All practice questions

Question 41Question →

Chief Okafor runs a retail store and maintains single-entry records. For the accounting year ended 31 December 2025, his cash transactions were summarized as follows:
- Opening cash balance: $\text{N}25,000$
- Cash received from debtors: $\text{N}340,000$
- Cash sales: $\text{N}275,000$
- Cash banked (paid into bank): $\text{N}410,000$
- General business expenses paid in cash: $\text{N}85,000$
- Cash wages paid: $\text{N}60,000$
- Closing cash balance: $\text{N}18,000$

Based on the cash summary analysis, what is the total amount of cash withdrawn by the proprietor for personal drawings during the year?

Show answer & explanation

Answer: 67000

Answer

The total amount of cash withdrawn by the proprietor for personal drawings is

\text{N}67,000

.

In single-entry accounting, a cash summary account is constructed to find missing cash transactions. Total cash inflows (opening cash

\text{N}25,000

+ receipts from debtors

\text{N}340,000

+ cash sales

\text{N}275,000

) amount to

\text{N}640,000

. Known cash payments and closing balance (cash banked

\text{N}410,000

+ general expenses

\text{N}85,000

+ wages

\text{N}60,000

+ closing balance

\text{N}18,000

) total

\text{N}573,000

. The balancing figure of

\text{N}67,000

represents cash withdrawn by the proprietor for personal use.

Step-by-Step Solution

1

Sum all cash receipts and the opening balance to get total cash available on the debit side.

Total debit side =

\text{N}25,000 + \text{N}340,000 + \text{N}275,000 = \text{N}640,000

.

The cash summary debit side records initial cash on hand plus all cash receipts during the accounting period.

2

Sum all documented cash payments and the unspent closing cash balance on the credit side.

Total accounted outflows and closing balance =

\text{N}410,000 + \text{N}85,000 + \text{N}60,000 + \text{N}18,000 = \text{N}573,000

.

The credit side of the cash summary lists disbursements (including cash banked) and remaining cash.

3

Deduct total accounted disbursements from total receipts to calculate the missing drawings figure.

Drawings =

\text{N}640,000 - \text{N}573,000 = \text{N}67,000

.

Any unexplained deficit on the credit side of a cash summary in single-entry bookkeeping represents unrecorded cash drawings by the owner.

Key Concept

Cash Summary Analysis for Missing Figures

Estimated Time:1m 30s

Question 42Question →

A morphometric study of a river basin reveals a total drainage area of $150\text{ km}^2$ . Quantitative stream analysis using Strahler's method identifies $18$ first-order streams, $7$ second-order streams, $4$ third-order streams, and $1$ fourth-order stream. What is the stream frequency ( $F_s$ ) of this river basin in streams per $\text{km}^2$ ?

Show answer & explanation

Answer: 0.2

Answer

The stream frequency of the river basin is

0.2\text{ streams/km}^2

.

Stream frequency (

F_s

) is defined as the total number of stream segments of all orders (

N

) per unit area (

A

) of a drainage basin, expressed as

F_s = \frac{N}{A}

. Summing all orders yields

N = 18 + 7 + 4 + 1 = 30\text{ streams}

. Dividing

30

streams by the basin area of

150\text{ km}^2

gives exactly

0.2\text{ streams/km}^2

.

Step-by-Step Solution

1

Calculate total number of stream segments (

N

) in the basin

N = 18 + 7 + 4 + 1 = 30

Stream frequency considers the total count of all stream channels of all orders within the basin.

2

Apply the stream frequency formula

F_s = \frac{N}{A}

using the total basin area

A = 150\text{ km}^2

F_s = \frac{30}{150} = 0.2\text{ streams/km}^2

Stream frequency measures the number of stream segments per unit basin area.

Key Concept

Stream Frequency (

F_s

)

Question 43Question →

During a meteorological observation at a weather station in Sokoto, Nigeria, the maximum air temperature recorded using a Six's maximum and minimum thermometer was $41.2^\circ\text{C}$ , while the minimum air temperature recorded was $24.8^\circ\text{C}$ . What is the diurnal range of temperature in $^\circ\text{C}$ for this observation period?

Show answer & explanation

Answer: 16.4

Answer

The diurnal temperature range is

16.4^\circ\text{C}

.

The diurnal range of temperature measures the net difference between the maximum temperature during the day and the minimum temperature at night. Subtracting

24.8^\circ\text{C}

from

41.2^\circ\text{C}

gives

16.4^\circ\text{C}

.

Step-by-Step Solution

1

Extract maximum and minimum temperature values from the observation data

Maximum temperature =

41.2^\circ\text{C}

, Minimum temperature =

24.8^\circ\text{C}

Diurnal range calculation depends strictly on the daily maximum and minimum readings.

2

Subtract the minimum temperature from the maximum temperature

41.2^\circ\text{C} - 24.8^\circ\text{C} = 16.4^\circ\text{C}

The diurnal range represents the difference between the highest and lowest temperatures within a 24-hour period.

Key Concept

Diurnal Range of Temperature

Estimated Time:1m 0s

Question 44Question →

Chief Ade keeps incomplete accounting records for his retail store. For the financial year ended 31st December 2025, his opening inventory was $\text{₦}15,000$ and total purchases amounted to $\text{₦}85,000$ . Total sales revenue recorded for the period was $\text{₦}120,000$ . If he maintains a uniform mark-up of $25\%$ on cost, what is the estimated value of his closing inventory in Naira?

Show answer & explanation

Answer: 4000

Answer

The estimated value of the closing inventory is ₦4,000.

The correct closing inventory figure is ₦4,000. With a mark-up of 25% on cost, the profit margin is 20% on sales revenue. Gross profit is therefore 20% of ₦120,000 = ₦24,000, leaving Cost of Goods Sold at ₦96,000. Deducting ₦96,000 from total goods available for sale (₦15,000 opening stock + ₦85,000 purchases = ₦100,000) leaves ₦4,000 as closing inventory.

Step-by-Step Solution

1

Convert the given mark-up on cost into profit margin on sales revenue.

Margin = 20% (or 1/5) on sales revenue.

Mark-up relates gross profit to cost of sales, whereas margin relates gross profit to sales revenue. Since sales revenue is known, margin must be used.

2

Calculate the Cost of Goods Sold (COGS).

COGS = ₦96,000.

COGS is calculated either as Sales Revenue minus Gross Profit (₦120,000 - ₦24,000) or by dividing Sales Revenue by (1 + Mark-up).

3

Calculate total Goods Available for Sale.

Goods Available for Sale = ₦100,000.

Goods available for sale is the sum of opening stock (₦15,000) and total purchases during the year (₦85,000).

4

Subtract Cost of Goods Sold from total Goods Available for Sale to find Closing Stock.

Closing Stock = ₦4,000.

Closing stock is the remaining portion of inventory available during the period that was not sold.

Key Concept

Application of Mark-up and Margin in Estimating Cost of Goods Sold and Stock

Question 45Question →

Chidi operates a retail store and compiled the following financial items at the end of his trading period:

- Fixtures and Fittings: $\text{₦}150,000$
- Office Equipment: $\text{₦}90,000$
- Stock of Goods: $\text{₦}45,000$
- Trade Debtors: $\text{₦}30,000$
- Cash in Hand: $\text{₦}5,000$
- Trade Creditors: $\text{₦}25,000$
- Bank Overdraft: $\text{₦}10,000$
- 5-Year Bank Loan: $\text{₦}60,000$

What is the total value of Chidi's working capital in Naira?

Show answer & explanation

Answer: 45000

Answer

The total value of Chidi's working capital is 45,000 Naira.

Working capital is calculated as Total Current Assets minus Total Current Liabilities. Total Current Assets consist of Stock of Goods (₦45,000), Trade Debtors (₦30,000), and Cash in Hand (₦5,000), which equal ₦80,000. Total Current Liabilities consist of Trade Creditors (₦25,000) and Bank Overdraft (₦10,000), which equal ₦35,000. Subtracting liabilities from assets yields ₦80,000 - ₦35,000 = ₦45,000.

Step-by-Step Solution

1

Classify current assets from the given financial items

Current Assets = Stock of Goods (₦45,000) + Trade Debtors (₦30,000) + Cash in Hand (₦5,000) = ₦80,000

Current assets are items held for less than one accounting year that can be converted into cash within the normal operating cycle.

2

Classify current liabilities from the given financial items

Current Liabilities = Trade Creditors (₦25,000) + Bank Overdraft (₦10,000) = ₦35,000

Current liabilities are short-term debts due within one accounting period. Bank overdraft is classified as a current liability, while the 5-year loan is a non-current liability.

3

Subtract Total Current Liabilities from Total Current Assets to determine Working Capital

Working Capital = ₦80,000 - ₦35,000 = ₦45,000

Working capital (net current assets) measures operational liquidity by showing the excess of liquid assets over short-term obligations.

Key Concept

Working Capital Calculation and Balance Sheet Classification

Question 46Question →

Ade Manufacturing Enterprise provided the following extract from its records for the year ended 31st December 2025:

Financial Item	Amount (\text{NGN})
Sales of finished goods	$650{}000$
Opening stock of finished goods	$50{}000$
Cost of production transferred from Manufacturing Account	$380{}000$
Closing stock of finished goods	$60{}000$
Rent paid	$40{}000$
General administrative expenses	$36{}000$
Selling and distribution expenses	$25{}000$

Additional Information:
At 31st December 2025, rent accrued amounted to $\text{NGN } 4{}000$ .

What is the Net Profit of Ade Manufacturing Enterprise for the year ended 31st December 2025?

Show answer & explanation

Answer: 175000

Answer

The Net Profit for Ade Manufacturing Enterprise for the year ended 31st December 2025 is NGN 175,000.

The correct Net Profit of NGN 175,000 is obtained by first deriving the Gross Profit of NGN 280,000 (Sales of NGN 650,000 minus COGS of NGN 370,000) and then deducting total period operating expenses of NGN 105,000 (Rent expense of NGN 44,000 including NGN 4,000 accrual, Administrative expenses of NGN 36,000, and Selling & distribution expenses of NGN 25,000).

Step-by-Step Solution

1

Calculate the Cost of Goods Sold (COGS)

\text{COGS} = \text{NGN } 50{}000 + \text{NGN } 380{}000 - \text{NGN } 60{}000 = \text{NGN } 370{}000

Cost of Goods Sold is obtained by adding the cost of production transferred from the manufacturing account to opening finished goods inventory and subtracting closing finished goods inventory.

2

Calculate the Trading Gross Profit

\text{Gross Profit} = \text{NGN } 650{}000 - \text{NGN } 370{}000 = \text{NGN } 280{}000

Gross profit is determined by deducting the cost of goods sold from total turnover (sales of finished goods).

3

Calculate Total Operating Expenses including accruals

\text{Total Expenses} = (\text{NGN } 40{}000 + \text{NGN } 4{}000) + \text{NGN } 36{}000 + \text{NGN } 25{}000 = \text{NGN } 105{}000

Accrued rent at year-end represents an unpaid expense of the current period and must be added to rent paid before aggregating administrative and selling expenses.

4

Calculate Net Profit

\text{Net Profit} = \text{NGN } 280{}000 - \text{NGN } 105{}000 = \text{NGN } 175{}000

Net Profit is computed by subtracting total operating expenses from the trading gross profit.

Key Concept

Preparation of Trading and Profit & Loss Account for Manufacturing Entities

Question 47Question →

An extract from the books of a sole trader shows Trade Debtors of $₦150,000$ and an existing Provision for Doubtful Debts of $₦6,000$ . At the end of the financial year, an additional bad debt of $₦10,000$ is to be written off, and the provision for doubtful debts is to be adjusted to $5\%$ of the remaining trade debtors. What is the net amount of Trade Debtors to be presented in the Statement of Financial Position (Balance Sheet)?

Show answer & explanation

Answer: 133000

Answer

The net amount of Trade Debtors to be presented in the Statement of Financial Position is

₦133,000

.

To compute net trade debtors for the Statement of Financial Position, first subtract the additional bad debt (

₦10,000

) from gross debtors (

₦150,000

) to get remaining debtors of

₦140,000

. Next, calculate the

5\%

closing provision on

₦140,000

, which equals

₦7,000

. Deducting

₦7,000

from

₦140,000

yields

₦133,000

.

Step-by-Step Solution

1

Deduct additional bad debts written off from gross trade debtors

₦150,000 - ₦10,000 = ₦140,000

Bad debts identified at the end of the period must be written off from trade debtors before computing the new percentage provision.

2

Calculate the required new provision for doubtful debts

5\% \times ₦140,000 = ₦7,000

The required closing provision rate of

5\%

applies to the remaining collectible trade debtors.

3

Subtract the new provision for doubtful debts from the remaining trade debtors

₦140,000 - ₦7,000 = ₦133,000

Net Trade Debtors shown in current assets is calculated as adjusted gross debtors minus the closing provision balance.

Key Concept

Adjustment for bad debts written off prior to calculating closing provision for doubtful debts and net debtors presentation.

Question 48Question →

Apex Cultural Society received $850,000$ in cash subscription fees during the financial year ended 31st December 2025. Additional subscription records reveal the following balances:

- Subscriptions in arrears on 1st January 2025: $60,000$
- Subscriptions received in advance on 1st January 2025: $40,000$
- Subscriptions in arrears on 31st December 2025: $75,000$
- Subscriptions received in advance on 31st December 2025: $50,000$

What is the exact amount in Naira to be credited to the Income and Expenditure Account as subscription income for the year ended 31st December 2025?

Show answer & explanation

Answer: 855000

Answer

The amount to be credited to the Income and Expenditure Account as subscription income for the year ended 31st December 2025 is

855,000

.

The Income and Expenditure Account operates on the accrual concept, taking into account income earned for the specific financial year regardless of cash timing. Calculating current subscription income requires adding opening prepaid subscriptions (

40,000

) and closing accrued subscriptions (

75,000

) to cash received (

850,000

), while subtracting opening accrued subscriptions (

60,000

) and closing prepaid subscriptions (

50,000

), yielding

855,000

.

Step-by-Step Solution

1

Identify cash received during the period

Subscriptions cash received =

850,000

This represents the total cash inflows recorded in the Receipts and Payments Account.

2

Add amounts attributable to the current period

Add opening advance (

40,000

) and closing arrears (

75,000

)

Subscriptions paid in advance at the beginning of the year belong to the current year's income, while subscriptions owing at the end of the year were earned during the current year.

3

Subtract amounts not attributable to the current period

Deduct opening arrears (

60,000

) and closing advance (

50,000

)

Subscriptions owing at the start of the year belong to the previous year, while subscriptions prepaid at the end of the year belong to the next accounting period.

4

Calculate net subscription revenue

850,000 + 40,000 + 75,000 - 60,000 - 50,000 = 855,000

Applies accrual concept rules to align income earned with the current financial period.

Key Concept

Accrual Accounting Adjustments for Subscription Revenue in Non-Profit Entities

Question 49Question →

A divided bar chart of total length $15\text{ cm}$ is constructed to illustrate the land use distribution of an agricultural zone covering a total area of $120,000\text{ hectares}$ . If livestock grazing occupies $32,000\text{ hectares}$ of this land, what is the length in $\text{cm}$ of the bar segment representing livestock grazing?

Show answer & explanation

Answer: 4

Answer

The length of the bar segment representing livestock grazing is

4\text{ cm}

.

The correct answer of

4\text{ cm}

is obtained by finding the fraction of total land devoted to grazing (

\frac{32,000}{120,000} = \frac{4}{15}

) and scaling it by the total bar length of

15\text{ cm}

.

Step-by-Step Solution

1

Calculate the proportion of the land area used for livestock grazing

Proportion =

\frac{32,000}{120,000} = \frac{4}{15} \approx 0.2667

Divided bar charts represent component parts proportionally relative to the whole dataset.

2

Multiply the proportion by the total bar length to find the length of the segment

Segment length =

\frac{4}{15} \times 15\text{ cm} = 4\text{ cm}

The total length of the bar represents

100\%

of the total area, so each sub-component's length is proportional to its contribution.

Key Concept

Calculating component segment lengths for divided bar charts using proportional ratios

Question 50Question →

During the year ended 31 December 2025, Midland Officers Club received $\text{N}240,000$ in cash as subscription payments from its members. Additional information regarding subscriptions is provided below:

- Subscriptions in arrears on 1 January 2025: $\text{N}25,000$
- Subscriptions in advance on 1 January 2025: $\text{N}18,000$
- Subscriptions in arrears on 31 December 2025: $\text{N}32,000$
- Subscriptions in advance on 31 December 2025: $\text{N}15,000$

What amount (in Naira) should be credited to the Income and Expenditure Account as subscription income for the year ended 31 December 2025?

Show answer & explanation

Answer: 250000

Answer

The amount credited to the Income and Expenditure Account as subscription income for the year ended 31 December 2025 is

\text{N}250,000

.

Under accrual accounting principles for non-profit entities, subscription income recognized in the Income and Expenditure Account represents the total revenue earned for that specific financial year. Starting with cash received of

\text{N}240,000

, we subtract opening arrears of

\text{N}25,000

(income from prior year), add opening advance of

\text{N}18,000

(income prepaid for current year), add closing arrears of

\text{N}32,000

(earned but unpaid current income), and subtract closing advance of

\text{N}15,000

(income prepaid for next year). This yields

\text{N}250,000

.

Step-by-Step Solution

1

Record total subscription cash received during the financial year.

Cash received =

\text{N}240,000

This forms the base figure for calculating the earned subscription income under accrual accounting.

2

Adjust for opening balances (1 January 2025).

Deduct opening arrears (

\text{N}25,000

) and add opening advance (

\text{N}18,000

).

Subscriptions in arrears at the start belong to the prior period, while subscriptions in advance at the start were paid for the current period.

3

Adjust for closing balances (31 December 2025).

Add closing arrears (

\text{N}32,000

) and deduct closing advance (

\text{N}15,000

).

Closing arrears represent current period income earned but uncollected, whereas closing advances represent money received for the next financial year.

4

Compute the net subscription income.

\text{N}240,000 - \text{N}25,000 + \text{N}18,000 + \text{N}32,000 - \text{N}15,000 = \text{N}250,000

The final adjusted figure reflects the total subscription revenue applicable strictly to the current accounting year.

Key Concept

Accrual concept applied to non-profit subscriptions account

Estimated Time:2m 0s

Question 51Question →

A sole trader evaluated three categories of unsold goods at the end of the accounting year as follows:

Inventory Category	Cost ($)	Estimated Selling Price ($)	Selling Expenses ($)
Category X	12,000	15,000	1,000
Category Y	18,000	16,000	1,500
Category Z	25,000	30,000	2,000

In accordance with the prudence concept, what is the total value of closing inventory to be credited to the Trading Account?

Show answer & explanation

Answer: 51500

Answer

The total value of closing inventory to be credited to the Trading Account is $51,500.

Closing inventory must be valued at the lower of cost and net realizable value (NRV) for each separate category. For Category X, cost (

12,000) is lower than NRV (

14,000). For Category Y, NRV (

14,500) is lower than cost (

18,000). For Category Z, cost (

25,000) is lower than NRV (

28,000). Summing these lower values (

12,000 +

14,500 +

25,000) yields

51,500.

Step-by-Step Solution

1

Calculate the Net Realizable Value (NRV) for each inventory category.

Category X NRV =

15,000 -

1,000 =

14,000; Category Y NRV =

16,000 -

1,500 =

14,500; Category Z NRV =

30,000 -

2,000 = $28,000.

Net Realizable Value represents the expected net cash receivable from selling the inventory after deducting estimated costs to complete and sell.

2

Determine the lower of Cost and NRV for each category individually.

Category X lower value =

12,000 (Cost); Category Y lower value =

14,500 (NRV); Category Z lower value = $25,000 (Cost).

According to the accounting principle of prudence (and IAS 2), inventory must be valued at the lower of cost and net realizable value evaluated line by line.

3

Sum the selected valuation amounts to obtain the total closing inventory value.

Total Closing Inventory =

12,000 +

14,500 +

25,000 =

51,500.

The total closing inventory reported in final accounts is the sum of the individual lower values.

Key Concept

Valuation of inventory at the lower of cost and net realizable value under the prudence concept

Question 52Question →

An urbanizing district in West Africa with an estimated mid-year population of $500{}000$ recorded $16{}000$ live births, $6{}000$ deaths, $3{}000$ in-migrants, and $1{}000$ out-migrants within a single year. What is the annual net population growth rate of this district expressed as a percentage?

Show answer & explanation

Answer: 2.4

Answer

The annual net population growth rate of the district is

2.4\%

.

Total net population change combines natural increase (

16{}000 \text{ births} - 6{}000 \text{ deaths} = 10{}000

) and net migration (

3{}000 \text{ in-migrants} - 1{}000 \text{ out-migrants} = 2{}000

), giving a total increase of

12{}000

. Expressing

12{}000

as a percentage of the total mid-year population of

500{}000

yields

\frac{12{}000}{500{}000} \times 100 = 2.4\%

.

Step-by-Step Solution

1

Calculate the natural increase in population.

Natural Increase =

16{}000 - 6{}000 = 10{}000

individuals.

Natural increase represents the population growth resulting strictly from the surplus of births over deaths.

2

Calculate the net migration for the year.

Net Migration =

3{}000 - 1{}000 = 2{}000

individuals.

Net migration is the net balance between incoming migrants (immigration) and outgoing migrants (emigration).

3

Determine the total annual population gain.

Total Population Addition =

10{}000 + 2{}000 = 12{}000

individuals.

Overall demographic growth incorporates both natural increase and net spatial mobility.

4

Compute the net growth rate percentage.

Annual Growth Rate =

\left(\frac{12{}000}{500{}000}\right) \times 100\% = 2.4\%

.

Dividing total net gain by the base mid-year population and scaling by 100 converts absolute growth into an annual percentage rate.

Key Concept

Total Net Population Growth Rate (Natural Increase + Net Migration)

Question 53Question →

A radio broadcast originates live from City A, located at longitude $10^\circ\text{W}$ , at 09:00 AM local time. What is the local time (expressed as an integer in 24-hour clock format, e.g., 14 for 14:00) at City B, located at longitude $65^\circ\text{E}$ , when the transmission begins?

Show answer & explanation

Answer: 14

Answer

14

City A (

10^\circ\text{W}

) and City B (

65^\circ\text{E}

) have a total angular separation of

75^\circ

. At the rate of

15^\circ

per hour, this equals a 5-hour time difference. Because City B is located to the east of City A, its local time is ahead, so 5 hours added to 09:00 AM gives 14:00 (14).

Step-by-Step Solution

1

Calculate the angular separation between the two longitudes

75^\circ

Since the points lie in different hemispheres (West and East), their longitudinal values must be added together (

10^\circ + 65^\circ = 75^\circ

).

2

Convert angular distance to time difference

5 hours

The Earth rotates

360^\circ

in 24 hours, which equals

15^\circ

per hour (

75^\circ \div 15^\circ = 5\text{ hours}

).

3

Adjust time for directional displacement

14

Travelling eastwards means gaining time because eastern locations experience solar movement earlier. Adding 5 hours to 09:00 AM yields 14:00.

Key Concept

Calculating local time difference across prime meridian boundaries

Question 54Question →

Two meteorological observatories, Station Alpha located at longitude $25^\circ\text{E}$ and Station Beta located in the Western Hemisphere, measure solar illumination simultaneously. When the local solar time at Station Alpha is 4:30 PM, the local solar time at Station Beta is 11:30 AM on the same day. What is the numerical value of the longitude of Station Beta in degrees West?

Show answer & explanation

Answer: 50

Answer

The longitude of Station Beta is

50^\circ\text{W}

.

Station Alpha is at longitude

25^\circ\text{E}

with local solar time 4:30 PM (16:30). Station Beta records a local solar time of 11:30 AM (11:30) on the same day. The time difference is

16:30 - 11:30 = 5\text{ hours}

. Since Earth rotates

15^\circ

per hour, a 5-hour time difference corresponds to an angular distance of

5 \times 15^\circ = 75^\circ

. Because Station Beta's local time is earlier than Station Alpha's local time, Station Beta is located to the west of Station Alpha. Subtracting

25^\circ

to reach the Prime Meridian (

0^\circ

) leaves

50^\circ

west, placing Station Beta at

50^\circ\text{W}

.

Step-by-Step Solution

1

Determine the time difference between the two stations.

Time difference = 16:30 (4:30 PM) - 11:30 (11:30 AM) = 5 hours.

Calculating time separation is the first step in finding angular longitudinal separation.

2

Convert the time difference to angular longitudinal distance using Earth's rotation rate (

15^\circ

per hour).

Angular distance =

5\text{ hours} \times 15^\circ/\text{hour} = 75^\circ

.

The Earth rotates

360^\circ

in 24 hours, which equals

15^\circ

per hour.

3

Determine the relative direction of Station Beta from Station Alpha.

Station Beta is west of Station Alpha.

Places located further west have earlier local times than places further east.

4

Compute the longitude of Station Beta relative to the Prime Meridian (

0^\circ

).

Longitude =

75^\circ - 25^\circ\text{E} = 50^\circ\text{W}

.

Moving

25^\circ

west from

25^\circ\text{E}

reaches

0^\circ

, and the remaining

50^\circ

extends west into the Western Hemisphere.

Key Concept

Calculating longitudinal position from local solar time difference across the Prime Meridian

Question 55Question →

A semi-urban local government area in Nigeria recorded a total mid-year population of $500,000$ in a specific census year. During this period, the area recorded $18,500$ in-migrants and $8,500$ out-migrants. What is the net migration rate per $1,000$ population for this area?

Show answer & explanation

Answer: 20

Answer

The net migration rate for the area is

20

per

1,000

population.

The net migration rate is calculated using the demographic formula

\text{Net Migration Rate} = \frac{\text{In-migrants} - \text{Out-migrants}}{\text{Total Population}} \times 1,000

. Substituting the given values gives

\frac{18,500 - 8,500}{500,000} \times 1,000 = \frac{10,000}{500,000} \times 1,000 = 20

per

1,000

population.

Step-by-Step Solution

1

Find the net migration volume

Net Migration Volume =

10,000

people

Net migration represents the net gain or loss of population through movement into and out of an area, calculated as in-migrants minus out-migrants (

18,500 - 8,500

).

2

Calculate the Net Migration Rate (NMR) per

1,000

inhabitants

NMR =

20

per

1,000

population

The rate standardizes population growth from migration per thousand people using the formula

\text{NMR} = \left(\frac{I - E}{P}\right) \times 1,000

, where

I = 18,500

,

E = 8,500

, and

P = 500,000

.

Key Concept

Net Migration Rate Calculation

Question 56Question →

Two chords $AB$ and $CD$ intersect at a point $P$ inside a circle. If $AP = 4\text{ cm}$ , $PB = 9\text{ cm}$ , and $CP = 3\text{ cm}$ , what is the length of segment $PD$ in centimeters?

Show answer & explanation

Answer: 12

Answer

The length of segment

PD

is

12\text{ cm}

.

According to the Intersecting Chords Theorem, for two chords intersecting inside a circle at point

P

, the relation

AP \times PB = CP \times PD

holds true. Substituting

AP = 4

,

PB = 9

, and

CP = 3

gives

4 \times 9 = 3 \times PD

, so

36 = 3 \times PD

, which yields

PD = 12\text{ cm}

.

Step-by-Step Solution

1

Apply the Intersecting Chords Theorem

AP \times PB = CP \times PD

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other.

2

Substitute the known values

4 \times 9 = 3 \times PD

, which gives

36 = 3 \times PD

Insert the values

AP = 4\text{ cm}

,

PB = 9\text{ cm}

, and

CP = 3\text{ cm}

.

3

Solve for

PD

PD = 12\text{ cm}

Divide both sides of the equation by 3.

Key Concept

Intersecting Chords Theorem

Question 57Question →

If $\log_4 x + \log_2 x = 6$ , find the value of $x$ .

Show answer & explanation

Answer: 16

Answer

The value of

x

is 16.

Using the change of base identity

\log_b a = \frac{\log_c a}{\log_c b}

, we convert

\log_4 x

into

\frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2}

. Substituting this into the equation yields

\frac{3}{2} \log_2 x = 6

, which simplifies to

\log_2 x = 4

, leading directly to

x = 2^4 = 16

.

Step-by-Step Solution

1

Apply the change of base formula to

\log_4 x

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2}

Converting all terms to a common base (base 2) simplifies addition of logarithmic terms.

2

Substitute the expression back into the original equation and collect like terms

\frac{1}{2}\log_2 x + \log_2 x = \frac{3}{2}\log_2 x = 6

Adding the coefficients of

\log_2 x

gives

\frac{3}{2}

.

3

Isolate

\log_2 x

\log_2 x = 6 \cdot \frac{2}{3} = 4

Multiplying both sides by

\frac{2}{3}

isolates the logarithmic term.

4

Convert from logarithmic form to exponential form

x = 2^4 = 16

If

\log_b a = c

, then

a = b^c

.

Key Concept

Change of base rule for logarithms

Question 58Question →

If $y = 2e^{4x} + \ln(2x + 1)$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Show answer & explanation

Answer: 10

Answer

10

Differentiating

y = 2e^{4x} + \ln(2x + 1)

with respect to

x

yields

\frac{dy}{dx} = 8e^{4x} + \frac{2}{2x + 1}

. Evaluating this derivative at

x = 0

gives

8(e^0) + \frac{2}{2(0) + 1} = 8(1) + 2 = 10

.

Step-by-Step Solution

1

Differentiate each term of

y = 2e^{4x} + \ln(2x + 1)

with respect to

x

\frac{dy}{dx} = 8e^{4x} + \frac{2}{2x + 1}

By the chain rule,

\frac{d}{dx}(ae^{kx}) = ak e^{kx}

and

\frac{d}{dx}(\ln(u(x))) = \frac{u'(x)}{u(x)}

.

2

Evaluate the derivative at

x = 0

\left.\frac{dy}{dx}\right|_{x=0} = 8e^{0} + \frac{2}{2(0) + 1} = 8(1) + \frac{2}{1} = 10

Substitute

x = 0

into the derived expression and simplify using

e^0 = 1

.

Key Concept

Differentiation of exponential and logarithmic functions using the chain rule

Question 59Question →

Find the value of $x$ that satisfies the exponential equation $27^{x - 1} = \frac{9^{x + 1}}{3^{x - 5}}$ .

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

Rewriting the terms with a common base of 3 transforms the equation into

3^{3x-3} = 3^{x+7}

. Equating exponents gives

3x - 3 = x + 7

, which yields

x = 5

.

Step-by-Step Solution

1

Convert all terms to base 3

27^{x-1} = 3^{3x-3}

and

9^{x+1} = 3^{2x+2}

Laws of indices require identical bases to manipulate exponents.

2

Apply division rule of indices to the right-hand side

\frac{3^{2x+2}}{3^{x-5}} = 3^{x+7}

When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Equate the exponents and solve for

x

3x - 3 = x + 7 \implies 2x = 10 \implies x = 5

If

a^f(x) = a^g(x)

for

a > 0

and

a \neq 1

, then

f(x) = g(x)

.

Key Concept

Solving exponential equations using base reduction and exponent laws

Estimated Time:1m 30s

Question 60Question →

A curve is defined by the equation $y = x^3 - 4x + 3$ . What is the $y$ -intercept of the tangent line to the curve at the point where $x = 2$ ?

Show answer & explanation

Answer: -13

Answer

The y-intercept of the tangent line is

-13

.

At

x = 2

, substituting into

y = x^3 - 4x + 3

yields

y = 3

. The derivative

\frac{dy}{dx} = 3x^2 - 4

evaluated at

x = 2

gives

m = 8

. The tangent line equation is

y - 3 = 8(x - 2)

, which simplifies to

y = 8x - 13

. Setting

x = 0

gives the

y

-intercept of

-13

.

Step-by-Step Solution

1

Calculate the y-coordinate of the point on the curve at

x = 2

y = 3

, giving the point

(2, 3)

The point of contact must lie on the curve.

2

Differentiate the curve equation to find the gradient function

\frac{dy}{dx} = 3x^2 - 4

The derivative of a function gives the gradient of the tangent at any point

x

.

3

Evaluate the derivative at

x = 2

to find the gradient of the tangent

m

m = 8

Substituting

x = 2

into

3x^2 - 4

yields

3(4) - 4 = 8

.

4

Formulate the linear equation of the tangent line using point

(2, 3)

and gradient

m = 8

y = 8x - 13

Applying

y - y_1 = m(x - x_1)

gives

y - 3 = 8(x - 2)

, which simplifies to

y = 8x - 13

.

5

Extract the y-intercept of the tangent line

y-intercept =

-13

In slope-intercept form

y = mx + c

, the constant term

c = -13

is the y-intercept.

Key Concept

Tangents to Curves and Line Intercepts

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