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Chemistry

What is the group 17 element with the highest electronegativity?

Fluorine is the Group 17 element with the highest electronegativity, and in fact, it is the most electronegative element in the entire periodic table. Electronegativity refers to how strongly an atom can pull the electrons in a covalent bond toward itself. According to Coulomb's law, this attraction gets stronger when the effective nuclear charge is higher and when the bonding electrons are closer to the nucleus (meaning the atom is smaller). As you go down Group 17 from fluorine to astatine, electronegativity decreases because the atoms get bigger. All halogens have seven valence electrons and a similar effective nuclear charge of about +7, but each step down the group adds an extra electron shell. This extra shell pushes the valence electrons farther from the nucleus and adds more shielding from inner electrons. Both effects reduce the pull on the shared electrons, so the larger halogens attract bonding pairs less strongly than fluorine does.

Mathematics

What are double angle identities?

Double angle identities appear in both the AA SL and AA HL courses, but do not appear in either the AI SL or AI HL courses. There are double angle identities for sine and cosine, which appear in both the AA SL and AA HL courses, and a double angle identity for tangent, which appears only in the AA HL course. There is one double angle identity for the sine ratio - $\sin 2\theta = 2 \sin \theta \cos \theta$ There are three double angle identities for the cosine ratio - $\cos 2\theta = \cos ^2 \theta - \sin ^2 \theta$ - $\cos 2\theta = 2 \cos ^2 \theta - 1$ - $\cos 2\theta = 1 - 2 \sin ^2 \theta$ There is one double angle identity for the tangent ratio - $\tan 2\theta = \dfrac{2 \tan \theta}{1 - \tan ^2 \theta}$ **The double angle identities give us a way of expressing a trigonometric ratio in another form that may make it easier to solve an equation, or help us to find the exact value of a new angle, particularly in paper 1 where we cannot use a G.D.C. to help us.** For example, suppose we are asked to solve the equation $\sin \theta = \dfrac{1}{4 \cos \theta}$ for $0 \leq \theta \leq 2\pi$. We could rearrange this by multiplying both sides by $2 \cos \theta$, giving us $$ \begin{align*} \sin \theta \times 2 \cos \theta &= \dfrac{1}{4 \cos \theta} \times 2 \cos \theta \\[8pt] 2 \sin \theta \cos \theta &= \dfrac{1}{2} \end{align*} $$ Now on the left hand side we can apply the double angle identity for sine so that we have $$ \begin{align*} \sin 2\theta &= \dfrac{1}{2} \end{align*} $$ From here we can proceed as though solving a “normal” trigonometric equation, finding that the two solutions are $\theta = \dfrac{\pi}{12}$ and $\theta = \dfrac{5\pi}{12}$. As a second example, suppose we are asked to find the exact value of $\tan \left(\dfrac{\pi}{12}\right)$. From ratio triangles, we know that $\tan \left(\dfrac{\pi}{6}\right)=\dfrac{1}{\sqrt 3}$ and $\dfrac{\pi}{6}$ is double the angle $\dfrac{\pi}{12}$ so we could use the double angle identity for tangent to help us. We can set up our equation as follows: $$ \begin{align*} \tan 2\theta &= \frac{2 \tan \theta}{1 - \tan ^2 \theta} \\[12pt] \tan \left(\frac{\pi}{6}\right) &= \frac{2 \tan \frac{\pi}{12}}{1 - \tan ^2 \frac{\pi}{12}} \\[12pt] \frac{1}{\sqrt 3} &= \frac{2 \tan \frac{\pi}{12}}{1 - \tan ^2 \frac{\pi}{12}} \end{align*} $$ Now we could rearrange this to make a quadratic equation. $$ \begin{align*} \tan ^2 \left(\frac{\pi}{12}\right) + 2\sqrt{3} \tan \left(\frac{\pi}{12}\right) - 1 = 0 \end{align*} $$ We could then complete the square, or use the quadratic formula to find that $\tan\left(\dfrac{\pi}{12}\right) = \pm 2 - \sqrt {3}$. Then we can reason that $\dfrac{\pi}{12}$ is in quadrant 1 where the tangent function is positive, hence $\tan\left(\dfrac{\pi}{12}\right)$ must equal $2 - \sqrt{3}$. If faced with a trigonometric equation or expression in an exam that doesn’t appear to be solvable or easy to simplify in its current form,, it is useful to investigate whether a substitution using a double angle identity makes the equation more easily solvable, or the expression more easy to simplify.

Physics

What is the gravitational field strength formula?

A gravitational field is a region in space where a mass will experience a gravitational force. The gravitational field strength at a location is defined as the force per unit mass experienced by a small test mass placed at that location. The field strength can be expressed as an equation as follows: $\hspace{3em} g = \dfrac{F}{m}$ Where:\ $\hspace{3em}$ $g$ is the gravitational field strength in units of N kg$^{-1}$\ $\hspace{3em}$ $F$ is the gravitational force experienced by the mass in units of N\ $\hspace{3em}$ $m$ is the mass of the object experiencing the gravitational force in units of kg. Any object with mass will create a gravitational field in the region around it. This field extends an infinite distance away; however, it decreases proportionally with the square of the distance. We can use Newton’s Law of Gravitation to develop the formula for the gravitational field strength at a given distance from an object. The law states that the force of gravity between two masses is directly proportional to the product of the masses and inversely proportional to the square of the distance of the separation of their centres of mass: $\hspace{3em} F_g = G\dfrac{m_1m_2}{r^2}$ Where:\ $\hspace{3em}$ $F_g$ is the force of gravity in units of N \ $\hspace{3em}$ $G$ is the gravitational constant 6.67 x 10$^{-11}$ N m$^2$ kg$^{-2}$\ $\hspace{3em}$ $m_1$ and $m_2$ are the masses of the two objects in units of kg\ $\hspace{3em}$ $r$ is the separation of their centers of mass in m. We can substitute this equation for gravitational force into the first question for gravitational field to derive a new equation $\hspace{3em} g = \dfrac{F}{m}$ $\hspace{3em} g = \dfrac{G\dfrac{m_1m_2}{r^2}}{m}$ $\hspace{3em} g = \dfrac{Gm_1}{r^2}$ This formula gives the gravitational field strength at a distance $r$ from a the centre of a mass $m_1$ . This mass is normally a planet, star or moon.

Economics

What is the difference between movement along the supply curve and a movement of the supply curve?

To understand the difference between movements along the supply curve and shifts of the supply curve, let’s start by understanding the law of supply. The law of supply states that if the price of a good or service increases, then the quantity supplied will also increase. To understand this fully, you have to think like a producer. Let’s assume that producers want to make as much money as possible. Also, remember that producers—like everyone else—make choices. For example, a T-shirt maker can choose to make T-shirts of various colours and styles, print different things on their T-shirts, or even make a different kind of clothing altogether. Putting these ideas together, we can better understand that the law of supply says that producers will allocate their productive resources towards whatever makes them the most money. So if the price of green T-shirts goes up—this doesn’t change the cost of making it—producers will be more interested in making (and selling) green T-shirts, because they can earn greater profits. The law of supply directly refers to the supply curve: the supply curve is upward-sloping because, based on the selling price, producers will produce more (or less) of a good. The supply schedule shows this relationship: :::center |Price |Quantity supplied| |:-:|:-:| |$0|0| |$10|50| |$20|100| |$30|150| |$40|200| ::: The supply schedule shows a movement **along** the supply curve. If this is true, then why don’t producers simply raise the price of a good and produce more? This is because, in the demand and supply model, price is determined by the interaction of the demand curve and the supply curve: consumers are less likely to buy a good at higher prices. Lastly, let’s look at a *shift* of the supply curve. As described above, the supply curve shows the relationship between price and quantity supplied, but it doesn’t give much insight into the production of the good itself. For a producer to make a given good or service, they will incur production costs. Among these are opportunity costs: the value of other goods or services the producer could make, and the best time to supply a product to the market. Along with the selling price, these factors influence how much of a product the producer will make. If one of these other factors changes, for example, the cost of raw materials increases, the relationship between price and quantity supplied changes. This is because the profit has decreased for every price, and the producers are less interested in producing the product. Together, these other factors are known as non-price determinants of supply. Changes to these determinants decrease supply (the supply curve shifts left) and increase supply (the supply curve shifts right). :br :::center |Non-price determinant of supply |Supply shifts right if…|Supply shifts left if…| |:-:|:-:|:-:| |Cost of raw materials/labour|costs decrease|costs increase| |Price of related goods (joint supply)|price of other goods increases|price of other goods decreases| |Price of related goods (competitive supply)|price of other goods decreases|price of other goods increases| |Taxes and subsidies|indirect taxes decrease/subsidies increase|indirect taxes increase/subsidies decrease| |Future expectations of price|prices are expected to fall|prices are expected to rise| |Technology|technology increases|technology decreases| |Number of firms|firms join market|firms leave market| :::

Mathematics

What is the geometric sequence formula?

There are two types of sequences in the IB mathematics course syllabi; arithmetic sequences and geometric sequences. A geometric sequence is one where the next term is obtained by multiplying the previous term by a fixed number called the common ratio. There are two formulas associated with geometric sequences. The first formula, shown below, is used to find a term value in the sequence. $$ \begin{align*} u_n&=u_1\times r^{n-1} \end{align*} $$ This means if we know the first value in a geometric sequence $(u_1)$ and the common ratio $(r)$ we can find any value in the sequence based on their position in the sequence (i.e. $3$rd term, $10$th term, etc.). For example, if we want to find the $5$th term in a geometric sequence with the first term of $2$ and a common ratio of $3$ we use the given formula with $n=5$, $u_1=2$ and $r=3$. $$ \begin{align*} u_5&=2(3)^{5-1}\\[6pt] &=2(3)^4\\[6pt] &=162 \end{align*} $$ This means that the $5$th term in the sequence is $162$. The other formula associated with geometric sequences is the sum formula. $$ \begin{align*} S_n&=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r} \end{align*} $$ We use this formula when we want to $\textbf{add}$ all the numbers in a geometric sequence. This sum of the geometric sequence is also called a geometric series. Notice that this formula is written in two ways. Both forms of the equation will get you to the correct answer but generally it is easier to use the first form when $|r|>1$ and the second form when $|r|<1$. Let’s look at the sequence above where $n=5$, $u_1=2$ and $r=3$. If we use the sum formula (choosing the first form, as we have $r>1$) $$ \begin{align*} S_5&=\frac{2(3^5-1)}{3-1}\\[6pt] &=\frac{\cancel{2}(243-1)}{\cancel{2}}\\[6pt] &=242 \end{align*} $$ We find that the sum of the first $5$ terms is $162$. This is the same result we would get if we generated the first $5$ terms in the sequence and then added them together. $$ \begin{align*} S_5&=2+6+18+54+162\\[6pt] &=242 \end{align*} $$ There is also a formula for the sum to infinity of a geometric sequence which is $$ \begin{align*} S_{\infty}&=\frac{u_1}{1-r}\hspace{1em}\text{where }|r|<1 \end{align*} $$ This formula appears in the Mathematics AA HL, AA SL and AI HL courses. It is not in the AI SL course. We use this formula when $|r|<1$ because in this case, each successive term added to the sum is smaller (so the sum will approach some value that we can find). Alternatively, when $|r|>1$ then we would be adding larger numbers each time and the sum to infinity would also be infinite.

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