WAEC SSCE - Physics - 2021

Maelezo ya Majibu

As the metallic sphere is heated, the temperature increases, causing the metal atoms to vibrate more rapidly. This increased vibration causes an increase in the space between the atoms, resulting in an increase in volume. Since the mass of the sphere remains constant, the increase in volume results in a decrease in density. Therefore, the correct option is: "Its volume increases and its density decreases".

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The correct option is: "It has dimension M\(^{-1}\)L\(^{3}\)T\(^{-2}\)." The universal gravitational constant, G, is a fundamental constant in physics that determines the strength of the gravitational force between two objects. Its value is approximately 6.674 × 10\(^{-11}\) N m\(^{2}\)kg\(^{-2}\). The dimension of G is M\(^{-1}\)L\(^{3}\)T\(^{-2}\), which means it has one unit of mass, negative one unit of length cubed, and two units of time in its dimensional formula. Its SI unit is N m\(^{2}\)kg\(^{-2}\). It is a scalar quantity, not a vector quantity, because it only has a magnitude and no direction.

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Maelezo ya Majibu

The set of quantities that have members which are all vectors is "Force, displacement and momentum." A vector is a quantity that has both magnitude and direction. Force, displacement, and momentum all have both magnitude and direction, making them vector quantities. Pressure, energy, distance, acceleration, work, density, volume, and weight are not all vectors because they do not have both magnitude and direction.

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Maelezo ya Majibu

Maelezo ya Majibu

Pascal's principle states that the pressure applied to a fluid in a closed container will be transmitted equally to all parts of the fluid and to the walls of the container. This means that the pressure in a fluid is transmitted equally in all directions, regardless of the density of the fluid. Therefore, the correct option is "transmitted equally in all directions".

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Out of the given options, mass is not an example of force. Mass is a measure of the amount of matter in an object and is typically measured in kilograms (kg). Force, on the other hand, is a quantity that describes the interaction between two objects and can cause a change in motion or deformation of an object. Tension, weight, and friction are all examples of forces. Tension is the force transmitted through a string, cable, or rope when it is pulled tight. Weight is the force exerted by gravity on an object with mass, and it is measured in Newtons (N). Friction is the force that opposes motion between two surfaces that are in contact with each other. To summarize, mass is not a force, whereas tension, weight, and friction are all examples of forces.

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Electric motor primarily converts electrical energy to mechanical energy. An electric motor is a device that converts electrical energy into mechanical energy by using the magnetic effect of current. When an electric current flows through a coil in a magnetic field, a force is exerted on the coil which causes it to rotate, thus converting electrical energy into mechanical energy.

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Maelezo ya Majibu

When an electric current flows through a wire, the wire heats up due to the resistance it offers to the flow of electrons. This heat can be dissipated to the surroundings by conduction, convection, and radiation. The amount of heat generated in the wire depends on the resistance of the wire, the current flowing through it, and the time it takes to flow. To minimize the heat generated in the wire, we need to reduce its resistance. The resistance of a wire depends on its length, cross-sectional area, and the resistivity of the material. Among the options given, the most effective way to minimize the heat generated in the wire is to increase the diameter of the wire. This is because the resistance of a wire is inversely proportional to its cross-sectional area. Therefore, a wire with a greater diameter will have less resistance and hence generate less heat than a wire with a smaller diameter. Thicker insulation will not help much in reducing the heat generated in the wire because insulation is used to prevent the electric current from flowing out of the wire, and it does not affect the flow of current in the wire itself. Thinner insulation or smaller diameter wire may even increase the heat generated because it will lead to a higher resistance and hence more heat generation. Therefore, the correct answer is "greater diameter" because it will reduce the resistance of the wire and hence minimize the heat generated in it.

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Maelezo ya Majibu

The inductive reactance (X_L) in a circuit can be calculated using the formula X_L = 2πfL, where f is the frequency of the circuit and L is the inductance of the inductor. Given that X_L = 1Ω and f = 100Hz, we can use this formula to calculate the inductance L: X_L = 2πfL 1Ω = 2π * 100Hz * L L = 1Ω / (2π * 100Hz) L = 1.59 x 10^-3H So, the inductance of the inductor is 1.59 x 10^-3 H.

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The opening in the eye through which light passes to the retina is called the "pupil". The pupil is a small, round hole located in the center of the iris (the colored part of the eye), which regulates the amount of light that enters the eye. The size of the pupil can change depending on the amount of light available and the focus required for near or far vision. The light that enters the eye through the pupil is then focused by the cornea and lens onto the retina at the back of the eye, which sends visual information to the brain through the optic nerve.

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The power of a person is defined as the amount of work done per unit time. In this case, the work done by the man is equal to his weight multiplied by the height he climbs. Therefore, the work done by the man is: Work = Force x Distance Work = 200 N x 9 m Work = 1800 J The time taken by the man to climb the stairs is 5 seconds. Therefore, the power of the man is: Power = Work / Time Power = 1800 J / 5 s Power = 360 W So the power of the man is 360 watts (W). This means that the man can perform work at a rate of 360 joules per second, which is a measure of his physical exertion during the climb.

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The electrons in a cathode-ray tube are produced by heating a metal filament. This process is called thermionic emission, which means that when a metal filament is heated, the electrons gain enough energy to escape from the surface of the metal and become free electrons. These free electrons are then accelerated towards the anode (positively charged plate) by applying an electric field to the x-plates, and they pass through a small hole in the anode, creating a narrow beam of electrons. This beam of electrons then passes through the tube and is focused by magnetic fields to produce an image on a fluorescent screen.

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When the air jet is directed onto the holes in the rotating disc, it causes the air to vibrate at a certain frequency, producing a sound wave. The frequency of the sound wave is determined by the number of holes on the disc and the speed of rotation of the disc. In this case, the note produced has a frequency of 480 Hz and there are 20 holes on the disc. This means that as each hole passes the air jet, it produces a sound wave of 480 Hz. Since there are 20 holes, the disc must rotate 20 times to produce 20 sound waves, which is equivalent to one cycle of the note. Therefore, the speed of rotation of the disc can be calculated by multiplying the frequency of the note by the number of holes and dividing by the number of cycles per second. In this case, we have: Speed of rotation = (480 Hz x 20 holes) / 1 cycle per second Speed of rotation = 9,600 revolutions per second However, the answer choices are given in revolutions per second (rev/s), so we need to convert the speed of rotation to rev/s by dividing by the number of holes: Speed of rotation = 9,600 / 20 Speed of rotation = 480 rev/s Therefore, the correct answer is 24 rev/s.

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Maelezo ya Majibu

To find the components of Q + R, we need to add the corresponding components of the two vectors. Given the components of Q and R, we have: Q = (-3.0, 5.5) R = (9.2, 4.4) Adding the corresponding components of Q and R, we get: Q + R = (-3.0 + 9.2, 5.5 + 4.4) = (6.2, 9.9) Therefore, the components of Q + R are (6.2, 9.9). To summarize, we can find the components of the sum of two vectors by adding the corresponding components of the two vectors. In this case, the components of Q + R are (6.2, 9.9).

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When an atom emits an alpha particle, it may become a different element. An alpha particle is a helium nucleus consisting of two protons and two neutrons. When an atom emits an alpha particle, it loses these two protons and two neutrons and therefore changes its atomic number and becomes a different element. The new element will have an atomic number that is two less than the original element. For example, if an atom of uranium-238 emits an alpha particle, it will become an atom of thorium-234. So, the correct statement is that an atom that emits an alpha particle may become a different element.

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Maelezo ya Majibu

To solve this problem, we can use the work-energy theorem, which states that the net work done on an object is equal to its change in kinetic energy. Mathematically, we can express this as: Net work = Change in kinetic energy In this problem, the force is acting continuously, which means that it is doing work on the object. The work done by a constant force can be calculated using the formula: Work = Force x Distance x cos(theta) where theta is the angle between the force vector and the displacement vector. Since the force is in the same direction as the displacement (i.e., they are both acting to the right), we can simplify the formula to: Work = Force x Distance Substituting the given values into this formula, we get: Work = 10 N x Distance Now, we can use the work-energy theorem to find the distance moved by the object. We know that the net work done on the object is equal to its change in kinetic energy, which is: Net work = 60 J - 20 J = 40 J Setting this equal to the work formula above, we get: 10 N x Distance = 40 J Solving for distance, we get: Distance = 4 meters Therefore, the distance moved by the object is 4 meters.

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When a ball is swung in a horizontal circle with a constant speed, it experiences a centripetal acceleration directed towards the center of the circle. The centripetal acceleration is the acceleration required to keep an object moving in a circular path, and it is given by the formula: a = v\(^{2}\) / r where v is the speed of the ball and r is the radius of the circle. In this case, the ball is moving in a horizontal circle, so the centripetal acceleration acts vertically towards the center of the circle (O). So, the acceleration of the ball is toward the center O.

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When gas molecules are compressed at constant temperature, the space they occupy decreases, resulting in more molecules per unit volume. As a result, the molecules make more impact per second on the walls of the container, causing an increase in pressure. Therefore, the correct option is: "The molecules make more impact per second on the walls of the container."

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The correct option is: "exhibit vibratory motion." The molecules of solids and liquids are constantly in motion, but the type of motion they exhibit differs. In solids, the molecules are packed closely together and vibrate in place. The vibrations are not enough to overcome the attractive forces between the molecules, so the molecules are held in a fixed position. In liquids, the molecules are not as closely packed together as in solids, and they have more freedom of movement. The molecules are in constant motion, but they are still attracted to each other, so they tend to stay close together. The motion of the molecules in liquids is more rapid and random than in solids, but they still exhibit a vibratory motion. Therefore, the statement "They both exhibit vibratory motion" is the correct statement about the molecules of solids and liquids.

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Maelezo ya Majibu

If the relative humidity of the atmosphere increases, the rate of evaporation of sweat from the human body decreases. This is because when the relative humidity is high, the air is already saturated with water vapor, and there is less capacity for more water to evaporate into the air. When sweat evaporates, it cools the skin and removes heat from the body. But if the air is already saturated with water vapor, sweat cannot evaporate as effectively, and the body will have a harder time cooling down. Therefore, high humidity can make it feel hotter and more uncomfortable because the body's natural cooling mechanism is less effective.

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The acceleration of a moving object can be determined from the slope of its velocity-time graph. This is because acceleration is defined as the rate of change of velocity over time. Therefore, the steeper the slope of the velocity-time graph, the greater the rate of change of velocity, and thus the greater the acceleration. On the other hand, the distance-time graph only provides information about the distance covered by an object over time, and the area under this graph only gives the total distance traveled. The velocity-time graph, however, shows how the velocity of the object changes over time, and the slope of this graph gives information about the acceleration. Therefore, the slope of the velocity-time graph is a more direct and reliable way to determine the acceleration of a moving object.

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To solve this problem, we need to use the formula for electrical energy: Energy = Power x Time where Power = Voltage x Current, and Time is in hours. First, we need to find the power of the bulb. Power = Voltage x Current = 440V x 2.0A = 880W. Since we want to find the cost of keeping the bulb on for two days, which is 48 hours, we can now find the energy used by the bulb: Energy = Power x Time = 880W x 48 hours = 42,240 Wh To convert watt-hours (Wh) to kilowatt-hours (kWh), we need to divide by 1000: 42,240 Wh ÷ 1000 = 42.24 kWh Finally, to find the cost of using 42.24 kWh at a rate of $0.40 per kWh, we multiply the two values: 42.24 kWh x $0.40/kWh = $16.90 Therefore, the answer is $16.90.

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Maelezo ya Majibu

To measure the potential difference across a resistor using a galvanometer, the student needs a high value resistor connected in series with the galvanometer. The high value resistor acts as a voltage divider, allowing the galvanometer to measure the potential difference without being damaged by excessive current. This is known as a voltmeter circuit, and it is commonly used in electronics to measure voltage. A low value resistor connected in parallel with the galvanometer would be used to measure current, not voltage.

Maelezo ya Majibu

Resistance is a measure of how much a material opposes the flow of an electric current. The resistance of a wire is influenced by four factors: length, cross-sectional area, material, and temperature. The length of the wire is directly proportional to its resistance, meaning that the longer the wire, the greater its resistance. The cross-sectional area of the wire is inversely proportional to its resistance, meaning that the larger the cross-sectional area, the lower the resistance. The material of the wire affects its resistance, with some materials having higher resistances than others. Finally, temperature affects the resistance of a wire, with most materials having a positive temperature coefficient of resistance, meaning that their resistance increases with temperature. Mass, on the other hand, is not a factor that determines the resistance of a wire. The mass of the wire is related to its density and volume, but it does not affect the flow of current through the wire or its resistance. Therefore, the correct answer is "mass."

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The resistance of a wire can be calculated using the formula: R = ρ * L / A where R is the resistance of the wire, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire. Given the resistivity of the wire (1.1 x 10\(^-{8}\) Ω m) and its resistance (21 Ω), we can calculate the length of the wire as follows: L = R * A / ρ = 21 Ω * (2 * π * 10\(^-{8}\) m\(^{2}\)) / (1.1 x 10\(^-{8}\) Ω m) = 2 * π * 21 / 1.1 = 120 m So, the length of the wire is 120 m.

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Maelezo ya Majibu

The scientist who suggested that moving particles exhibit wave properties was Louis de Broglie. De Broglie was a French physicist who developed the theory of wave-particle duality. In his theory, de Broglie proposed that all particles, not just light, have wave-like properties, and that their wavelength is directly proportional to their momentum. This idea challenged the prevailing view of the time, which held that particles and waves were distinct entities, and helped to lay the foundation for the development of quantum mechanics. So, Louis de Broglie is the scientist who suggested that moving particles exhibit wave properties.

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Maelezo ya Majibu

The capacitance of a parallel plate capacitor is defined as the ratio of the charge stored on the plates to the potential difference between the plates. Mathematically, we can express this as: C = Q/V where C is capacitance, Q is charge, and V is potential difference. Now, let's consider each of the options in turn and determine how they affect the capacitance of a parallel plate capacitor. I. Decreasing the distance between the plates When the distance between the plates is decreased, the electric field between the plates increases. This, in turn, increases the charge that can be stored on the plates for a given potential difference. As a result, the capacitance of the capacitor increases. II. Increasing the distance between the plates When the distance between the plates is increased, the electric field between the plates decreases. This, in turn, decreases the charge that can be stored on the plates for a given potential difference. As a result, the capacitance of the capacitor decreases. III. Increasing the area of the plates overlap When the area of the plates overlap is increased, the capacitance of the capacitor increases. This is because there is more surface area for the charges to accumulate on, which increases the amount of charge that can be stored on the plates for a given potential difference. IV. Avoiding the use of dielectric between the plates A dielectric is an insulating material that is placed between the plates of a capacitor to increase its capacitance. Without a dielectric, the capacitance of the capacitor will be smaller. Based on the above analysis, we can see that options I and III will increase the capacitance of a parallel plate capacitor. Option II will decrease capacitance and option IV will not have any effect on capacitance. Therefore, the correct answer is: I and III only.

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The factor that does not affect the rate of evaporation of a liquid is the volume of the liquid. Evaporation is the process of a liquid turning into a gas or vapor, and it occurs when the molecules of the liquid gain enough energy to break away from the surface and enter the surrounding air. The rate of evaporation is affected by several factors, including temperature, wind, and surface area. Temperature is the most important factor that affects the rate of evaporation. As the temperature of the liquid increases, the molecules of the liquid gain more energy and move faster, increasing the chances of the molecules escaping the surface and entering the air. Wind can also affect the rate of evaporation. When there is wind, it removes the humid air around the liquid and replaces it with drier air. This leads to a faster rate of evaporation, as there is more room in the air for the liquid molecules to enter. Surface area is another factor that affects the rate of evaporation. The larger the surface area of the liquid, the more molecules are exposed to the air, leading to a faster rate of evaporation. However, the volume of the liquid does not affect the rate of evaporation. A larger volume of liquid will take longer to evaporate than a smaller volume, but the rate of evaporation is not affected by the volume of the liquid.

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A heat sensitive resistor made of a semiconductor is called a thermistor. A thermistor is a type of resistor whose resistance changes in response to changes in temperature. They are used in a variety of applications, such as temperature sensing and control, temperature compensation, and over-temperature protection. They work by exploiting the property of certain materials to change their electrical resistance in response to changes in temperature. When the temperature of a thermistor increases, its resistance decreases, and when the temperature decreases, its resistance increases. This relationship can be used to measure temperature with high accuracy.

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To calculate the energy required to raise the temperature of an ice cube from 5°C to 0°C, we need to use the specific heat capacity of ice, which is 2.1 J/g°C. The formula for calculating the energy required to change the temperature of a substance is: Energy = mass x specific heat capacity x change in temperature We are given that the mass of the ice cube is 20 g, and we want to raise its temperature by 5°C (from 5°C to 0°C). Plugging in the values into the formula, we get: Energy = 20 g x 2.1 J/g°C x 5°C Simplifying the equation, we get: Energy = 210 J Therefore, the energy required to raise the temperature of the ice cube from 5°C to 0°C is 210 J. The correct option is 4) 210 J.

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The statement that best describes the particles in a solid at room temperature is "close together and vibrating". In a solid, the particles are tightly packed and held together by strong forces of attraction, so they cannot move freely. However, they do have some kinetic energy, causing them to vibrate around their fixed positions. This means that the particles do not move randomly like in a gas, but rather they maintain their positions while vibrating.

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The reason for laminating the soft iron core of a transformer is to reduce eddy currents. Eddy currents are circulating currents that flow in the core of the transformer, and they generate heat and waste energy. By laminating the core into thin sheets, the eddy currents are reduced, which helps to increase the efficiency of the transformer. In other words, laminating the core helps to prevent energy loss and make the transformer run more efficiently.

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a) (i) The root-mean-square (RMS) value of an alternating current (AC) is a measure of the effective value of the current, which is equal to the DC value that would produce the same amount of heat dissipation in a resistor. It is defined as the square root of the mean of the squares of the current values.

(ii) Impedance is the total opposition offered by a circuit to the flow of AC, and it is a complex quantity that includes both resistance and reactance. Reactance is the opposition to AC caused by the capacitance or inductance in the circuit, while resistance is the opposition to AC caused by the resistance of the conductors.

b) To find the capacitance of the capacitor, we need to use the formula for power in an AC circuit, which is given by P = VIcos(Φ). Here, V is the RMS voltage, I is the RMS current, and Φ is the phase angle between the voltage and current. If we know the power and voltage, we can calculate the current, and then use the formula for impedance (Z = V/I) to find the reactance of the capacitor. The reactance of a capacitor is given by X_C = 1/(2πfC), where f is the frequency of the AC and C is the capacitance. Solving for C, we get C = 1/(2πfX_C).

c) (i) Capacitance of a capacitor is a measure of its ability to store electrical energy in an electric field between its plates. It is defined as the ratio of the charge stored on one plate of a capacitor to the potential difference across the plates.

(ii) The total capacitance of the capacitors connected in series is given by C_total = C_1 + C_2. The potential difference across each capacitor can be found using the formula V = Q/C, where Q is the charge stored on each capacitor. If we know the potential difference across the whole circuit, we can calculate the charge stored on each capacitor, and then find the potential difference across each capacitor by dividing the charge by the capacitance.

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(a) Two artificial satellites. A communication satellite and a weather (meteorological) satellite. (Other acceptable examples: a navigation/GPS satellite, a spy/reconnaissance satellite, the International Space Station.)

(b) Speed of the geostationary satellite. A geostationary satellite has the same period as the earth's rotation, so its period is

Radius of orbit \(r = 6300\ \text{km} = 6.3\times10^{6}\ \text{m}\).

The satellite moves once round the circular orbit (circumference \(2\pi r\)) in time T, so its orbital speed is

The satellite moves in its orbit with a speed of about 458 m s\(^{-1}\).

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The circuit shown is a simple half-wave rectifier with a smoothing (filter) stage. An a.c. supply feeds a diode, which passes current in one direction only, and the output is then smoothed before it reaches the load.

(a) Functions of the labelled parts

• A - This is the a.c. source (the secondary winding of the transformer). Its function is to supply the alternating current (a.c.) that is to be rectified.
• B - This is the smoothing (reservoir) capacitor connected across the load. Its function is to smoothen the rectified current. It charges up near the peak of each pulse and then discharges slowly through the load during the gaps, so it fills in the dips and greatly reduces the ripple, giving an almost steady d.c. output.

(b) Sketch of the output signal

Before smoothing, the rectifier delivers a series of separate voltage pulses (broken line). After the capacitor B is added, these pulses are smoothed into a nearly steady d.c. voltage that carries only a small saw-tooth ripple (solid line). The output is therefore unidirectional (one-way) and almost constant, as sketched below.

Thus the final output is a steady, one-directional (d.c.) voltage with a small ripple, rather than the alternating voltage supplied by A.

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(a)(i) Focal length. The focal length of a converging lens is the distance from the optical centre of the lens to its principal focus (the point on the principal axis to which rays travelling parallel to the axis converge after refraction).

(a)(ii) Optical centre. The optical centre is the point at the middle of the lens through which a ray of light passes without being deviated (it travels straight on).

(a)(iii) Ray diagram: virtual image formed by a converging lens. When the object is placed between the lens and its principal focus (\(u < f\)) the lens acts as a magnifying glass: the emergent rays diverge and, produced backwards, meet on the same side as the object to form a virtual, erect and magnified image.

Two standard construction rays are used:

• Ray 1 leaves the top of the object parallel to the principal axis and, after refraction, passes through the far principal focus \(F'\).
• Ray 2 passes through the optical centre \(C\) of the lens and continues undeviated.

After the lens the two emergent rays diverge, so no real image is formed. Extending them backwards (dashed) they intersect on the same side as the object, locating the tip of the virtual image \(I\).

(b)(i) Primary colours of light. Red, Green and Blue.

(b)(ii) Complementary pairs. Each primary colour pairs with the colour obtained by mixing the other two primaries:

(c) Ray passing symmetrically through a 60° prism, \(n = 1.5\). Symmetric passage means the ray traverses the prism at minimum deviation \(D_m\), so the refraction is described by

(c)(i) Angle of incidence. At symmetric (minimum-deviation) passage the two refracting angles inside the prism are equal, each \(=\tfrac{A}{2}=30^{\circ}\), so the angle of incidence at the first face is

(c)(ii) Angle of minimum deviation. Using the prism formula with \(i=\tfrac{A+D_m}{2}\):

Angle of incidence \(i \approx 48.6^{\circ}\); angle of minimum deviation \(D_m \approx 37.2^{\circ}\).

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(a) Wave-particle duality of light. Light shows a dual nature. In some experiments it behaves as a wave (it undergoes interference, diffraction and polarisation, as in Young's double-slit experiment), while in other experiments it behaves as a stream of particles called photons, each carrying energy \(E=hf\) (as in the photoelectric effect and the Compton effect). Light is therefore neither purely a wave nor purely a particle; it exhibits whichever behaviour the experiment probes. The two aspects are linked by the de Broglie relation \(\lambda = h/p\).

(b) Finding Planck's constant. The de Broglie relation connects wavelength and momentum:

With \(\lambda = 4.2\times10^{-11}\ \text{m}\) and \(p = 1.6\times10^{-23}\ \text{kg m/s}\):

Planck's constant is about \(6.72\times10^{-34}\ \text{J s}\), in good agreement with the accepted value \(6.63\times10^{-34}\ \text{J s}\).

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(a)(i) The stability of a radioactive nucleus is determined by its neutron to proton ratio. If the ratio is too high or too low, the nucleus will become unstable and undergo decay. The relative stability of a nucleus is also influenced by the binding energy per nucleon, which is the energy required to separate the nucleus into its individual protons and neutrons.

(ii) To determine the order of stability of the given nuclei, X, W and Y, we need to examine their neutron to proton ratios. However, the information given is not enough to do so. We need to know the atomic numbers of these nuclei.

(b)(i) Ionization potential refers to the minimum amount of energy required to remove an electron from an atom or ion.

(ii) Information given is not enough to answer the question.

(c)(i) Work function is the minimum amount of energy required to remove an electron from a solid surface. In this case, the work function of sodium is 2.0 eV, meaning that 2.0 eV of energy is required to remove an electron from the surface of a sodium metal.

(ii) The energy of light can be calculated using the formula E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the light. The energy of the light in this case is 1.97 eV. If the energy of the light is greater than the work function of sodium, which is 2.0 eV, photoelectrons will be emitted. In this case, the energy of the light is not enough to remove electrons from the surface of sodium, so photoelectrons will not be emitted.

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Method. The work done in stretching an elastic material is stored as elastic potential energy and is equal to the area under the load-extension graph (load on the vertical axis, extension on the horizontal axis).

For the linear (Hooke's-law) region, the graph is a straight line from the origin, so the area is a triangle:

If the material stretches beyond the elastic limit, the line curves; then the work done is found by counting squares under the curve (or by adding the triangular and rectangular/trapezoidal areas).

Worked illustration. If, for example, the graph shows a load of \(F = 20\ \text{N}\) producing an extension of \(e = 0.10\ \text{m}\) at the end of the straight line, then

Read the actual final load and extension (and any change of gradient) from the printed graph and substitute into the area calculation to obtain the work done.

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Three observable phenomena in which particles behave like waves.

• Electron diffraction: a beam of electrons passed through a thin crystal (or graphite film) produces diffraction rings, just as X-rays do (Davisson-Germer experiment).
• Interference of electrons: electrons passed through a double slit build up an interference pattern of bright and dark fringes.
• The electron microscope: electrons are focused and form images because they behave like short-wavelength waves; neutron diffraction by crystals is a further example.

Scientific principle underlying fibre optics. Fibre optics works on the principle of total internal reflection of light. When light inside the denser core strikes the core-cladding boundary at an angle greater than the critical angle, it is totally reflected back into the core, so light is guided along the fibre with very little loss even when the fibre bends.

(b) Terms used in fibre optics.

• (i) Core: the central, thin strand of very pure glass (or plastic) of higher refractive index along which the light signal actually travels by repeated total internal reflection.
• (ii) Cladding: the outer layer of glass of lower refractive index surrounding the core. Because it is optically less dense than the core, it ensures that light striking the boundary is totally internally reflected and kept inside the core.

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(a)(i) Hooke's law. Provided the elastic limit is not exceeded, the extension of an elastic material is directly proportional to the force (load) producing it. \(F = k e\).

(a)(ii) Spring calculation. (The second length is 0.75 m.)

Force when 0.30 kg hangs: \(F_1 = 0.30\times10 = 3\ \text{N}\), length \(L_1 = 0.20\ \text{m}\).

Force when 1.95 kg hangs: \(F_2 = 1.95\times10 = 19.5\ \text{N}\), length \(L_2 = 0.75\ \text{m}\).

Force constant:

Unloaded (natural) length \(L_0\): using \(F_1 = k(L_1-L_0)\),

Force constant \(= 30\ \text{N/m}\); natural length \(= 0.10\ \text{m}\).

(b)(i) Diffusion. Diffusion is the net movement of particles (molecules or ions) of a substance from a region of higher concentration to a region of lower concentration until they are evenly spread.

(b)(ii) Two factors affecting rate of diffusion. Temperature (higher temperature gives faster diffusion); the density or molar mass of the substance (lighter/less dense substances diffuse faster). (Also the concentration gradient.)

(b)(iii) Relationship with density. The rate of diffusion of a gas is inversely proportional to the square root of its density (Graham's law):

(c) Period-radius relationship for a satellite. The gravitational pull provides the centripetal force:

The satellite covers the circumference \(2\pi R\) in one period, so \(v = \dfrac{2\pi R}{T}\). Substituting:

Hence \(T^{2} \propto R^{3}\) (Kepler's third law).

Maelezo ya Majibu

(a)(i) Dew point

Dew point is the temperature at which the water vapour present in air is just sufficient to saturate the air, so that condensation begins to occur.

(a)(ii) Why dew forms faster on metal than on rubber

Metal is a better conductor of heat than rubber. At night, the metal parts of the bicycle lose heat more rapidly and become colder faster than the rubber parts. Their temperature therefore falls below the dew point earlier, causing water vapour in the surrounding air to condense as dew on the metal surfaces. Rubber is a poor conductor of heat, so it cools more slowly and dew forms on it later.

(b)(i) Meaning of the specific heat capacity of copper being 400 J kg^-1 K^-1

It means that 400 J of heat energy is required to raise the temperature of 1 kg of copper by 1 K (or 1°C).

(b)(ii) Ratio of temperature rise

For each metal,

\[Q = mc\Delta\theta\]

Since the same quantity of heat is supplied to both metals:

\[m_Pc_P\Delta\theta_P = m_Qc_Q\Delta\theta_Q\]

Given:

\[c_P:c_Q = 3:1\]

\[m_P:m_Q = 1:2\]

Therefore,

\[\frac{\Delta\theta_P}{\Delta\theta_Q} = \frac{m_Qc_Q}{m_Pc_P}\]

\[\frac{\Delta\theta_P}{\Delta\theta_Q} = \frac{2\times 1}{1\times 3} = \frac{2}{3}\]

Hence,

\[\boxed{\Delta\theta_P:\Delta\theta_Q = 2:3}\]

(c)(i) Coefficient of thermal conductivity

The coeff

Maelezo ya Majibu

Resolve the initial velocity. With \(u = 40\ \text{m/s}\) at \(30^{\circ}\) to the horizontal:

After t = 1 s. The horizontal component is unchanged (no horizontal force):

The vertical component is reduced by gravity:

Resultant velocity.

Direction above the horizontal.

The velocity after 1 s is about 36.0 m/s directed at about 16° above the horizontal.