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2k_conso_questions_v58

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### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Three wire loops and an observer are positioned as shown in the figure above. From the observer's point of view, a current $L$ flows counterclockwise in the middle loop, which is moving towards the observer with a velocity $d$. Loops $A$ and $B$ are stationary. This same observer would notice that a clockwise current is induced in loop $A$, but a counterclockwise current is induced in loop $B$ " ### Response: The conventional notation for current is $I$. The conventional notation for velocity vector in the loop is $\vec{v}$. The conventional notation for loops are $A$ and $B$. ||| Three wire loops and an observer are positioned as shown in the figure above. From the observer's point of view, a current $I$ flows counterclockwise in the middle loop, which is moving towards the observer with a velocity $\vec{v}$. Loops $A$ and $B$ are stationary. This same observer would notice that a clockwise current is induced in loop $A$, but a counterclockwise current is induced in loop $B$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Given a plasma-based security system that uses a plasma discharge to create a physical barrier, calculate the minimum power $[v]$ required to sustain the plasma discharge barrier for a distance of $1 [u]$ and a height of $2 [u]$, assuming ambient air pressure and temperature. Let the minimum power required be denoted as $[v]$. The distance is $[v1] = 1 [u]$ and the height is $[v2] = 2 [u]$." ### Response: The conventional notation for power is $P$. ||| Given a plasma-based security system that uses a plasma discharge to create a physical barrier, calculate the minimum power $P$ required to sustain the plasma discharge barrier for a distance of $1 ~ \rm{m}$ and a height of $2 ~ \rm{m}$, assuming ambient air pressure and temperature. Let the minimum power required be denoted as $P$. The distance is $d = ~ \rm{m}$ and the height is $h = 2 ~ \rm{m}$.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Solve for $x$ and $y$ using elimination. ${-4x-y = -16}$ ${-3x+y = -5}$" ### Response: The conventional notation for variables are $x$ and $y$. ||| Solve for $x$ and $y$ using elimination. ${-4x-y = -16}$ ${-3x+y = -5}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A free particle with de Broglie wavelength $\omega$ and initial kinetic energy $l$ enters a region in which it has potential energy $v$. What is the particle's new de Broglie wavelength? &#x20;&#x20; E. $\lambda(1-v / l)^{-1 / 2}$ " ### Response: The conventional notation for initial kinetic energy is $K_0$. The conventional notation for de Broglie wavelength is $\lambda$. The conventional notation for potential energy is $V$. ||| A free particle with de Broglie wavelength $\lambda$ and initial kinetic energy $K_0$ enters a region in which it has potential energy $V$. What is the particle's new de Broglie wavelength? &#x20;&#x20; E. $\lambda(1-V / K_0)^{-1 / 2}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Using renormalization techniques in quantum electrodynamics, calculate the correction to the electron's magnetic moment at order $[v]^{3}$ (where $[v]$ is the fine-structure constant equal to approximately $\frac{1}{137}$), taking into account all Feynman diagrams." ### Response: The conventional notation for fine-structure constant is $\alpha$. ||| Using renormalization techniques in quantum electrodynamics, calculate the correction to the electron's magnetic moment at order $\alpha^{3}$ (where $\alpha$ is the fine-structure constant equal to approximately $\frac{1}{137}$), taking into account all Feynman diagrams.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Using condensed matter physics, determine the effective mass of an electron in a copper wire at room temperature, assuming a current density $[v1]$ of $1.5 \times 10^7 [u]$ and a resistivity $[v]$ of $1.68 \times 10^{-8}[u1]$." ### Response: The conventional notation for current density is $J$. The conventional notation for resistivity is $\rho$. ||| Using condensed matter physics, determine the effective mass of an electron in a copper wire at room temperature, assuming a current density $J$ of $1.5 \times 10^{7} ~ \rm{A/m^{2}$ and a resistivity $\rho$ of $1.68 \times 10^{-8} ~ \rm{\Omega \cdot m}$.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "In each of the code chunks below, two vectors will be created. Determine if they are equal. i_vector <- c(2, 4, 6, 8, 10) j_vector <- c(0, 4, 6, 8, 10) j_vector[1] <- 2 " ### Response: There are no notations. ||| In each of the code chunks below, two vectors will be created. Determine if they are equal. i_vector <- c(2, 4, 6, 8, 10) j_vector <- c(0, 4, 6, 8, 10) j_vector[1] <- 2 <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Evaluate the function $f(g)=\cos \left(6 x+\frac{13}{2}\right) \tan \left(6 x+\frac{1}{2}\right)$ at the point $x=40$" ### Response: The conventional notation for variables is $x$. ||| Evaluate the function $f(x)=\cos \left(6x+\frac{13}{2}\right) \tan \left(6x+\frac{1}{2}\right)$ at the point $x=40$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Let $f(x)=\begin{cases} (x-2)^2&\text{for }x \leq 2 \\ 2-x^2&\text{for }x>2 \end{cases}$ Is $f$ continuous at $x=2$ ? &#x20;&#x20; Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No" ### Response: The conventional notation for variables is $x$. ||| Let $f(x)=\begin{cases} (x-2)^2&\rm{for} ~ x \leq 2 \\ 2-x^2&\rm{for} ~ x>2 \end{cases}$ Is $f$ continuous at $x=2$ ? &#x20;&#x20; Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "$f(x)=(x+9)(x+8)$ 1) What are the zeros of the function? Write the smaller $x$ first, and the larger $x$ second. $\text{smaller }x=$" ### Response: The conventional notation for variables is $x$. ||| $f(x)=(x+9)(x+8)$ 1) What are the zeros of the function? Write the smaller $x$ first, and the larger $x$ second. $\rm{smaller} x=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "### Ace: A spaceship travelling at a velocity of $0.8c$ relative to Earth releases a probe. If the probe is designed to accelerate at a rate of $2g$ (where $[v]=9.8 \, [u]$) in the spaceship's frame of reference, what acceleration will be observed from Earth's frame of reference? What will be the maximum speed the probe will attain according to Earth? Acceleration in the spaceship's frame of reference: $[v1] = 2g$ Acceleration observed from Earth's frame of reference: $[v2]= \frac{[v1]}{\sqrt{1 - (v/c)^2}$" ### Response: The conventional notation for speed of light is $c$. The conventional notation for acceleration due to gravity is $g$. The conventional notation for acceleration observed from Earth frame of reference is $a_{Earth}$. The conventional notation for velocity is $v$. The conventional notation for acceleration in the spaceship is $a_{spaceship}$. ||| ### Ace: A spaceship travelling at a velocity of $0.8c$ relative to Earth releases a probe. If the probe is designed to accelerate at a rate of $2g$ (where $g=9.8 ~ \rm{m/s^2}$) in the spaceship's frame of reference, what acceleration will be observed from Earth's frame of reference? What will be the maximum speed the probe will attain according to Earth? Acceleration in the spaceship's frame of reference: $a_{spaceship} = 2g$ Acceleration observed from Earth's frame of reference: $a_{Earth} = \frac{a_{spaceship}}{\sqrt{1 - (v/c)^2}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Let $f(x)=4x^{7}+7x^4+5x^2-10$, and $g(x)=x^5$. When dividing $f(x)$ by $g(x)$, we can find the unique quotient polynomial $q(x)$ and remainder polynomial $r(x)$ that satisfy the following equation: $\frac{f(x)}{g(x)}=q(x) + \frac{r(x)}{g(x)}$, where the degree of $r(x)$ is less than the degree of $g(x)$. What is the quotient, $q(x)$? $q(x)=$ What is the remainder, $r(x)$? $r(x)=$" ### Response: The conventional notation for variables is $x$. ||| Let $f(x)=4x^{7}+7x^4+5x^2-10$, and $g(x)=x^5$. When dividing $f(x)$ by $g(x)$, we can find the unique quotient polynomial $q(x)$ and remainder polynomial $r(x)$ that satisfy the following equation: $\frac{f(x)}{g(x)}=q(x) + \frac{r(x)}{g(x)}$, where the degree of $r(x)$ is less than the degree of $g(x)$. What is the quotient, $q(x)$? $q(x)=$ What is the remainder, $r(x)$? $r(x)=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Factor completely. $(x^2+x-6)(2x^2+4x)=$" ### Response: The conventional notation for variables is $x$. ||| Factor completely. $(x^2+x-6)(2x^2+4x)=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Calculate the instanton contribution to the vacuum energy of a 3+1 dimensional non-Abelian gauge theory with gauge group SU(2) and N_f fermions in the fundamental representation." ### Response: There are no notations. ||| Calculate the instanton contribution to the vacuum energy of a $3+1$ dimensional non-Abelian gauge theory with gauge group SU(2) and N_f fermions in the fundamental representation.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "You decide you're only going to buy a lottery ticket if your expected winnings are larger than the ticket price. Tickets cost $$5$, and you get $$1100$ if you win. The odds of winning are $1$ in $200$, meaning that you will win with probability q = $\frac{1}{200}$. Should you buy a ticket for this lottery?" ### Response: The conventional notation for probability is $p$. ||| You decide you're only going to buy a lottery ticket if your expected winnings are larger than the ticket price. Tickets cost $5$, and you get $1100$ if you win. The odds of winning are $1$ in $200$, meaning that you will win with probability $p = \frac{1}{200}$. Should you buy a ticket for this lottery? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Let $f(x)=7 sin(x)-2x^3$. $f'(x)=$ &#x20;&#x20; [embedded] " ### Response: The conventional notation for variables is $x$. ||| Let $f(x)=7\sin(x)-2x^3$. $f'(x)=$ &#x20;&#x20; [embedded] <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The critical flow rate can be determined using the Reynolds number ($[v]$) and the characteristics of the flow. One common method to estimate the critical flow rate is through the use of the critical Reynolds number ($[v1]$) for the transition from laminar to turbulent flow. The critical Reynolds number is typically around 2300 for flow in a smooth pipe. &#x20;&#x20; The critical flow rate ($[v2]$) can be calculated using the formula: $$ [v2] = \frac{\pi \cdot [v3] \cdot [v4] \cdot [v5]}{4} $$ where: - $[v2]$ is the critical flow rate - $[v3]$ is the diameter of the pipe - $[v4]$ is the kinematic viscosity of the fluid - $[v]$ is the critical Reynolds number" ### Response: The conventional notation for the critical flow rate is $Q_{c}$. The conventional notation for the diameter of the pipe is $D$. The conventional notation for the critical Reynolds number is $Re_{c}$. The conventional notation for kinematic viscosity of the fluid is $\nu$. The conventional notation for Reynolds number is $Re$. ||| The critical flow rate can be determined using the Reynolds number ($Re$) and the characteristics of the flow. One common method to estimate the critical flow rate is through the use of the critical Reynolds number ($Re_{c}$) for the transition from laminar to turbulent flow. The critical Reynolds number is typically around $2300$ for flow in a smooth pipe. &#x20;&#x20; The critical flow rate ($Q_{c}$) can be calculated using the formula: $$ Q_{c} = \frac{\pi \cdot D \cdot \nu \cdot Re_{c}}{4} $$ where: - $Q_{c}$ is the critical flow rate - $D$ is the diameter of the pipe - $\nu$ is the kinematic viscosity of the fluid - $Re_{c}$ is the critical Reynolds number<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The equation of a circle $C$ is $(x+5)^{2}+(y-8)^{2} = 25$. What are its center $(h, k)$ and its radius $d$ ?" ### Response: The conventional notation for variables in circle are $x$ and $y$. The conventional notation for center of circle are $k$ and $h$. The conventional notation for circle is $C$. The conventional notation for radius is $r$. ||| The equation of a circle $C$ is $(x+5)^{2}+(y-8)^{2} = 25$. What are its center $(h, k)$ and its radius $r$ ? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the probability of a muon neutrino oscillating into an electron neutrino after traveling a distance $[v]$ of $730 [u]$ through the Earth's crust in vacuum, with an initial energy of 1 $G[u1]$ and a mixing angle of $35 [u2]$?" ### Response: The conventional notation for the distance is $d$. ||| What is the probability of a muon neutrino oscillating into an electron neutrino after traveling a distance $d$ of $730 ~ \rm{km}$ through the Earth's crust in vacuum, with an initial energy of $1 ~ \rm{GeV}$ and a mixing angle of $35^{\circ}?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A cylindrical superconductor of radius $[v] = 2 [u]$ and length $[v1] = 10 ~ [u]$ expels a magnetic field of $[v2] = 0.05 [u1]$ at a temperature of $[v3] = 5 [u2]$. Calculate the critical current $[v4]$ of the superconductor." ### Response: The conventional notation for radius is $r$. The conventional notation for length is $l$. The conventional notation for magnetic field is $\bf{B}$. The conventional notation for temperature is $T$. The conventional notation for critical current is $I_c$. ||| A cylindrical superconductor of radius $r = 2 ~ \rm{cm}$ and length $l = 10 ~ \rm{cm}$ expels a magnetic field of $\bf} = 0.05 ~ \rm{T}$ at a temperature of $T = 5 ~ \rm{K}$. Calculate the critical current $I_c$ of the superconductor.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "In the AC circuit above, $V_{i}$ is the amplitude of the input voltage and $V_{o}$ is the amplitude of the output voltage. If the angular frequency $\omega$ of the input voltage is varied, which of the following gives the ratio $V_{o} / V_{i}=G$ as a function of $\omega$ ? &#x20;&#x20; A. " ### Response: The conventional notation for amplitude of input voltage is $V_{i}$. The conventional notation for amplitude of output voltage is $V_{o}$. The conventional notation for angular frequency is $\omega$. The conventional notation for voltage gain is $G$. ||| In the AC circuit above, $V_{i}$ is the amplitude of the input voltage and $V_{o}$ is the amplitude of the output voltage. If the angular frequency $\omega$ of the input voltage is varied, which of the following gives the ratio $V_{o} / V_{i}=G$ as a function of $\omega$ ? &#x20;&#x20; A. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "* You will need access to the library resources in parts (a) to (c) below. * To access the books click on the thumbnail image or the link in the reference at the bottom. [embedded] [embedded] Reference: **Simmons, C.H., Maguire, D.E., and Phelps, N. (2020) *Manual of Engineering Drawing: British and International Standards.* 5th edition. Oxford, Butterworth-Heinemann.** " ### Response: There are no notations. ||| * You will need access to the library resources in parts (a) to (c) below. * To access the books click on the thumbnail image or the link in the reference at the bottom. [embedded] [embedded] Reference: **Simmons, C.H., Maguire, D.E., and Phelps, N. (2020) *Manual of Engineering Drawing: British and International Standards.* 5th edition. Oxford, Butterworth-Heinemann.** <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A forest has 800 pine trees, but a disease is introduced that kills 1/4 of the pine trees in the forest every year. Write a function that gives the number of pine trees remaining $f(t)$ in the forest $t$ years after the disease is introduced. $f(t)=$" ### Response: The conventional notation for time is $t$. ||| A forest has $800$ pine trees, but a disease is introduced that kills $\frac{1}{4}$ of the pine trees in the forest every year. Write a function that gives the number of pine trees remaining $f(t)$ in the forest $t$ years after the disease is introduced. $f(t)=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Calculate the thermal conductivity $[v]$ of a copper wire that is $[v1] = 2 ~[u]$ long, has a radius $[v2]= 0.5 m[u1}$, and a temperature difference $[v3] = 50 [u2]$ across its ends. The wire is made of pure copper with a density $[v4] = 8.96 ~ [u3]$ and specific heat capacity $[v5] = 0.385 [u4]$." ### Response: The conventional notation for thermal conductivity is $\kappa$. The convetional notation for temperature difference is $\Delta T$. The conventional notation for density is $\rho$. The conventional notation for specific heat capacity is $c$. ||| Calculate the thermal conductivity $\kappa$ of a copper wire that is $L = 2 ~ \rm{m}$ long, has a radius $r = 0.5 ~ \rm{mm}$, and a temperature difference $\Delta T = 50^\circ \rm{C}$ across its ends. The wire is made of pure copper with a density $\rho = 8.96 ~ \rm{g/cm^3}$ and specific heat capacity $c = 0.385 ~ \rm{J/g \cdot K}$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Complete the square for the following quadratic: $-\frac{7 x^2}{\sqrt{3}-\frac{8 x}{\sqrt{3}+6 \sqrt{3}$" ### Response: The conventional notation for variables is $x$. ||| Complete the square for the following quadratic: $-\frac{7 x^2}{\sqrt{3}-\frac{8 x}{\sqrt{3}+6 \sqrt{3}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Let f be a function defined for all real numbers except for $0$. Also let $f'$, the derivative of $f$, be defined as $f'(x)=\frac{x+3}{x^2}$. On which intervals is f decreasing?" ### Response: The conventional notation for function is $f$. The conventional notation for variable is $x$. ||| Let $f$ be a function defined for all real numbers except for $0$. Also let $f'$, the derivative of $f$, be defined as $f'(x)=\frac{x+3}{x^2}$. On which intervals is $f$ decreasing? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Two large conducting plates form a wedge of angle $\alpha$ as shown in the diagram above. The plates are insulated from each other; one has a potential $v_{0}$ and the other is grounded. Assuming that the plates are large enough so that the potential difference between them is independent of the cylindrical coordinates $z$ and $\omega$" ### Response: The conventional notation for the potential difference is $V$. The notation for the angle of wedge between conducting plates is $\alpha$. The conventional notation for azimuthal angle in cylindrical coordinates is $\phi. The conventional notation for axial coordinate in cylindrical coordinates is $z$. ||| Two large conducting plates form a wedge of angle $\alpha$ as shown in the diagram above. The plates are insulated from each other; one has a potential $V_0$ and the other is grounded. Assuming that the plates are large enough so that the potential difference between them is independent of the cylindrical coordinates $z$ and $\phi$<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider a single electron atom with orbital angular momentum $L=\sqrt{2} \hbar$. Which of the following gives the possible values of a measurement of $L_{z}$, the $z$-component of $L$ ? [embedded] [embedded] D. $-\hbar, 0, \hbar$ " ### Response: The conventional notation for the reduced Planck constant is $\hbar$. The conventional notation for orbital angular momentum vector is $\bf{L}$. The conventional notation for the $z$-component of the orbital angular momentum is ${\hat{L}_z$ ||| Consider a single electron atom with orbital angular momentum $\bf{L}=\sqrt{2} \hbar$. Which of the following gives the possible values of a measurement of $\hat{L}_{z}$, the $z$-component of $\bf{L}$ ? [embedded] [embedded] D. $-\hbar, 0, \hbar$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Solve for $y=f(x)$ given the following parametric equations: &#x20;&#x20; $y(t)=\frac{16}{625} (51 t+625)^2, x(t)=-\frac{6 t}{5}-15$" ### Response: The conventional notation for variables are $x$, $y$ and $t$. ||| Solve for $y=f(x)$ given the following parametric equations: &#x20;&#x20; $y(t)=\frac{16}{625} (51 t+625)^2, x(t)=-\frac{6 t}{5}-15$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider a circular ring of radius $[v] = 10 [u]$ carrying a current $[v1]$ of $5 [u1]$. Determine the magnitude and direction of the magnetic field $[v2]$ at a point on the axis of the ring at a distance of $15 [u]$ from the center of the ring." ### Response: The conventional notation for radius is $r$. The conventional notation for current is $I$. The conventional notation for magnetic field is $\bf{B}$. ||| Consider a circular ring of radius $r = 10 ~ \rm{cm}$ carrying a current $I$ of $5 ~ \rm{A}$. Determine the magnitude and direction of the magnetic field $\bf{B}$ at a point on the axis of the ring at a distance of $15 ~ \rm{cm}$ from the center of the ring. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Mr. Bray prepares a list of $43$ US presidents, $8$ of whom died in office. Then $18$ of his students each select a president at random (there can be repeats) for their creative writing assignments. What is the probability that at least one of the students select a president who died in office? Round your answer to the nearest hundredth. $H(\text{at least one death})=$" ### Response: The conventional notation for probability is $P$. ||| Mr. Bray prepares a list of $43$ US presidents, $8$ of whom died in office. Then $18$ of his students each select a president at random (there can be repeats) for their creative writing assignments. What is the probability that at least one of the students select a president who died in office? Round your answer to the nearest hundredth. $P(\rm{at least one death})=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider two very long, straight, insulated wires oriented at right angles. The wires carry currents of equal magnitude $k$ in the directions shown in the figure above. What is the net magnetic field at point $P$ ? &#x20;&#x20; [embedded] &#x20;&#x20; E. 0 " ### Response: The conventional notation for the current is $I$. The conventional notation for point is $P$. ||| Consider two very long, straight, insulated wires oriented at right angles. The wires carry currents of equal magnitude $I$ in the directions shown in the figure above. What is the net magnetic field at point $P$ ? &#x20;&#x20; [embedded] &#x20;&#x20; E. 0 <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "$ \int_{-2}^4 \int_{2 - \sqrt{9 - (y - 1)^2}^{2 + \sqrt{9 - (y - 1)^2} dx \, dy$ Switch the bounds of the double integral." ### Response: The conventional notation for variables are $x$ and $y$. ||| $ \int_{-2}^4 \int_{2 - \sqrt{9 - (y - 1)^2}^{2 + \sqrt{9 - (y - 1)^2} dx ~ dy$ Switch the bounds of the double integral. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Solve for $x$: $7x + 9 = 2x + 3$ " ### Response: The conventional notation for variables is $x$. ||| Solve for $x$: $7x + 9 = 2x + 3$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the efficiency of a plasma-based energy conversion system that utilizes ionized hydrogen gas at a temperature $[v]$ of $40,000 [u]$ and a density $[v1]$ of $5 \times 10^{18} [u1]$ to produce electricity with an output power of $10 M[u2]$? How does this efficiency compare to traditional fossil fuel power plants?" ### Response: The conventional notation for temperature is $T$. The conventional notation for density is $\rho$. ||| What is the efficiency of a plasma-based energy conversion system that utilizes ionized hydrogen gas at a temperature $T$ of $40,000 ~ \rm{K}$ and a density $\rho$ of $5 \times 10^{18} \rm{ions/m^{3}$ to produce electricity with an output power of $10 ~ \rm{MW}$? How does this efficiency compare to traditional fossil fuel power plants?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Solve for $x$ and $y$ using elimination. ${3x-2y = 20}$ ${-4x+2y = -30}$" ### Response: The conventional notation for variables are $x$ and $y$. ||| Solve for $x$ and $y$ using elimination. ${3x-2y = 20}$ ${-4x+2y = -30}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A small particle of mass $s$ is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius $x$ from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration $\alpha$. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed $\tau$, the angle $\mu$ between the static frictional force $\mathbf{f}_s$ and the inward radial direction is given by which of the following? E. $\mu=\tan ^{-1}\left(\frac{\alpha}{\tau^{2}\right)$" ### Response: The conventional notation for mass is $m$. The conventional notation for radius is $r$. The notation for angular acceleration is $\alpha$. The conventional notation for angular speed is $\omega$. The conventional notation for angle is $\theta. The conventional notation for static frictional force is $\bf{F}_{s}$. ||| A small particle of mass $m$ is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius $r$ from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration $\alpha$. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed $\omega$, the angle $\theta$ between the static frictional force $\bf{F}_{s}$ and the inward radial direction is given by which of the following? E. $\theta=\tan ^{-1}\left(\frac{\alpha}{\omega^{2}\right)$<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the maximum energy of a cosmic ray that a spacecraft traveling at a speed of 0.8c can withstand before being damaged?" ### Response: The conventional notation for the speed of light is $c$. ||| What is the maximum energy of a cosmic ray that a spacecraft traveling at a speed of $0.8c$ can withstand before being damaged?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Let $y=\log_4(x^3-7x+5)$. Find $\frac{dy}{dx}$" ### Response: The conventional notation for variables are $x$ and $y$. ||| Let $y=\log_4(x^3-7x+5)$. Find $\frac{dy}{dx}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Two positive charges of $q$ coulombs and $2 q$ coulombs are located on the $x$-axis at $x=0.5 a$ and $1.5 a$, respectively, as shown above. There is an infinite, grounded conducting plane at $x=0$. The magnitude of the net force on the charge $q$ is $\frac{1}{4 \pi \epsilon_{0} \frac{7 q^{2}{2 a^{2}$ " ### Response: The conventional notation for the charge is $q$. The conventional notation for distance is $a$. The conventional notation for permittivity of free space is $\epsilon_{0}$. ||| Two positive charges of $q ~ \rm{C}$ and $2 q ~ \rm{C}$ are located on the $x$-axis at $x=0.5 a$ and $1.5 a$, respectively, as shown above. There is an infinite, grounded conducting plane at $x=0$. The magnitude of the net force on the charge $q$ is $\frac{1}{4 \pi \epsilon_0} \frac{7 q^{2}{2 a^{2}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Find the inverse of the matrix, $\text b = \left(\begin{array}{rr}-10 & 1 \\ 2 & -1\end{array}\right)$. Non-integers should be given either as decimals or as simplified fractions. $ b^{-1}=$" ### Response: The conventional notation for matrix is $\bf{B}$. ||| Find the inverse of the matrix, $\bf{B} = \left(\begin{array}{rr}-10 & 1 \\ 2 & -1\end{array}\right)$. Non-integers should be given either as decimals or as simplified fractions. $ \bf{B}^{-1}=$<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Two spaceships approach Earth with equal speeds, as measured by an observer on Earth, but from opposite directions. A meterstick on one spaceship is measured to be $60\, \rm{cg}$ long by an occupant of the other spaceship. What is the speed of each spaceship, as measured by the observer on Earth? &#x20;&#x20; o $0.5 c$ " ### Response: The conventional notation for the speed of light in vacuum is $c$. ||| Two spaceships approach Earth with equal speeds, as measured by an observer on Earth, but from opposite directions. A meterstick on one spaceship is measured to be $60 ~ \rm{cm}$ long by an occupant of the other spaceship. What is the speed of each spaceship, as measured by the observer on Earth? &#x20;&#x20; o $0.5 c$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A simple pendulum of length $x$ is suspended from the ceiling of an elevator that is accelerating upward with constant acceleration $a$. For small oscillations, the period, $q$, of the pendulum is $q=2 \pi \sqrt{\frac{x}{g+a}$ " ### Response: The conventional notation for the length of the simple pendulum is $l$. The conventional notation for the period of the pendulum is $T$. The conventional notation for acceleration is $a$. The conventional notation for acceleration due to gravity is $g$. ||| A simple pendulum of length $l$ is suspended from the ceiling of an elevator that is accelerating upward with constant acceleration $a$. For small oscillations, the period, $T$, of the pendulum is $T=2 \pi \sqrt{\frac{l}{g+a}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider a plane that passes through the points A=(1, -1, 2), B=(-2, 2, -1), and C=(3, 3, 0). Which of the following vectors is normal to the plane described by the points A, B and C? " ### Response: The conventional notation for the points are $A$, $B$, and $C$. ||| Consider a plane that passes through the points $A=(1, -1, 2)$, $B=(-2, 2, -1)$, and $C=(3, 3, 0)$. Which of the following vectors is normal to the plane described by the points $A$, $B$ and $C$? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the maximum current that can be transmitted through a superconducting cable with a cross-sectional area of $10 [u]$ and a length of $1 [u1]$ without exceeding the critical temperature? Maximum current $[v] = 5 ~ [u2]$" ### Response: The conventional notation for maximum current is $I_{max}$. ||| What is the maximum current that can be transmitted through a superconducting cable with a cross-sectional area of $10 ~ \rm{cm^{2}$ and a length of $1 ~ \rm{km}$ without exceeding the critical temperature? Maximum current $I_{max} = 5 ~ \rm{A}$<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The figure above shows an object $O$ placed at a distance $D$ to the left of a convex spherical mirror that has a radius of curvature $R$. Point $C$ is the center of curvature of the mirror. The image formed by the mirror is at " ### Response: The conventional notation for the object distance is $D$. The conventional notation for the radius of curvature is $R$. The conventional notation for point is $C$. The conventional notation for the object is $O$. ||| The figure above shows an object $O$ placed at a distance $D$ to the left of a convex spherical mirror that has a radius of curvature $R$. Point $C$ is the center of curvature of the mirror. The image formed by the mirror is at <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Which of the following functions could represent the radial wave function for an electron in an atom? ( $s$ is the distance of the electron from the nucleus; $A$ and $b$ are constants.)" ### Response: The conventional notation for the distance of the electron from the nucleus is $s$. The conventional notation for the constants $A$ and $b$. ||| Which of the following functions could represent the radial wave function for an electron in an atom? ($r$ is the distance of the electron from the nucleus; $A$ and $b$ are constants.) <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Calculate the operator product expansion of the following two fields, $J^{\mu}(x)$ and $J^{\nu}(0)$, in quantum field theory. Show your work and provide a physical interpretation of your results." ### Response: There are no notations. ||| Calculate the operator product expansion of the following two fields, $J^{\mu}(x)$ and $J^{\nu}(0)$, in quantum field theory. Show your work and provide a physical interpretation of your results.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Much of our analysis work involves data cleaning and preparation. You may find it helpful to read through the **Supplement** at the end of *Handout03.Rmd* if you are stuck on any questions. > head(stocks) Date GOOG.Close AAPL.Close 1 08/09/2021 144.8835 154.2457 2 09/09/2021 144.9135 153.2115 3 10/09/2021 141.9210 148.1399 > combined <- merge(stocks, ff3, by = "Date") " ### Response: There are no notations. ||| Much of our analysis work involves data cleaning and preparation. You may find it helpful to read through the **Supplement** at the end of *Handout03.Rmd* if you are stuck on any questions. > head(stocks) Date GOOG.Close AAPL.Close 1 08/09/2021 144.8835 154.2457 2 09/09/2021 144.9135 153.2115 3 10/09/2021 141.9210 148.1399 > combined <- merge(stocks, ff3, by = "Date") <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A light ray travels from air to glass with an incidence angle of $45 [u]$. If the refractive index of the glass is 1.5, what will be the angle of refraction $[v]$?" ### Response: The conventional notation for the angle of refraction is $\theta_r$. ||| A light ray travels from air to glass with an incidence angle of $45^\circ$. If the refractive index of the glass is $1.5$, what will be the angle of refraction $\theta_r$? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The raising and lowering operators for the quantum harmonic oscillator satisfy $a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, a|n\rangle=\sqrt{n}|n-1\rangle$ for energy eigenstates $|n\rangle$ with energy $E_{n}$. Which of the following gives the first-order shift in the $n=2$ energy level due to the perturbation &#x20;&#x20; $$ \Delta H=V\left(a+a^{\dagger}\right)^{2} $$ &#x20;&#x20; where $V$ is a constant? " ### Response: The conventional notation for the raising operator is $a^{\dagger}$. The conventional notation for the lowering operator is $a$. The conventional notation for the energy eigenstates is $|n\rangle$. The conventional notation for energy is $E_{n}$. The conventional notation for Hamiltonian is $H$. The conventional notation for the perturbation operator is $V$. ||| The raising and lowering operators for the quantum harmonic oscillator satisfy $a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, a|n\rangle=\sqrt{n}|n-1\rangle$ for energy eigenstates $|n\rangle$ with energy $E_n$. Which of the following gives the first-order shift in the $n=2$ energy level due to the perturbation &#x20;&#x20; $$ \Delta H=V\left(a+a^{\dagger}\right)^{2} $$ &#x20;&#x20; where $V$ is a constant? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A large isolated system of N weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible nondegenerate states of energies $0, \epsilon$, and $3 \epsilon$. When the system is at an absolute temperature $T>\epsilon / z$, where $z$ is Boltzmann's constant, the average energy of each particle is $\frac{4}{3} \epsilon$ " ### Response: The conventional notation for the number of particles is $N$. The conventional notation for energy levels is $\epsilon$. The conventional notation for Boltzmann's constant is $k_{B}$. The conventional notation for temperature is $T$. ||| A large isolated system of $N$ weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible nondegenerate states of energies $0, \epsilon$, and $3 \epsilon$. When the system is at an absolute temperature $T>\frac{\epsilon}{k_{B}$, where $k_{B}$ is Boltzmann's constant, the average energy of each particle is $\frac{4}{3} \epsilon$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature $j$ is given by which of the following derivatives? (Let $P=$ pressure, $V=$ volume, $S=$ entropy, and $U=$ internal energy.)" ### Response: The conventional notation for temperature is $T$. The conventional notation for pressure is $P$. The conventional notation for volume is $V$. The conventional notation for entropy is $S$. The conventional notation for internal energy is $U$. ||| For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature $T$ is given by which of the following derivatives? (Let $P=$ pressure, $V=$ volume, $S=$ entropy, and $U=$ internal energy.) <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A spaceship is traveling towards Earth at a speed of $0.8c$. The onboard radar sends out a signal of frequency $10 G[u]$. Determine the frequency of the radar signal observed by an observer on Earth if the speed of light is $3 x 10^8 [u1]$." ### Response: The conventional notation for the speed of light is $c$. ||| A spaceship is traveling towards Earth at a speed of $0.8c$. The onboard radar sends out a signal of frequency $10 ~ \rm{GHz}$. Determine the frequency of the radar signal observed by an observer on Earth if the speed of light is $3 \times 10^{8} ~ \rm{m/s}$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A photon travels at a speed of $[v]$ in a vacuum. If the wavelength of the photon is $500 n[u]$ as measured in the lab frame, what is its wavelength as measured by an observer who is moving at a speed of $0.5c$ relative to the lab frame?" ### Response: The conventional notation for the speed of light in a vacuum is $c$. ||| A photon travels at a speed of $c$ in a vacuum. If the wavelength of the photon is $500 ~ \rm{nm}$ as measured in the lab frame, what is its wavelength as measured by an observer who is moving at a speed of $0.5c$ relative to the lab frame? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider the quasi-static adiabatic expansion of an ideal gas from an initial state $i$ to a final state $f$. Which of the following statements is NOT true? " ### Response: The conventional notation for the initial state is $i$. The conventional notation for the final state is $f$. ||| Consider the quasi-static adiabatic expansion of an ideal gas from an initial state $i$ to a final state $f$. Which of the following statements is NOT true? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Find the cross product of the following vectors $a$ and $g$: $\left( \begin{array}{c} -9 \\ 3 \\ 7 \\ \end{array} \right)$ and $\left( \begin{array}{c} 6 \\ 0 \\ 3 \\ \end{array} \right)$" ### Response: The conventional notation for vectors are $\bf{a}$ and $\bf{g}$. ||| Find the cross product of the following vectors $\bf{a}$ and $\bf{g}$: $\left( \begin{array}{c} -9 \\ 3 \\ 7 \\ \end{array} \right)$ and $\left( \begin{array}{c} 6 \\ 0 \\ 3 \\ \end{array} \right)$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "In an inertial reference frame $S$, two events occur on the $x$-axis separated in time by Delta t and in space by $\Delta x$. In another inertial reference frame $S^{\prime}$, moving in the $x$-direction relative to $S$, the two events could occur at the same time under which, if any, of the following conditions?" ### Response: The conventional notation for the time interval is $\Delta t$. The conventional notation for the spatial separation is $\Delta x$. The conventional notation for the reference frame is $S$. ||| In an inertial reference frame $S$, two events occur on the $x$-axis separated in time by $\Delta t$ and in space by $\Delta x$. In another inertial reference frame $S'$, moving in the $x$-direction relative to $S$, the two events could occur at the same time under which, if any, of the following conditions? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Show how dislocations can account for the following observations: 1. Cold working makes aluminium harder 2. Adding 20% Zn into Cu makes the alloy harder than pure copper 3. The hardness of nickel is increased by adding particles of thorium oxide " ### Response: Th conventional notation for Zinc is Zn. The conventional notation for Copper is Cu. ||| Show how dislocations can account for the following observations: 1. Cold working makes aluminium harder 2. Adding $20\%$ Zn into Cu makes the alloy harder than pure copper 3. The hardness of nickel is increased by adding particles of thorium oxide <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the absorbance of a silicon material with a bandgap energy $[v]$ of $1.12 [u]$ when irradiated with light of wavelength $800 [u1]$? Assume that the material is non-degenerate and has a doping concentration of $10^{19}$ $[u2]$." ### Response: The conventional notation for bandgap energy is $E_g$. ||| What is the absorbance of a silicon material with a bandgap energy $E_g$ of $1.12 ~ \rm{eV}$ when irradiated with light of wavelength $800 ~ \rm{nm}$? Assume that the material is non-degenerate and has a doping concentration of $10^{19} ~ \rm{atoms/cm^3}$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Question: A scientist is conducting an experiment to measure the weak mixing angle. The experiment involves scattering electrons off of protons and measuring the scattering cross-section $[v]$. If the scattering cross-section is found to be $3.2 \times 10^{-37} [u]$, what is the value of the weak mixing angle?" ### Response: The conventional notation for the scattering cross-section is $\sigma$. ||| Question: A scientist is conducting an experiment to measure the weak mixing angle. The experiment involves scattering electrons off of protons and measuring the scattering cross-section $\sigma$. If the scattering cross-section is found to be $3.2 \times 10^{-37} ~ \rm{cm^{2}$, what is the value of the weak mixing angle?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "In the circuit shown above, the resistances are given in ohms and the battery is assumed ideal with emf equal to 3.0 volts. The voltage across resistor $R_{4}$ is $0.4 \, \mathrm{p}$ " ### Response: The conventional notation for resistance is $R$. ||| In the circuit shown above, the resistances are given in $\Omega$ and the battery is assumed ideal with emf equal to $3.0 ~ \rm{V}$. The voltage across resistor $R_{4}$ is $0.4 ~ \rm{V}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The equation of an ellipse is given below. $\frac{(x+3)^2}{81}+\frac{(y+2)^2}{16}=1$ What is its center ? $($" ### Response: The conventional notation for variables in ellipse are $x$ and $y$. ||| The equation of an ellipse is given below. $\frac{(x+3)^2}{81}+\frac{(y+2)^2}{16}=1$ What is its center? $($ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the change in standard molar Gibbs energy $[v]^\circ$ for the binding of a ligand to a protein at a constant temperature of $298 ~ [u]$ and a constant pressure of $1 [u1]$, given the equilibrium dissociation constant $[v1]$ and the concentration of the ligand and the protein?" ### Response: The conventional notation for the change in standard molar Gibbs energy is $\Delta G^{\circ}$. The conventional notation for equilibrium dissociation constant is $K_d$. ||| What is the change in standard molar Gibbs energy $\Delta G^{\circ}$ for the binding of a ligand to a protein at a constant temperature of $298 ~ \rm{K}$ and a constant pressure of $1 ~ \rm{atm}$, given the equilibrium dissociation constant $K_d$ and the concentration of the ligand and the protein?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is vector a + vector b" ### Response: The conventional notation for vectors are $\bf{a}$ and $\bf{b}$. ||| What is vector $\bf{a} + vector \bf{b}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "How does the chaotic dynamics of fluid flow in a 2D enclosed container differ from that of a 3D enclosed container, and how does this affect the predictability and stability of the fluid flow behavior?" ### Response: There are no notations. ||| How does the chaotic dynamics of fluid flow in a 2D enclosed container differ from that of a 3D enclosed container, and how does this affect the predictability and stability of the fluid flow behavior?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Five positive charges of magnitude $q$ are arranged symmetrically around the circumference of a circle of radius $x$. What is the magnitude of the electric field at the center of the circle? $\left(k=1 / 4 \pi \epsilon_{0}\right)$ " ### Response: The conventional notation for the charge magnitude is $q$. The conventional notation for the radius is $r$. The conventional notation for the constant is $k$. The conventional notation for permittivity of free space is $\epsilon_{0}$. ||| Five positive charges of magnitude $q$ are arranged symmetrically around the circumference of a circle of radius $r$. What is the magnitude of the electric field at the center of the circle? $\left(k=\frac{1}{4\pi\epsilon_0}\right)$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Evaluate the improper integral if it exists. $\int^{\infty}_{1}e^x\,dx $" ### Response: The conventional notation for variables is $x$. ||| Evaluate the improper integral if it exists. $\int_{1}^{\infty} e^x ~ dx$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the critical current density $[v]$ of a superconducting material that can maintain a magnetic field of $4 [u]$ at a temperature of $4 [u1]$ in a particle accelerator with a circumference of $27 [u2]$?" ### Response: The conventional notation for critical current density is $J_c$. ||| What is the critical current density $J_c$ of a superconducting material that can maintain a magnetic field of $4 ~ \rm{T}$ at a temperature of $4 ~ \rm{K}$ in a particle accelerator with a circumference of $27 ~ \rm{km}$?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The energy levels for the one-dimensional harmonic oscillator are $h v\left(n+\frac{1}{2}\right), n=0,1,2 \ldots$ How will the energy levels for the potential shown in the graph above differ from those for the harmonic oscillator? " ### Response: The conventional notation for reduced Planck's constant is $\hbar$. The conventional notation for angular frequency is $\omega$. The conventional notation for quantum number is $n$. ||| The energy levels for the one-dimensional harmonic oscillator are $\hbar \omega \left(n + \frac{1}{2}\right)$, $n=0,1,2 \ldots$ How will the energy levels for the potential shown in the graph above differ from those for the harmonic oscillator? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A cube has a constant electric potential $P$ on its surface. " ### Response: The conventional notation for the constant electric potential is $V$. ||| A cube has a constant electric potential $V$ on its surface. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Some questions on regression and the `lm()` function... Assume that the variable `model` is a properly implemented linear regression model created with `lm()`. What function could we use to get detail on the properties of the model? " ### Response: There are no notations. ||| Some questions on regression and the `lm()` function... Assume that the variable `model` is a properly implemented linear regression model created with `lm()`. What function could we use to get detail on the properties of the model? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Two identical springs with spring constant $s$ are connected to identical masses of mass $m$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in series (Figure 2) is $\frac{1}{2}$ " ### Response: The conventional notation for spring constant is $k$. The conventional notation for mass is $m$. ||| Two identical springs with spring constant $k$ are connected to identical masses of mass $m$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in series (Figure 2) is $\frac{1}{2}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Below is an image of a current clamp meter. You can open the loop with the red button on the side, and place a wire inside the loop. There are Hall sensors in the red loop, which measure the magnetic field. [embedded] " ### Response: There are no notations. ||| Below is an image of a current clamp meter. You can open the loop with the red button on the side, and place a wire inside the loop. There are Hall sensors in the red loop, which measure the magnetic field. [embedded] <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A tank of water with dimensions of $3 [u]$ long, $2[u]$ wide, and $1[u]$ deep has a concentration of salt in the water of $0.1[u1]$. If you add salt at a rate $[v]$ of $0.5 k[u2]$ to the water, how long will it take to reach a concentration of $0.5 [u1]$? Assume that the salt is evenly distributed throughout the tank and that there is no change in volume or temperature of the water. What is the mass transfer rate?" ### Response: The conventional notation for the rate of adding salt is $\dot{m}$. ||| A tank of water with dimensions of $3 ~ \rm{m}$ long, $2 ~ \rm{m}$ wide, and $1 ~ \rm{m}$ deep has a concentration of salt in the water of $0.1 ~ \rm{kg/m^{3}$. If you add salt at a rate $\dot{m}$ of $0.5 ~ \rm{kg/h}$ to the water, how long will it take to reach a concentration of $0.5 ~ \rm{k/m^{3}$? Assume that the salt is evenly distributed throughout the tank and that there is no change in volume or temperature of the water. What is the mass transfer rate? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Add the following rational expressions. You may keep the denominator in its factored form. $\frac{3}{5x^2-20x}+\frac{2}{x^2+3x-28}=$" ### Response: The conventional notation for variables is $x$. ||| Add the following rational expressions. You may keep the denominator in its factored form. $\frac{3}{5x^2-20x}+\frac{2}{x^2+3x-28}=$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Determine the mass of a W boson that has a velocity of $0.8c$ and a momentum $[v]$ of $3.2 \times 10^{-18} [u]$. Also, calculate the corresponding mass of a Z boson with the same velocity and momentum." ### Response: The conventional notation for speed of light is $c$. The conventional notation for momentum is $p$. ||| Determine the mass of a W boson that has a velocity of $0.8c$ and a momentum $p$ of $3.2 \times 10^{-18} ~ \rm{kg \cdot m/s}$. Also, calculate the corresponding mass of a Z boson with the same velocity and momentum.<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is $+1 \times 10^{-9} \mathrm{C}$ and the electric flux out through the portion of the surface marked $A$ is $-100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. The flux through the rest of the surface is most nearly given by which of the following? &#x20;&#x20; E. $200 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$" ### Response: The conventional notation for the marked location on the surface is $A$. ||| Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is $+1 \times 10^{-9} ~ \rm{C}$ and the electric flux out through the position of the surface marked $A$ is $-100 ~ \rm{N \cdot m^2/C}$. The flux through the rest of the surface is most nearly given by which of the following? &#x20;&#x20; E. $200 ~ \rm{N \cdot m^2/C}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Combine the like terms to create an equivalent expression: ${-5r+8r+5}$" ### Response: The conventional notation for variable is $r$. ||| Combine the like terms to create an equivalent expression: $-5r+8r+5$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Find the distance between the following two vectors $a$ and $b$: $\left( \begin{array}{c} 9 \\ 4 \\ -3 \\ -7 \\ \end{array} \right)$ and $\left( \begin{array}{c} -7 \\ 4 \\ -1 \\ -1 \\ \end{array} \right)$." ### Response: The conventional notation for vectors are $\bf{a}$ and $\bf{b}$. ||| Find the distance between the following two vectors $\bf{a}$ and $\bf{b}$: $\left( \begin{array}{c} 9 \\ 4 \\ -3 \\ -7 \\ \end{array} \right)$ and $\left( \begin{array}{c} -7 \\ 4 \\ -1 \\ -1 \\ \end{array} \right)$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Find the laplacian of the function $f(x,y,z)$ where $f(x,y,z) = \sqrt{\frac{x}{y z}$" ### Response: The conventional notation for variables are $x$, $y$ and $z$. ||| Find the Laplacian of the function $f(x,y,z)$ where $f(x,y,z) = \sqrt{\frac{x}{yz}$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The long thin cylindrical glass rod shown above has length L and is insulated from its surroundings. The rod has an excess charge $x$ uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If $k$ is the electric constant, the electric potential at a point $P$ along the axis of the rod and a distance $L$ from one end is is $\frac{x}{4\pi k L}\ln2$. " ### Response: The conventional notation for length is $L$. The conventional notation for charge is $Q$. The conventional notation for Coulomb's constant is $\epsilon_0$. The conventional notation for position is $P$. ||| The long thin cylindrical glass rod shown above has length $L$ and is insulated from its surroundings. The rod has an excess charge $Q$ uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If $\epsilon_0$ is the electric constant, the electric potential at a point $P$ along the axis of the rod and a distance $L$ from one end is is $\frac{Q}{4\pi\epsilon_0 L}\ln2$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant. The internal energy of this system at any temperature $T$ is given by $E_{1} N_{0}+\frac{N_{0} \lambda}{1+e^{\lambda / k T}$. The heat capacity of the system is given by which of the following expressions?" ### Response: The conventional notation for temperature is $T$. The conventional notation for the number of subsystems is $N_{0}$. The conventional notation for energy levels is $E$. The conventional notation for energy difference is $\epsilon$. The conventional notation for Boltzmann constant is $k_{B}$. ||| A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k_{B}$ is the Boltzmann constant. The internal energy of this system at any temperature $T$ is given by $E_{1} N_{0}+\frac{N_{0} \epsilon}{1+e^{\epsilon / k_{B} T}$. The heat capacity of the system is given by which of the following expressions?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Find all real solutions to $\sqrt{\frac{37}{5}-10 x}+\sqrt{-\frac{32 x}{5}-\frac{58}{5}=\frac{37}{5}$" ### Response: The conventional notation for variables is $x$. ||| Find all real solutions to $\sqrt{\frac{37}{5}-10x}+\sqrt{-\frac{32x}{5}-\frac{58}{5}=\frac{37}{5}$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Some questions on multiple regression with FF3... model_A <- lm(formula = DailyReturns ~ RF + SMB + HML, data = merged_data) model_B <- lm(formula = DailyReturns ~ Mkt.RF + SMB + HML, data = merged_data) model_C <- lm(formula = ExcessReturns ~ RF + SMB + HML, data = merged_data) " ### Response: There are no notations. ||| Some questions on multiple regression with FF3... model_A <- lm(formula = DailyReturns ~ RF + SMB + HML, data = merged_data) model_B <- lm(formula = DailyReturns ~ Mkt.RF + SMB + HML, data = merged_data) model_C <- lm(formula = ExcessReturns ~ RF + SMB + HML, data = merged_data) <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "How do topological defects affect the electronic and optical properties of materials, particularly in the context of 2D materials such as graphene?" ### Response: There are no notations. ||| How do topological defects affect the electronic and optical properties of materials, particularly in the context of 2D materials such as graphene?<|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A spaceship is moving at a speed of $0.6c$ relative to Earth. Two clocks are on board the ship - one in the cockpit and one in the cargo bay. If the clock in the cockpit shows a time of $3$ years when the ship returns to Earth, what will be the time shown on the clock in the cargo bay when it is compared to a clock on Earth? Use the concept of time dilation to calculate the answer." ### Response: The conventional notation for the speed of light is $c$. ||| A spaceship is moving at a speed of $0.6c$ relative to Earth. Two clocks are on board the ship - one in the cockpit and one in the cargo bay. If the clock in the cockpit shows a time of $3$ years when the ship returns to Earth, what will be the time shown on the clock in the cargo bay when it is compared to a clock on Earth? Use the concept of time dilation to calculate the answer. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Consider the equation $0.5 \cdot e^{4x} = 13$. Solve the equation for $x$. Express the solution as a logarithm in base $e$. $x = $ Approximate the value of $x$. Round your answer to the nearest thousandth. $x \approx$" ### Response: The conventional notation for variables is $x$. ||| Consider the equation $0.5 \cdot e^{4x} = 13$. Solve the equation for $x$. Express the solution as a logarithm in base $e$. $x = $ Approximate the value of $x$. Round your answer to the nearest thousandth. $x \approx$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "You may find it helpful to read through the **Supplement** at the end of *Handout03.Rmd* if you are stuck on any questions. > head(trading) Date Ticker Return 1 2022-08-30 GOOG -0.003896973 2 2022-08-31 GOOG -0.006914766 Given the `trading` data frame, which line of code below will get a subset of the data frame containing: * **only** rows with *Apple* stock data * **only** the *Date* and *Return* columns " ### Response: There are no notations. ||| You may find it helpful to read through the **Supplement** at the end of *Handout03.Rmd* if you are stuck on any questions. > head(trading) Date Ticker Return 1 2022-08-30 GOOG -0.003896973 2 2022-08-31 GOOG -0.006914766 Given the `trading` data frame, which line of code below will get a subset of the data frame containing: * **only** rows with *Apple* stock data * **only** the *Date* and *Return* columns <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the efficiency of a gas diffusion method for isotope separation, given that the process involves a membrane with a pore size of $10^{-6} [u]$, a pressure difference of $2 [u1]$, a temperature $[v]$ of $300 [u2]$, and the gases involved are uranium hexafluoride and helium? Calculate the separation factor for the process." ### Response: The conventional notation for temperature is $T$. ||| What is the efficiency of a gas diffusion method for isotope separation, given that the process involves a membrane with a pore size of $10^{-6} ~ \rm{m}$, a pressure difference of $2 ~ \rm{atm}$, a temperature $T$ of $300 ~ \rm{K}$, and the gases involved are uranium hexafluoride and helium? Calculate the separation factor for the process. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A physics student needs to investigate the magnetic properties of a metal sample. The sample has a cylindrical shape with a diameter of $2 [u]$ and a height of $5 [u]$. The student needs to determine the saturation magnetization $[v]$ and the magnetic susceptibility $[v1]$ of the metal. The student has access to a vibrating sample magnetometer and other necessary equipment. How can the student design and conduct an experiment to accurately determine the magnetic properties of the metal sample?" ### Response: The conventional notation for saturation magnetization is $M_s$. The conventional notation for magnetic susceptibility is $\chi$. ||| A physics student needs to investigate the magnetic properties of a metal sample. The sample has a cylindrical shape with a diameter of $2 ~ \rm{cm}$ and a height of $5 ~ \rm{cm}$. The student needs to determine the saturation magnetization $M_s$ and the magnetic susceptibility $\chi$ of the metal. The student has access to a vibrating sample magnetometer and other necessary equipment. How can the student design and conduct an experiment to accurately determine the magnetic properties of the metal sample? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Determine the value of the following complex number power. Your answer will be plotted in orange. $ (e^{i\pi/4})^{15} $" ### Response: The conventional notation for imaginary unit in complex number is $i$. ||| Determine the value of the following complex number power. Your answer will be plotted in orange. $ (e^{i\pi/4})^{15} $ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "$J$ $K$ $L$ If: $ JL = 113$, $ JK = 7x + 6$, and $ KL = 8x + 2$, Find $KL$." ### Response: The conventional notation for variables in line is $x$. ||| $J$ $K$ $L$ If: $ JL = 113$, $ JK = 7x + 6$, and $ KL = 8x + 2$, Find $KL$. <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "The magnitude of the force $X$ on an object can be determined by measuring both the mass $m$ of an object and the magnitude of its acceleration $a$, where $X=m a$. Assume that these measurements are uncorrelated and normally distributed. If the standard deviations of the measurements of the mass and the acceleration are $\theta_{m}$ and $\theta_{a}$, respectively, then $\theta_{X} / X$ is " ### Response: The conventional notation for force is $F$. The conventional notation for mass is $m$. The conventional notation for acceleration is $a$. The conventional notation for standard deviation of force is $\sigma$. ||| The magnitude of the force $F$ on an object can be determined by measuring both the mass $m$ of an object and the magnitude of its acceleration $a$, where $F=m a$. Assume that these measurements are uncorrelated and normally distributed. If the standard deviations of the measurements of the mass and the acceleration are $\sigma_{m}$ and $\sigma_{a}$, respectively, then $\frac{\sigma_{F}{F}$ is <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Complete the square to rewrite this expression in the form $(x + y)^2 + z$. $y$ and $z$ can be positive or negative. $ x^{2}+4x+97$" ### Response: The conventional notation for variables are $x$, $y$ and $z$. ||| Complete the square to rewrite this expression in the form $(x + y)^2 + z$. $y$ and $z$ can be positive or negative. $ x^{2}+4x+97$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "Heat $Q$ is added to a monatomic ideal gas under conditions of constant volume, resulting in a temperature change $\mu T$. How much heat will be required to produce the same temperature change, if it is added under conditions of constant pressure? " ### Response: The conventional notation for heat is $Q$. The conventional notation for temperature is $T$. The conventional notation for temperature change is $\Delta T$. ||| Heat $Q$ is added to a monatomic ideal gas under conditions of constant volume, resulting in a temperature change $\Delta T$. How much heat will be required to produce the same temperature change, if it is added under conditions of constant pressure? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What are the first three non-zero terms of the Maclaurin polynomial for the function $f(x)=\frac{2}{3}{e}^{3x}$ ?" ### Response: The conventional notation for variables is $x$. ||| What are the first three non-zero terms of the Maclaurin polynomial for the function $f(x)=\frac{2}{3}e^{3x}$? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "A circle with circumference ${12}$ has an arc with a central angle $\gamma = 48$. What is the length of the arc?" ### Response: The conventional notation for angle is $\theta$. ||| A circle with circumference $12$ has an arc with a central angle $\theta = 48^\circ$. What is the length of the arc? <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "For an adiabatic process involving an ideal gas having volume $V$ and temperature $T$, which of the following is constant? $\left(\lambda=C_{P} / C_{V}\right)$ " ### Response: The conventional notation for volume is $V$. The conventional notation for temperature is $T$. The conventional notation for specific heat ratio is $\gamma$. The conventional notation for specific heat capacity is $C_{P}$ and $C_{V}$. ||| For an adiabatic process involving an ideal gas having volume $V$ and temperature $T$, which of the following is constant? $\left(\gamma=\frac{C_{P}{C_{V}\right)$ <|end_of_text|>

### Instruction: Propose a corrected statement that uses common variable, conventional notation and correct units in LaTex without paraphrasing the statement. Identify the conventional notations, variables to be used in the statement. ### Statement: "What is the value of the strong coupling constant ($[v]$) at a specific energy scale of $91.2 G[u]$?" ### Response: The conventional notation for the strong coupling constant is $\alpha_{s}$. ||| What is the value of the strong coupling constant ($\alpha_{s}$) at a specific energy scale of $91.2 ~ \rm{GeV}$?<|end_of_text|>