All practice questions

Question 4421Question →

A $150\text{ V}$ (RMS) AC generator is connected across a series combination of a $12\ \Omega$ resistor, an inductor with an inductive reactance of $20\ \Omega$ , and a capacitor with a capacitive reactance of $11\ \Omega$ . What is the average electrical power consumed by this circuit?

900\text{ W}

1200\text{ W}

1500\text{ W}

1875\text{ W}

Show answer & explanation

Answer: $1200\text{ W}$

Answer

The average electrical power consumed by the circuit is

1200\text{ W}

.

To find the average real power dissipated in an AC series circuit, first determine the net reactance

X = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega

. Next, calculate total impedance using

Z = \sqrt{R^2 + X^2} = \sqrt{12^2 + 9^2} = 15\ \Omega

. The RMS current in the circuit is

I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}

. Since energy is dissipated only by resistance, the average real power is

P = I_{\text{rms}}^2 R = (10)^2 \times 12 = 1200\text{ W}

.

Step-by-Step Solution

1

Calculate the net reactance of the circuit.

X = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega

Inductive and capacitive reactances oppose each other in phase by

180^\circ

.

2

Calculate the total impedance of the series RLC circuit.

Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\ \Omega

Resistance and net reactance add vectorially at a right angle.

3

Determine the RMS current flowing through the circuit.

I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}

Ohm's law for AC circuits states that current is the supply RMS voltage divided by total impedance.

4

Calculate the average real power dissipated by the circuit.

P_{\text{avg}} = I_{\text{rms}}^2 R = (10\text{ A})^2 \times 12\ \Omega = 1200\text{ W}

In an AC circuit, average power is only dissipated by resistive components, as ideal inductors and capacitors consume zero net real power over a complete cycle.

Key Concept

Power Dissipation in AC Series Circuits

Question 4422Question →

The sum of the first three terms of an increasing geometric progression of positive real numbers is $21$ , and the sum of their squares is $189$ . What is the common ratio of this progression?

Show answer & explanation

Answer: 2

Answer

The common ratio of the geometric progression is 2.

Let the terms be

a

,

ar

, and

ar^2

. The given conditions yield

a(1+r+r^2) = 21

and

a^2(1+r^2+r^4) = 189

. Squaring the first equation gives

a^2(1+r+r^2)^2 = 441

. Dividing the sum of squares equation by this squared equation gives

\frac{1-r+r^2}{1+r+r^2} = \frac{189}{441} = \frac{3}{7}

. Simplifying

7(1-r+r^2) = 3(1+r+r^2)

results in

2r^2 - 5r + 2 = 0

, which factors into

(2r-1)(r-2) = 0

. Because the progression is increasing,

r > 1

, making

r = 2

the correct common ratio.

Step-by-Step Solution

1

Formulate algebraic expressions for the sum of terms and sum of squares.

a(1 + r + r^2) = 21

and

a^2(1 + r^2 + r^4) = 189

The first three terms of any geometric progression can be expressed as

a

,

ar

, and

ar^2

.

2

Eliminate the first term

a

by squaring the first equation and dividing.

\frac{a^2(1 + r^2 + r^4)}{a^2(1 + r + r^2)^2} = \frac{189}{441} \implies \frac{(1 + r + r^2)(1 - r + r^2)}{(1 + r + r^2)^2} = \frac{3}{7}

Using the algebraic factorization

1 + r^2 + r^4 = (1 + r + r^2)(1 - r + r^2)

allows cancellation of

a^2

and

(1 + r + r^2)

.

3

Solve the resulting equation for the common ratio

r

.

7(1 - r + r^2) = 3(1 + r + r^2) \implies 4r^2 - 10r + 4 = 0 \implies 2r^2 - 5r + 2 = 0

Cross-multiplying reduces the ratio to a standard quadratic equation.

4

Factor the quadratic equation and select the correct root.

(2r - 1)(r - 2) = 0 \implies r = 2 \text{ or } r = 0.5

Since the geometric progression is specified as increasing, the common ratio must be greater than 1 (

r = 2

).

Key Concept

Geometric progression term representations, sum formulas, and algebraic identity factorization.

Alternative Method

Find

a

and

r

by testing factors of

21

:

21 = 3 \times 7

, so the terms could be

3, 6, 12

(

a=3, r=2

). Check squares:

3^2 + 6^2 + 12^2 = 9 + 36 + 144 = 189

, which confirms

r = 2

.

Estimated Time:1m 30s

Question 4423Question →

Two equal positive point charges, each of magnitude $2.0 \times 10^{-6}\text{ C}$ , are placed $0.2\text{ m}$ apart in a vacuum. What is the magnitude of the net electric field intensity at the midpoint between the two charges? (Take Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ )

0\text{ N C}^{-1}

1.8 \times 10^6\text{ N C}^{-1}

3.6 \times 10^6\text{ N C}^{-1}

7.2 \times 10^6\text{ N C}^{-1}

Show answer & explanation

Answer: $0\text{ N C}^{-1}$

Answer

The net electric field intensity at the midpoint is

0\text{ N C}^{-1}

.

At the midpoint between two identical positive charges, the electric field created by each charge has the exact same magnitude because the charges and distances are equal. Because electric field lines point away from positive charges, the two field vectors at the midpoint point in directly opposite directions. Taking vector superposition gives a net electric field of zero.

Step-by-Step Solution

1

Determine the distance from each charge to the midpoint.

The midpoint distance

r = \frac{0.2\text{ m}}{2} = 0.1\text{ m}

.

Electric field calculation requires the distance from the point charge to the point of evaluation.

2

Calculate the magnitude of the electric field due to one charge.

E = \frac{k Q}{r^2} = \frac{9.0 \times 10^9 \times 2.0 \times 10^{-6}}{(0.1)^2} = 1.8 \times 10^6\text{ N C}^{-1}

.

Electric field intensity magnitude is given by Coulomb's field formula

E = \frac{k Q}{r^2}

.

3

Apply vector addition to find the net electric field at the midpoint.

E_{\text{net}} = E_1 - E_2 = 1.8 \times 10^6 - 1.8 \times 10^6 = 0\text{ N C}^{-1}

.

Electric field is a vector quantity. Since both charges are positive, the field vectors point away from each charge and act in opposite directions at the midpoint.

Key Concept

Vector superposition of electric fields

Question 4424Question →

A solid sphere of mass $0.40\text{ kg}$ and relative density $2.5$ is held fully submerged in a liquid of density $800\text{ kg/m}^3$ by a light string attached to a fixed support. What is the tension in the string? (Take $g = 10\text{ m/s}^2$ )

2.72\text{ N}

3.60\text{ N}

1.28\text{ N}

2.40\text{ N}

Show answer & explanation

Answer: $2.72\text{ N}$

Answer

The tension in the string is

2.72\text{ N}

.

The sphere has a weight of

4.0\text{ N}

acting downward. When fully submerged, it displaces a volume of liquid equal to its own volume (

1.6 \times 10^{-4}\text{ m}^3

). The displaced liquid of density

800\text{ kg/m}^3

exerts an upward buoyant force (upthrust) of

1.28\text{ N}

. Tension balances the remaining downward force:

T = 4.0\text{ N} - 1.28\text{ N} = 2.72\text{ N}

.

Step-by-Step Solution

1

Calculate the weight of the sphere

W = mg = 0.40\text{ kg} \times 10\text{ m/s}^2 = 4.0\text{ N}

The force of gravity acting downward on the mass.

2

Determine the volume of the sphere using its relative density

Density of sphere

\rho_s = 2.5 \times 1000\text{ kg/m}^3 = 2500\text{ kg/m}^3

. Volume

V = \frac{m}{\rho_s} = \frac{0.40}{2500} = 1.6 \times 10^{-4}\text{ m}^3

.

Relative density is the ratio of the substance's density to the density of water (

1000\text{ kg/m}^3

).

3

Calculate the upthrust exerted by the liquid

U = V \cdot \rho_{\text{liquid}} \cdot g = (1.6 \times 10^{-4}) \times 800 \times 10 = 1.28\text{ N}

By Archimedes' principle, upthrust equals the weight of the fluid displaced by the submerged volume.

4

Calculate tension in the string

T = W - U = 4.0\text{ N} - 1.28\text{ N} = 2.72\text{ N}

For vertical equilibrium of the submerged body, weight acts downward while upthrust and tension act upward.

Key Concept

Archimedes' Principle and Apparent Weight

Estimated Time:1m 30s

Question 4425Question →

A merchant bought a power generator for $\text{₦}120,000$ and sold it to a retailer at a profit of $20\%$ . The retailer later sold the generator to a customer at a loss of $15\%$ . How much did the customer pay for the generator?

\text{₦}122,400

\text{₦}126,000

\text{₦}110,400

\text{₦}144,000

Show answer & explanation

Answer: $\text{₦}122,400$

Answer

The customer paid

\text{₦}122,400

for the generator.

The merchant sells the generator for

120\%

of

\text{₦}120,000

, which equals

\text{₦}144,000

. The retailer then sells it at

85\%

of

\text{₦}144,000

, yielding

\text{₦}122,400

.

Step-by-Step Solution

1

Calculate the selling price of the generator from the merchant to the retailer.

\text{Selling Price}_1 = \text{₦}120,000 \times \left(1 + \frac{20}{100}\right) = \text{₦}120,000 \times 1.20 = \text{₦}144,000

The merchant makes a

20\%

profit on the initial cost price of

\text{₦}120,000

.

2

Calculate the selling price from the retailer to the final customer.

\text{Selling Price}_2 = \text{₦}144,000 \times \left(1 - \frac{15}{100}\right) = \text{₦}144,000 \times 0.85 = \text{₦}122,400

The retailer incurs a

15\%

loss calculated relative to their purchase price of

\text{₦}144,000

.

Key Concept

Successive Percentage Profit and Loss

Question 4426Question →

A transverse wave propagating through a stretched string has a frequency of $250\text{ Hz}$ and a wavelength of $1.4\text{ m}$ . What is the speed of propagation of the wave in $\text{m/s}$ ?

Show answer & explanation

Answer: 350

Answer

The speed of propagation of the wave is

350\text{ m/s}

.

The speed of propagation of a progressive wave is determined using the wave equation

v = f \lambda

. Substituting the given values

f = 250\text{ Hz}

and

\lambda = 1.4\text{ m}

gives

v = 250 \times 1.4 = 350\text{ m/s}

.

Step-by-Step Solution

1

Identify the given physical quantities

Frequency

f = 250\text{ Hz}

, Wavelength

\lambda = 1.4\text{ m}

.

These parameters are required to calculate the speed of propagation.

2

Apply the fundamental wave equation

v = f \lambda

The speed of a progressive wave equals the product of its frequency and its wavelength.

3

Perform the numerical calculation

v = 250 \times 1.4 = 350\text{ m/s}

Multiplying

250\text{ Hz}

by

1.4\text{ m}

yields the wave velocity in meters per second.

Key Concept

Wave Propagation Speed Equation (

v = f \lambda

)

Question 4427Question →

A box of mass $2\text{ kg}$ slides down a rough inclined plane from a height of $5\text{ m}$ . If it reaches the bottom of the incline with a speed of $6\text{ m s}^{-1}$ , what is the work done against friction during the descent? (Take $g = 10\text{ m s}^{-2}$ )

64\text{ J}

36\text{ J}

100\text{ J}

136\text{ J}

Show answer & explanation

Answer: $64\text{ J}$

Answer

64\text{ J}

According to the principle of conservation of energy, the work done against friction equals the loss in total mechanical energy. The initial potential energy is

mgh = 2 \times 10 \times 5 = 100\text{ J}

, and the final kinetic energy is

\frac{1}{2}mv^2 = \frac{1}{2} \times 2 \times 6^2 = 36\text{ J}

. Subtracting final kinetic energy from initial potential energy yields

100\text{ J} - 36\text{ J} = 64\text{ J}

.

Step-by-Step Solution

1

Calculate the initial potential energy (

E_p

) at height

h = 5\text{ m}

.

E_p = mgh = 2\text{ kg} \times 10\text{ m s}^{-2} \times 5\text{ m} = 100\text{ J}

At the top of the incline, all mechanical energy is stored as gravitational potential energy.

2

Calculate the final kinetic energy (

E_k

) at the bottom where

v = 6\text{ m s}^{-1}

.

E_k = \frac{1}{2}mv^2 = \frac{1}{2} \times 2\text{ kg} \times (6\text{ m s}^{-1})^2 = 36\text{ J}

At the bottom of the incline, the remaining mechanical energy is kinetic energy.

3

Apply the work-energy theorem to find the work done against friction (

W_f

).

W_f = E_p - E_k = 100\text{ J} - 36\text{ J} = 64\text{ J}

The non-conservative friction force reduces the total mechanical energy by doing work against motion.

Key Concept

Work-Energy Theorem and Conservation of Energy with Friction

Question 4428Question →

A uniform metre rule of mass $120\text{ g}$ is balanced horizontally on a pivot placed at the $40\text{ cm}$ mark when an unknown mass $m$ is suspended at the $10\text{ cm}$ mark. What is the value of the mass $m$ in grams?

Show answer & explanation

Answer: 40

Answer

The mass

m

required to balance the metre rule horizontally is

40\text{ g}

.

According to the Principle of Moments, a system is in rotational equilibrium when the sum of anticlockwise moments equals the sum of clockwise moments about the pivot. The weight of the

120\text{ g}

uniform metre rule acts at its center of gravity (

50\text{ cm}

mark), which is

10\text{ cm}

to the right of the pivot at

40\text{ cm}

. This creates a clockwise moment of

120\text{ g} \times 10\text{ cm} = 1200\text{ g}\cdot\text{cm}

. The mass

m

is suspended at the

10\text{ cm}

mark, which is

30\text{ cm}

to the left of the pivot, creating an anticlockwise moment of

m \times 30\text{ cm}

. Setting

30m = 1200

yields

m = 40\text{ g}

.

Step-by-Step Solution

1

Determine the position of the center of gravity of the metre rule.

Center of gravity is at the

50\text{ cm}

mark.

A uniform metre rule has its weight concentrated at its geometric midpoint.

2

Calculate perpendicular distances from the pivot at

40\text{ cm}

to all acting forces.

Distance to rule weight =

50\text{ cm} - 40\text{ cm} = 10\text{ cm}

; Distance to mass

m

=

40\text{ cm} - 10\text{ cm} = 30\text{ cm}

.

Moments are calculated by multiplying force (or mass) by perpendicular distance from the turning point.

3

Apply the Principle of Moments for equilibrium.

Sum of anticlockwise moments = Sum of clockwise moments

\implies m \times 30 = 120 \times 10

.

For rotational equilibrium, the total clockwise moment must equal the total anticlockwise moment about the pivot.

4

Solve for the unknown mass

m

.

m = 40\text{ g}

.

Dividing

1200\text{ g}\cdot\text{cm}

by

30\text{ cm}

gives

40\text{ g}

.

Key Concept

Principle of Moments and Center of Gravity of a Uniform Body

Estimated Time:50s

Question 4429Question →

The ratio of the measure of an interior angle to an exterior angle of a regular polygon is $5 : 1$ . How many sides does the polygon have?

10

12

6

14

Show answer & explanation

Answer: 12

Answer

The polygon has 12 sides.

The interior angle and exterior angle of a regular polygon are supplementary, adding up to

180^\circ

. For a ratio of

5:1

, the exterior angle is

\frac{1}{5+1} \times 180^\circ = 30^\circ

. Using the exterior angle formula

n = \frac{360^\circ}{\text{exterior angle}}

, the number of sides is

\frac{360^\circ}{30^\circ} = 12

.

Step-by-Step Solution

1

Set up an equation using the linear pair relationship between interior and exterior angles.

5x + 1x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ

At any vertex of a regular polygon, the interior angle and exterior angle lie on a straight line and are supplementary (

180^\circ

).

2

Identify the measure of the exterior angle.

Exterior angle =

30^\circ

The exterior angle corresponds to

1

part of the

5:1

ratio, which equals

x = 30^\circ

.

3

Calculate the number of sides

n

of the polygon.

n = \frac{360^\circ}{30^\circ} = 12

The sum of the exterior angles of any convex polygon is

360^\circ

, so

n = \frac{360^\circ}{\text{exterior angle}}

.

Key Concept

Interior and exterior angles of regular polygons

Question 4430Question →

An infrared sensor detects electromagnetic radiation with a frequency of $4.0 \times 10^{14}\text{ Hz}$ . A second sensor detects ultraviolet radiation with a wavelength of $300\text{ nm}$ . Given that the speed of light in a vacuum is $c = 3.0 \times 10^8\text{ m/s}$ , what is the ratio of the wavelength of the infrared radiation to the wavelength of the ultraviolet radiation?

2.5

0.4

25.0

1.33

Show answer & explanation

Answer: $2.5$

Answer

The ratio of the wavelength of the infrared radiation to the wavelength of the ultraviolet radiation is 2.5.

Using

c = f \lambda

, the infrared wavelength is

\lambda = \frac{3.0 \times 10^8}{4.0 \times 10^{14}} = 7.5 \times 10^{-7}\text{ m}

. Comparing this to the ultraviolet wavelength of

300\text{ nm} = 3.0 \times 10^{-7}\text{ m}

gives a ratio of

\frac{7.5 \times 10^{-7}}{3.0 \times 10^{-7}} = 2.5

.

Step-by-Step Solution

1

Calculate the wavelength of the infrared radiation using the wave equation

c = f \lambda

.

\lambda_{\text{IR}} = \frac{3.0 \times 10^8\text{ m/s}}{4.0 \times 10^{14}\text{ Hz}} = 7.5 \times 10^{-7}\text{ m}

The speed of all electromagnetic waves in a vacuum is constant (

c = 3.0 \times 10^8\text{ m/s}

).

2

Convert the ultraviolet wavelength to meters for consistent units.

\lambda_{\text{UV}} = 300\text{ nm} = 300 \times 10^{-9}\text{ m} = 3.0 \times 10^{-7}\text{ m}

Units must be in standard meters before calculating the dimensionless ratio.

3

Compute the ratio of the infrared wavelength to the ultraviolet wavelength.

\text{Ratio} = \frac{7.5 \times 10^{-7}\text{ m}}{3.0 \times 10^{-7}\text{ m}} = 2.5

Dividing the two wavelengths in the same units yields the desired ratio.

Key Concept

Wave Equation and Electromagnetic Spectrum Properties

Estimated Time:2m 0s

Question 4431Question →

Match each state of matter with the kinetic theory postulate that correctly describes its microscopic particle behavior and arrangement.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Solid state matches with particles held tightly in fixed positions vibrating about mean positions; Liquid state matches with particles close together sliding past one another; Gaseous state matches with particles moving rapidly, randomly, and independently.

According to the kinetic theory of matter, solid particles vibrate in fixed positions due to strong intermolecular forces; liquid particles slide past one another because they have enough kinetic energy to break rigid bonds while remaining in proximity; gas particles move rapidly and randomly in all directions because their kinetic energy far exceeds any attractive forces.

Step-by-Step Solution

1

Analyze the particle behavior for the solid state according to the kinetic theory.

Particles in solids have minimal kinetic energy and strong attractive forces, fixing them in position and permitting only vibrational motion.

This maintains a fixed shape and fixed volume.

2

Analyze the particle behavior for the liquid state.

Liquid particles have sufficient energy to overcome rigid spatial constraints and slide over each other while maintaining contact.

This allows liquids to take the shape of their container while keeping a constant volume.

3

Analyze the particle behavior for the gaseous state.

Gas particles possess high kinetic energy that completely overcomes intermolecular attractions, leading to continuous, rapid, random motion.

This explains why gases fill the entire volume of any container.

Key Concept

Postulates of Kinetic Theory and States of Matter

Question 4432Question →

A student performed a paper chromatography experiment to analyze the purity of an organic dye sample. On the chromatogram, the solvent front traveled $8.0\text{ cm}$ from the baseline, and a single distinct spot traveled $5.0\text{ cm}$ . Which of the following gives the correct $R_f$ value and the accurate deduction about the purity of the dye sample?

R_f = 0.625

, indicating that the sample is a pure substance.

R_f = 1.60

, indicating that the sample is a pure substance.

R_f = 0.625

, indicating that the sample is a mixture of several components.

R_f = 1.60

, indicating that the sample is an impure mixture.

Show answer & explanation

Answer: $R_f = 0.625$ , indicating that the sample is a pure substance.

Answer

The retention factor is

R_f = 0.625

, which indicates that the dye sample is a pure substance because only a single spot appeared on the chromatogram.

The retention factor (

R_f

) is determined by dividing the distance moved by the solute spot (

5.0\text{ cm}

) by the distance moved by the solvent front (

8.0\text{ cm}

), giving

R_f = 0.625

. Furthermore, a single spot on the paper chromatogram serves as a standard criterion of chemical purity, confirming that the sample is a pure substance.

Step-by-Step Solution

1

Calculate the retention factor (

R_f

) using the formula

R_f = \frac{\text{Distance moved by solute}}{\text{Distance moved by solvent front}}

.

R_f = \frac{5.0\text{ cm}}{8.0\text{ cm}} = 0.625

The

R_f

value represents the relative distance traveled by the compound compared to the solvent front.

2

Interpret the number of spots observed on the developed chromatogram.

A single spot confirms the presence of only one chemical component.

Pure substances produce a single spot under specified solvent conditions, whereas mixtures resolve into two or more distinct spots.

Key Concept

Criteria of purity in chromatography: pure substances yield a single spot with a characteristic

R_f

value between 0 and 1.

Estimated Time:1m 0s

Question 4433Question →

A coil consisting of $150$ turns is placed in a magnetic field. The magnetic flux passing through the coil decreases uniformly from $4.0 \times 10^{-3}\text{ Wb}$ to $1.0 \times 10^{-3}\text{ Wb}$ over a time interval of $0.015\text{ s}$ . What is the magnitude of the average induced electromotive force in the coil in volts?

Show answer & explanation

Answer: 30

Answer

The magnitude of the average induced electromotive force in the coil is

30\text{ V}

.

According to Faraday's law of electromagnetic induction, the magnitude of the average induced electromotive force in a coil with

N

turns is given by

E = N \frac{\Delta \Phi}{\Delta t}

. Given

N = 150

,

\Delta \Phi = 3.0 \times 10^{-3}\text{ Wb}

, and

\Delta t = 0.015\text{ s}

, the magnitude of the induced e.m.f. is

E = 150 \times \frac{3.0 \times 10^{-3}}{0.015} = 30\text{ V}

.

Step-by-Step Solution

1

Calculate the magnitude of the change in magnetic flux through the coil

\Delta \Phi = 4.0 \times 10^{-3}\text{ Wb} - 1.0 \times 10^{-3}\text{ Wb} = 3.0 \times 10^{-3}\text{ Wb}

Faraday's law relates induced e.m.f. directly to the rate of change of magnetic flux.

2

Apply Faraday's Law of Electromagnetic Induction equation for an N-turn coil

E = N \frac{\Delta \Phi}{\Delta t}

The total induced electromotive force in a coil is proportional to the number of turns and the rate of change of flux.

3

Substitute the given numerical values to compute the magnitude of the induced e.m.f.

E = 150 \times \frac{3.0 \times 10^{-3}\text{ Wb}}{0.015\text{ s}} = 150 \times 0.2\text{ V} = 30\text{ V}

Performing clean calculation without needing a calculator.

Key Concept

Faraday's Law of Electromagnetic Induction

Question 4434Question →

Read the sentence below carefully and identify the word that ensures proper grammatical alignment. What pronoun correctly completes the given sentence?

Fill in the blanks below

Every one of the female research fellows presented findings at the international bioethics symposium.

Show answer & explanation

Answer

her

The correct answer is 'her'. The grammatical subject of the sentence is the singular indefinite expression 'Every one'. The prepositional phrase 'of the female research fellows' specifies gender while leaving the subject singular. Therefore, the singular feminine possessive pronoun 'her' maintains correct agreement.

Step-by-Step Solution

1

Identify the grammatical antecedent of the pronoun in the sentence.

The simple subject and true antecedent is the singular indefinite expression 'Every one', modified by the prepositional phrase 'of the female research fellows'.

Indefinite pronoun constructions beginning with 'every one' are grammatically singular, regardless of any plural nouns within the modifying prepositional phrase.

2

Determine the appropriate number and gender for the possessive pronoun.

Since the modifying phrase explicitly identifies a female group ('female research fellows'), the singular feminine possessive pronoun 'her' is required.

A pronoun must agree with its antecedent in both number (singular) and gender (feminine).

Key Concept

Pronoun-Antecedent Agreement with Singular Indefinite Expressions

Question 4435Question →

An object is placed $12\text{ cm}$ in front of a concave mirror having a focal length of $20\text{ cm}$ . What is the distance of the image from the mirror and its nature?

30\text{ cm}

behind the mirror, virtual

30\text{ cm}

in front of the mirror, real

7.5\text{ cm}

in front of the mirror, real

7.5\text{ cm}

behind the mirror, virtual

Show answer & explanation

Answer: $30\text{ cm}$ behind the mirror, virtual

Answer

The image is formed

30\text{ cm}

behind the mirror and is virtual.

Applying the mirror formula

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

with

f = 20\text{ cm}

and

u = 12\text{ cm}

yields

\frac{1}{v} = \frac{1}{20} - \frac{1}{12} = -\frac{1}{30}\text{ cm}^{-1}

, resulting in

v = -30\text{ cm}

. The negative sign confirms that the image is virtual and located

30\text{ cm}

behind the mirror.

Step-by-Step Solution

1

Identify given quantities and apply standard mirror sign conventions.

Object distance

u = +12\text{ cm}

, focal length

f = +20\text{ cm}

.

For a concave mirror, real objects and real focal points carry positive values.

2

Set up the mirror equation

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

.

\frac{1}{20} = \frac{1}{12} + \frac{1}{v}

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

3

Rearrange the equation to isolate

\frac{1}{v}

and solve.

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

Finding a common denominator of 60 allows exact calculation of the negative reciprocal.

4

Determine image characteristics based on the sign of

v

.

The image distance magnitude is

30\text{ cm}

behind the mirror, and the image is virtual.

A negative value for image distance

v

signifies a virtual image located behind the mirror surface.

Key Concept

Concave Mirror Formula and Virtual Image Formation

Question 4436Question →

An atom in a gas discharge tube has a ground state energy level of $-10.4\text{ eV}$ and an excited energy level of $-3.8\text{ eV}$ . An electron absorbs a single photon to undergo a transition directly from the ground state to this excited level. If the frequency of the absorbed photon is expressed as $x \times 10^{15}\text{ Hz}$ , what is the numerical value of $x$ ? (Take Planck's constant $h = 6.6 \times 10^{-34}\text{ J}\cdot\text{s}$ and $1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$ )

Show answer & explanation

Answer: 1.6

Answer

The numerical value of

x

is 1.6.

The energy of the absorbed photon is equal to the difference between the excited state and ground state energies:

\Delta E = -3.8\text{ eV} - (-10.4\text{ eV}) = 6.6\text{ eV}

. Converting this to Joules yields

6.6 \times 1.6 \times 10^{-19}\text{ J} = 1.056 \times 10^{-18}\text{ J}

. Using Einstein's photon relation

E = hf

, the frequency is

f = \frac{1.056 \times 10^{-18}}{6.6 \times 10^{-34}} = 1.6 \times 10^{15}\text{ Hz}

, which gives

x = 1.6

.

Step-by-Step Solution

1

Determine the energy absorbed during transition

\Delta E = 6.6\text{ eV}

The energy of the absorbed photon equals the energy difference between the initial and final states:

\Delta E = -3.8\text{ eV} - (-10.4\text{ eV}) = 6.6\text{ eV}

.

2

Convert energy from eV to Joules

\Delta E = 1.056 \times 10^{-18}\text{ J}

Multiply by

1.6 \times 10^{-19}\text{ J/eV}

to convert energy to standard SI units.

3

Calculate photon frequency

f = 1.6 \times 10^{15}\text{ Hz}

Use Planck's energy equation

f = \frac{\Delta E}{h} = \frac{1.056 \times 10^{-18}\text{ J}}{6.6 \times 10^{-34}\text{ J}\cdot\text{s}} = 1.6 \times 10^{15}\text{ Hz}

.

Key Concept

Photon energy and atomic transition frequency

Estimated Time:2m 0s

Question 4437Question →

An electric immersion heater rated at $200\text{ W}$ is used to heat $0.8\text{ kg}$ of a liquid contained in a vessel of heat capacity $120\text{ J K}^{-1}$ . The initial temperature of the liquid and vessel is $20^\circ\text{C}$ . If the heater is operated for $3.5\text{ minutes}$ and the final temperature reaches $50^\circ\text{C}$ , calculate the specific heat capacity of the liquid in $\text{J kg}^{-1}\text{ K}^{-1}$ , assuming no thermal energy is lost to the surroundings.

Show answer & explanation

Answer: 1600

Answer

The specific heat capacity of the liquid is

1600\text{ J kg}^{-1}\text{ K}^{-1}

.

Total energy supplied by the heater is

Q = 200\text{ W} \times 210\text{ s} = 42,000\text{ J}

. The temperature increase is

\Delta T = 30\text{ K}

. The energy absorbed by the container is

Q_{\text{vessel}} = C \Delta T = 120 \times 30 = 3,600\text{ J}

. The remaining energy

38,400\text{ J}

is absorbed by the liquid. Dividing this value by the product of the mass of liquid (

0.8\text{ kg}

) and temperature rise (

30\text{ K}

) gives the specific heat capacity

1600\text{ J kg}^{-1}\text{ K}^{-1}

.

Step-by-Step Solution

1

Convert the heating time to seconds and compute total heat energy supplied by the heater.

Q = P \times t = 200\text{ W} \times (3.5 \times 60\text{ s}) = 42,000\text{ J}

Heat energy supplied by an electric source equals electrical power multiplied by duration in seconds.

2

Determine the change in temperature of the system.

\Delta T = 50^\circ\text{C} - 20^\circ\text{C} = 30\text{ K}

Both the vessel and liquid experience the same initial and final temperatures.

3

Calculate the heat energy absorbed by the vessel.

Q_{\text{vessel}} = C \times \Delta T = 120\text{ J K}^{-1} \times 30\text{ K} = 3,600\text{ J}

Heat capacity

C

represents heat required per unit temperature rise for the container as a whole.

4

Subtract vessel absorption from total heat supplied to find heat absorbed by the liquid.

Q_{\text{liquid}} = 42,000\text{ J} - 3,600\text{ J} = 38,400\text{ J}

Conservation of energy dictates

Q_{\text{total}} = Q_{\text{vessel}} + Q_{\text{liquid}}

.

5

Calculate the specific heat capacity

c

of the liquid.

c = \frac{Q_{\text{liquid}}}{m \Delta T} = \frac{38,400}{0.8 \times 30} = 1600\text{ J kg}^{-1}\text{ K}^{-1}

Specific heat capacity isolates heat absorbed per unit mass per Kelvin.

Key Concept

Principle of conservation of thermal energy combining heat capacity of a vessel (

C

) and specific heat capacity of a liquid (

c

).

Estimated Time:1m 30s

Question 4438Question →

Which of the following sentences correctly avoids a dangling modifier?

After analyzing the network diagnostic reports, the core routers were upgraded by the chief technician.

After analyzing the network diagnostic reports, the chief technician upgraded the core routers.

After analyzing the network diagnostic reports, the upgrade was carried out complimentary to the main server installation.

After analyzing the network diagnostic reports, the technician executed the repairs exceptional well.

Show answer & explanation

Answer: After analyzing the network diagnostic reports, the chief technician upgraded the core routers.

Answer

The sentence 'After analyzing the network diagnostic reports, the chief technician upgraded the core routers.' correctly avoids a dangling modifier.

The introductory participial phrase 'After analyzing the network diagnostic reports' describes an action performed by an individual. Placing 'the chief technician' directly after the comma establishes a clear and grammatically correct connection between the modifier and its logical subject.

Step-by-Step Solution

1

Identify the introductory modifier in the sentence

The phrase 'After analyzing the network diagnostic reports' is a participial modifier requiring an agent capable of performing the action of analyzing.

Participial modifiers must modify the noun phrase that directly follows them.

2

Evaluate the main clause subjects in each option

In the correct sentence, 'the chief technician' is the immediate subject following the comma, which logically pairs with 'analyzing'.

An inanimate object like 'the core routers' or an abstract concept like 'the upgrade' cannot perform the action of analyzing reports.

3

Verify modifier agreement and vocabulary correctness

The proper adverbial modification and accurate word choices are maintained without grammatical errors or homophone confusions.

Ensures the chosen option satisfies all syntactic and lexical constraints.

Key Concept

Dangling Modifiers and Modifier Placement

Estimated Time:1m 0s

Question 4439Question →

Two candidates, Ada and Babatunde, independently sit for an aptitude test. The probability that Ada passes the test is $\frac{1}{3}$ and the probability that Babatunde passes is $\frac{2}{5}$ . What is the probability that at least one of them passes the test?

\frac{2}{15}

\frac{2}{5}

\frac{3}{5}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The probability that at least one candidate passes the test is

\frac{3}{5}

.

For independent events, the probability of both occurring is given by

P(A \cap B) = P(A) \cdot P(B) = \frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15}

. Applying the addition law

P(A \cup B) = P(A) + P(B) - P(A \cap B)

yields

\frac{1}{3} + \frac{2}{5} - \frac{2}{15} = \frac{9}{15} = \frac{3}{5}

. Alternatively, calculating

1 - P(\text{neither passes}) = 1 - (1 - \frac{1}{3})(1 - \frac{2}{5}) = 1 - (\frac{2}{3} \cdot \frac{3}{5}) = 1 - \frac{2}{5} = \frac{3}{5}

gives the identical correct outcome.

Step-by-Step Solution

1

Calculate the probability that both Ada and Babatunde pass the test.

P(\text{Ada} \cap \text{Babatunde}) = \frac{1}{3} \times \frac{2}{5} = \frac{2}{15}

Since the two events are independent, the multiplication law of probability applies:

P(A \cap B) = P(A) \times P(B)

.

2

Apply the general addition law of probability to find the union of the two events.

P(\text{Ada} \cup \text{Babatunde}) = P(\text{Ada}) + P(\text{Babatunde}) - P(\text{Ada} \cap \text{Babatunde})

The probability of at least one event occurring corresponds to the union of the compound events.

3

Substitute the probabilities and solve for the outcome.

P(\text{Ada} \cup \text{Babatunde}) = \frac{1}{3} + \frac{2}{5} - \frac{2}{15} = \frac{5 + 6 - 2}{15} = \frac{9}{15} = \frac{3}{5}

Find a common denominator (15) and simplify the resulting fraction to its lowest terms.

Key Concept

Probability Laws for Compound and Independent Events

Estimated Time:1m 30s

Question 4440Question →

A rigid rod $AB$ of length $2.0\text{ m}$ is hinged at end $A$ . A force of $50\text{ N}$ is applied at end $B$ at an angle of $30^\circ$ to the rod. What is the moment of this force about the hinge $A$ ?

25\text{ N m}

50\text{ N m}

86.6\text{ N m}

100\text{ N m}

Show answer & explanation

Answer: $50\text{ N m}$

Answer

The moment of the force about hinge

A

is

50\text{ N m}

.

The moment of a force about a pivot is calculated by multiplying the force magnitude by the perpendicular distance to the pivot line of action (

F \times L \sin\theta

). Substituting

F = 50\text{ N}

,

L = 2.0\text{ m}

, and

\sin(30^\circ) = 0.5

gives

50 \times 2.0 \times 0.5 = 50\text{ N m}

.

Step-by-Step Solution

1

Identify the formula for the moment of a force applied at an angle.

\text{Moment} = F \times d_\perp = F \times L \sin\theta

Moment is defined as the product of the magnitude of the force and the perpendicular distance from the line of action of the force to the pivot point.

2

Substitute the given values into the formula.

\text{Moment} = 50\text{ N} \times 2.0\text{ m} \times \sin(30^\circ)

The force

F = 50\text{ N}

, distance

L = 2.0\text{ m}

, and angle

\theta = 30^\circ

.

3

Calculate the final moment value.

\text{Moment} = 50 \times 2.0 \times 0.5 = 50\text{ N m}

Since

\sin(30^\circ) = 0.5

, the perpendicular distance is

1.0\text{ m}

, giving a moment of

50\text{ N m}

.

Key Concept

Moment of a Force at an Angle

Estimated Time:45s

Answer

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