All practice questions

Question 4041Question →

An AC series circuit consists of a resistor of resistance $R = 6\ \Omega$ connected in series with an inductor of inductive reactance $X_L = 8\ \Omega$ . What is the total impedance of the circuit?

10\ \Omega

14\ \Omega

2\ \Omega

100\ \Omega

Show answer & explanation

Answer: $10\ \Omega$

Answer

The total impedance of the circuit is

10\ \Omega

.

The net impedance

Z

in an AC series circuit containing a resistor and an inductor is given by

Z = \sqrt{R^2 + X_L^2}

. Substituting

R = 6\ \Omega

and

X_L = 8\ \Omega

yields

Z = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\ \Omega

.

Step-by-Step Solution

1

Identify the formula for impedance in an RL series circuit

Z = \sqrt{R^2 + X_L^2}

In an AC circuit, resistance and inductive reactance add vectorially at a

90^\circ

phase angle.

2

Substitute the given values

R = 6\ \Omega

and

X_L = 8\ \Omega

into the formula

Z = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100}

Squaring each component gives the terms required under the radical.

3

Calculate the square root

Z = 10\ \Omega

The square root of 100 gives the net opposition to current flow (impedance).

Key Concept

Impedance of an RL Series AC Circuit

Estimated Time:45s

Question 4042Question →

What quantity of heat energy is required to completely melt $0.50\text{ kg}$ of ice at $0^\circ\text{C}$ into water at $0^\circ\text{C}$ ? (Specific latent heat of fusion of ice = $3.3 \times 10^5\text{ J kg}^{-1}$ )

1.65 \times 10^5\text{ J}

3.30 \times 10^5\text{ J}

6.60 \times 10^5\text{ J}

1.65 \times 10^7\text{ J}

Show answer & explanation

Answer: $1.65 \times 10^5\text{ J}$

Answer

The quantity of heat energy required is

1.65 \times 10^5\text{ J}

.

The quantity of heat required for melting at constant temperature depends only on mass and specific latent heat of fusion:

Q = mL_f = 0.50 \times 3.3 \times 10^5 = 1.65 \times 10^5\text{ J}

.

Step-by-Step Solution

1

Identify the given values and formula for change of state

Mass

m = 0.50\text{ kg}

, Specific latent heat of fusion

L_f = 3.3 \times 10^5\text{ J kg}^{-1}

. The formula is

Q = m L_f

.

During a phase change at constant temperature, latent heat is absorbed without a change in temperature.

2

Calculate total heat energy

Q

Q = 0.50\text{ kg} \times 3.3 \times 10^5\text{ J kg}^{-1} = 1.65 \times 10^5\text{ J}

.

Multiplying mass by the specific latent heat gives the total energy transferred.

Key Concept

Latent Heat of Fusion

Estimated Time:45s

Question 4043Question →

Find the sum, in degrees, of all solutions to the trigonometric equation $\sqrt{3}\tan(2x) = 3$ in the interval $0^\circ \le x \le 180^\circ$ .

Show answer & explanation

Answer: 150

Answer

The sum of all solutions to the equation in the given interval is 150 degrees.

Isolating

\tan(2x)

gives

\sqrt{3}

. For

0^\circ \le 2x \le 360^\circ

,

\tan(2x) = \sqrt{3}

yields solutions at

2x = 60^\circ

and

2x = 240^\circ

. Dividing by 2 gives

x = 30^\circ

and

x = 120^\circ

. Adding these solutions yields

30^\circ + 120^\circ = 150^\circ

.

Step-by-Step Solution

1

Isolate the trigonometric function

\tan(2x) = \frac{3}{\sqrt{3}} = \sqrt{3}

Dividing both sides by \sqrt{3} simplifies the expression to a standard special angle ratio.

2

Determine the domain for the argument

2x

Since

0^\circ \le x \le 180^\circ

, multiplying the inequality by 2 gives

0^\circ \le 2x \le 360^\circ

.

This establishes the range of angles to search for

2x

within one complete turn.

3

Find all values of

2x

where tangent equals

\sqrt{3}

2x = 60^\circ

(1st quadrant) and

2x = 180^\circ + 60^\circ = 240^\circ

(3rd quadrant)

The tangent function is positive in Quadrants I and III with a reference angle of

60^\circ

.

4

Solve for

x

x = \frac{60^\circ}{2} = 30^\circ

and

x = \frac{240^\circ}{2} = 120^\circ

Dividing each angle by 2 yields the values of

x

lying within the domain

0^\circ \le x \le 180^\circ

.

5

Calculate the sum of the solutions

30^\circ + 120^\circ = 150^\circ

The question specifically requests the sum of all valid solutions.

Key Concept

Solving trigonometric equations using reference angles and domain transformation

Estimated Time:2m 0s

Question 4044Question →

In $\Delta ABC$ , side $a = 6\text{ cm}$ , side $b = 10\text{ cm}$ , and the included angle $\angle C = 120^\circ$ . What is the length of side $c$ ?

14\text{ cm}

2\sqrt{19}\text{ cm}

2\sqrt{34}\text{ cm}

16\text{ cm}

Show answer & explanation

Answer: $14\text{ cm}$

Answer

The length of side

c

is

14\text{ cm}

.

According to the Cosine Rule

c^2 = a^2 + b^2 - 2ab \cos C

, substituting the given values

a = 6

,

b = 10

, and

\cos 120^\circ = -\frac{1}{2}

yields

c^2 = 36 + 100 - 2(6)(10)\left(-\frac{1}{2}\right) = 136 + 60 = 196

. Taking the positive square root gives

c = 14\text{ cm}

.

Step-by-Step Solution

1

Identify the given values and state the relevant Cosine Rule formula

Given:

a = 6\text{ cm}

,

b = 10\text{ cm}

,

\angle C = 120^\circ

. Formula:

c^2 = a^2 + b^2 - 2ab \cos C

.

Since two sides and the included angle (SAS configuration) are known, the Cosine Rule must be used to find the third side.

2

Evaluate

\cos 120^\circ

and substitute all values into the formula

\cos 120^\circ = -\frac{1}{2}

. Thus,

c^2 = 6^2 + 10^2 - 2(6)(10)\left(-\frac{1}{2}\right)

.

Cosine of an obtuse angle in the second quadrant is negative.

3

Simplify the algebraic expression

c^2 = 36 + 100 + 60 = 196

.

Multiplying

-2(60)\left(-\frac{1}{2}\right)

yields

+60

.

4

Take the principal square root to solve for

c

c = \sqrt{196} = 14\text{ cm}

.

Length must be a positive real number.

Key Concept

Cosine Rule for finding an unknown side in SAS triangle configurations

Question 4045Question →

In how many different ways can the letters of the word SUCCESS be arranged such that the three 'S's do not all come together?

360

420

60

35

Show answer & explanation

Answer: 360

Answer

The letters of the word SUCCESS can be arranged in 360 ways such that the three 'S's do not all come together.

The correct answer is calculated using complementary counting. First, the total unrestricted permutations of SUCCESS (7 letters with 3 'S's and 2 'C's) is

\frac{7!}{3!2!} = 420

. Next, treating the three 'S's as one single block leaves 5 items to arrange with 2 'C's, giving

\frac{5!}{2!} = 60

ways where the 'S's are together. Subtracting 60 from 420 yields 360.

Step-by-Step Solution

1

Calculate the total number of unrestricted arrangements of the word SUCCESS.

The word SUCCESS has 7 letters in total: 3 'S's, 2 'C's, 1 'U', and 1 'E'. Total arrangements

N_{total} = \frac{7!}{3! \times 2!} = \frac{5040}{6 \times 2} = 420

.

Repeated letters must be accounted for by dividing the factorial of the total count by the factorials of the counts of repeated letters.

2

Calculate the number of arrangements where the three 'S's are all together.

Treat the three 'S's as a single entity (SSS). We now arrange 5 entities: (SSS), U, C, C, E. Since 'C' appears twice,

N_{together} = \frac{5!}{2!} = \frac{120}{2} = 60

.

Grouping restricted identical items into a single block allows us to find the subset of arrangements where they stay together.

3

Subtract the number of 'together' arrangements from the total arrangements.

N_{not\_together} = N_{total} - N_{together} = 420 - 60 = 360

.

The complementary counting principle gives the number of ways where the restriction is satisfied.

Key Concept

Permutations with Repeated Elements and Complementary Counting

Estimated Time:1m 30s

Question 4046Question →

A metal rod of length $12.5\text{ cm}$ is divided into two parts in the ratio $2 : 3$ . If the length of the smaller part is measured as $5.2\text{ cm}$ , what is the percentage error in the measurement of the smaller part?

4.0\%

3.8\%

1.6\%

2.7\%

Show answer & explanation

Answer: $4.0\%$

Answer

The percentage error in the measurement of the smaller part is

4.0\%

.

The true length of the smaller part of the rod is

\frac{2}{5} \times 12.5 = 5.0\text{ cm}

. Subtracting this from the measured length

5.2\text{ cm}

gives an error of

0.2\text{ cm}

. Dividing the error

0.2\text{ cm}

by the true value

5.0\text{ cm}

and multiplying by

100\%

yields

4.0\%

.

Step-by-Step Solution

1

Calculate the actual length of the smaller part using the given ratio.

Total ratio parts =

2 + 3 = 5

. Actual length of smaller part =

\frac{2}{5} \times 12.5\text{ cm} = 5.0\text{ cm}

.

The total length is shared according to the ratio

2 : 3

, so the smaller piece corresponds to 2 out of 5 equal parts.

2

Find the absolute error in measurement.

Absolute error =

|5.2\text{ cm} - 5.0\text{ cm}| = 0.2\text{ cm}

.

Absolute error is the magnitude of the difference between the measured value and the true value.

3

Compute the percentage error.

Percentage error =

\frac{\text{Absolute Error}}{\text{True Value}} \times 100\% = \frac{0.2}{5.0} \times 100\% = 4.0\%

.

Percentage error is defined as the absolute error divided by the true value, multiplied by 100%.

Key Concept

Ratio partitioning and Percentage Error calculation

Estimated Time:1m 30s

Question 4047Question →

A trader invested $\text{₦}50,000$ in a savings scheme that pays compound interest at a rate of $10\%$ per annum compounded annually. What is the total compound interest earned at the end of $2$ years?

\text{₦}10,500

\text{₦}60,500

\text{₦}10,000

\text{₦}5,000

Show answer & explanation

Answer: $\text{₦}10,500$

Answer

The total compound interest earned at the end of

2

years is

\text{₦}10,500

.

The compound interest is obtained by subtracting the principal from the total accumulated amount. Using

A = P(1 + r)^n

, the total amount is

\text{₦}60,500

. Subtracting the principal of

\text{₦}50,000

yields

\text{₦}10,500

.

Step-by-Step Solution

1

Calculate the total accumulated amount using the compound interest formula

A = P(1 + r)^n

A = 50,000 \times (1 + 0.10)^2 = 50,000 \times 1.21 = \text{₦}60,500

Determines the total value of the investment at the end of the duration.

2

Calculate the compound interest earned using

CI = A - P

CI = 60,500 - 50,000 = \text{₦}10,500

Subtracts the original principal from the total accumulated amount to find the interest portion.

Key Concept

Compound interest calculation

Question 4048Question →

Given the universal set $U = \{x \in \mathbb{Z} : 1 \le x \le 20\}$ . Let $P = \{x \in U : x \text{ is a prime number}\}$ , $Q = \{x \in U : x \text{ is an odd integer}\}$ , and $R = \{x \in U : x \text{ is a multiple of } 3\}$ . What is the value of $n((P \cup Q) \cap R')$ ?

7

8

11

14

Show answer & explanation

Answer: 8

Answer

8

The correct value is 8. The union

P \cup Q

yields

\{1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

. Intersecting this set with

R'

means removing any element that is a multiple of 3. The multiples of 3 in

P \cup Q

are 3, 9, and 15. Removing these 3 elements from the 11 elements of

P \cup Q

leaves exactly 8 elements.

Step-by-Step Solution

1

Identify the elements of sets P, Q, and R within the universal set U.

U = \{1, 2, 3, \dots, 20\}

,

P = \{2, 3, 5, 7, 11, 13, 17, 19\}

,

Q = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

, and

R = \{3, 6, 9, 12, 15, 18\}

.

Listing elements helps accurately compute set unions and intersections.

2

Find the union of sets P and Q, denoted as P ∪ Q.

P \cup Q = \{1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

, so

n(P \cup Q) = 11

.

Combining elements of both P and Q without repetition gives their union.

3

Find the complement of set R with respect to U, denoted as R'.

R' = \{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20\}

.

The complement set R' contains all elements in U that are not multiples of 3.

4

Determine the intersection of (P ∪ Q) and R'.

(P \cup Q) \cap R' = \{1, 2, 5, 7, 11, 13, 17, 19\}

. The number of elements is 8.

This removes the multiples of 3 (namely 3, 9, and 15) from the set

P \cup Q

.

Key Concept

Set operations including union, intersection, and set complementation.

Estimated Time:1m 30s

Question 4049Question →

Which of the following physical quantities is classified as a fundamental quantity in the International System of Units (SI)?

Electric current

Force

Velocity

Density

Show answer & explanation

Answer: Electric current

Answer

Electric current

Electric current is one of the seven fundamental physical quantities defined by the International System of Units (SI). It serves as a base quantity from which other electrical quantities (such as charge and potential difference) are derived.

Step-by-Step Solution

1

Identify the definition of a fundamental quantity

Fundamental quantities are independent physical quantities that cannot be expressed in terms of other quantities.

SI defines seven basic quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

2

Evaluate the options against the list of SI fundamental quantities

Electric current is a basic quantity (measured in amperes,

A

), whereas force, velocity, and density are derived from mass, length, and time.

Only electric current is an independent base quantity.

Key Concept

Fundamental quantities are basic physical quantities that are independent of other quantities.

Question 4050Question →

Which of the following represents the complete set of real values of $x$ that satisfy the inequality $\frac{x^2 - x - 6}{x - 1} \le 0$ ?

x \le -2

or

1 < x \le 3

x \le -2

or

1 \le x \le 3

-2 \le x < 1

or

x \ge 3

-2 \le x \le 3

with

x \neq 1

Show answer & explanation

Answer: $x \le -2$ or $1 < x \le 3$

Answer

The complete solution set is

x \le -2

or

1 < x \le 3

.

The expression

\frac{(x-3)(x+2)}{x-1}

evaluates to a non-positive value (

\le 0

) when the numerator and denominator have opposite signs or when the numerator is zero. Evaluating across the critical boundaries

x = -2, 1, 3

while excluding

x = 1

yields the solution set

x \le -2

or

1 < x \le 3

.

Step-by-Step Solution

1

Factor the quadratic numerator and state the domain restriction.

The numerator factors as

x^2 - x - 6 = (x - 3)(x + 2)

, giving the rational inequality

\frac{(x - 3)(x + 2)}{x - 1} \le 0

with

x \neq 1

.

Factoring isolates the critical boundary points where the expression can change sign.

2

Identify all critical numbers.

The critical values are

x = -2

,

x = 1

, and

x = 3

.

These points partition the real number line into sub-intervals.

3

Test points in each interval to determine the sign of the rational function

f(x) = \frac{(x - 3)(x + 2)}{x - 1}

.

For

x < -2

,

f(x) \le 0

; for

-2 < x < 1

,

f(x) > 0

; for

1 < x < 3

,

f(x) \le 0

; for

x > 3

,

f(x) > 0

.

Determining where the function is negative or zero identifies the regions satisfying

\le 0

.

4

Combine intervals and include non-undefined endpoints.

Endpoints

x = -2

and

x = 3

make the numerator zero (included), while

x = 1

makes the denominator zero (excluded). Thus,

x \le -2

or

1 < x \le 3

.

Division by zero must be excluded from the solution set.

Key Concept

Solving rational inequalities by finding critical values, using test intervals, and respecting domain restrictions.

Question 4051Question →

Read the sentence below carefully and complete the literary analysis by filling in the missing term. What figure of speech is demonstrated in the sentence?

Fill in the blanks below

In the sentence, 'The exhausted detective spent a sleepless night searching for the missing evidence,' the transfer of the modifier 'sleepless' from the detective to the noun 'night' is an example of .

Show answer & explanation

Answer

transferred epithet (or hypallage)

The correct response is 'transferred epithet' (or 'hypallage'). This figure of speech takes place when an epithet or qualifying adjective ('sleepless') is transferred from the person experiencing the condition (the detective) to a related noun ('night').

Step-by-Step Solution

1

Analyze the modifier and its modified noun in the sentence.

The adjective 'sleepless' is syntactically attached to the noun 'night'.

Logically, a period of time such as a 'night' cannot experience sleep or sleeplessness; only a living person (the detective) can.

2

Identify the literary device where a modifier is shifted from its logical subject to another associated word.

The device is a transferred epithet (also known in classical rhetoric as hypallage).

A transferred epithet occurs when an adjective (epithet) qualifies a noun other than the person or thing it logically describes.

Key Concept

Transferred Epithet (Hypallage)

Question 4052Question →

Find the real value of

x

that satisfies the exponential equation

\frac{8^{x + 2}}{4^{x - 1}} = 16^{x - 1}

Show answer & explanation

Answer: 4

Answer

The value of

x

is

4

.

Rewriting the terms in base 2 gives

2^{3(x+2)} / 2^{2(x-1)} = 2^{4(x-1)}

. Applying the quotient rule gives an exponent of

(3x + 6) - (2x - 2) = x + 8

on the left. Equating the exponents yields

x + 8 = 4x - 4

, which solves cleanly to

x = 4

.

Step-by-Step Solution

1

Express all terms using a common prime base of 2

The equation becomes

\frac{(2^3)^{x + 2}}{(2^2)^{x - 1}} = (2^4)^{x - 1}

.

Converting to a common base enables the use of index laws to simplify the equation.

2

Apply the power-of-a-power and quotient laws of indices

The left side simplifies to

2^{3(x+2) - 2(x-1)} = 2^{x + 8}

and the right side is

2^{4x - 4}

.

When dividing powers of the same base, exponents are subtracted:

a^m \div a^n = a^{m-n}

.

3

Equate exponents and solve the linear equation

x + 8 = 4x - 4 \implies 3x = 12 \implies x = 4

.

Because the bases on both sides are identical, their respective exponents must be equal.

Key Concept

Solving exponential equations using common base conversion and laws of indices

Question 4053Question →

A physical quantity $X$ is defined by the expression $X = \frac{P \cdot V}{m \cdot t}$ , where $P$ represents pressure, $V$ represents volume, $m$ represents mass, and $t$ represents time. Which of the following statements correctly classifies $X$ and expresses its unit strictly in terms of fundamental SI base units?

X

is a derived quantity, and its unit in SI base units is

\text{m}^2 \cdot \text{s}^{-3}

.

X

is a fundamental quantity, and its unit in SI base units is

\text{m}^2 \cdot \text{s}^{-3}

.

X

is a derived quantity, and its unit in fundamental units is

\text{N} \cdot \text{m} \cdot \text{kg}^{-1} \cdot \text{s}^{-1}

.

X

is a derived quantity, and its unit in SI base units is

\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}

.

Show answer & explanation

Answer: $X$ is a derived quantity, and its unit in SI base units is $\text{m}^2 \cdot \text{s}^{-3}$ .

Answer

The physical quantity

X

is a derived quantity, and its unit expressed strictly in fundamental SI base units is

\text{m}^2 \cdot \text{s}^{-3}

.

The quantity

X

is defined via a mathematical formula involving pressure, volume, mass, and time, which classifies it as a derived quantity. Replacing each component with its fundamental SI base units gives pressure as

\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}

and volume as

\text{m}^3

, making the numerator

\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}

. Dividing by the denominator (

m \cdot t = \text{kg} \cdot \text{s}

) cancels out kilograms and leaves

\text{m}^2 \cdot \text{s}^{-3}

, consisting purely of fundamental base units.

Step-by-Step Solution

1

Classify the physical quantity

X

X

is a derived physical quantity.

Fundamental physical quantities in the SI system are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Because

X

is calculated from a combination of other quantities, it is derived.

2

Express pressure (

P

) and volume (

V

) in terms of fundamental SI base units

P = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}

and

V = \text{m}^3

.

Pressure is defined as force per unit area

\left(\frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}^2}\right)

, and volume has the base unit

\text{m}^3

.

3

Calculate the base unit expression for the numerator

P \cdot V

P \cdot V = (\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}) \cdot (\text{m}^3) = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}

.

Combining the powers of length (metres) gives

-1 + 3 = 2

.

4

Divide by the denominator

m \cdot t

to obtain the base units of

X

X = \frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{kg} \cdot \text{s}} = \text{m}^2 \cdot \text{s}^{-3}

.

The unit of mass (

\text{kg}

) cancels completely, and dividing by time (

\text{s}

) reduces the exponent of seconds from

-2

to

-3

.

Key Concept

Fundamental quantities are independent base quantities defined by the SI system, whereas derived quantities are defined algebraically from fundamental quantities. Reducing derived units to SI base units requires breaking down all non-base units into metres (m), kilograms (kg), seconds (s), amperes (A), kelvins (K), moles (mol), or candelas (cd).

Estimated Time:2m 0s

Question 4054Question →

A copper calorimeter of heat capacity $300\text{ J K}^{-1}$ contains $0.5\text{ kg}$ of water at an initial temperature of $25^\circ\text{C}$ . An electric heater rated at $800\text{ W}$ is immersed in the water to heat the system for $4\text{ minutes}$ . If heat is lost to the surrounding environment at a constant rate of $200\text{ W}$ throughout the heating duration, what is the final temperature of the water-calorimeter system in $^\circ\text{C}$ ? (Take the specific heat capacity of water as $4200\text{ J kg}^{-1}\text{K}^{-1}$ )

Show answer & explanation

Answer: 85

Answer

The final temperature of the system is

85^\circ\text{C}

.

The net thermal energy added to the system accounts for both the supplied electrical energy and the heat energy lost to the surroundings:

Q_{\text{net}} = (800 - 200)\text{ W} \times 240\text{ s} = 144,000\text{ J}

. The total heat capacity of the water-calorimeter system is

C_{\text{total}} = 300\text{ J K}^{-1} + (0.5\text{ kg} \times 4200\text{ J kg}^{-1}\text{K}^{-1}) = 2400\text{ J K}^{-1}

. The temperature increase is

\Delta T = \frac{144,000}{2400} = 60^\circ\text{C}

. Adding this to the initial temperature of

25^\circ\text{C}

gives the final temperature of

85^\circ\text{C}

.

Step-by-Step Solution

1

Convert heating time to standard SI units (seconds)

t = 4 \times 60\text{ s} = 240\text{ s}

Power is measured in Joules per second (Watts), so time must be in seconds.

2

Calculate the net rate of heat energy input to the system

P_{\text{net}} = 800\text{ W} - 200\text{ W} = 600\text{ W}

The net heating power is the input power minus the power dissipated as heat loss.

3

Calculate total net heat energy transferred to the system

Q_{\text{net}} = P_{\text{net}} \times t = 600\text{ W} \times 240\text{ s} = 144,000\text{ J}

Thermal energy transferred equals net power multiplied by time.

4

Compute the total heat capacity of the combined system (water + calorimeter)

C_{\text{total}} = C_{\text{calorimeter}} + (m_{\text{water}} \times c_{\text{water}}) = 300\text{ J K}^{-1} + (0.5\text{ kg} \times 4200\text{ J kg}^{-1}\text{K}^{-1}) = 2400\text{ J K}^{-1}

Heat capacity of water is mass multiplied by specific heat capacity, added to the calorimeter's given heat capacity.

5

Calculate the temperature rise of the system

\Delta T = \frac{Q_{\text{net}}}{C_{\text{total}}} = \frac{144,000\text{ J}}{2400\text{ J K}^{-1}} = 60^\circ\text{C}

Temperature change is total net heat supplied divided by total heat capacity.

6

Find the final temperature of the system

T_{\text{final}} = T_{\text{initial}} + \Delta T = 25^\circ\text{C} + 60^\circ\text{C} = 85^\circ\text{C}

Final temperature equals initial temperature plus temperature increase.

Key Concept

Conservation of thermal energy in calorimeter systems with continuous power loss

Estimated Time:1m 30s

Question 4055Question →

A factory uses $8$ identical machines to produce $480$ items in $6$ hours. If $3$ of the machines break down, how many hours will it take the remaining machines to produce $600$ items at the same rate?

Show answer & explanation

Answer: 12

Answer

The remaining machines will take 12 hours to produce 600 items.

Each machine produces

\frac{480}{8 \times 6} = 10

items per hour. With

5

active machines, the combined rate is

50

items per hour. To reach

600

items, the time required is

\frac{600}{50} = 12

hours.

Step-by-Step Solution

1

Calculate the output rate per machine per hour

10 items per machine per hour

The combined rate of 8 machines is

480 \div 6 = 80

items per hour. Dividing by 8 machines gives

10

items/hour per machine.

2

Determine the new combined production rate

50 items per hour

With 3 machines out of service,

8 - 3 = 5

machines remain, working at a total rate of

5 \times 10 = 50

items per hour.

3

Calculate total hours required for the new target quantity

12 hours

Dividing the target quantity of

600

items by the rate of

50

items/hour yields

\frac{600}{50} = 12

hours.

Key Concept

Compound Proportion and Work Rate

Estimated Time:1m 30s

Question 4056Question →

An electron of mass $9.1 \times 10^{-31}\text{ kg}$ and charge $1.6 \times 10^{-19}\text{ C}$ enters perpendicularly into a uniform magnetic field of flux density $2.0 \times 10^{-3}\text{ T}$ with a speed of $3.2 \times 10^6\text{ m/s}$ . What is the radius of the circular path followed by the electron, expressed in millimeters (mm)?

Show answer & explanation

Answer: 9.1

Answer

The radius of the circular path followed by the electron is 9.1 mm.

When a charge

q

enters a magnetic field

B

perpendicularly at speed

v

, the magnetic force

qvB

supplies the centripetal force

\frac{mv^2}{r}

. Rearranging for radius yields

r = \frac{mv}{qB}

. Substituting

m = 9.1 \times 10^{-31}\text{ kg}

,

v = 3.2 \times 10^6\text{ m/s}

,

q = 1.6 \times 10^{-19}\text{ C}

, and

B = 2.0 \times 10^{-3}\text{ T}

gives

r = 9.1 \times 10^{-3}\text{ m} = 9.1\text{ mm}

.

Step-by-Step Solution

1

Equate the magnetic force to the centripetal force for circular motion

qvB = \frac{mv^2}{r}

A charged particle moving perpendicularly to a magnetic field experiences a magnetic force that acts entirely as a centripetal force.

2

Rearrange the equation to make the orbital radius

r

the subject

r = \frac{mv}{qB}

Cancelling one factor of velocity

v

from both sides allows direct computation of

r

.

3

Substitute the physical values into the formula

r = \frac{(9.1 \times 10^{-31})(3.2 \times 10^6)}{(1.6 \times 10^{-19})(2.0 \times 10^{-3})} = 9.1 \times 10^{-3}\text{ m}

Calculates the radius in standard SI units (meters).

4

Convert the resulting radius from meters to millimeters

r = 9.1 \times 10^{-3}\text{ m} \times 1000\text{ mm/m} = 9.1\text{ mm}

The question explicitly requests the answer in millimeters.

Key Concept

Motion of a charged particle in a uniform magnetic field

Question 4057Question →

A gas cylinder fitted with a frictionless piston contains a fixed mass of ideal gas occupying a volume of $0.040\text{ m}^3$ at a pressure of $1.50 \times 10^5\text{ Pa}$ and a temperature of $27^\circ\text{C}$ . The gas is heated to $127^\circ\text{C}$ while expanding to a new volume of $0.080\text{ m}^3$ . What is the final pressure of the gas?

1.00 \times 10^5\text{ Pa}

3.53 \times 10^5\text{ Pa}

7.50 \times 10^4\text{ Pa}

4.00 \times 10^5\text{ Pa}

Show answer & explanation

Answer: $1.00 \times 10^5\text{ Pa}$

Answer

The final pressure of the gas is

1.00 \times 10^5\text{ Pa}

.

By converting temperatures to absolute zero scale (

300\text{ K}

and

400\text{ K}

) and applying the Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

, the final pressure

P_2

evaluates directly to

1.00 \times 10^5\text{ Pa}

.

Step-by-Step Solution

1

Convert temperatures from Celsius to Kelvin.

T_1 = 27 + 273 = 300\text{ K}

and

T_2 = 127 + 273 = 400\text{ K}

.

Gas laws require absolute temperatures measured on the Kelvin scale.

2

Apply the Combined Gas Law formula

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

.

\frac{(1.50 \times 10^5\text{ Pa})(0.040\text{ m}^3)}{300\text{ K}} = \frac{P_2 (0.080\text{ m}^3)}{400\text{ K}}

.

The mass of the gas is fixed while pressure, volume, and temperature all change.

3

Rearrange and solve for final pressure

P_2

.

P_2 = 1.50 \times 10^5 \times \left(\frac{0.040}{0.080}\right) \times \left(\frac{400}{300}\right) = 1.00 \times 10^5\text{ Pa}

.

Simplifying the numerical expression yields the final equilibrium pressure.

Key Concept

Combined Gas Law

Estimated Time:2m 0s

Question 4058Question →

Match each derived SI unit listed on the left with its corresponding fundamental (base) SI unit representation on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Joule (

\text{J}

) matches

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

; Pascal (

\text{Pa}

) matches

\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

; Watt (

\text{W}

) matches

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}

; Newton (

\text{N}

) matches

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}

.

Each derived unit is systematically expressed in terms of the fundamental SI units of mass (kg), length (m), and time (s) by substituting definitions: Newton is

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}

, Joule is

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

, Pascal is

\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

, and Watt is

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}

.

Step-by-Step Solution

1

Express Newton (N) in fundamental units

\text{Force} = \text{mass} \times \text{acceleration} \Rightarrow 1\text{ N} = 1\text{ kg}\cdot\text{m}\cdot\text{s}^{-2}

Force is defined as mass multiplied by acceleration.

2

Express Joule (J) in fundamental units

\text{Work} = \text{Force} \times \text{distance} \Rightarrow 1\text{ J} = (\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{m} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-2}

Work done is force multiplied by displacement in the direction of the force.

3

Express Pascal (Pa) in fundamental units

\text{Pressure} = \frac{\text{Force}}{\text{Area}} \Rightarrow 1\text{ Pa} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = 1\text{ kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

Pressure is defined as force applied perpendicular to a surface per unit area.

4

Express Watt (W) in fundamental units

\text{Power} = \frac{\text{Work}}{\text{time}} \Rightarrow 1\text{ W} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{s}} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-3}

Power is the rate at which work is done or energy is transferred.

Key Concept

Expressing derived SI units in terms of base SI fundamental units

Question 4059Question →

Which qualitative characteristic of accounting information ensures that financial statements contain data capable of making a difference in the economic decisions of users?

Relevance

Verifiability

Understandability

Business Entity Concept

Show answer & explanation

Answer: Relevance

Answer

Relevance is the primary qualitative characteristic that makes accounting information capable of influencing user decisions.

Accounting information possesses relevance when it is capable of making a difference in the decisions made by users. It achieves this by helping users evaluate past, present, or future events (predictive value) or confirming/correcting previous expectations (confirmatory value).

Step-by-Step Solution

1

Identify the core requirement of the question.

The question asks for the qualitative characteristic that directly makes financial information capable of making a difference in decision-making.

Understanding the definition of fundamental qualitative characteristics is key to selecting the correct accounting attribute.

2

Distinguish between fundamental and enhancing qualitative characteristics.

Fundamental characteristics are Relevance and Faithful Representation. Enhancing characteristics are Comparability, Verifiability, Timeliness, and Understandability.

Relevance directly addresses decision-usefulness through predictive and confirmatory value.

Key Concept

Fundamental Qualitative Characteristics of Accounting Information

Question 4060Question →

The following trial balance extract was taken from the books of a sole proprietor as at 31st December 2025:

Ledger Account	Amount ( $\text{₦}$ )
Sales	200,000
Returns inwards	10,000
Purchases	130,000
Returns outwards	6,000
Opening inventory	30,000
Carriage inwards	4,000
Carriage outwards	8,000

Additional Information:
1. Goods costing $\text{₦}4,000$ were withdrawn by the proprietor for personal use during the financial year, but no entry has been made in the accounting records.
2. The business sells goods at a uniform mark-up of $25\%$ on cost of goods sold.

What is the value of closing inventory to be recorded in the trading account?

₦6,000

₦2,000

₦10,000

₦11,500

Show answer & explanation

Answer: ₦2,000

Answer

The value of closing inventory to be recorded in the trading account is ₦2,000.

Net sales equal ₦190,000 (Gross sales ₦200,000 less Returns inwards ₦10,000). Convert the 25% mark-up (

\frac{1}{4}

on cost) to margin (

\frac{1}{5}

or 20% on sales), yielding Gross Profit of ₦38,000 (

20\% \times \text{₦}190,000

) and Cost of Goods Sold (COGS) of ₦152,000 (

\text{₦}190,000 - \text{₦}38,000

). Cost of goods available for sale is ₦154,000 (Opening stock ₦30,000 + Purchases ₦130,000 - Returns outwards ₦6,000 - Goods withdrawn ₦4,000 + Carriage inwards ₦4,000). Subtracting COGS (₦152,000) from Cost of Goods Available for Sale (₦154,000) gives a closing inventory of ₦2,000.

Step-by-Step Solution

1

Calculate Net Sales

Net Sales = ₦200,000 - ₦10,000 = ₦190,000

Returns inwards must be deducted from gross sales to obtain net sales revenue.

2

Convert Mark-up to Margin and compute Gross Profit and Cost of Goods Sold (COGS)

Margin = 20%; Gross Profit = ₦38,000; COGS = ₦152,000

A mark-up of

25\%

(

\frac{1}{4}

) on cost corresponds to a margin of

\frac{1}{1 + 4} = \frac{1}{5} = 20\%

on net sales. Gross Profit =

20\% \times \text{₦}190,000 = \text{₦}38,000

. COGS = Net Sales - Gross Profit =

\text{₦}190,000 - \text{₦}38,000 = \text{₦}152,000

.

3

Calculate Adjusted Net Purchases

Adjusted Net Purchases = ₦130,000 - ₦6,000 - ₦4,000 = ₦120,000

Returns outwards and owner's inventory drawings must be deducted from gross purchases.

4

Calculate Cost of Goods Available for Sale

Goods Available = ₦30,000 + ₦120,000 + ₦4,000 = ₦154,000

Carriage inwards is added to opening inventory and net purchases as a direct cost of getting goods into the business. Carriage outwards is a selling expense and is excluded.

5

Determine Closing Inventory

Closing Inventory = ₦154,000 - ₦152,000 = ₦2,000

Closing Inventory = Cost of Goods Available for Sale - Cost of Goods Sold.

Key Concept

Trading Account, COGS Determination, and Mark-up/Margin Conversion

Estimated Time:3m 0s

Answer

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