skip=10pt, labelfont=bf, labelsep=space]{caption}
\usepackage{tikz}
\usepackage[authoryear]{natbib}
\usepackage{har2nat}
\usepackage{hyperref}

\newcommand{\mytab}[1]{%
\begin{tabular}{@{}c@{}}
#1
\end{tabular}
}

\begin{document}

\begin{figure} [h!]
\centering
\begin{tikzpicture}
\draw (0,0) -- (11,0);
\foreach \x in {0.8,4,5.5,7,10.2}
\draw(\x cm,3pt) -- (\x cm, -3pt);
\draw (0.8,0) node[below=3pt] {$T_0$};
\draw (4,0) node[below=3pt] {$T_1$};
\draw (5.5,0) node[below=3pt] {$0$};
\draw (7,0) node[below=3pt] {$T_2$};
\draw (10.2,0) node[below=3pt] {$T_3$};
\draw (2.35,0) node[above=12pt, align=center] {
$\left(\mytab{estimation \\ window}\right]$};
\draw (5.5,0) node[above=12pt, align=center]{
$\left(\mytab{event \\ window}\right]$};
\draw(8.65,0) node[above=12pt, align=center]{
$\left(\mytab{post-event \\ window}\right]$};
\end{tikzpicture}
\caption{Time Line for an Event Study \protect\citep[p.~157]{campbell1996}}
\label{fig:timeline}
\end{figure}

A cross-reference to \autoref{fig:timeline}.

\bibliographystyle{agsm}
\bibliography{./references}
\end{document}


text:
# Give an example of a series such that $\sum a_n$ is convergent but $\sum a_{3n}$ is divergent [duplicate]

This question already has an answer here:

Give an example of a series such that $\sum a_n$ is convergent but $\sum a_{3n}$ is divergent.

I am trying to construct an example such that $\sum a_n$ is convergent but $\lim a_{3n}$ is not zero. But I could not find such example. Give me some hints.

## marked as duplicate by Micah, Namaste, Dando18, B. Goddard, StrantsMay 31 '18 at 23:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• By "convergent" do you mean absolutely convergent, or what? – kimchi lover Dec 10 '17 at 16:32
• @Rwitam Jama, your point it would not be useful, since the series converges $a_n$ tends to zero, so is every subsequence. – DonQuixote Dec 10 '17 at 16:32
• Hint: start with the alternating harmonic sequence, so $a_{3n}=\frac {(-1)^n}n$. That diverges. Now add terms to make the full sequence converge (conditionally). – lulu Dec 10 '17 at 16:33
• @lulu If $a_{3n} = (-1)^n/n$ then $\sum a_{3n}$ converges. – zhw. Dec 10 '17 at 16:59
• @zhw Right, of course. So make $a_{3n}=\frac 1n$, even simpler. – lulu Dec 10 '17 at 17:03

## 1 Answer

$$1/2 + 1/2 -2/2 +1/3 +1/3 -2/3 + 1/4 + 1/4- 2/4 + 1/5 + 1/5 -2/5 + \cdots$$


text:
## chain event graphs [RSS Midlands seminar]

Posted in pictures, Statistics, University life with tags , , , , , , , , , , on October 16, 2013 by xi'an

Last evening, I attended the RSS Midlands seminar here in Warwick. The theme was chain event graphs (CEG), As I knew nothing about them, it was worth my time listening to both speakers and discussing with Jim Smith afterwards. CEGs are extensions of Bayes nets with originally many more nodes since they start with the probability tree involving all modalities of all variables. Intensive Bayesian model comparison is then used to reduce the number of nodes by merging modalities having the same children or removing variables with no impact on the variable of interest. So this is not exactly a new Bayes net based on modality dummies as nodes (my original question). This is quite interesting, esp. in the first talk illustration of using missing value indicators as a supplementary variable (to determine whether or not data is missing at random). I also wonder how much of a connection there is with variable length Markov chains (either as a model or as a way to prune the tree). A last vague idea is a potential connection with lumpable Markov chains, a concept I learned from Kemeny & Snell (1960): a finite Markov chain is lumpable if by merging two or more of its states it remains a Markov chain. I do not know if this has ever been studied from a statistical point of view, i.e. testing for lumpability, but this sounds related to the idea of merging modalities of some variables in the probability tree…

## A repulsive random walk

Posted in R, Statistics with tags , , , , on May 28, 2010 by xi'an

Matt Asher posted an R experiment on R-bloggers yesterday simulating the random walk

$x_{t+1} = x_t + \varepsilon_t / x_t$

which has the property of avoiding zero by quickly switching to a large value as soon as $x_t$ is small. He was then wondering about the “convergence” of the random walk given that it moves very little once $x_t$ is large enough. The values he found for various horizons t seemed to indicate a stable regime.

I reran the same experiment as Matt in a Monte Carlo perspective, using the R program

resu=matrix(0,ncol=100,nrow=25)
sampl=rnorm(100)
for (i in 1:25){
for (t in 2^(i-1):2^i) sampl=sampl+rnorm(100)/sampl
resu[i,]=sampl
}
boxplot(as.data.frame(t(abs(resu))),name=as.character(1:25),col="wheat3")

The outcome of this R code plotted above shows that the range and the average of the 100 replications is increasing with t. This behaviour indicates a transient behaviour of the Markov chain, which almost surely goes to infinity and never comes back (because at infinity the variance is zero). Another indication for transience is shown by the fact that $x_t$ comes back to the interval (-1,1) with probability $\Phi(-|x_t|)$, a probability which goes to zero with $x_t$. As suggested to me by Randal Douc, this transience can be established rigorously by considering

$x_{t+1}^2 = x_t^2 + 2\epsilon_t + \epsilon_t^2/x_t^2 > x_t^2 + 2\epsilon_t>2\sum_{i=1}^t \epsilon_t$

which is thus bounded from below by a null recurrent process, which almost surely goes to infinity. Therefore the above Markov chain cannot have a stationary distribution or even a stationary measure: it almost surely goes to (plus or minus) infinity.

## Computational Statistics

Posted in Books, R, Statistics with tags , , , , , , , , , , , , , , , , , , on May 10, 2010 by xi'an

Do not resort to Monte Carlo methods unnecessarily.

When I received this 2009 Springer-Verlag book, Computational Statistics, by James Gentle a while ago, I briefly took a look at the table of contents and decided to have a better look later… Now that I have gone through the whole book, I can write a short review on its scope and contents (to be submitted). Despite its title, the book aims at covering both computational statistics and statistical computing. (With 752 pages at his disposal, Gentle can afford to do both indeed!)

The book Computational Statistics is separated into four parts:

• Part I: Mathematical and statistical preliminaries.
• Part II: Statistical Computing (Computer storage and arithmetic.- Algorithms and programming.- Approximation of functions and numerical quadrature.- Numerical linear algebra.- Solution of nonlinear equations and optimization.- Generation of random numbers.)
• Part III: Methods of Computational Statistics (Graphical methods in computational statistics.- Tools for identification of structure in data.- Estimation of functions.- Monte Carlo methods for statistical inference.- Data randomization, partitioning, and augmentation.- Bootstrap methods.)
• Part IV: Exploring Data Density and Relationship (Estimation of probability density functions using parametric models.- Nonparametric estimation of probability density functions.- Statistical learning and data mining.- Statistical models of dependencies.)

Computational inference, together with exact inference and asymptotic inference, is an important component of statistical methods.

The first part of Computational Statistics is indeed a preliminary containing essentials of math and probability and statistics. A reader unfamiliar with too many topics within this chapter should first consider improving his or her background in the corresponding area! This is a rather large chapter, with 82 pages, and it should not be extremely useful to readers, except to signal deficiencies in their background, as noted above. Given this purpose, I am not certain the selected exercises of this chapter are necessary (especially when considering that some involve tools introduced much later in the book).

The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different .

The second part of Computational Statistics is truly about computing, meaning the theory of computation, i.e. of using computers for numerical approximation, with discussions about the representation of numbers in computers, approximation errors, and of course random number generators. While I judge George Fishman’s Monte Carlo to provide a deeper and more complete coverage of those topics, I appreciate the need for reminding our students of those hardware subtleties as they often seem unaware of them, despite their advanced computer skills. This second part is thus a crash course of 250 pages on numerical methods (like function approximations by basis functions and …) and on random generators, i.e. cover the same ground as Gentle’s earlier books, Random Number Generation and Monte Carlo Methods and Numerical Linear Algebra for Applications in Statistics, while the more recent Elements of Computational Statistics looks very much like a shorter entry on the same topics as those of Parts III IV of Computational Statistics. This part could certainly sustain a whole semester undergraduate course while only advanced graduate students could be expected to gain from a self-study of those topics. It is nonetheless the most coherent and attractive part of the book. It constitutes a must-read for all students and researchers engaging into any kind of serious programming. Obviously, some notions are introduced a bit too superficially, given the scope of this section (as for instance Monte Carlo methods, in particular MCMC techniques that are introduced in less than six pages), but I came to realise this is the point of the book, which provides an entry into “all” necessary topics, along with links to the relevant literature (if missing Monte Carlo Statistical Methods!). I however deplore that the important issue of Monte Carlo experiments, whose construction is often a hardship for students, is postponed till the 100 page long appendix. (I suspect that students do not read appendices is another of those folk theorems!)

Monte Carlo methods differ from other methods of numerical analysis in yielding an estimate rather than an approximation.

The third and fourth parts of the book cover methods of computational statistics, including Monte Carlo methods, randomization and cross validation, the bootstrap, probability density estimation, and statistical learning. Unfortunately, I find the level of Part III to be quite uneven, where all chapters are short and rather superficial because they try to be all-encompassing. (For instance, Chapter 8 includes two pages on the RGB colour coding.) Part IV does a better job of presenting machine learning techniques, if not with the thoroughness of Hastie et al.’s The Elements of Statistical Learning: Data Mining, Inference, and Prediction… It seems to me that the relevant sections of Part III would have fitted better where they belong in Part IV. For instance, Chapter 10 on estimation of functions only covers the evaluation of estimators of functions, postponing the construction of those estimators till Chapter 15. Jackknife is introduced on its own in Chapter 12 (not that I find this introduction ultimately necessary) without the bootstrap covered in eight pages in Chapter 13 (bootstrap for non-iid data is dismissed rather quickly, given the current research in the area). The first chapter of Part IV covers some (non-Bayesian) estimation approaches for parametric families, but I find this chapter somehow superfluous as it does not belong to the computational statistic domain (except as an approximation method, as stressed in Section 14.4). While Chapter 16 is a valuable entry on clustering and data-analysis tools like PCA, the final section on high dimensions feels out of context (and mentioning the curse of dimensionality only that close to the end of the book does not seem appropriate). Chapter 17 on dependent data is missing the rich literature on graphical models and their use in the determination of dependence structures.

Programming is the best way to learn programming (read that again) .

In conclusion, Computational Statistics is a very diverse book that can be used at several levels as textbook, as well as a reference for researchers (even if as an entry towards further and deeper references). The book is well-written, in a lively and personal style. (I however object to the reduction of the notion of Markov chains to discrete state-spaces!) There is no requirement for a specific programming language, although R is introduced in a somewhat dismissive way (R most serious flaw is usually lack of robustness since some [packages] are not of high-quality) and some exercises start with Design and write either a C or a Fortran subroutine. BUGS is not mentioned at all. The appendices of Computational Statistics also contain the solutions to some exercises, even though the level of detail is highly variable, from one word (“1″) to one page (see, e.g., Exercise 11.4). The 20 page list of references is preceded by a few pages on available journals and webpages, which could get obsolete rather quickly. Despite the reservations raised above about some parts of Computational Statistics that would benefit from a deeper coverage, I think this book is a reference book that should appear in the shortlist of any computational statistics/statistical computing graduate course as well as on the shelves of any researcher supporting his or her statistical practice with a significant dose of computing backup.


text:
# Math Help - differential equations

1. ## differential equations

How would you do this:
solve:
y^2dy - 3x^5dx = 0
the answer in the book is 2y^3 = 3x^6 + C
I am clueless how to do these types of problems. The book doesnt explain it clearly and my notes arent helping me. Thanks.

2. So $y^2 \ dy = 3x^{5} \ dx$.

Then $\frac{y^3}{3} = \frac{x^6}{2} + c$

Or $2y^3 = 3x^6 + 6c$

Let $6c = C$ and we get

$2y^3 = 3x^6 + C$

This is also separation of variables.

3. Originally Posted by davecs77
How would you do this:
solve:
y^2dy - 3x^5dx = 0
the answer in the book is 2y^3 = 3x^6 + C
I am clueless how to do these types of problems. The book doesnt explain it clearly and my notes arent helping me. Thanks.
Exactly like the other one:

$
y^2dy - 3x^5dx = 0
$

so:

$
\int y^2dy - \int 3x^5dx = K
$

$
\frac{y^3}{3} - \frac{x^6}{2} = K
$

Now multiply through by 6:

$
2y^3 - 3x^6 = C
$

(the constant $K$ has now been replaced by the constant $C=6K$)

Reaarange and you have the result.

RonL


text:
# Reciprocal Trig Functions

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

## Reciprocal Trig Functions

Remember how we said that $\sin ^{-1}$, $\cos ^{-1}$ and $\tan ^{-1}$ aren’t quite the same as $\dfrac{1}{\textcolor{blue}{\sin}}$, $\dfrac{1}{\textcolor{limegreen}{\cos}}$ and $\dfrac{1}{\textcolor{red}{\tan}}$? Here’s why…

A Level

## Introducing $\mathbf{\cosec}$, $\mathbf{\sec}$ and $\mathbf{\cot}$

We have three definitions, now:

• Cosecant: $\textcolor{blue}{\cosec x} = \dfrac{1}{\textcolor{blue}{\sin x}}$
• Secant: $\textcolor{limegreen}{\sec x} = \dfrac{1}{\textcolor{limegreen}{\cos x}}$
• Cotangent: $\textcolor{orange}{\cot x} = \dfrac{1}{\textcolor{red}{\tan x}}$

We’ve given these three reciprocal functions a special name so we don’t get them confused with the inverse functions.

The easiest way to remember which is which, is to look at the third letter of each function, so, Cosecant $= \dfrac{1}{\textcolor{blue}{\sin}}$, Secant $= \dfrac{1}{\textcolor{limegreen}{\cos}}$ and Cotangent $= \dfrac{1}{\textcolor{red}{\tan}}$.

A Level

## In Graphical Form

Now, we’ll represent the graph of $\textcolor{blue}{y = \cosec x}$:

Notice how there are now asymptotes at $x = -180°, 0°, 180°$.

Well, $\textcolor{blue}{\cosec x} = \dfrac{1}{\sin x}$, so it will not have values where $\sin x = 0°$, i.e. where $x = -180°, 0°, 180°$.

Notice, also, how there are no values between $y = 1$ and $y = -1$. Think on that for a while…

Anyway, here’s the graph for $\textcolor{limegreen}{y = \sec x}$:

See how the asymptotes are now at $x = -270°, -90°, 90°, 270°$. Again, this is because we’ll find that $\cos x = 0$ at these points.

Again, there are no values between $y = 1$ and $y = -1$

To complete the set, here’s $\textcolor{orange}{y = \cot x}$:

This time, there are no undefined values for $y$, but we do have vertical asymptotes at $x = -180°, 0°, 180°$ again.

A Level

## Domains, Ranges and Key Values

You might want to reference from this table of ranges and domains:

and here’s some key points that you might need to know.

In case you were wondering, $(-1,1)$ means the value is between but not including $-1$ and $1$.

A Level

## Example Questions

$\sec x = 2$ means that $\cos x = \dfrac{1}{2}$, which has solutions at $x = 60°, 300°, 420°, 660°$.

$\cosec x = \dfrac{1}{\sin x}$ and $\sec x = \dfrac{1}{\cos x}$.

Since $\sin x$ and $\cos x$ have a minimum of $-1$ and maximum of $1$, we conclude that $\cosec x$ and $\sec x$ have a positive minimum of $\dfrac{1}{1} = 1$, and a negative maximum of $\dfrac{1}{-1} = -1$.

To have $\cosec x$ or $\sec x$ between $-1$ and $1$, we require $\sin x$ or $\cos x$ to be greater than $1$, or less than $-1$, which is not possible.

Rewrite this as

$\dfrac{\sqrt{3}}{\cos x} = \dfrac{1}{\sin x}$

We can rearrange to get

$\tan x = \dfrac{1}{\sqrt{3}}$

This gives

$x = \tan ^{-1}\dfrac{1}{\sqrt{3}} = \dfrac{-5\pi}{6}, \dfrac{\pi}{6}$

A Level

A Level

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text:
mucosa

0th

Percentile

Cells in Gastric Mucosa

A bivariate inhomogeneous point pattern, giving the locations of the centres of two types of cells in a cross-section of the gastric mucosa of a rat.

Keywords
datasets
Usage
data(mucosa)
Details

This point pattern dataset gives the locations of cell centres in a cross-section of the gastric mucosa (mucous membrane of the stomach) of a rat. The rectangular observation window has been scaled to unit width. The lower edge of the window is closest to the outside of the stomach.

The cells are classified into two types: ECL cells (enterochromaffin-like cells) and other cells. There are 86 ECL cells and 807 other cells in the dataset. ECL cells are a type of neuroendocrine cell which synthesize and secrete histamine. One hypothesis of interest is whether the spatially-varying intensities of ECL cells and other cells are proportional.

The data were originally collected by Dr Thomas Berntsen. The data were discussed and analysed in Moller and Waagepetersen (2004, pp. 2, 169).

The associated object mucosa.subwin is the smaller window to which the data were restricted for analysis by Moller and Waagepetersen.

format

An object of class "ppp", see ppp.object. This is a multitype point pattern with two types of points, ECL and other.

source

Dr Thomas Berntsen and Prof Rasmus Waagepetersen.

References

Moller, J. and Waagepetersen, R. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC.

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• mucosa
• mucosa.subwin
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plot(mucosa, chars=c(1,3), cols=c("red", "green"))
plot(mucosa.subwin, add=TRUE, lty=3)
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## samtab Solve for b2 in A = 1/2v h(b1+b2), if A = 16, h = 4, and b1 = 3. b2 =_______ a0 please help me:( one year ago one year ago

1. kirbykirby

$A=\frac{1}{2v}h(b _{1}+b _{2})?$

2. kirbykirby

I want to be sure that the equation is right first...


text:
# Derivation of momentum & energy formulas

by DrGreg
Tags: derivation, energy, formulas, momentum
 Sci Advisor PF Gold P: 1,841 Most of the texts I've seen (admittedly not a large number) which introduce the concept of relativistic momentum usually just pull a formula out of nowhere and then prove the formula must be correct. There is nothing technically wrong with this approach, but it's aesthetically unsatisfying, as it doesn't show how you would get the formula if you didn't already know the answer. The few occasions where I've seen a derivation that didn't assume an answer (e.g. in Einstein's own papers) contain some rather ugly and tedious mathematics. I thought I'd have a go at a more elegant method, and after some weeks of effort in my spare time, here it is. I wouldn't be at all surprised if someone else hasn't published something similar to this before, so I claim no originality. I do assume a fair amount of background knowledge. You will need to know about rapidity and how to add it. You need familiarity with hyperbolic functions and their derivatives, partial differentiation and the solution of second-order differential equations. I am confident of the logic that leads from my assumptions to my conclusions. The reason I am posting this is to ask whether the assumptions that I make are reasonable in this context. Could I simplify my argument still further with even better assumptions? I will restrict consideration to a single dimension of space, i.e. motion along a single straight line in the absence of gravity. We already know the Newtonian definitions of mass, momentum, force and energy. We want to derive relativistic versions which are compatible with the Newtonian versions at low speeds, but which are also consistent with the postulates of relativity. General Assumptions (A1) The mass of a particle is invariant (i.e. independent of relative motion of particle and observer). (A2) Total energy is conserved in a closed system. (A3) Total momentum is conserved in a closed system.Otherwise neither of these concepts would be useful. (A4) The energy of any particle is proportional to its mass. (A5) The momentum of any particle is proportional to its mass.This follows from conservation. Consider the limit as any two particles collide and coalesce at a relative speed approaching zero. In this Newtonian limit, mass is additive, and so too are momentum and energy. We will also assume that the Newtonian relationships between force, momentum and energy (A6)....$$F = \frac{dp}{dt}$$and (A7)....$$E = \int F dx$$are still valid (relative to any inertial frame) in relativity. The specific scenario Consider two particles of equal mass which collide and coalesce to form a single particle, with no loss of energy to the surroundings. (A8) Measured in the rest frame of the final particle, the two initial particles have equal but opposite velocities.On grounds of symmetry. Note also that as mass is invariant, the mass of the final particle depends only on the two initial particles and their relative velocity and is independent of the motion of any observer. Mathematical formulation I will measure motion using rapidity $\phi$ instead of velocity $v = c \tanh \phi$. This will simplify the maths because the composition law for rapidity (in one spatial dimension) is just ordinary addition. Thus, from (A4) and (A5), we postulate that momentum p and energy E are given by (F1).....$$p = m c F(\phi)$$ (F2).....$$E = m c^2 G(\phi)$$for some smooth functions F and G to be determined. In the Newtonian limit as $\phi \rightarrow 0$ we know these functions are approximately $F(\phi) = \phi + O(\phi^2)$ and $G(\phi) = K + \tfrac{1}{2}\phi^2 + O(\phi^3)$, for some unknown constant K. To be more precise: (F3).....$$F(0) = 0$$ (F4).....$$F'(0) = 1$$ (F5).....$$G'(0) = 0$$ (F6).....$$G''(0) = 1$$So, considering our two colliding particles of mass m as described above, moving with rapidities $\phi$ and $-\phi$ relative to the final rest frame, let the mass of the final particle be (F7).....$$M = 2 m H(\phi)$$for some smooth function H to be determined. In the Newtonian limit, mass is conserved so (F8).....$$H(0) = 1$$Note also that (F9).....$$H(\phi) > 0$$We have, relative to the final rest frame, conservation of momentum and energy (F10).....$$m c F(\phi) + m c F(-\phi) = 2 m c H(\phi) F(0)$$ (F11).....$$m c^2 G(\phi) + m c^2 G(-\phi) = 2 m c^2 H(\phi) G(0)$$Note that either of the above equations implies that H is an even function, and so, in particular, (F12).....$$H'(0) = 0$$More generally, relative to a frame in which the final particle has rapidity $\psi$, we have (F13).....$$m c F(\psi + \phi) + m c F(\psi - \phi) = 2 m c H(\phi) F(\psi)$$ (F14).....$$m c^2 G(\psi + \phi) + m c^2 G(\psi - \phi) = 2 m c^2 H(\phi) G(\psi)$$making use of the additivity of rapidities. To summarise, both F and G are solutions for f of the equation (F15).....$$f(\psi + \phi) + f(\psi - \phi) = 2 H(\phi) f(\psi)$$each satisfying the appropriate boundary conditions as above. (Due to length, the derivation will continue in the next post) Attached Thumbnails
 Sci Advisor PF Gold P: 1,841 (Continued from previous post) Relation between momentum and energy From (A6) and (A7) we have $$\frac{dp}{dt} = F = \frac{dE}{dx} = \frac{dE/dt}{dx/dt}$$ $$\frac{dp}{d\phi} \frac{d\phi}{dt} = \frac{(dE/d\phi) (d\phi/dt) }{c \tanh \phi}$$Hence $$\frac{dp}{d\phi} c \tanh \phi = \frac{dE}{d\phi}$$whenever $d\phi/dt \neq 0$. On grounds of continuity it must always be true. Substituting (F1) and (F2) we have (F16).....$$F'(\phi) \tanh \phi = G'(\phi)$$ Solving the equations This section is just maths, no physics involved. Apply $\partial^2/\partial\phi^2$ to (F15): (M1).....$$f''(\psi + \phi) + f''(\psi - \phi) = 2 H''(\phi) f(\psi)$$Apply $\partial^2/\partial\psi^2$ to (F15): (M2).....$$f''(\psi + \phi) + f''(\psi - \phi) = 2 H(\phi) f''(\psi)$$Equate the two and divide by $H(\phi)$ which is never zero. (M3)....$$f''(\psi) = \frac{H''(\phi)}{H(\phi)} f(\psi)$$As the left hand side is independent of $\phi$ then $$\frac{H''(\phi)}{H(\phi)}$$must be some constant k i.e. (M4)....$$H''(\phi) = k H(\phi)}$$ (M5)....$$f''(\psi) = k f(\psi)}$$As the solution for f = G must satisfy G''(0) = 1, we can reject k = 0. If k < 0 the solution for H is (M6)....$$H(\phi) = H(0) \cos(\phi\sqrt{|k|}) + \frac{H'(0)}{\sqrt{|k|}} \sin(\phi\sqrt{|k|})$$which is incompatible with the requirement that $H(\phi) > 0$. Therefore $k = \alpha^2 > 0$, for some non-zero constant $\alpha$ yet to be determined, giving the solutions (M7)....$$H(\phi) = H(0) \cosh(\alpha\phi) + \frac{H'(0)}{\alpha}\sinh(\alpha\phi) = \cosh(\alpha\phi)$$ (M8)....$$f(\psi) = f(0) \cosh(\alpha\psi) + \frac{f'(0)}{\alpha} \sinh(\alpha\psi)$$When solving for momentum f = F, we have F(0)= 0 and F'(0) = 1, whence (M9)....$$F(\psi) = \frac{1}{\alpha} \sinh (\alpha \psi)$$When solving for momentum f = G, we have G'(0)= 0 and G''(0) = 1, so $G(0) = 1/\alpha^2$, whence (M10)....$$G(\psi) = \frac{1}{\alpha^2} \cosh (\alpha \psi)$$All that remains is to determine the value of $\alpha$, and for this we use (F16) which gives (M11)....$$\cosh (\alpha \phi) \tanh \phi = \frac{1}{\alpha} \sinh (\alpha \phi)$$i.e. (M12)....$$\tanh \phi = \frac{1}{\alpha} \tanh (\alpha \phi)$$and so $\alpha = 1$. Conclusion $$p = m c \sinh \phi$$ $$E = m c^2 \cosh \phi$$which are the standard equations for momentum and energy expressed as functions of rapidity. Converted into velocity they take the more familiar form $$p = \gamma m v$$ $$E = \gamma m c^2$$ Reminder My main question is, are assumptions (A1) to (A8) above sensible?


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Below is direct formula for finding roots of quadratic equation. The concept of solving a quadratic equation through the use of factorisation is explained. from simplified quadratic formula to original equati. I’m giving you the true answers because I don’t want you to listen to these stupid people lying to you. #3035: Do NOT solve the equation, use the discriminant to answer the question. Introduction to Quadratic Equations. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Download and Practice Quadratic equations CAT Problems PDF. (b) Find the vertex of the parabola. Section 8: Summary of Functions Section 8 – Topic 1 Comparing Linear, Quadratic, and Exponential Functions – Part 1 ! Complete the table below to describe the characteristics of linear functions. a can't be 0. NCERT solutions for Maths Quadratic Equations Download as PDF. Derive the quadratic formula from this form. 1 Answer to Determine the pH of a 0. Mathematics NCERT Grade 10, Chapter 4 Quadratic Equations. Quadratic Equations 50 Qns PDF Download : Practice and Master the Topic. Solving quadratic equations worksheets pdf questions with answers included. Format of the Quadratic Formula. Solving quadratic equations worksheet 1 works at grade 4 for foundation gcse aimed at year 9 students. In order to learn how to solve quadratic equations by four different methods, please follow this tutorial; it's detailed and offers plenty of worked examples and practice questions. By using this website, you agree to our Cookie Policy. Solution: Given quadratic equation is 4x 2 + 8x – k = 0 …. The quadratic formula sta _ tes the solutions. The four methods of factorisation are revised and how to solve for an unknown variable once the quadratic equation is factorised. Select your options in the form below and click on the 'Make Worksheet' button. “For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. Please watch the video where I will do questions 1 and 2 and YOU ARE TO DO #3,4,5,6 and check your answers for understanding. Using the quadratic formula features several examples with a, b and c colour coded to make it more visual. Mathematics (Linear) - 1MA0 SIMULTANEOUS EQUATIONS WITH A QUADRATIC Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. Representing quadratic functions, factoring quadratic expressions, patterns of change, effect of parameters. Solve real-life problems. Completing the square starts with a multiple choice activity where students identify which is the correct factorisation given a variety of questions. This function passes through the point ( ) The factored format of a quadratic equation is: ( )( ). 0* Students solve a quadratic equation by factoring or completing the square. 200 Important Quadratic Equation Questions PDF Free Download for IBPS PO, SBI PO, RRB PO and Various Other Bank PO & Clerk Exam. A mathematician has derived an easier way to solve quadratic equation problems, according to MIT's Technology Review. Maths Class 10 Important Questions are very helpful to score high marks in board exams. b)we have a quadratic to solve, we can factorise, complete the square or use the quadratic formula. Unit 2 Test Multiple Choice Identify the choice that best completes the statement or answers the question. 5: Quadratic Applications Use the following information to answer questions 1 and 2. So that you can easily get the logic of question. A quadratic equation contains terms. 114 Equation of a straight line F and H C 106 115 Simultaneous Equations Graphs F and H C 107 116 Drawing Quadratic Graphs F and H C 108 117 Real-life Graphs F and H C 109 118 Pythagoras' Theorem F and H C 110 119 Pythagoras - line on a graph F and H C 111 120 Surface area of cuboids F and H C 112 121 Surface area of triangular prisms F and H C 113. 12x2 2+ 7x = 12 → 12x + 7x - 12 = 0 Step 2: Identify the values of a, b, and c, then plug them into the quadratic formula. Worksheet Solving Quadratic Inequalities and Quadratic Word Problems. Questions include solving Quadratics questions involving: – find the roots of an equation – solving word problems – converting to vertex form – writing the quadratic equation given the roots – determine the quadratic equation given the vertex and compression factor. Format of the Quadratic Formula. Give answer (2), If x>y, Give answer (3), If xI usually print these questions as an A5 booklet and issue them in class or give them out as a homework. Past paper exam questions organised by topic for Edexcel GCSE Maths. A real number is said to be a root of the quadratic equation ax2 + bx + c = 0, a 0. Show that α^4-65α+114=0. (b) Find the vertex of the parabola. Can solve quadratic equations by: o Factorising o Using the quadratic formula o Using the K-substitution Can analyse the nature of the roots of a quadratic equation. Coordinate Geometry Expansions & Factorisation - pdf Financial Arithmetic - pdf. Quadratic Equations [email protected] Get NCERT Solutions for all exercise questions and examples of Chapter 4 Class 10 Quadratic Equations free at Teachoo. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. In order to learn how to solve quadratic equations by four different methods, please follow this tutorial; it's detailed and offers plenty of worked examples and practice questions. a Worksheet by Kuta Software LLC. There are many ways to solve quadratics. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they. We've seen factorising a bunch of terms into one bracket by taking all the common factors outside the bracket - if you haven't seen this, you should revise factorising. 5×2 + 6x = 3. Note that. 12x 2 t 40x + 32 = 0 II. Solution : Short Answer Type Questions I [2 Marks] Question 31. The theory involved in this topic is very simple and students should be comfortable with the some basic formulas and concepts. docx), PDF File (. These Worksheets for Grade 10 Quadratic Equation, class assignments and. For instance, physicists can model the height of an object over time t with quadratic equations. Jun 25 2020 kuta-software-infinite-algebra-1-using-the-quadratic-formula-answers 1/6 PDF Drive - Search and download PDF files for free. Lectures #4. Quadratic Equations: Very Difficult Problems with Solutions. In the function fx 2 2 53 3 2 3 xx xx (a) Use the quadratic formula to find the x-intercepts of the function, and then use a calculator to round these answers to the nearest tenth. Quadratic Equation Questions is an an essential part for Competitive Exams like Banking, Insurance, SSC and Railways Exams. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Topics like Quadratic Equation can assist you to surge throughout safely if you master on this topic. I recommend Gauss's third proof with modifications by Eisenstein. Generally, we use the quadratic formula to find the solutions when we are unable to find them by factoring or graphing (decimal answers). Find the solutions to the quadratic equation $$x^2-13x+12=0$$. Format of the Quadratic Formula. 5×2 + 6x – 3 = 0 Since the left side cannot be factored, use the quadratic formula to solve the equation, which is written in the form ax2 + bx + c = 0. Solve Applications Modeled by Quadratic Equations. A (0, 1); maximum C (0, 1); minimum B (0, 0); minimum D (0, 0); maximum ____ 2 Which of the quadratic functions has the narrowest graph? A. A quadratic equation is an equation of degree 2, which means it has at least one squared term, and no term has an exponent greater than 2. Remember that log a M =x means exactly the same thing as a x = M , that is, "log a M is the number to which you raise a in order to get M. Indicate the answer choice that best completes the statement or answers the question. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. 2 Introduction A quadratic equation is one which can be written in the form ax2 + bx + c = 0 where a, b and c are numbers, a 6= 0 , and x is the unknown whose value(s) we wish to ﬁnd. [2]:202–207 If one is given a quadratic equation in the form x2 + bx. ASSESSSMENT TASK OVERVIEW & PURPOSE: The student will examine the path of a projectile and explain the motion using a quadratic function. Then I have students solve the same Quadratic Equation without a calculator to verify that their solution is correct (MP3). This worksheet is a PDF document. However, these problems lead to quadratic equations. Identify the quadratic, linear, and constant terms. The same method can also be applied to non‑monic quadratic equations. We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Write the equation in standard form. Quadratic Equation Quiz Quadratic Equation Quiz. (i) (ii) (iii) Ans. Download and Practice Quadratic equations CAT Problems PDF. These roots are either REAL, EQUAL or COMPLEX *. y 2&'x 32 6 b. JEE Main & Advanced Mathematics Quadratic Equations Question Bank done Solution of quadratic inequations and Miscellaneous equations question_answer 1) The number of real values of x for which the equality $\left| \,3{{x}^{2}}+12x+6\, \right|=5x+16$ holds good is [AMU 1999]. x2 = 4x The pupils need to use their own strategies to solve these equations. Questions and Answers 1. This quadratic equations formulas for CAT pdf covers all the important formulas and concepts related to Quadratic Equations. The other student made a mistake in the coefficient of the constant term got roots of -1 and 4. Quadratic equations. This would be a great lesson to review, as you will see a lot of vocabulary that relates to graphing parabolas. Meanings of solutions discussed with images of graphs included. Keep in mind that the questions below in. Many quadratic equations cannot be solved by factoring. Since y = mx + b is an equation of degree one, the quadratic function, y = ax2 + bx + c represents the next level of algebraic complexity. 5 Explain, using examples, why the graph of a quadratic function may have zero, one or two x-intercepts. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks. When these functions are graphed, they create a parabola which looks like a curved "U" shape on the graph. While you might occassionally get an easy question that provides the equation, many a times questions from. In this unit, we discovered how to use a table of values in order to graph a quadratic function. The most basic parabola is obtained from the function. Each one has model problems worked out step by step practice problems challenge proglems quadratic equation worksheets with answer keys. Below is direct formula for finding roots of quadratic equation. If the quadratic equation px 2 – 2√5px + 15 = 0 has two equal roots, then find the value of p. ____ 1 Identify the vertex of the graph. Extra Questions for Class 10 Maths Chapter 4 Quadratic Equations. Quadratic Exam 2018 Quadratic Formula: = − ±√ 2−4 2 Quadratic Equations Name _____ November 2018 Multiple Choice & Numeric Response. com provides you Free PDF download of NCERT Exemplar of Class 10 Maths chapter 4 Quadratic Equations solved by expert teachers as per NCERT (CBSE) book guidelines. Introduction - Algebra I Following the questions is a table that gives the correct answer for each question, the content standard that each question is measuring, and the year each question last appeared on the test. x(x − 3) = 4x − 6 2. For every quadratic equation, there can be one or more than one solution. 2 points each Answer is correct. Passport to Advanced Math questions require a high comfort level working with quadratic equations and expressions, including foiling and factoring. In this chapter, you will study quadratic equations, and various ways of finding their roots. Concepts questions; Calculation problems; Multiple choice questions; Long answer questions; Fill in the blank's; Short Answer type. Simultaneous equations worksheet 1 includes questions involving the elimination method. Session PDF for "Solution of a Quadratic Equation by Factorisation"  https: This session will mainly be around the Quadratic Equation Questions and Answers with proper Explanation. These Quadratic Equation Questions are important for Bank, SSC, SEBI. C language interview questions solution for freshers beginners placement tricky good pointers answers explanation operators data types arrays structures functions recursion preprocessors looping file handling strings switch case if else printf advance linux objective mcq faq online written test prime numbers Armstrong Fibonacci series factorial palindrome code programs examples on c++. Examples of Quadratic Equation A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. Mathematics NCERT Grade 10, Chapter 4 Quadratic Equations. Example 2: Solve: 2 x x + = 2 7 Solution: Remember, that we need to write the equation in. Ordered pair(s): c. Quadratic Formula Worksheets with Answers Feet and Inches Measurement Conversion Worksheets. I expect that many of the students answers will not be the same. Lesson 6 Complex Numbers. Neglecting air resistance, projectiles follow the path of a parabola in nature. Given below are the Class 10 Maths Important Questions for Quadratic Equation. Solve quadratic equations algebraically. A worksheet can then come any subject. The pupils may, for example, establish a set of equivalent quadratic equations through the balancing. You need to understand how to project cash flow. In each question, one or two equations are provided. Once you have completed the activity sheet Exploring Quadratics in Factored Form , answer the following questions. ax 2 + bx + c = 0 are − b ± √ b. The highest power of the unknown is 2 Examples 1. Solving Systems of Linear Equations by Graphing Example 2 Solve the system of linear equations by graphing. 4: Solve Quadratic Equations Using. He throws a ball straight up into the air. 4 Explain the relationships among the roots of an equation, the zeros of the corresponding function, and the x-intercepts of the graph of the function. Facing this Quadratic Equations Online Test candidates can rate among themselves. y 30 12x y x2 11x 12 In Lesson 10-7, you used the discriminant to ﬁnd the number of solutions of a quadratic equation. Solving Quadratics by the Quadratic Formula - Pike Page 2 of 4 Example 1: Solve 12x2 + 7x = 12 Step 1: Simplify the problem to get the problem in the form ax2 + bx + c = 0. Corbettmaths - Videos, worksheets, 5-a-day and much more. A squirrel in a tree drops an acorn. Download the PDF of RD Sharma For Class 10 Maths Chapter 8 Quadratic Equations here Chapter 8 of RD Sharma Class 10 is Quadratic Equations, and it contains thirteen exercises. The quadratic equation topic is very basic but typically asked in the set of five questions in various bank exams. How To Prepare A Quadratic Equation For The Jee Quora. Give the exact answer and, if it's irrational, a decimal approximation (to the nearest tenth of a second). 74 and x = -5. 7c Applying the quadratic formula Decimal answers Topic: Applying the Quadratic Formula: Decimal Answers This is a topic. Frogs and Fleas and Painted Cubes: Quadratic Functions. As with solving quadratics before, we will generally end up with two solutions. Examples of Quadratic Equations (a) 5x 2 − 3x − 1 = 0 is a quadratic equation in quadratic form where a = 5, b = -3, c = -1. NCERT Solutions for Class 10 Maths Chapter 4 - Quadratic Equations are provided here to help out the students in their exam preparation. NCERT Solutions for class 10 Maths Chapter 4 are given below in PDF form or view online. (a) Determine if the function opens upward or downward. Class: _____ Date: _____ PostAssessment Quadratic Unit Multiple Choice Identify the choice that best completes the statement or answers the question. Factor x2-6x-40. Quadratic Equations for Bank Exams: In each problem, two equations are given, Solve these equations to find x and y values and establish the relationship between them by using Quadratic Equation Shortcut Tricks to compare two equations and then if we get x and y value in fractions then use shortcut method for comparing two fractions. define unknown variables 3. Examples of Quadratic Equation A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. Other Results for Emathinstruction Answer Key Unit 8: Answer Keys - eMathInstruction. It is a formula that solves for any quadratic equation of the form ax bx c 2 ++ = 0 by using only the coefficients a , b and c. Can solve quadratic equations by: o Factorising o Using the quadratic formula o Using the K-substitution Can analyse the nature of the roots of a quadratic equation. In every exam you will get at least 3-4 questions from this topic. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The other student made a mistake in the coefficient of the constant term got roots of -1 and 4. 2x 2 + 8x - m 3 = 0 Problem 8 : If the sum of roots of the quadratic equation given below is 0, then find the value of p. Regardless of which option you choose, the ultimate goal is to […]. online or www. If the equation required a substitution to solve, be sure to re-substitute and finish solving after using the quadratic formula. To solve quadratic equations in one variable, you have three options: Factor the quadratic. Quadratic equations refer to equations with at least one squared variable, with the most standard form being ax² + bx + c = 0. U worksheet by kuta software llc kuta software infinite algebra 2 name using the quadratic formula date period solve each equation with the quadratic formula. As abnormal as it may appear, the announcement holds valid for some reasons. 2 Determine solutions to quadratic equations (with real roots) by graphing, factoring, completing the square, or using the quadratic formula; Packet 12. Write an equation for the parabola whose vertex is at ä3,6 æ åô ö õ and which passes through ä4,4 æ åô ö õ. By looking at , a = 7, b = -4, and c = 13. Mathematics (Linear) – 1MA0 SIMULTANEOUS EQUATIONS WITH A QUADRATIC Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. For the following equation, solve using the quadratic formula or state that there are no real number solutions: 5x 2 - 3x - 1 = 0. Quadratic Equations Test Solving Quadratic Equations By The Quadratic Formula Quadratic Equations 8-4 Quadratic Equations Unit 8 Quadratic Equations Simultaneous Quadratic Equations Questions On Quadratic Equations Solving Quadratic And Other Equations 3. Quantitative Aptitude: Quadratic Equations Questions Set 59. For quadratic equations that cannot be solved by factorising, we use a method which can solve ALL quadratic equations called completing the square. Question 14 Find the equation of the quadratic function f whose graph increases over the interval (- infinity , -2) and decreases over the interval (-2 , + infinity), f(0) = 23 and f(1. In this chapter, you will study quadratic equations, and various ways of finding their roots. As we solve proportions we may end up with a quadratic that we will have to solve. Embedded Assessment 1: Graphing Quadratic. I’m giving you the true answers because I don’t want you to listen to these stupid people lying to you. This is our answer by the first merthod. Quadratic Formula Use the quadratic formula to solve each equation If the equation has no real solutions, write no real solutionsIf necessary, round to the nearest hundredth 1 x2 +8x +5 =0 2 2-36 =0 3 d2-4d-96 =0 4 a2-3 -154 =0 5 4 p2-12 -91 =0 6 5m2 +9m. NCERT Notes For Mathematics Class 11 Chapter 5 :- Quadratic Equations. After you have studied lesson, you must be looking for answers of. The quadratic formula to find the roots of a quadratic equation is: x 1,2 = (-b ± √Δ) / 2a where Δ = b 2 - 4ac and is called the discriminant of the quadratic equation. Anything with a * next to it represents content that is brand new to GCSE Maths for the 2017 exams. The form ax bx c. To answer the questions, we must find both and. For example, we can describe linear equations with one unknown as equations that contain only first powers of the unknown. 2x2 - 5x – 10 = 0 Solve using the quadratic formula. A quadratic equation contains terms. Two differentiated exercises. A the quadratic formula find out how long the rocket will take to hit the ground after it is. These numbers are called coefficients. The concept of solving a quadratic equation through the use of factorisation is explained. 0* Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Writing Quadratic Equations 27) The square of a positve number is 24 more than 5 times the number. The standard form is ax² + bx + c = 0 with a, b, and c being constants, or numerical coefficients, and x is an unknown variable. Quadratic functions have important applications in science and engineering. quadratic system, quadratic term, nonlinear system, linear-quadratic system, quadratic-quadratic system (AII. Online Library Quadratic Formula Answers answers Common Calculator Errors with Quadratic Formula This video will hopefully steer you away from two common mistakes made when using a Texas Instruments TI-30 series calculator. 73205 If b*b == 4*a*c, then roots are real and both roots are same. So, the answer keys arrived yesterday. X1 X2 The Height, H, In Feet Of An Object Above The Ground Is Given By H(t) = -16t2 +64t + 190,t 20 Where T Is The Time In Seconds. Theorem (Euler’s. Re-write the second equation like this:y = 3 + xNow you can replace y with 3 + x in the first equation:x^2 + (3 + x)^2 = 29Expand this out and simplify:x^2 + 9 + 6x + x^2 = 292x^2 + 6x = 20x^2 + 3x - 10 = 0Solve this (you can factorise, complete the square, or use the quadratic formula):You will find x = -5 or x = 2. The Standard Form of a Quadratic Equation looks like this:. Quadratic formula Khan Academy New 0d solving Quadratic Equations by from quadratic formula worksheet with answers pdf , source:sakeaji-kinoji. Students may need to be. Solving quadratic equations worksheets pdf questions with answers included. If b2 – 4ac = 0 WHY is there only 1 root? c. Scan the QR-code with a smartphone app for more resources. Mathematics (Linear) - 1MA0 QUADRATIC FORMULA Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. Quadratic Formula Another method for finding roots to a quadratic equation is the quadratic formula. 0 points each Assignment: Solve the problems below. A quadratic equation is one of the form ax 2 + bx + c = 0, where a, b, and c are numbers, and a is not equal to 0. ax^2 + bx + c. 4x 2 - 2x - 6 = 0 Step 1: Since the equation is set equal to 0, we ca 3. Quadratic equations math tests for GCSE maths. tt Read each question carefully before you start to answer it. from simplified quadratic formula to original equati. Multiple Choice. Past paper exam questions organised by topic for Edexcel GCSE Maths. In this tutorial, I assume you have already got a good skill in solving quadratic equations as explained in the above tutorial. This quadratic equations formulas for CAT pdf covers all the important formulas and concepts related to Quadratic Equations. by Priya / March 9, 2018 March 20, 2018. Find the solutions to the quadratic equation $$x^2-13x+12=0$$. 4x 2 11x 20 0 2. Factorising quadratics is a useful technique to learn. Kevin Regardie. Given a quadratic equation, the student will use graphical methods to solve the equation. 4x2 +16x 3. Example 4 requires careful translation of a word problem into an algebraic equation. • Determine what you are asked to find. By solving two equations and find the values of x andy and find the relation between x and y. Simultaneous equations worksheet 1 includes questions involving the elimination method. What to PROCESS: Activity 5: Is the Formula Effective? Directions: Find the solutions of each of the following quadratic equations using the quadratic formula. Past paper exam questions organised by topic for Edexcel GCSE Maths. In the function fx 2 2 53 3 2 3 xx xx (a) Use the quadratic formula to find the x-intercepts of the function, and then use a calculator to round these answers to the nearest tenth. several ways in which quadratic equations can be solved. This quadratic equations formulas for CAT pdf covers all the important formulas and concepts related to Quadratic Equations. 2x2 - 5x – 10 = 0 Solve using the quadratic formula. This product is suitable for Preschool, kindergarten and Grade 1. If you have any doubts, please comment below. What is the maximum profit the company can achieve per day, according to this mode? a. 5: Quadratic Applications Use the following information to answer questions 1 and 2. 114 Equation of a straight line F and H C 106 115 Simultaneous Equations Graphs F and H C 107 116 Drawing Quadratic Graphs F and H C 108 117 Real-life Graphs F and H C 109 118 Pythagoras' Theorem F and H C 110 119 Pythagoras - line on a graph F and H C 111 120 Surface area of cuboids F and H C 112 121 Surface area of triangular prisms F and H C 113. 1) x2 − 3x − 10 = 0 2) x2 − 4 = 0 3) v2 + 10v + 16 = 0 4) n2 + 3n − 10 = 0 5) x2 = −4 − 5x 6) x2 + 14 = −9x 7) r2 = 35 + 2r 8) −3n2 − 2n = −4n2 9) 3p2 − 11p + 6 = 0 10) 3m2 + 4m − 4 = 0. Solutions are in Hindi and English Medium. Solving quadratic equations worksheet 1 works at grade 4 for foundation gcse aimed at year 9 students. Now a definition of the linear and the quadratic equation with one unknown can be introduced. Quadratic Equations Aptitude Questions The quadratic Equations Aptitude Questions quiz helps the candidates to speed up themselves at the time of actual examination. Solving Quadratic Equations by Factoring. These are core topics and will be a part of the quantitative reasoning / aptitude sections of TANCET, CAT, and GMAT. A (0, 1); maximum C (0, 1); minimum B (0, 0); minimum D (0, 0); maximum ____ 2 Which of the quadratic functions has the narrowest graph? A. apply the solving techniques learned in the unit to real-life situations and how to decide what technique would be most appropriate in a situation. Thus, for a < -1, the parabola would open down and be stretched vertically away from the x-axis in a negative direction. PDF Pass Chapter 4 iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 4 Resource Masters The Chapter 4 Resource Masters includes the core materials needed for Chapter 4. Maths Class 10 Important Questions are very helpful to score high marks in board exams. Use the quadratic formula. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Quadratic Equations Puzzle - Free download as PDF File (. 2 − x − 3 = 0. quadratic simultaneous equations questions and answers Example Solve the quadratic. The equation can be solved by using the quadratic formula or by completing the square. However, unlike quadratics you are used to, at least some of the “constants” will not actually be constant but will in fact involve $$x$$’s. Correct Answer Your Answer; 2: f(x) = 2 x 2 + 2 x + 7: x-intercepts: Solution To find the x-intercepts, set the trinomial equal to 0. This formula can be used on any quadratic with the form ax2 + bx + c = 0. Record the answer for each question on the answer sheet. Facing this Quadratic Equations Online Test candidates can rate among themselves. Quadratic Equation Questions Hello Aspirants, As we all know that Quadratic Equation Questions is a important part of quantitative aptitude section for every competitive exams. C I copied someone off here and I got a 9/21 because of it. A quadratic function has x-intercepts of (-3,0) and (5,0). Using the quadratic formula features several examples with a, b and c colour coded to make it more visual. NCERT Solutions for Class 10 Maths Quadratic Equations 1. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions. Grade 9 Quadratic Equations Worksheet - Free download as Word Doc (. Note that that the quadratic formula, while always guaranteeing the answers, is not always the best method to solve a quadratic equation. Explain WHY… a. com Mobile: 9999 249717 Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051 (One Km from ‘Welcome’ Metro Station) Question 10: Express the given complex number in the form a + ib:. C language interview questions solution for freshers beginners placement tricky good pointers answers explanation operators data types arrays structures functions recursion preprocessors looping file handling strings switch case if else printf advance linux objective mcq faq online written test prime numbers Armstrong Fibonacci series factorial palindrome code programs examples on c++. Quadratic equations are actually used in everyday life, as when calculating areas, determining a product's profit or formulating the speed of an object. Example 2: Solve: 2 x x + = 2 7 Solution: Remember, that we need to write the equation in. Xmin:_____. Show that α^4-65α+114=0. Answer the following questions and provide an explanation. If b2 – 4ac < 0 WHY are. Elementary Algebra Skill Solving Quadratic Equations by Factoring Solve each equation by factoring. Wilson's Math Classes. Example 4 requires careful translation of a word problem into an algebraic equation. x2 - 8x - 29 º (x + a)2 + b, where a and b are constants. Since in the Scholar Worksheet about 90% of the contents of the entire guide are issues, equally multiple selection and solution issues that are not available. Quadratic equations are also used in other situations such as avalanche control, setting the best ticket prices for concerts, designing roller coasters, and planning gardens. A quadratic equation is an equation where the highest exponent power of a variable is 2 (ie, x 2). ____ 1 Identify the vertex of the graph. Indicate the answer choice that best completes the statement or answers the question. x-intercept A. The answers for these pages appear at the back of this booklet. Quadratic Equation Calculator The calculator will solve the quadratic equation step by step either by completing the square or using the quadratic formula. In Section 12. Solve real-life problems. Expand And Factorize Quadratic Expressions Expanding Quadratic Expressions: Quadratic expressions are algebraic expressions where the variable has an exponent of 2. Can solve quadratic equations by: o Factorising o Using the quadratic formula o Using the K-substitution Can analyse the nature of the roots of a quadratic equation. Solve the following equation for using the quadratic formula: x² +3x-5=0 Solution. See page 14 for more on the new grading, including how it relates to the current A*–G grading. To form a quadratic equation when you are given the vertex, focus or directrix, click here: Calculating a Quadratic Equation from the Vertex, Focus or Directrix. Quadratic formula Khan Academy New 0d solving Quadratic Equations by from quadratic formula worksheet with answers pdf , source:sakeaji-kinoji. In this Section we describe several ways in which quadratic equations can be solved. linear function. Example 4 requires careful translation of a word problem into an algebraic equation. The proof makes no use of any mathematical discipline other than elementary number theory. Mathematics (Linear) - 1MA0 SIMULTANEOUS EQUATIONS WITH A QUADRATIC Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. You can rewrite the quadratic function as a quadratic equation set equal to zero to find the zeros of the function 0 = -16t2 + 80t + 0. The Quadratic Formula (AQA Higher) These questions are suitable for Higher Tier students Questions 1 to 3 should be done without a calculator Questions 4 to 15 can be done with a calculator Model Answers. Question: What is the the purpose of calibration curve? Answer: Most analytical instruments generate an electrical output signal such as a current or a voltage. (See note below on question 7. Short Answer Type Questions I [2 Marks] Question 1. Download free printable worksheets for CBSE Class 10 Quadratic Equation with important topic wise questions, students must practice the NCERT Class 10 Quadratic Equation worksheets, question banks, workbooks and exercises with solutions which will help them in revision of important concepts Class 10 Quadratic Equation. It’s also a good idea, if you have a minus number in front of your term, to multiply everything by -1 , as this will make your calculations easier. Quadratic Equations is one of the important topics for CAT. A quadratic equation contains a variable such as the x 2 term as the highest power. Download latest questions with multiple choice answers for Class 10 Quadratic Equation in pdf free or read online in online reader free. PostAssessment Quadratic Unit Multiple Choice Identify the choice that best completes the statement or answers the question. (c) Find the maximum value. To help you identify and help students who have the difficulty of solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring. (x+8)(x-4) = 0. There are many ways to solve quadratics. Get NCERT Solutions for all exercise questions and examples of Chapter 4 Class 10 Quadratic Equations free at Teachoo. For the following problems, practice choosing the best method by solving for x in the quadratic equation. Quadratic Equations. com provides you Free PDF download of NCERT Exemplar of Class 10 Maths chapter 4 Quadratic Equations solved by expert teachers as per NCERT (CBSE) book guidelines. Sample answers and a scoring rubric are included for evaluation. solve by adding solve by subtracting a mix of the above two require one multiplication step general system solve by substitution. In this section, we will solve quadratic equations by a process called completing the square, which is important for our work on conics later. This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula. Well, it's given oversized importance in high school math class because we know basically everything about it, so it's easy for teachers to understand and it's easy to teach by rote memorization methods. On this page you can read or download gina wilson all things algebra 2015 solving quadratic equations homework 10 answers in PDF format. wilson unit 8 quadratic equation answers PDF may not make exciting reading, but gina wilson unit 8 quadratic equation answers is packed with valuable instructions, information and warnings. 1 y 2x2 12x 17 x y 3 2 11234567 8 7 6 5 4 3 2 1 1 2 2 y x2 6x 8 x y 112345 2 15 1 05 05 1. Write them separated by commas in the answer box. 19) If a quadratic equation can be factored and each factor contains only real numbers then there cannot be an imaginary solution. Solving Quadratics by the Quadratic Formula - Pike Page 2 of 4 Example 1: Solve 12x2 + 7x = 12 Step 1: Simplify the problem to get the problem in the form ax2 + bx + c = 0. Question: ACTIVITY 7 QUADRATIC FORMULA. 4 Explain the relationships among the roots of an equation, the zeros of the corresponding function, and the x-intercepts of the graph of the function. An indeﬁnite quadratic form will notlie completely above or below the plane but will lie above for somevalues of x and belowfor other values of x. a can't be 0. Maths Questions and Answers with Full Working on Factorising Quadratic Equations that range in difficulty from easy to hard. The easiest way to write a generic algorithm is to simply use the quadratic formula. Powered by Create your own unique website with customizable templates. For each of these equations, there is no real number solution because a solution requires that we find the square root of the discriminant and no real number can be the square root of a negative number. The three main ways to solve quadratic equations are: to factor, to use the quadratic formula, or to complete the square. (i) (ii) (iii) Ans. Mathematics (Linear) - 1MA0 SIMULTANEOUS EQUATIONS WITH A QUADRATIC Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. ) H =-16t2 - 10t + 30 b. The Standard Form of a Quadratic Equation looks like this:. Zero, because the discriminant is not a perfect square. Quadratic Equations Test Solving Quadratic Equations By The Quadratic Formula Quadratic Equations 8-4 Quadratic Equations Unit 8 Quadratic Equations Simultaneous Quadratic Equations Questions On Quadratic Equations Solving Quadratic And Other Equations 3. C1N , (3 5,) 2 4 P Question 12 (***+) Use the discriminant of a suitable quadratic equation to show that the graphs of the curves with equations 1 y 2 x. These are called the roots of the quadratic equation. Quadratic equations are also used in other situations such as avalanche control, setting the best ticket prices for concerts, designing roller coasters, and planning gardens. ! Linear Functions Equation !=#\$+& Shape linear Rate of Change constant Number of ’-intercepts (or ) or infinitely many Number of *-intercepts ).