All practice questions

What is the area of the region bounded by the curve $y = x^2 - 4$ , the $x$ -axis, and the lines $x = 0$ and $x = 3$ ?

\frac{23}{3}

square units

3

square units

\frac{16}{3}

square units

\frac{25}{3}

square units

Show answer & explanation

Answer: $\frac{23}{3}$ square units

Answer

\frac{23}{3}

square units

The curve

y = x^2 - 4

intersects the x-axis at

x = 2

. To find the total enclosed area between

x = 0

and

x = 3

, the integral must be split into two parts:

[0, 2]

, where the curve is below the x-axis (yielding an area of

\frac{16}{3}

), and

[2, 3]

, where the curve is above the x-axis (yielding an area of

\frac{7}{3}

). Summing these absolute values gives

\frac{16}{3} + \frac{7}{3} = \frac{23}{3}

square units.

Step-by-Step Solution

1

Find the x-intercept of the curve

y = x^2 - 4

within the interval

[0, 3]

.

x^2 - 4 = 0 \implies x = 2

. The curve lies below the x-axis for

0 \le x < 2

and above the x-axis for

2 < x \le 3

.

Total geometric area requires evaluating regions below and above the x-axis separately so negative integral values do not cancel positive area.

2

Calculate the area

A_1

of the region below the x-axis from

x = 0

to

x = 2

.

A_1 = \left| \int_{0}^{2} (x^2 - 4) \, dx \right| = \left| \left[ \frac{x^3}{3} - 4x \right]_{0}^{2} \right| = \left| \frac{8}{3} - 8 \right| = \left| -\frac{16}{3} \right| = \frac{16}{3}

square units.

The curve is below the x-axis, so taking the absolute value gives the true physical area.

3

Calculate the area

A_2

of the region above the x-axis from

x = 2

to

x = 3

.

A_2 = \int_{2}^{3} (x^2 - 4) \, dx = \left[ \frac{x^3}{3} - 4x \right]_{2}^{3} = (9 - 12) - \left( \frac{8}{3} - 8 \right) = -3 - \left(-\frac{16}{3}\right) = \frac{7}{3}

square units.

The curve lies above the x-axis on this interval, yielding a positive definite integral.

4

Sum the areas of the two regions to find the total bounded area.

\text{Total Area} = A_1 + A_2 = \frac{16}{3} + \frac{7}{3} = \frac{23}{3}

square units.

Adding the individual positive areas yields the total bounded area.

Key Concept

Area bounded by a curve that crosses the x-axis

Question 8022Question →

Match each physical quantity listed on the left with its corresponding classification and physical description on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric potential difference matches derived quantity defined as work done per unit electric charge; Thermodynamic temperature matches fundamental quantity representing thermal state, measured in kelvins; Impulse matches derived quantity defined as force times time interval; Luminous intensity matches fundamental quantity measuring perceived light power per unit solid angle.

Thermodynamic temperature and luminous intensity are two of the seven fundamental SI quantities. Electric potential difference and impulse are derived quantities because they are expressed through equations involving base physical quantities.

Step-by-Step Solution

1

Identify fundamental physical quantities

Thermodynamic temperature and luminous intensity are basic independent physical quantities defined by SI standards.

Fundamental quantities cannot be defined in terms of other physical quantities.

2

Identify derived physical quantities and their defining expressions

Electric potential difference (

V = \frac{W}{Q}

) and impulse (

I = F \Delta t

) are derived from basic quantities.

Derived quantities are defined by mathematical combinations of fundamental quantities.

Key Concept

Fundamental and Derived Quantities

Question 8023Question →

A gas sample enclosed in a rigid container of fixed volume has a root-mean-square (r.m.s.) speed of $500\text{ m s}^{-1}$ at a temperature of $127^\circ\text{C}$ . If the gas is heated until its pressure is quadrupled, what is the new r.m.s. speed of the gas molecules in $\text{m s}^{-1}$ ?

Show answer & explanation

Answer: 1000

Answer

1000

According to kinetic theory, the pressure of a fixed volume of gas is directly proportional to its absolute temperature (

P \propto T

), and the root-mean-square speed of its molecules is proportional to the square root of absolute temperature (

v_{\text{rms}} \propto \sqrt{T}

). Quadrupling the pressure quadruples the absolute temperature in Kelvin from

400\text{ K}

to

1600\text{ K}

. Since the speed scales as

\sqrt{4} = 2

, the initial r.m.s. speed of

500\text{ m s}^{-1}

doubles to

1000\text{ m s}^{-1}

.

Step-by-Step Solution

1

Convert the initial temperature to absolute temperature (Kelvin)

T_1 = 127^\circ\text{C} + 273 = 400\text{ K}

Kinetic theory relationships and gas laws require temperature in absolute units (Kelvin).

2

Determine the new absolute temperature based on the pressure change at constant volume

T_2 = 4 \times T_1 = 1600\text{ K}

For a fixed volume of gas, pressure is directly proportional to absolute temperature (

P \propto T

). Therefore, quadrupling the pressure quadruples the absolute temperature.

3

Calculate the new root-mean-square speed using the square root relationship

v_2 = v_1 \sqrt{\frac{T_2}{T_1}} = 500 \times \sqrt{4} = 1000\text{ m s}^{-1}

Root-mean-square speed is proportional to the square root of absolute temperature (

v_{\text{rms}} \propto \sqrt{T}

).

Key Concept

Relationship between microscopic kinetic parameters (r.m.s. speed) and macroscopic state variables (pressure and absolute temperature)

Question 8024Question →

A cell of electromotive force $E = 6.0\text{ V}$ and internal resistance $r = 1.0\text{ }\Omega$ is connected in series with a resistor $R$ and a shunted galvanometer. The galvanometer has a resistance of $90\text{ }\Omega$ and produces a full-scale deflection for a current of $2.0\text{ mA}$ . If the shunt resistance connected across the galvanometer is $10\text{ }\Omega$ , calculate the value of the series resistor $R$ , in ohms $(\Omega)$ , required for the galvanometer to show full-scale deflection.

Show answer & explanation

Answer: 290

Answer

The required value of the series resistor

R

is

290\text{ }\Omega

.

At full-scale deflection, a current of

2.0\text{ mA}

passes through the galvanometer, resulting in a potential difference of

V_g = 2.0 \times 10^{-3} \times 90 = 0.18\text{ V}

. Since the shunt is connected in parallel with the galvanometer, the current through the shunt is

I_s = \frac{0.18}{10} = 0.018\text{ A} = 18\text{ mA}

. Thus, the total current supplied by the cell is

I = 2\text{ mA} + 18\text{ mA} = 20\text{ mA} = 0.02\text{ A}

. The total equivalent resistance of the shunted galvanometer is

R_p = \frac{90 \times 10}{90 + 10} = 9.0\text{ }\Omega

. Applying Ohm's law to the total loop including internal resistance

r

, we have

E = I(R + R_p + r) \implies 6.0 = 0.02(R + 9.0 + 1.0) \implies R + 10.0 = 300 \implies R = 290\text{ }\Omega

.

Step-by-Step Solution

1

Calculate voltage across the galvanometer at full-scale deflection

V_g = 0.18\text{ V}

The potential difference across parallel branches is equal, and for full-scale deflection

I_g = 2.0\text{ mA}

.

2

Calculate the current passing through the shunt resistor

I_s = 18.0\text{ mA} = 0.018\text{ A}

Using Ohm's law across the shunt resistor

S = 10\text{ }\Omega

with

V_s = V_g = 0.18\text{ V}

.

3

Calculate the total circuit current provided by the cell

I = 20.0\text{ mA} = 0.020\text{ A}

By Kirchhoff's current law, the main current splits between the galvanometer and shunt.

4

Find the equivalent resistance of the shunted galvanometer and total circuit resistance

R_p = 9.0\text{ }\Omega

and total circuit resistance

R_{\text{total}} = 300\text{ }\Omega

Parallel resistance formula gives

R_p = 9.0\text{ }\Omega

, and

R_{\text{total}} = \frac{E}{I} = \frac{6.0\text{ V}}{0.020\text{ A}} = 300\text{ }\Omega

.

5

Solve for the unknown external series resistance

R

R = 290\text{ }\Omega

R_{\text{total}} = R + r + R_p \implies 300 = R + 1.0 + 9.0 \implies R = 290\text{ }\Omega

.

Key Concept

Galvanometer Shunting and Electric Circuit Analysis with Internal Resistance

Question 8025Question →

A $0.020\text{ kg}$ sample of a liquid metal at its melting point of $500^\circ\text{C}$ solidifies completely as thermal energy is extracted from it at a constant rate of $10\text{ W}$ . If the specific latent heat of fusion of the metal is $2.0 \times 10^4\text{ J kg}^{-1}$ , how long does the solidification process take?

40\text{ s}

400\text{ s}

2000\text{ s}

20000\text{ s}

Show answer & explanation

Answer: $40\text{ s}$

Answer

The solidification process takes

40\text{ s}

.

The thermal energy released when a substance solidifies at its melting point is given by

Q = m L_f

. Substituting

m = 0.020\text{ kg}

and

L_f = 2.0 \times 10^4\text{ J kg}^{-1}

gives

Q = 400\text{ J}

. Dividing this energy by the constant rate of heat removal (

10\text{ W}

) yields

t = \frac{400\text{ J}}{10\text{ W}} = 40\text{ s}

.

Step-by-Step Solution

1

Calculate the total thermal energy (

Q

) released during phase change at constant temperature

Q = m L_f = 0.020\text{ kg} \times 2.0 \times 10^4\text{ J kg}^{-1} = 400\text{ J}

Phase change occurs at a constant temperature, so thermal energy depends solely on mass and specific latent heat of fusion.

2

Calculate time (

t

) required using the power rate (

P

)

t = \frac{Q}{P} = \frac{400\text{ J}}{10\text{ W}} = 40\text{ s}

Power is defined as energy transferred per unit time (

P = Q / t

).

Key Concept

Latent Heat of Fusion and Energy Balance

Question 8026Question →

A side-view convex mirror on a bus has a radius of curvature of $40\text{ cm}$ . If a motorcycle is located $30\text{ cm}$ in front of the mirror, what is the location of the image formed relative to the mirror?

12\text{ cm}

behind the mirror

12\text{ cm}

in front of the mirror

60\text{ cm}

behind the mirror

60\text{ cm}

in front of the mirror

Show answer & explanation

Answer: $12\text{ cm}$ behind the mirror

Answer

The image is formed

12\text{ cm}

behind the mirror.

For a convex mirror, the focal length is virtual, so

f = -20\text{ cm}

. With an object distance of

u = +30\text{ cm}

, applying the mirror equation

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

gives

\frac{1}{v} = -\frac{1}{20} - \frac{1}{30} = -\frac{1}{12}

, leading to

v = -12\text{ cm}

. The negative sign specifies that the virtual image is located

12\text{ cm}

behind the mirror.

Step-by-Step Solution

1

Determine the focal length of the mirror from its radius of curvature

f = -\frac{R}{2} = -\frac{40\text{ cm}}{2} = -20\text{ cm}

For spherical mirrors, focal length is half the radius of curvature. Convex mirrors have a negative focal length by sign convention.

2

Set up the mirror formula using the given object distance

u = +30\text{ cm}

\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \implies -\frac{1}{20} = \frac{1}{30} + \frac{1}{v}

The mirror equation relates focal length, object distance, and image distance.

3

Solve for the image distance

v

\frac{1}{v} = -\frac{1}{20} - \frac{1}{30} = -\frac{3 + 2}{60} = -\frac{5}{60} = -\frac{1}{12} \implies v = -12\text{ cm}

Algebraic manipulation yields a negative image distance.

4

Interpret the physical meaning of the calculated value

The negative sign indicates a virtual image located

12\text{ cm}

behind the mirror.

Under standard optical sign conventions, negative image distances correspond to virtual images formed behind the mirror.

Key Concept

Mirror equation and sign conventions for convex spherical mirrors

Question 8027Question →

A rigid scuba diving cylinder contains a fixed mass of compressed air at an initial absolute pressure of $2.00 \times 10^7\text{ Pa}$ and a temperature of $27^\circ\text{C}$ . The cylinder is left under direct sunlight on a boat deck, causing its temperature to rise to $77^\circ\text{C}$ . Assuming the volume of the cylinder remains constant, what is the new pressure of the air inside the cylinder?

2.33 \times 10^7\text{ Pa}

5.70 \times 10^7\text{ Pa}

1.71 \times 10^7\text{ Pa}

5.13 \times 10^6\text{ Pa}

Show answer & explanation

Answer: $2.33 \times 10^7\text{ Pa}$

Answer

The final pressure of the air inside the cylinder is

2.33 \times 10^7\text{ Pa}

.

According to the Pressure Law, the pressure of a fixed mass of gas at constant volume is directly proportional to its absolute temperature (

P \propto T

). Converting the given temperatures yields

T_1 = 300\text{ K}

and

T_2 = 350\text{ K}

. Substituting these into

P_2 = P_1(T_2 / T_1)

gives

2.00 \times 10^7 \times (350 / 300) = 2.33 \times 10^7\text{ Pa}

.

Step-by-Step Solution

1

Convert given temperatures from Celsius to Kelvin.

T_1 = 27^\circ\text{C} + 273 = 300\text{ K}

, and

T_2 = 77^\circ\text{C} + 273 = 350\text{ K}

.

All gas laws strictly require absolute temperature measured in Kelvin.

2

State and rearrange the Pressure Law (Gay-Lussac's Law) for constant volume.

\frac{P_1}{T_1} = \frac{P_2}{T_2} \implies P_2 = P_1 \times \frac{T_2}{T_1}

For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature.

3

Substitute the known values into the equation to calculate the final pressure.

P_2 = 2.00 \times 10^7\text{ Pa} \times \frac{350\text{ K}}{300\text{ K}} \approx 2.33 \times 10^7\text{ Pa}

.

Evaluates the final pressure accurately after heating.

Key Concept

Pressure Law (Gay-Lussac's Law) for Ideal Gases

Question 8028Question →

A tungsten filament lamp operating at a room temperature of $20\,^\circ\text{C}$ draws a current of $0.50\,\text{A}$ when connected to a $120\,\text{V}$ power source. When the lamp reaches its steady operating temperature, the current drops to $0.10\,\text{A}$ under the same voltage. If the temperature coefficient of resistance of tungsten is $4.0 \times 10^{-3}\,^\circ\text{C}^{-1}$ , what is the operating temperature of the filament?

1020\,^\circ\text{C}

1000\,^\circ\text{C}

270\,^\circ\text{C}

200\,^\circ\text{C}

Show answer & explanation

Answer: $1020\,^\circ\text{C}$

Answer

The operating temperature of the tungsten filament is

1020\,^\circ\text{C}

.

Using Ohm's law (

R = V/I

), the initial resistance at

20\,^\circ\text{C}

is

240\,\Omega

and the operating resistance is

1200\,\Omega

. Substituting these into

R_2 = R_1[1 + \alpha(T_2 - T_1)]

yields

1200 = 240[1 + 4.0\times 10^{-3}(T_2 - 20)]

. Solving gives

T_2 - 20 = 1000\,^\circ\text{C}

, so the operating temperature is

1020\,^\circ\text{C}

.

Step-by-Step Solution

1

Calculate the resistance of the filament at room temperature (

20\,^\circ\text{C}

) and at the operating temperature using Ohm's Law (

R = V/I

).

Cold resistance

R_1 = \frac{120\,\text{V}}{0.50\,\text{A}} = 240\,\Omega

; Hot resistance

R_2 = \frac{120\,\text{V}}{0.10\,\text{A}} = 1200\,\Omega

.

Ohm's law relates voltage, current, and resistance for a conductor.

2

Apply the temperature dependence equation of electrical resistance:

R_2 = R_1[1 + \alpha(T_2 - T_1)]

.

\frac{1200}{240} = 1 + (4.0 \times 10^{-3})(T_2 - 20) \Rightarrow 5 = 1 + (4.0 \times 10^{-3})(T_2 - 20)

.

Resistance increases linearly with temperature according to the material's temperature coefficient.

3

Solve for the temperature difference

\Delta T = T_2 - 20

and find the final temperature

T_2

.

4.0 \times 10^{-3}(T_2 - 20) = 4 \Rightarrow T_2 - 20 = \frac{4}{4.0 \times 10^{-3}} = 1000\,^\circ\text{C} \Rightarrow T_2 = 1020\,^\circ\text{C}

.

Adding the initial temperature to the temperature change gives the final absolute operating temperature.

Key Concept

Temperature Dependence of Electrical Resistance and Ohm's Law

Question 8029Question →

Which of the following conditions must be satisfied for light to undergo total internal reflection at the boundary between two transparent media?

The light ray must travel from an optically denser medium into an optically less dense medium, and the angle of incidence must exceed the critical angle.

The light ray must travel from an optically less dense medium into an optically denser medium, and the angle of incidence must exceed the critical angle.

The light ray must travel from an optically denser medium into an optically less dense medium, and the angle of incidence must be less than the critical angle.

The light ray must travel from an optically less dense medium into an optically denser medium at an angle of incidence equal to

0^\circ

.

Show answer & explanation

Answer: The light ray must travel from an optically denser medium into an optically less dense medium, and the angle of incidence must exceed the critical angle.

Answer

The light ray must travel from an optically denser medium into an optically less dense medium, and the angle of incidence must exceed the critical angle.

Total internal reflection occurs only when light passes from a medium of higher optical density into a medium of lower optical density, and the angle of incidence exceeds the critical angle for that boundary.

Step-by-Step Solution

1

Identify the boundary requirement for wave propagation direction in total internal reflection.

Light must travel from an optically denser medium (higher refractive index

n_1

) toward an optically rarer medium (lower refractive index

n_2

).

This direction allows the ray to bend away from the normal, increasing the angle of refraction relative to the angle of incidence.

2

Determine the required angle of incidence at the boundary.

The angle of incidence

i

must be strictly greater than the critical angle

\theta_c

(where

\sin \theta_c = n_2 / n_1

).

When

i > \theta_c

, no refraction can take place because

\sin r > 1

, forcing all light energy to reflect back into the initial denser medium.

Key Concept

Conditions for Total Internal Reflection

Question 8030Question →

A $24.6\text{ g}$ sample of hydrated magnesium tetraoxosulfate(VI), $\text{MgSO}_4 \cdot x\text{H}_2\text{O}$ , was heated strongly until all the water of crystallization was driven off, leaving behind $12.0\text{ g}$ of anhydrous $\text{MgSO}_4$ . What is the value of $x$ ? $[\text{Mg} = 24, \text{S} = 32, \text{O} = 16, \text{H} = 1]$

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

Heating the sample evaporates

12.6\text{ g}

of water of crystallization (

24.6\text{ g} - 12.0\text{ g}

). Converting the masses to moles gives

0.1\text{ mol}

of anhydrous

\text{MgSO}_4

(

12.0\text{ g} / 120\text{ g/mol}

) and

0.7\text{ mol}

of

\text{H}_2\text{O}

(

12.6\text{ g} / 18\text{ g/mol}

). The mole ratio of water to salt is

0.7 / 0.1 = 7

, which means

x = 7

.

Step-by-Step Solution

1

Calculate the mass of water of crystallization lost during heating.

Mass of water =

24.6\text{ g} - 12.0\text{ g} = 12.6\text{ g}

.

The decrease in mass after heating represents the mass of water driven off from the hydrated salt.

2

Determine the molar masses of

\text{MgSO}_4

and

\text{H}_2\text{O}

.

Molar mass of

\text{MgSO}_4 = 24 + 32 + (4 \times 16) = 120\text{ g/mol}

; Molar mass of

\text{H}_2\text{O} = (2 \times 1) + 16 = 18\text{ g/mol}

.

Molar masses are required to convert the measured masses into chemical amounts (moles).

3

Calculate the number of moles of anhydrous salt and water.

Moles of

\text{MgSO}_4 = \frac{12.0\text{ g}}{120\text{ g/mol}} = 0.1\text{ mol}

; Moles of

\text{H}_2\text{O} = \frac{12.6\text{ g}}{18\text{ g/mol}} = 0.7\text{ mol}

.

The chemical formula stoichiometry is determined by the molar ratio of components.

4

Determine the mole ratio of water to anhydrous salt (

x

).

x = \frac{\text{Moles of } \text{H}_2\text{O}}{\text{Moles of } \text{MgSO}_4} = \frac{0.7\text{ mol}}{0.1\text{ mol}} = 7

.

The coefficient

x

is the integer ratio of moles of water of crystallization per mole of anhydrous salt.

Key Concept

Water of Crystallization Stoichiometry

Question 8031Question →

A convex spherical mirror produces an image that is one-third the size of an object placed in front of it. If the distance of the object from the mirror is doubled, what is the new linear magnification of the image?

\frac{1}{5}

\frac{1}{4}

\frac{1}{6}

\frac{1}{2}

Show answer & explanation

Answer: $\frac{1}{5}$

Answer

The new linear magnification of the image is

\frac{1}{5}

.

For a convex mirror, the linear magnification

m

relates object distance

u

and focal magnitude

f

by

m = \frac{f}{f + u}

. Given

m = \frac{1}{3}

, solving

\frac{1}{3} = \frac{f}{f + u}

yields

u = 2f

. When the object distance is doubled to

u' = 4f

, the new magnification becomes

m' = \frac{f}{f + 4f} = \frac{1}{5}

.

Step-by-Step Solution

1

Express linear magnification in terms of object distance and focal length for a convex mirror

m = \frac{f}{f + u}

For a convex mirror, the focal length is negative under the Cartesian sign convention, making the virtual image distance

v = -\frac{f u}{u + f}

, so magnification

m = -\frac{v}{u} = \frac{f}{f + u}

.

2

Substitute the initial magnification

m = \frac{1}{3}

to express initial object distance

u

in terms of focal length

f

\frac{1}{3} = \frac{f}{f + u} \implies f + u = 3f \implies u = 2f

This establishes that the object was originally located at a distance equal to twice the focal length of the mirror.

3

Calculate the new object distance

u'

when distance is doubled

u' = 2u = 2(2f) = 4f

The problem states the object distance from the mirror is doubled.

4

Compute the new linear magnification

m'

m' = \frac{f}{f + u'} = \frac{f}{f + 4f} = \frac{f}{5f} = \frac{1}{5}

Substituting

u' = 4f

into the magnification formula yields the final reduced magnification.

Key Concept

Linear magnification and sign convention for convex mirrors

Estimated Time:2m 0s

Question 8032Question →

Match each physical quantity or concept from the kinetic theory of gases on the left with its corresponding microscopic description or mathematical relation on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Root-mean-square speed corresponds to

\sqrt{\frac{3kT}{m}}

; Average translational kinetic energy per molecule corresponds to

\frac{3}{2}kT

; Gas pressure corresponds to

\frac{1}{3}\rho v_{\text{rms}}^2

; and Absolute temperature corresponds to the macroscopic measure proportional to mean translational kinetic energy.

Each kinetic theory quantity correctly matches its corresponding microscopic formula and definition derived from fundamental assumptions of ideal gas particle behavior.

Step-by-Step Solution

1

Analyze the microscopic derivation of root-mean-square speed

From kinetic theory,

\bar{E}_k = \frac{1}{2}m v_{\text{rms}}^2 = \frac{3}{2}kT

, which yields

v_{\text{rms}} = \sqrt{\frac{3kT}{m}}

.

This establishes the relationship between molecular speed, temperature, and mass.

2

Identify the relationship for average translational kinetic energy per molecule

The average translational kinetic energy per molecule is given directly by

\bar{E}_k = \frac{3}{2}kT

.

The mean kinetic energy per degree of freedom is

\frac{1}{2}kT

, summing to

\frac{3}{2}kT

for three translational dimensions.

3

Relate macroscopic gas pressure to microscopic particle collisions

Gas pressure is expressed as

P = \frac{1}{3}\rho v_{\text{rms}}^2

based on continuous elastic collisions of gas molecules with the container walls.

Pressure represents the average force exerted per unit area by molecular collisions.

4

Define absolute temperature in terms of molecular kinetic energy

Absolute temperature

T

is the macroscopic physical property directly proportional to the average kinetic energy of the molecules.

This provides the thermodynamic definition of temperature from kinetic theory.

Key Concept

Microscopic properties of ideal gas molecules and kinetic derivation of pressure and temperature

Question 8033Question →

Treasury bills are short-term money market instruments that yield returns to investors through periodic coupon interest payments prior to maturity.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False. Treasury bills do not yield periodic interest payments; instead, they are issued at a discount to face value and redeemed at par upon maturity.

Treasury bills are zero-coupon money market instruments. They do not make periodic interest payments during their tenure. Instead, they are issued at a discount to face value and redeemed at par (full face value) upon maturity, with the discount serving as the holder's earned interest.

Step-by-Step Solution

1

Analyze the features and income structure of Treasury bills as short-term government debt instruments.

Treasury bills are zero-coupon money market instruments with maturity periods typically ranging from 91 to 364 days.

Understanding the yield structure helps distinguish discounted instruments from coupon-bearing instruments.

2

Evaluate how investors earn a return on Treasury bills.

An investor buys the bill at a price below face value (at a discount) and receives the full face value upon maturity.

The difference between the discounted purchase price and the face value represents the investor's total gain.

3

Determine the validity of the statement based on whether periodic interest is paid.

Since returns are earned exclusively via the discount mechanism at maturity without intermediate interest payments, the statement is false.

Periodic interest payments are characteristic of long-term capital market instruments like government bonds, not short-term Treasury bills.

Key Concept

Treasury Bill Issuance and Yield Mechanism

Question 8034Question →

Two capacitors of capacitances $8.0\text{ }\mu\text{F}$ and $4.0\text{ }\mu\text{F}$ are connected in parallel. This parallel combination is then connected in series with a single $4.0\text{ }\mu\text{F}$ capacitor across a $36.0\text{ V}$ d.c. voltage source. What is the potential difference across the single $4.0\text{ }\mu\text{F}$ capacitor?

9.0\text{ V}

14.4\text{ V}

27.0\text{ V}

36.0\text{ V}

Show answer & explanation

Answer: $27.0\text{ V}$

Answer

The potential difference across the single

4.0\text{ }\mu\text{F}

capacitor is

27.0\text{ V}

.

The two parallel capacitors (

8.0\text{ }\mu\text{F}

and

4.0\text{ }\mu\text{F}

) combine directly to give an equivalent capacitance of

12.0\text{ }\mu\text{F}

. This equivalent capacitor is in series with the single

4.0\text{ }\mu\text{F}

capacitor across the

36.0\text{ V}

source. Using the voltage divider rule for series capacitors, the voltage across the single

4.0\text{ }\mu\text{F}

capacitor is

V = 36.0 \times \frac{12.0}{4.0 + 12.0} = 27.0\text{ V}

.

Step-by-Step Solution

1

Calculate the equivalent capacitance of the two parallel capacitors.

C_p = 8.0\text{ }\mu\text{F} + 4.0\text{ }\mu\text{F} = 12.0\text{ }\mu\text{F}

Capacitors in parallel add algebraically.

2

Calculate the total equivalent capacitance of the entire circuit.

C_T = \frac{4.0 \times 12.0}{4.0 + 12.0} = \frac{48.0}{16.0} = 3.0\text{ }\mu\text{F}

The single

4.0\text{ }\mu\text{F}

capacitor and the

12.0\text{ }\mu\text{F}

parallel combination are in series.

3

Find the total charge supplied by the battery.

Q = C_T \times V = 3.0\text{ }\mu\text{F} \times 36.0\text{ V} = 108.0\text{ }\mu\text{C}

The total charge is equal to the total capacitance multiplied by the total voltage.

4

Calculate the potential difference across the single

4.0\text{ }\mu\text{F}

capacitor.

V_1 = \frac{Q}{C_1} = \frac{108.0\text{ }\mu\text{C}}{4.0\text{ }\mu\text{F}} = 27.0\text{ V}

In a series connection, the charge on the single capacitor equals the total charge.

Key Concept

Potential difference distribution in mixed series-parallel capacitor networks

Estimated Time:1m 30s

Question 8035Question →

In the industrial production of cement in Nigeria, such as at the factory in Ewekoro, Ogun State, proximity to the bulk raw material is the primary factor influencing site selection. Which of the following is the principal raw material extracted at the site for this process?

Limestone

Bauxite

Coal

Gypsum

Show answer & explanation

Answer: Limestone

Answer

Limestone is the principal raw material for cement production and determines factory siting due to its bulk and heavy transport cost.

Limestone is the chief calcareous raw material needed in large quantities for burning in kilns to produce cement clinker. Because of its weight and bulk, cement factories are located directly at limestone quarry sites to minimize transportation costs.

Step-by-Step Solution

1

Identify the industrial chemical process

The process described is cement manufacturing in Nigeria.

Establishing the main product helps determine the required starting materials.

2

Analyze raw material requirements for cement production

Limestone (

CaCO_3

) forms the primary raw material component (calcareous material) used in massive volume.

Heavy industries sit close to raw materials when transport costs of heavy raw inputs outweigh other factors.

3

Evaluate Nigerian industrial geography context

Ewekoro in Ogun State and Nkalagu in Ebonyi State were selected for major cement plants specifically due to large natural limestone deposits.

This confirms limestone as the critical siting raw material.

Key Concept

Raw material availability as a major siting factor for heavy chemical industries.

Estimated Time:45s

Question 8036Question →

Match each physical scenario or effect involving magnetic forces listed on the left with its corresponding physical characteristic or outcome on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair stationary charges with zero force; perpendicular entry with circular motion; anti-parallel currents with mutual repulsion; and force direction with mutual perpendicularity to velocity and magnetic field vectors.

Each physical scenario correctly pairs with its corresponding principle: stationary charges experience zero magnetic force; perpendicular charge motion forms a circular orbit; anti-parallel currents produce repulsion; and magnetic force is always mutually perpendicular to velocity and field vectors.

Step-by-Step Solution

1

Evaluate the magnetic force formula for a static charge

Using

F = qvB\sin\theta

, with

v = 0

,

F = 0

.

A magnetic field does not exert force on a stationary electric charge.

2

Determine the path of a particle moving perpendicular to a magnetic field

The magnetic force acts continuously at right angles to the velocity vector, providing centripetal acceleration and forming a circular orbit.

A perpendicular force of constant magnitude changes motion direction continuously without altering speed.

3

Apply Ampere's law and the right-hand rule to parallel currents in opposite directions

The magnetic field generated by each wire exerts an outward force on the other, producing repulsion.

Opposite currents create reinforcing field lines between the wires, driving them apart.

4

Analyze vector cross product orientation for magnetic force on a charge

The force vector

\vec{F}

is oriented perpendicular to the plane containing vectors

\vec{v}

and

\vec{B}

.

By definition of vector cross-product

\vec{F} = q(\vec{v} \times \vec{B})

, the resultant vector is orthogonal to both input vectors.

Key Concept

Magnetic Force on Moving Charges and Current-Carrying Conductors

Question 8037Question →

Car $A$ , traveling at a constant speed of $20\text{ m/s}$ along a straight horizontal road, passes a landmark $50\text{ m}$ ahead of car $B$ , which is initially stationary. At $t = 0\text{ s}$ , car $B$ starts moving in the same direction, accelerating uniformly at $3.0\text{ m/s}^2$ until it reaches a top speed of $30\text{ m/s}$ , after which it continues at this constant top speed. How many seconds after $t = 0\text{ s}$ does car $B$ catch up with car $A$ ?

Show answer & explanation

Answer: 20

Answer

Car B catches up with car A after 20 seconds.

Car B accelerates for

10\text{ s}

covering

150\text{ m}

to reach

30\text{ m/s}

. During these

10\text{ s}

, car A reaches a position of

250\text{ m}

(taking into account its

50\text{ m}

head start). Car B then closes the remaining

100\text{ m}

gap at a relative speed of

10\text{ m/s}

(

30\text{ m/s} - 20\text{ m/s}

), taking an extra

10\text{ s}

. The total elapsed time is

10\text{ s} + 10\text{ s} = 20\text{ s}

.

Step-by-Step Solution

1

Calculate the duration

t_1

of car B's acceleration phase to reach

30\text{ m/s}

.

t_1 = \frac{v_{max} - u}{a} = \frac{30 - 0}{3.0} = 10\text{ s}

Car B accelerates uniformly from rest at

3.0\text{ m/s}^2

until reaching its top speed limit.

2

Determine the distance

s_B

covered by car B and the position

s_A

of car A at

t = 10\text{ s}

.

s_B = \frac{1}{2}a t_1^2 = \frac{1}{2}(3.0)(10)^2 = 150\text{ m}

;

s_A = 50 + v_A t_1 = 50 + (20)(10) = 250\text{ m}

Car A starts

50\text{ m}

ahead and moves continuously at

20\text{ m/s}

.

3

Find the separation distance between the two cars at

t = 10\text{ s}

and compute the time

\Delta t

required to close it.

\text{Separation} = 250 - 150 = 100\text{ m}; \Delta t = \frac{100}{30 - 20} = 10\text{ s}

Beyond

t = 10\text{ s}

, car B travels at a constant relative velocity of

10\text{ m/s}

faster than car A.

4

Sum the acceleration time and constant speed time to find the total time taken.

t_{total} = t_1 + \Delta t = 10\text{ s} + 10\text{ s} = 20\text{ s}

Combining both phases yields the exact instant car B overtakes car A.

Key Concept

Multi-stage relative motion with acceleration limits

Question 8038Question →

What are all the values of $\theta$ in the interval $0^\circ \le \theta \le 360^\circ$ that satisfy the trigonometric equation $4\sin^2\theta - 3 = 0$ ?

60^\circ, 120^\circ, 240^\circ, 300^\circ

60^\circ \text{ and } 120^\circ

30^\circ, 150^\circ, 210^\circ, 330^\circ

60^\circ \text{ and } 300^\circ

Show answer & explanation

Answer: $60^\circ, 120^\circ, 240^\circ, 300^\circ$

Answer

60^\circ, 120^\circ, 240^\circ, 300^\circ

Solving

4\sin^2\theta - 3 = 0

gives

\sin^2\theta = \frac{3}{4}

, so

\sin\theta = \pm\frac{\sqrt{3}}{2}

. The reference angle for which

\sin\theta = \frac{\sqrt{3}}{2}

is

60^\circ

. The positive root

\sin\theta = +\frac{\sqrt{3}}{2}

gives solutions in Quadrants I and II:

60^\circ

and

180^\circ - 60^\circ = 120^\circ

. The negative root

\sin\theta = -\frac{\sqrt{3}}{2}

gives solutions in Quadrants III and IV:

180^\circ + 60^\circ = 240^\circ

and

360^\circ - 60^\circ = 300^\circ

. Combining these yields all four angles:

60^\circ, 120^\circ, 240^\circ, 300^\circ

.

Step-by-Step Solution

1

Isolate the squared trigonometric term in the equation

4\sin^2\theta = 3 \implies \sin^2\theta = \frac{3}{4}

Rearranging the equation allows solving for

\sin\theta

directly.

2

Take the square root of both sides, keeping both positive and negative roots

\sin\theta = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}

Taking the square root of a squared quantity yields both positive and negative values.

3

Find solutions for

\sin\theta = +\frac{\sqrt{3}}{2}

in Quadrants I and II

\theta = 60^\circ \text{ and } \theta = 180^\circ - 60^\circ = 120^\circ

Sine is positive in Quadrants I and II.

4

Find solutions for

\sin\theta = -\frac{\sqrt{3}}{2}

in Quadrants III and IV

\theta = 180^\circ + 60^\circ = 240^\circ \text{ and } \theta = 360^\circ - 60^\circ = 300^\circ

Sine is negative in Quadrants III and IV.

5

Combine all solutions within the domain

0^\circ \le \theta \le 360^\circ

\theta \in \{60^\circ, 120^\circ, 240^\circ, 300^\circ\}

All four angles satisfy the original quadratic trigonometric equation.

Key Concept

Solving quadratic trigonometric equations by finding reference angles and evaluating solutions across all four quadrants

Estimated Time:1m 30s

Question 8039Question →

A factory employs $12$ workers to produce $180$ identical wooden chairs in $5$ days, working $8$ hours per day. If $4$ workers are reassigned to another department, how many days will the remaining workers take to produce $210$ such chairs if they work $7$ hours per day?

10\text{ days}

20\text{ days}

15\text{ days}

8\text{ days}

Show answer & explanation

Answer: $10\text{ days}$

Answer

The remaining workers will take

10\text{ days}

to complete the task.

The correct answer is

10\text{ days}

. Using compound proportion, the relationship is given by

\frac{W_1 \times D_1 \times H_1}{C_1} = \frac{W_2 \times D_2 \times H_2}{C_2}

. Substituting the given values

\frac{12 \times 5 \times 8}{180} = \frac{8 \times D_2 \times 7}{210}

yields

\frac{480}{180} = \frac{56 D_2}{210}

, which simplifies to

\frac{8}{3} = \frac{56 D_2}{210}

. Solving for

D_2

gives

D_2 = 10\text{ days}

.

Step-by-Step Solution

1

Calculate the total man-hours required for the initial production batch.

Total man-hours =

12\text{ workers} \times 5\text{ days} \times 8\text{ hours/day} = 480\text{ man-hours}

.

Determining total labor input needed for

180

chairs.

2

Find the man-hours required per chair.

Man-hours per chair =

\frac{480}{180} = \frac{8}{3}\text{ hours/chair}

.

Establishing the unit rate of work.

3

Calculate the total man-hours required for the new target of

210

chairs.

Total man-hours needed =

210 \times \frac{8}{3} = 560\text{ man-hours}

.

Scaling the unit work rate to the new quantity.

4

Determine the new workforce size and daily labor capacity.

Remaining workers =

12 - 4 = 8\text{ workers}

. Daily man-hours =

8 \times 7 = 56\text{ man-hours/day}

.

Accounting for the reduced workforce and new daily hours.

5

Calculate the number of days required.

Days =

\frac{560\text{ man-hours}}{56\text{ man-hours/day}} = 10\text{ days}

.

Dividing total required work by daily capacity.

Key Concept

Compound Proportion and Work-Rate Relationships

Estimated Time:1m 30s

Question 8040Question →

A radioactive mixture initially contains two radioisotopes, $P$ and $Q$ , such that the initial number of undecayed nuclei of $P$ is $8$ times that of $Q$ . If the half-life of isotope $P$ is $2\text{ hours}$ and the half-life of isotope $Q$ is $6\text{ hours}$ , calculate the time, in hours, after which the number of undecayed nuclei of both isotopes will be equal.

Show answer & explanation

Answer: 9

Answer

The time after which the number of undecayed nuclei of both isotopes will be equal is 9 hours.

By applying the radioactive decay law

N(t) = N_0(1/2)^{t/T_{1/2}}

to both isotopes with initial ratio

N_{P0} = 8N_{Q0}

and equating

N_P(t) = N_Q(t)

, we obtain

8 = 2^{t/3}

, which gives

t = 9\text{ hours}

.

Step-by-Step Solution

1

Write the decay equations for isotopes P and Q based on their half-lives

N_P(t) = N_{P0}\left(\frac{1}{2}\right)^{t/2}

and

N_Q(t) = N_{Q0}\left(\frac{1}{2}\right)^{t/6}

Radioactive decay follows the exponential relationship

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}

.

2

Apply the initial condition

N_{P0} = 8 N_{Q0}

and set the two expressions equal

8 N_{Q0} \left(\frac{1}{2}\right)^{t/2} = N_{Q0} \left(\frac{1}{2}\right)^{t/6}

The problem asks for the time

t

when both isotopes have equal remaining undecayed nuclei.

3

Divide both sides by

N_{Q0} \left(\frac{1}{2}\right)^{t/2}

and simplify the exponents

8 = \frac{(1/2)^{t/6}}{(1/2)^{t/2}} = \left(\frac{1}{2}\right)^{-t/3} = 2^{t/3}

Applying exponent laws simplifies the ratio of powers of one-half into a single base-two exponent.

4

Solve for time

t

using powers of 2

2^3 = 2^{t/3} \implies \frac{t}{3} = 3 \implies t = 9\text{ hours}

Equating the exponents of identical base 2 gives the exact time.

Key Concept

Radioactive Decay Law and Half-life for Isotope Mixtures

Estimated Time:2m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept