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Submitting answer for FRQ problems:

You may use plus, minus sign, decimal point, exponential form e or E, and numbers in your submitted answer. For example, the following numerical answers are accepted: 3.14159, 0.314159e1, 31.4159e-2, -1.6e-19, -1.6E-19, -0.16E-18

No other alphabetical letter or symbol is accepted in the numerical answer. Fractions are not accepted. You will NOT need to specify any unit for FRQ problem.

How to upload your work

Email mindquest2018@gmail.com a single zip file within 1 hour after the exam with your name and some random letter in the format firstname_lastname_NNNN to avoid name clash, e.g. john_doe_2315.zip

You should practice scanning your work, zip your files and email them before exam. It should not take long once you know how to do it.

Notations used in the problems

$\sin^2 x = \sin x \cdot \sin x$ and similarly for other trig functions. $\ln x$ stands for $e$ based natural log, $\log x$ stands for 10 based log.

The following constants are given

Symbol	Name	Value
$c$	Speed of light in vacuum	$299,792,458 m/s \approx 3.00 \times 10^8 \text{m/s}$
$G$	Gravitational constant	$6.67 \times 10^{-11} \;\text{N} \cdot \text{m}^2 / \text{kg}^2$
$g$	g near Earth surface	$10 \;\text{m} \cdot \text{s}^{-2}$
$N_A$	Avogadro’s number	$6.02 \times 10^{23}$
$k$	Boltzmann’s constant	$1.38 \times 10^{-23} \;\text{J} / \text{K}$
$R$	Gas constant	$8.31 \;\text{J} / \text{mol} \cdot \text{K} = 1.99 \text{cal} / \text{mol} \cdot \text{K} = 0.0821 \text{atm} \cdot \text{L} / \text{mol} \cdot \text{K}$
$\sigma$	Stefan-Boltzmann constant	$5.67 \times 10^{-8} \;\text{W} / \text{m}^2 \cdot \text{K}$
$b$	Wien's constant	$2.898 \times 10^{-3} \;\text{K } \cdot \text{m}$
$k$	Coulomb force constant	$8.99 \times 10^9 \;\text{N} \cdot \text{m}^2 / \text{C}^2$
$q_e$	Charge on electron	$-1.60 \times 10^{-19} \;\text{C}$
$\varepsilon _0$	Permittivity of free space	$8.85 \times 10^{-12} \;\text{C}^2 / \text{N} \text{m}^2$
$\mu _0$	Permeability of free space	$4 \pi \times 10^{-7} \;\text{T} \cdot \;\text{m}/ \text{A}$
$h$	Planck’s constant	$6.63 \times 10^{-34} \;\text{J} \cdot \text{s}$
$m_e$	Electron mass	$9.11 \times 10^{-31} \text{kg}$
$m_p$	Proton mass	$1.6726 \times 10^{-27} \text{kg}$
$m_n$	Neutron mass	$1.6749 \times 10^{-27} \text{kg}$
$\text{u}$	Atomic mass unit	$1.6605 \times 10^{-27} \text{kg}$
$M_\odot$	Mass of the Sun	$1.989 \times 10^{30} \text{kg}$
$R_\odot$	Radius of the Sun	$6.96 \times 10^{8} \text{m}$
$T_\odot$	Solar temperature	$5770 \text{K}$
$M_\odot$	Absolute magnitude	4.83
$M_\oplus$	Mass of the Earth	$5.976 \times 10^{24} \text{kg}$
$R_\oplus$	Radius of the Earth	$6.371 \times 10^{6} \text{m}$
AU	Astronomical Unit	$1.496 \times 10^{11} \text{m}$
ly	light year	$9.461 \times 10^{15} \text{m}$
pc	parsec	$3.086 \times 10^{16} \text{m}$
$H_0$	Hubble's constant	$70 \text{(km/s)/Mpc}$
$R_y$	Rydberg constant	$13.6 \text{eV}$
$F$	Faraday's constant	$96500 \text{C/mol}$
$c_p$	Specific Heat water	$4.184 \text{J/g/K}$

The following integrals are given

• $\int \frac{dx}{x^2 + r^2} = \frac{1}{r} \tan^{-1} (\frac{x}{r})$
• $\int \frac{dx}{\sqrt{1 - x^2} } = \sin^{-1} x$
• $\int \frac{dx}{\sqrt{1 + x^2} } = \sinh^{-1} x$
• $\int \frac{dx}{1 - x^2 } = \tanh^{-1} x$
• $\int \frac{dx}{1 + x^2 } = \tan^{-1} x$
• $\int \frac{dx}{(1 - x^2)^{3/2} } = \frac{x}{\sqrt{1 - x^2}}$
• $\int \frac{dx}{(1 + x^2)^{3/2} } = \frac{x}{\sqrt{1 + x^2}}$
• $\int \sqrt{ \frac{ 1 + x}{ 1 - x} } dx = - \sqrt{1 - x^2} - 2 \sin^{-1} (\sqrt{ \frac{1 - x}{2} } )$
• $\int \sqrt{ \frac{ 1 + x}{ (1 - x)^3 } } dx = 2\sqrt{ \frac{1 + x}{1 - x}} + 2 \sin^{-1} (\sqrt{ \frac{1 - x}{2} } )$
• $\int \frac{dx}{ (1 - x)^{3/2} (1 + x)^{1/2} } = \sqrt{ \frac{1 + x}{1 - x}}$
• $\int \frac{dx}{ (1 - x)^{3/2} (1 + x)^{3/2} } = \frac{ x}{\sqrt{1 - x^2}}$
• $\int \frac{dx}{ (1 - x)^{5/2} (1 + x)^{1/2} } = \frac{ (2-x) \sqrt{1 + x} }{3 \sqrt{1 - x}^{3/2} }$
• $\int \frac{dx}{ (1 - x)^{5/2} (1 + x)^{3/2} } = \frac{ 1-2x-2x^2}{3 \sqrt{1 - x}^{3/2} (1+x)^{1/2} }$
• $\int \frac{dx}{\sqrt{x^2 - 1} } = \ln (x + \sqrt{x^2 - 1})$
• $\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln ( \sqrt{x^2 + a^2} + x)$
• $\int \frac{dx}{(a^2 + x^2)^{3/2}} = \frac{x}{a^2 ( a^2 + x^2)^{1/2} }$
• $\int \ln x dx = x \ln x - x$
• $\int x^n \ln ( \frac{a}{x} ) dx = \frac{ x^{n+1} }{(n+1)^2 } + \frac{x^{n+1} }{n + 1} \ln ( \frac{a}{x} )$
• $\int x e^{-x} dx = - (x+1) e^{-x}$
• $\int x^2 e^{-x} dx = - (x^2+2x+2) e^{-x}$
• $\int \sin^3 x dx = - \cos x + \frac{\cos^3 x}{3}$
• $\int \cos^3 x dx = \sin x - \frac{\sin^3 x}{3}$
• $\int \frac{dx}{\cos x} = \ln ( \frac{ 1 + \sin x}{\cos x})$
• $\int \frac{dx}{\sin x} = \ln ( \frac{ 1 - \cos x}{\sin x})$
• $\int \frac{\cos x dx}{ (1 - a^2 \cos^2 x)^{3/2} } = \frac{ \sin x}{ ( 1 - a^2) \sqrt{ 1- a^2 \cos^2 x} }$
• $\int \frac{\sin x dx}{ (1 - a^2 \cos^2 x)^{3/2} } = \frac{ -\cos x}{ \sqrt{ 1- a^2 \cos^2 x} }$
• $\int \frac{\sin x dx}{ (1 - a^2 \sin^2 x)^{3/2} } = \frac{ - \cos x}{ ( 1 - a^2) \sqrt{ 1- a^2 \sin^2 x} }$
• $\int \frac{\cos x dx}{ (1 - b^2 \sin^2 (x-a) )^{3/2} } = \frac{ (2-b^2) \sin x + b^2 \sin(2a -x) }{2 ( 1 - b^2) \sqrt{ 1- b^2 \sin^2 (a-x)} }$
• $\int \frac{\sin x (a \cos x - b) dx}{ (a^2 + b^2 - 2 ab \cos x)^{3/2} } = \frac{ - a + b \cos x}{b^2 \sqrt{a^2 + b^2 - 2 ab \cos x} }$
• $\int_0^\pi \frac{ x - \cos \theta} { 1 + x^2 - 2 x \cos \theta} d \theta = 0$

The following vector identities are given

• $\nabla \cdot (\nabla \times \vec A) = 0$
• $\nabla \cdot (f \vec A) = f \nabla \cdot \vec A + \vec A \cdot \nabla f$
• $\nabla \cdot (\vec A \times \vec B) = \vec B \cdot (\nabla \times \vec A) - \vec A \cdot (\nabla \times \vec B)$
• $\nabla \times (\nabla f) = 0$
• $\nabla \times (f \vec A) = f \nabla \times \vec A + (\nabla f) \times \vec A$
• $\nabla \times (\nabla \times \vec A) = \nabla (\nabla \cdot \vec A) - \nabla^2 \vec A$
• $\nabla \times (\vec A \times \vec B) = \vec A (\nabla \cdot \vec B) - \vec B(\nabla \cdot \vec A) + (\vec B \cdot \nabla) \vec A - (\vec A \cdot \nabla) \vec B$
• $\vec A \times (\vec B \times \vec C) = \vec B (\vec A \cdot \vec C) - \vec C (\vec A \cdot \vec B)$
• $\nabla (\vec A \cdot \vec B) = (\vec A \cdot \nabla) \vec B + (\vec B \cdot \nabla) \vec A + \vec A \times (\nabla \times \vec B) + \vec B \times (\nabla \times \vec A)$

The following Taylor Series are given

• $(1+x)^n = 1 + n x + \frac{n (n-1)}{2} x^2 + \frac{n (n-1) (n-2)}{3!} x^3 + ...$
• $\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + ...$
• $\frac{1}{\sqrt{1+x}} = 1 - \frac{x}{2} + \frac{3x^2}{8} + ...$
• $\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$
• $e^x = 1 + x + \frac{1}{2!} x^2 + \frac{1}{3!} x^3 + ...$
• $\ln (1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$
• $\cos x = 1 - \frac{1}{2!} x^2 + \frac{1}{4!} x^4 + ...$
• $\sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 + ...$
• $\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$

The following formula are given

• $E = \frac{q}{4 \pi \epsilon_0 r^2} \frac{ 1 - \beta^2}{(1 - \beta^2 \sin^2 \theta)^{3/2} }$