1Let $S = (0,1) \cup (1,2) \cup (3,4)$ and $T = \{0,1,2,3\}$. Let $f: S \to T$ be a function. Which of the following statements is(are) true?
2Two satellites P and Q move in circular orbits around Earth (radius $R$). Heights of P and Q are $h_P = R/3$ and $h_Q$, respectively. If the ratio of accelerations due to gravity $g_P/g_Q = 36/25$, find the height $h_Q$.
3A gas has a compressibility factor of 0.5 and a molar volume of $0.4 \text{ dm}^3 \text{ mol}^{-1}$ at 800 K and pressure $x$ atm. If its ideal molar volume at the same $T$ and $P$ is $y \text{ dm}^3 \text{ mol}^{-1}$, calculate the value of $x/y$. (Use $R = 0.08 \text{ L atm K}^{-1} \text{ mol}^{-1}$)
4Let $T_1$ and $T_2$ be distinct common tangents to the ellipse $x^2/6 + y^2/3 = 1$ and the parabola $y^2 = 12x$. If $A_1, A_4$ are points on the parabola and $A_2, A_3$ are points on the ellipse, which statements are correct?
5In a photoelectric effect experiment, photons emitted from a Hydrogen-like atom (atomic number $Z$) during the $n=4$ to $n=3$ transition hit a metal target. If the threshold wavelength is 310 nm and the maximum kinetic energy of photoelectrons is 1.95 eV, what is the value of $Z$?
6For a monobasic weak acid HX, a plot of $1/\Lambda_m$ vs $c\Lambda_m$ gives a straight line. What is the ratio of the y-axis intercept $P$ to the slope $S$?
7Consider the sum $S = 77 + 757 + 7557 + \dots + \underbrace{75\dots57}_{98} = (\underbrace{75\dots57}_{99} + m)/n$. If $m, n < 3000$, find the value of $m + n$.
8A series LCR circuit with $L = 50$ mH has a resonant angular frequency of $10^5 \text{ rad s}^{-1}$. At resonance, the current amplitude is $I_0$. At $\omega = 8 \times 10^4 \text{ rad s}^{-1}$, the amplitude is $0.05 I_0$. Which of the following is correct?
9The solubility of a sparingly soluble salt MX of a weak acid HX increases from $10^{-4} \text{ mol L}^{-1}$ to $10^{-3} \text{ mol L}^{-1}$ when pH is decreased from 7 to 2. What is the $pK_a$ of HX?
10Let $V$ be the volume of a parallelepiped determined by three distinct unit vectors $\mathbf{u, v, w}$ that lie at a distance of 3.5 from the plane $\sqrt{3}x + 2y + 3z = 16$ and satisfy $|\mathbf{u}-\mathbf{v}| = |\mathbf{v}-\mathbf{w}| = |\mathbf{w}-\mathbf{u}|$. Find the value of $\frac{80}{\sqrt{3}}V$.
