2025 HKDSE Physics Practice Paper | DSE Mock

Two particles \(P\) and \(Q\) in a progressive transverse wave are separated by a distance of \(0.6\text{ m}\). The wave travels at a speed of \(12\text{ m s}^{-1}\) with a frequency of \(10\text{ Hz}\). What is the phase difference (in radians) between \(P\) and \(Q\)?

A string of length \(L\) fixed at both ends vibrates at its third harmonic (second overtone) with a frequency of \(f_3\). If the tension in the string is quadrupled while its length remains unchanged, what will be the frequency of the first harmonic (fundamental frequency)?

Two identical resistors of resistance \(R\) are connected in parallel with each other, and this combination is connected in series with a third identical resistor of resistance \(R\). A switch \(S\) is connected in parallel across the third resistor. What is the ratio of the total equivalent resistance of the circuit when \(S\) is open to that when \(S\) is closed?

Two light bulbs \(X\) and \(Y\) are rated as "\(12\text{ V}, 24\text{ W}\)" and "\(12\text{ V}, 12\text{ W}\)" respectively. If they are connected in series across a \(12\text{ V}\) ideal d.c. voltage source, find the total electrical power consumed by the two bulbs. (Assume the resistances of the bulbs are constant.)

A light ray enters a right-angled triangular glass prism normally through the side opposite to the \(60^\circ\) angle. It then strikes the hypotenuse of the prism. In order for total internal reflection to occur at the hypotenuse, what is the minimum refractive index of the glass?

An object is placed in front of a thin converging lens of focal length \(f\), forming a real image on a screen. If the distance between the object and the screen is \(4.5f\), what are the two possible magnifications of the image?

A metal rod of length \(0.5\text{ m}\) slides along two parallel conducting rails at a constant speed of \(4\text{ m s}^{-1}\) in a direction perpendicular to a uniform magnetic field of \(0.8\text{ T}\) pointing into the page. A resistor of resistance \(2\ \Omega\) is connected across the rails to form a closed loop. What is the induced current in the loop and the magnetic force acting on the rod?

A proton (charge \(+e\), mass \(m\)) enters a region of uniform magnetic field \(B\) pointing vertically upwards with a horizontal velocity \(v\). Which of the following statements about the subsequent motion of the proton in the magnetic field is/are correct?

(1) The magnetic force does no work on the proton.
(2) The speed of the proton increases.
(3) The kinetic energy of the proton remains constant.

Two satellites, \(A\) and \(B\), travel around the Earth in circular orbits of radii \(R_A\) and \(R_B\) respectively. If the orbital period of satellite \(A\) is \(8\text{ times}\) that of satellite \(B\), what is the ratio of their orbital speeds, \(\frac{v_A}{v_B}\)?

In a hydrogen atom, an electron transitions from the energy level \(n = 3\) to \(n = 1\), emitting a photon of wavelength \(\lambda_1\). When an electron transitions from \(n = 2\) to \(n = 1\), a photon of wavelength \(\lambda_2\) is emitted. What is the ratio \(\frac{\lambda_1}{\lambda_2}\)?

A string of length \(L\) is fixed at both ends and vibrates in its third harmonic with frequency \(f\). If the tension in the string is quadrupled while its length and linear mass density remain unchanged, what is the new fundamental frequency?

Three identical resistors of resistance \(R\) are connected to an ideal battery of voltage \(V\). In Circuit 1, two resistors are connected in parallel, and this combination is connected in series with the third resistor. The total power dissipated is \(P_1\). In Circuit 2, the resistors are rearranged such that two are in series, and this combination is in parallel with the third. The total power dissipated is \(P_2\). What is the ratio \(P_2 / P_1\)?

A ray of monochromatic light enters a semi-circular glass block (refractive index \(n = 1.50\)) normally through the curved surface and is directed towards the centre \(O\) of the flat surface. The flat surface is in contact with a liquid. When the angle of incidence at \(O\) exceeds \(60^\circ\), total internal reflection occurs. What is the refractive index of the liquid?

A square metal loop of side length \(L\) and resistance \(R\) is pulled with a constant speed \(v\) out of a region of uniform magnetic field \(B\) which is perpendicular to the plane of the loop. What is the magnitude of the external force required to maintain this constant speed?

Two satellites, \(X\) and \(Y\), orbit the Earth in circular orbits. The orbital radius of \(X\) is twice that of \(Y\) (i.e., \(r_X = 2r_Y\)). Which of the following statements is/are correct?

(1) The ratio of their orbital speeds is \(v_X / v_Y = 1 / \sqrt{2}\).
(2) The ratio of their orbital periods is \(T_X / T_Y = 2\sqrt{2\nu}\).
(3) If they have equal mass, the ratio of their kinetic energies is \(K_X / K_Y = 1/2\).

In a hydrogen atom, the energy levels are given by \(E_n = -13.6 / n^2\text{ eV}\), where \(n = 1, 2, 3, \dots\). An electron makes a transition from \(n = 3\) to \(n = 1\), emitting a photon of frequency \(f_1\). Another transition from \(n = 2\) to \(n = 1\) emits a photon of frequency \(f_2\). What is the ratio \(f_1 / f_2\)?

An insulated container contains \(0.20\text{ kg}\) of water at \(20^\circ\text{C}\). A metal block of mass \(0.50\text{ kg}\) at \(80^\circ\text{C}\) is placed into the water. The specific heat capacity of water is \(4200\text{ J kg}^{-1\,\circ}\text{C}^{-1}\) and that of the metal block is \(400\text{ J kg}^{-1\,\circ}\text{C}^{-1}\). Assuming no heat is lost to the surroundings or container, what is the final equilibrium temperature of the mixture?

A car of mass \(m\) starts from rest and accelerates along a straight horizontal road. The engine of the car delivers a constant power \(P\). Assuming resistive forces are negligible, what is the speed \(v\) of the car as a function of time \(t\)?

A ball of mass \(m\) moving due East with speed \(u\) collides with another ball of mass \(2m\) moving due North with speed \(u\). After the collision, the two balls stick together and move as a single combined mass. What is the magnitude of the velocity of the combined mass after the collision?

An ideal gas is contained in a rigid, sealed container of fixed volume. The temperature of the gas is increased from \(27^\circ\text{C}\) to \(327^\circ\text{C}\). Which of the following statements about the gas is/are correct?

(1) The root-mean-square (r.m.s.) speed of the gas molecules is doubled.
(2) The pressure of the gas is doubled.
(3) The average kinetic energy of the gas molecules is doubled.

A sinusoidal transverse wave of wavelength \(\lambda\) propagates along the positive \(x\)-direction. At a certain instant, the displacement of a particle at \(x = 2\text{ cm}\) is at its positive maximum. Which of the following statements about the motion of the particles is/are correct?

(1) The particle at \(x = 2\text{ cm}\) is momentarily at rest.
(2) The phase difference between the particle at \(x = 2\text{ cm}\) and the particle at \(x = 2 + 0.5\lambda\text{ cm}\) is \(\pi\text{ rad}\).
(3) The particle at \(x = 2 + 0.25\lambda\text{ cm}\) is moving in the positive \(y\)-direction (upwards) at this instant.

A light ray is incident from medium X into medium Y with an angle of incidence \(\theta\). The refractive indices of medium X and Y are \(n_X\) and \(n_Y\) respectively, where \(n_X > n_Y\). If the angle of refraction is \(r\), and the angle of deviation of the ray is \(d = r - \theta\), what is the maximum possible value of \(d\)?

Three identical resistors, each of resistance \(R\), are connected to a cell of e.m.f. \(\mathcal{E}\) and internal resistance \(r\). When they are connected in series, the total power dissipated in the external circuit is \(P_s\). When they are connected in parallel, the total power dissipated in the external circuit is \(P_p\). If \(r = R\), what is the ratio \(P_s / P_p\)?

A rigid, rectangular conducting loop of mass \(m\), width \(w\), and resistance \(R\) is released from rest and falls vertically under gravity. A uniform horizontal magnetic field \(B\) is directed perpendicular to the plane of the loop, but exists only in a region of height \(h\). As the bottom side of the loop enters the magnetic field, it is observed to fall at a constant terminal velocity \(v\). What is the expression for \(v\)? (Ignore air resistance and self-inductance.)

A satellite of mass \(m\) is in a circular orbit of radius \(r\) around the Earth (mass \(M\)). Due to atmospheric drag, the satellite experiences a small, resistive force. As a result, the orbital radius slowly decreases. Over a short period of time, the radius decreases from \(r\) to \(r - \Delta r\) (where \(\Delta r \ll r\)). What happens to the kinetic energy \(K\) and gravitational potential energy \(U\) of the satellite?

In a hydrogen atom, when an electron transitions from energy level \(n = 3\) to \(n = 2\), a photon of wavelength \(\lambda_0\) is emitted. What is the wavelength of the photon emitted when the electron transitions from \(n = 4\) to \(n = 3\)?

In a Young's double-slit experiment, monochromatic light of wavelength \(\lambda_1\) is used. The fringe separation on a screen at a distance \(D\) is \(y_1\). When the wavelength is changed to \(\lambda_2\), and the slit separation is halved while the screen distance is doubled, the new fringe separation is \(y_2\). If \(y_2 = 3 y_1\), what is the ratio \(\lambda_2 / \lambda_1\)?

A real object is placed at a distance \(u\) from a convex lens of focal length \(f\). A real image is formed at a distance \(v\) with magnification \(m\). A graph of \(m\) against \(v\) is plotted. Which of the following is correct?

A real battery with e.m.f. \(\mathcal{E}\) and internal resistance \(r\) is connected to a variable resistor of resistance \(R\). As \(R\) increases from a very small value to a very large value, which of the following statements is/are correct?

(1) The terminal voltage across the battery increases.
(2) The power dissipated in the internal resistance increases.
(3) The efficiency of the circuit (defined as the ratio of power delivered to \(R\) to the total power supplied by the battery) increases.

An ideal transformer has a primary coil of \(N_P\) turns and a secondary coil of \(N_S\) turns. The primary coil is connected to an AC source of constant root-mean-square (r.m.s.) voltage \(V\). A resistor of resistance \(R\) is connected across the secondary coil. What is the r.m.s. current in the primary circuit?

A progressive wave of frequency \(10\text{ Hz}\) travels along a stretched string. Two particles \(P\) and \(Q\) on the string are separated by a distance of \(0.15\text{ m}\). The minimum phase difference between the oscillations of \(P\) and \(Q\) is \(\pi/3\text{ rad}\). Find the wave speed.

A resistor \(R_1 = 10\ \Omega\) is connected in series with a parallel network. This parallel network consists of two branches: one branch contains a resistor \(R_2 = 20\ \Omega\), and the other contains a resistor \(R_3\) in series with a switch \(S\). A cell of e.m.f. \(12\text{ V}\) and negligible internal resistance is connected across the entire combination. An ideal voltmeter is connected across \(R_1\). When the switch \(S\) is open, the voltmeter reads \(4\text{ V}\). When the switch \(S\) is closed, the voltmeter reading becomes \(6\text{ V}\). Find the resistance of \(R_3\).

Two satellites, \(X\) and \(Y\), undergo uniform circular motion around the Earth. The orbital radius of \(X\) is \(R\) and that of \(Y\) is \(4R\). The mass of \(Y\) is twice that of \(X\). If the kinetic energy of \(X\) is \(E\), what is the kinetic energy of \(Y\)?

A string of length \(0.80\ \text{m}\) is fixed at both ends and vibrates in its third harmonic with a frequency of \(150\ \text{Hz}\).\n(a) Determine the wave speed in the string.\n(b) If the tension in the string is quadrupled, find the new fundamental frequency of the string.

Two in-phase coherent sound sources, \(S_1\) and \(S_2\), are separated by \(2.0\ \text{m}\). A detector at point \(P\) is \(4.50\ \text{m}\) from \(S_1\) and \(5.25\ \text{m}\) from \(S_2\). The speed of sound is \(340\ \text{m s}^{-1}\) and the frequency of the sound is \(680\ \text{Hz}\).\n(a) Show whether constructive or destructive interference occurs at \(P\).\n(b) If the frequency of both sources is gradually increased, find the next frequency at which the nature of interference at \(P\) changes.

A real battery of emf \(\varepsilon\) and internal resistance \(r\) is connected to a variable resistor \(R\). When \(R = 4.0\ \Omega\), the terminal voltage is \(6.0\ \text{V}\). When \(R\) is increased to \(10.0\ \Omega\), the terminal voltage becomes \(7.5\ \text{V}\).\n(a) Calculate \(r\) and \(\varepsilon\).\n(b) What is the maximum power that can be delivered to the variable resistor \(R\)?

An electric kettle rated at "\(220\ \text{V}, 1800\ \text{W}\)" and a microwave oven rated at "\(220\ \text{V}, 1200\ \text{W}\)" are connected in parallel to a \(220\ \text{V}\) mains socket protected by a \(15\ \text{A}\) fuse.\n(a) Calculate the total current drawn from the mains when both appliances are operating at their rated values. Hence, explain whether the fuse will blow.\n(b) If the mains voltage drops by \(10\%\) due to a power fluctuation, estimate the percentage decrease in the heat power generated by the electric kettle. (Assume the resistance of the kettle remains constant.)

A semicircular glass block has a refractive index of \(1.62\).\n(a) Calculate the critical angle for the glass-air interface.\n(b) If the glass block is now submerged in a liquid of refractive index \(1.35\), find the new critical angle.\n(c) State, with a brief explanation, how the critical angle changes if the frequency of the light is increased when submerged in the liquid. (Assume normal dispersion: refractive index increases with frequency).

An object of height \(3.0\ \text{cm}\) is placed \(12.0\ \text{cm}\) in front of a thin converging lens. A real image of height \(6.0\ \text{cm}\) is formed on a screen on the other side of the lens.\n(a) Calculate the focal length of the lens.\n(b) The object is now moved \(4.0\ \text{cm}\) closer to the lens. State and explain whether the screen should be moved closer to or further away from the lens to capture a sharp image.

A square coil of \(50\) turns with side length \(0.10\ \text{m}\) and resistance \(2.0\ \Omega\) is placed in a uniform magnetic field directed perpendicularly into the plane of the coil. The magnetic field strength \(B\) decreases uniformly from \(0.80\ \text{T}\) to \(0.20\ \text{T}\) in \(0.30\ \text{s}\).\n(a) Calculate the magnitude of the induced emf and the induced current in the coil.\n(b) State the direction of the induced current (clockwise or anticlockwise) looking down at the plane of the coil. Explain your answer using Lenz's Law.

A satellite of mass \(m\) is orbiting the Earth in a circular geostationary orbit of radius \(r\).\n(a) Explain why a geostationary satellite must orbit directly above the Earth's equator.\n(b) Given that the radius of the geostationary orbit is \(4.2 \times 10^7\ \text{m}\), calculate the mass of the Earth. (Take gravitational constant \(G = 6.67 \times 10^{-11}\ \text{N m}^2\ \text{kg}^{-2}\).)

The energy levels of a hydrogen atom can be represented by \(E_n = -\frac{13.6}{n^2}\ \text{eV}\). An electron makes a transition from the \(n=3\) energy level (\(-1.51\ \text{eV}\)) to the \(n=1\) ground state (\(-13.6\ \text{eV}\)).\n(a) Calculate the frequency of the photon emitted during this transition. (Take \(1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J}\) and \(h = 6.63 \times 10^{-34}\ \text{J s}\).)\n(b) Explain whether a photon of energy \(11.5\ \text{eV}\) can be absorbed by a hydrogen atom initially in its ground state.

A space probe of mass \(800\text{ kg}\) is in a circular orbit around a newly discovered exoplanet.

(a) Derive an expression relating the orbital period \(T\) of the probe to the radius of the circular orbit \(r\) and the mass of the exoplanet \(M\). State any ONE major assumption made in the derivation. (3 marks)

(b) The probe orbits at an altitude of \(1.5 \times 10^6\text{ m}\) above the exoplanet's surface. The radius of the exoplanet is \(3.5 \times 10^6\text{ m}\). If the orbital period is \(4.0\text{ hours}\), calculate:
    (i) the mass of the exoplanet \(M\). (3 marks)
    (ii) the acceleration due to gravity on the surface of the exoplanet. (2 marks)

(c) A small landing module is to be launched from the probe to land on the exoplanet. The module is ejected backwards relative to the probe's direction of motion. Explain how the orbital parameters of the module change immediately after ejection and why this action helps it descend. (3 marks)

A flat square coil of \(N = 150\) turns, resistance \(R = 2.5\ \Omega\), and side length \(L = 0.20\text{ m}\) enters a region of a uniform magnetic field \(B = 0.40\text{ T}\) at a constant horizontal speed \(v = 3.0\text{ m s}^{-1}\). The plane of the coil is horizontal and perpendicular to the vertical magnetic field, which points upwards (out of the page).

(a) During the time interval when the coil is entering the magnetic field:
    (i) Show that the induced electromotive force (e.m.f.) in the coil is \(36\text{ V}\). (2 marks)
    (ii) Calculate the induced current in the coil and state its direction (clockwise or counter-clockwise) as viewed from above. Explain your answer with Lenz's law. (3 marks)

(b) Calculate the magnitude and direction of the magnetic force acting on the leading edge of the coil as it enters the magnetic field. (3 marks)

(c) Explain why an external horizontal force must be applied to keep the coil moving at a constant speed, and calculate the mechanical power delivered by this external force. (3 marks)

An optical fiber consists of a cylindrical core of refractive index \(n_1 = 1.48\) surrounded by a cladding of refractive index \(n_2 = 1.42\).

(a) (i) Define the term critical angle. (1 mark)
    (ii) Calculate the critical angle \(\theta_c\) for light traveling from the core to the cladding interface. (2 marks)

(b) Light is launched from air (refractive index \(n_0 = 1.00\)) into the core at an angle of incidence \(\theta_a\) relative to the axis of the fiber.
    (i) Show that the maximum angle of incidence \(\theta_{a,\text{max}}\) in air for which light can undergo total internal reflection at the core-cladding interface is given by:
\(\sin \theta_{a,\text{max}} = \sqrt{n_1^2 - n_2^2}\). (4 marks)
    (ii) Hence, calculate this maximum angle \(\theta_{a,\text{max}}\). (2 marks)

(c) State one advantage and one disadvantage of using optical fibers over traditional copper wires for long-distance telecommunication. (2 marks)

Two planets, P and Q, orbit a distant star in circular orbits. The orbital radius of P is \(R\), and its orbital period is \(T\). If the orbital period of Q is \(8T\), what is the orbital radius of Q?

A satellite of mass \(m\) is orbiting the Earth of mass \(M\) in a circular orbit of radius \(r\) with kinetic energy \(K\). If a thruster moves the satellite to a higher stable circular orbit of radius \(2r\), find the work done on the satellite by the thruster.

Star X is a red giant with surface temperature \(3000\text{ K}\) and luminosity \(10^4 L_{\odot}\). Star Y is a white dwarf with surface temperature \(12000\text{ K}\) and luminosity \(10^{-4} L_{\odot}\). What is the ratio of the radius of Star X to that of Star Y, \(\frac{R_X}{R_Y}\)?

A distant galaxy is observed to have a hydrogen absorption line of rest wavelength \(\lambda_0 = 656.3\text{ nm}\) redshifted to \(\lambda = 671.6\text{ nm}\). If the Hubble constant is \(H_0 = 70\text{ km s}^{-1}\text{ Mpc}^{-1}\), estimate the distance to this galaxy. (Take speed of light \(c = 3.0 \times 10^5\text{ km s}^{-1}\))

Which of the following statements about the evolution of stars is/are correct?
(1) Stars with initial mass much greater than the Sun (e.g., \(15 M_{\odot}\)) will end their lives as white dwarfs.
(2) Main sequence stars generate energy primarily through the fusion of hydrogen into helium in their cores.
(3) The lifespan of a high-mass star on the main sequence is longer than that of a low-mass star because it has more hydrogen fuel.

Two stars, A and B, form a binary system orbiting about their common center of mass (barycenter) with a period \(T\). The mass of star A is \(M_A\) and the mass of star B is \(M_B = 2 M_A\). Which of the following statements is/are correct?
(1) The orbital radius of star A is twice that of star B.
(2) The linear speed of star A is twice that of star B.
(3) The gravitational force acting on star A by star B is twice that acting on star B by star A.

A star has a parallax angle of \(0.04\text{ arcseconds}\). What is the distance of the star from Earth in light-years? (Given: \(1\text{ pc} = 3.26\text{ light-years}\))

Which of the following is/are evidence supporting the Big Bang theory?
(1) The cosmic microwave background radiation (CMB) is highly isotropic with a temperature of approximately \(2.7\text{ K}\).
(2) The abundance of light elements (such as helium and deuterium) in the universe matches the predictions of Big Bang nucleosynthesis.
(3) The observation that almost all distant galaxies are moving away from us, with their recessional speed proportional to their distance.

An exoplanet orbits a distant star in a circular orbit of radius \(6.0 \times 10^{10} \text{ m}\) with an orbital period of \(90\) days.\n\n(a) Calculate the mass of the star. (3 marks)\n\n(b) The star is a main-sequence star. Its luminosity \(L\) is related to its mass \(M\) by the mass-luminosity relation \(L \propto M^{3.5}\). Given that the Sun has a mass of \(2.0 \times 10^{30} \text{ kg}\) and a luminosity of \(3.8 \times 10^{26} \text{ W}\), calculate the luminosity of this star. (3 marks)\n\n(c) Calculate the radiant flux (intensity of stellar radiation) received by the exoplanet from its host star. (2 marks)\n\n(d) State the spectroscopic method commonly used to detect such an exoplanet due to the stellar motion, and briefly explain how it works. (2 marks)

When monochromatic light of frequency \(f\) is incident on a metal plate, the stopping potential of the photoelectrons is \(V\). When monochromatic light of frequency \(1.5f\) is incident on the same metal plate, the stopping potential becomes \(2V\). Find the work function of the metal plate.

According to Bohr's model of the hydrogen atom, how does the de Broglie wavelength \(\lambda\) of the orbiting electron in the \(n\)-th state depend on the principal quantum number \(n\)?

An electron (mass \(m_e\)) and a proton (mass \(m_p\)) are accelerated from rest through the same potential difference \(V\). What is the ratio of the de Broglie wavelength of the electron to that of the proton, \(\frac{\lambda_e}{\lambda_p}\)?

In a photoelectric experiment, three light beams P, Q, and R are incident on the same metal plate. The variation of the photocurrent \(I\) with the applied voltage \(V\) for each beam is described as follows:
- Beam P and Beam Q have the same stopping potential, but Beam P has a higher saturation current than Beam Q.
- Beam Q and Beam R have different stopping potentials, where the stopping potential of Beam R is more negative than that of Beam Q, but they have the same saturation current.
Which of the following statements is/are correct?
(1) Beam P and Beam Q have the same frequency, but Beam P has a higher intensity.
(2) Beam R has a higher frequency than Beam Q.
(3) The maximum kinetic energy of photoelectrons emitted by Beam R is greater than that by Beam P.

An X-ray tube operates at a tube voltage \(V\). The resulting X-ray spectrum shows a continuous spectrum with a minimum wavelength \(\lambda_{\min}\) and characteristic peaks. If the tube voltage is increased to \(1.5V\), which of the following statements is/are correct?
(1) The minimum wavelength of the continuous spectrum becomes \(\frac{2}{3}\lambda_{\min}\).
(2) The wavelengths of the characteristic peaks shift to shorter wavelengths.
(3) The overall intensity of the X-rays increases.

Which of the following statements about the Scanning Tunneling Microscope (STM) is/are correct?
(1) The physical principle of STM is based on the wave nature of electrons and the quantum tunneling effect.
(2) The tunneling current increases exponentially when the distance between the metallic probe tip and the sample surface increases.
(3) STM can be used to manipulate individual atoms on a surface.

In an electron diffraction experiment, a beam of electrons with kinetic energy \(E\) passes through a thin polycrystalline graphite film to produce a series of concentric diffraction rings on a screen. If the kinetic energy of the electrons is increased to \(4E\), how does the radius of the diffraction rings change?

In a hydrogen atom, an electron transitions from the orbit \(n = 3\) to \(n = 2\), emitting a photon of frequency \(f_1\). It then transitions from \(n = 2\) to \(n = 1\), emitting a photon of frequency \(f_2\). What is the ratio \(\frac{f_1}{f_2}\)?

A binary star system consists of Star A of mass \(M_1\) and Star B of mass \(M_2\) orbiting around their common centre of mass in circular orbits. The orbital plane lies along our line of sight from Earth.

(a) Explain why the spectral lines of this system, as observed from Earth, exhibit periodic splitting. (3 marks)

(b) The distance between the two stars is \(d\), and the orbital radii of Star A and Star B about the centre of mass are \(r_1\) and \(r_2\) respectively, so that \(d = r_1 + r_2\).

(i) Show that \(r_1 = \frac{M_2}{M_1 + M_2} d\). (2 marks)

(ii) Show that the orbital period \(T\) of the system satisfies:

\[M_1 + M_2 = \frac{4\pi^2 d^3}{G T^2}\]

(3 marks)

(iii) If the separation \(d\) is \(8.0 \times 10^{11} \text{ m}\) and the orbital period \(T\) is \(1.26 \times 10^8 \text{ s}\), calculate the total mass \((M_1 + M_2)\) of the binary system. (2 marks)