Tüm alıştırma soruları

Soru 21Soru →

A car traveling along a straight road at an initial speed of $20\text{ m/s}$ applies its brakes, causing a uniform deceleration of $5\text{ m/s}^2$ until it comes to a complete stop. What is the total distance traveled by the car during this braking period?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The total distance traveled by the car while coming to a stop is

40\text{ m}

.

Using the third equation of linear motion

v^2 = u^2 + 2as

with

u = 20\text{ m/s}

,

v = 0\text{ m/s}

, and acceleration

a = -5\text{ m/s}^2

, we obtain

0 = 400 - 10s

, which simplifies directly to

s = 40\text{ m}

.

Adım Adım Çözüm

1

Identify the given kinematic parameters from the problem statement.

u = 20\text{ m/s}

,

v = 0\text{ m/s}

,

a = -5\text{ m/s}^2

The car decelerates to a stop, so final velocity is zero and acceleration is negative relative to initial direction of motion.

2

Select the appropriate equation of motion linking

u

,

v

,

a

, and displacement

s

.

v^2 = u^2 + 2as

This equation directly relates initial velocity, final velocity, acceleration, and distance without requiring time.

3

Substitute the known values into the equation and solve for distance

s

.

0^2 = 20^2 + 2(-5)s \implies 10s = 400 \implies s = 40\text{ m}

Algebraic simplification yields the stopping distance.

Anahtar Kavram

Uniformly Accelerated Motion and Stopping Distance

Soru 22Soru →

A body of mass $50\text{ kg}$ is suspended from a spring balance attached to the ceiling of a lift. Calculate the reading registered by the spring balance, in Newtons, when the lift accelerates upward at $2.5\text{ m/s}^2$ . (Take acceleration due to gravity, $g = 10\text{ m/s}^2$ ).

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Cevap: 625

Cevap

The reading of the spring balance is

625\text{ N}

.

When a lift accelerates upward, the effective acceleration experienced by an object relative to the lift is

g + a

. A spring balance measures the tension required to support and accelerate the object, which is

T = m(g + a)

. Substituting

m = 50\text{ kg}

,

g = 10\text{ m/s}^2

, and

a = 2.5\text{ m/s}^2

gives

T = 50 \times (10 + 2.5) = 625\text{ N}

.

Adım Adım Çözüm

1

Identify the given physical quantities

Mass

m = 50\text{ kg}

, acceleration due to gravity

g = 10\text{ m/s}^2

, and upward acceleration

a = 2.5\text{ m/s}^2

.

Spring balances measure weight (tension or normal force), which varies in an accelerating reference frame.

2

Apply Newton's second law to determine apparent weight in an upward accelerating lift

Apparent weight (scale reading)

T = m(g + a)

.

The upward force from the spring must overcome gravitational force

mg

and provide net upward accelerating force

ma

.

3

Compute the numerical value

T = 50 \times (10 + 2.5) = 50 \times 12.5 = 625\text{ N}

.

Multiplying mass by effective acceleration yields the correct balance reading.

Anahtar Kavram

Apparent weight measurement in an accelerating frame of reference using a spring balance

Soru 23Soru →

A consumer's total utility ( $TU$ ) schedule from consuming successive bottles of soft drink is presented in the table below:

Quantity of Soft Drink (Bottles)	Total Utility ( $TU$ in utils)
1	40
2	75
3	102
4	120
5	130

Based on the table above, what is the marginal utility ( $MU$ ) derived from consuming the 4th bottle of soft drink?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The marginal utility derived from consuming the 4th bottle of soft drink is 18 utils.

Marginal Utility (

MU

) is defined as the additional utility derived from the consumption of one extra unit of a commodity. From the schedule, total utility increases from 102 utils to 120 utils when moving from 3 to 4 bottles. Thus,

MU_4 = 120 - 102 = 18

utils.

Adım Adım Çözüm

1

Find total utility values from the schedule for

Q = 4

and

Q = 3

TU_4 = 120

utils and

TU_3 = 102

utils

Marginal utility measures the extra satisfaction gained from consuming one additional unit.

2

Apply the Marginal Utility formula

MU = \Delta TU / \Delta Q

MU_4 = 120 - 102 = 18

utils

Subtracting previous total utility from current total utility gives the additional utility contributed by the 4th unit.

Anahtar Kavram

Marginal Utility Calculation from Total Utility Schedule

Soru 24Soru →

A wave generator in a ripple tank creates periodic surface waves. An observer counts 45 complete wave crests passing a fixed marker in $15\text{ seconds}$ . If the distance measured between the first crest and the seventh crest is $1.2\text{ m}$ , what is the speed of propagation of the wave in $\text{m/s}$ ?

Cevabı ve açıklamayı göster

Cevap: 0.6

Cevap

The speed of propagation of the wave is

0.6\text{ m/s}

.

The frequency is calculated as

f = \frac{45\text{ crests}}{15\text{ s}} = 3\text{ Hz}

. The distance between the 1st and 7th crest spans 6 complete wavelengths (

6\lambda = 1.2\text{ m}

), giving

\lambda = 0.2\text{ m}

. Applying the wave speed formula

v = f\lambda

yields

v = 3 \times 0.2 = 0.6\text{ m/s}

.

Adım Adım Çözüm

1

Calculate the frequency of the wave

Frequency

f = 3\text{ Hz}

Frequency is the rate at which complete wave cycles pass a fixed point, calculated as

f = \frac{45}{15\text{ s}} = 3\text{ Hz}

.

2

Calculate the wavelength from crest spacing

Wavelength

\lambda = 0.2\text{ m}

The distance between

n

successive crests contains

(n-1)

full wavelength intervals. Thus, between the 1st and 7th crest there are

7 - 1 = 6

wavelengths, so

6\lambda = 1.2\text{ m} \implies \lambda = 0.2\text{ m}

.

3

Calculate wave speed using the fundamental wave equation

Speed

v = 0.6\text{ m/s}

Wave propagation speed is the product of frequency and wavelength:

v = f\lambda = 3\text{ Hz} \times 0.2\text{ m} = 0.6\text{ m/s}

.

Anahtar Kavram

Wave speed calculation using frequency and wavelength derived from crest counting (

v = f\lambda

).

Soru 25Soru →

If $\log_2 x + \log_4 x + \log_{16} x = \frac{21}{4}$ , find the value of $x$ .

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Cevap: 8

Cevap

8

Applying the change of base formula

\log_b x = \frac{\log_2 x}{\log_2 b}

allows

\log_4 x

and

\log_{16} x

to be rewritten as

\frac{1}{2}\log_2 x

and

\frac{1}{4}\log_2 x

. Summing

(1 + \frac{1}{2} + \frac{1}{4})\log_2 x

yields

\frac{7}{4}\log_2 x = \frac{21}{4}

, which simplifies to

\log_2 x = 3

. Converting to exponential form gives

x = 2^3 = 8

.

Adım Adım Çözüm

1

Express all logarithmic terms in terms of base 2 using the change of base formula

\log_4 x = \frac{1}{2}\log_2 x \text{ and } \log_{16} x = \frac{1}{4}\log_2 x

Converting all terms to a common base allows for algebraic simplification.

2

Substitute the expressions back into the equation and factor out \log_2 x

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x = \frac{21}{4}

Combining the fractional coefficients simplifies the left side of the equation.

3

Solve for \log_2 x and evaluate x using the definition of logarithm

\log_2 x = 3 \implies x = 2^3 = 8

Multiplying both sides by \frac{4}{7} isolates \log_2 x, and converting to exponential form gives the value of x.

Anahtar Kavram

Logarithms and Change of Base

Soru 26Soru →

During an ecological survey of a savanna ecosystem in Nigeria, 40 grasshoppers were captured, marked with non-toxic paint, and released back into their habitat. A second sample of 50 grasshoppers captured two days later contained 10 marked individuals. What is the estimated total population size of grasshoppers in this habitat?

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Cevap: 200

Cevap

The estimated total population size of grasshoppers is 200.

Applying the Lincoln Index formula

N = \frac{M \times C}{R}

with

M = 40

marked initially,

C = 50

in the second capture, and

R = 10

recaptured marked individuals gives

N = \frac{40 \times 50}{10} = 200

grasshoppers.

Adım Adım Çözüm

1

Identify the values from the capture-recapture sampling data.

Initial marked individuals

M = 40

, second sample size

C = 50

, recaptured marked individuals

R = 10

.

These parameters are required for the Lincoln Index population size estimation formula.

2

Calculate the population size using

N = \frac{M \times C}{R}

.

N = \frac{40 \times 50}{10} = 200

.

Multiplying the size of the first sample by the size of the second sample and dividing by the number of recaptured marked individuals yields the total population estimate.

Anahtar Kavram

Lincoln Index (Capture-Mark-Recapture Method)

Soru 27Soru →

An ecologist used a $2\text{ m} \times 2\text{ m}$ quadrat frame thrown randomly 15 times across a $1.2\text{ hectare}$ ( $12,000\text{ m}^2$ ) savanna habitat in Yankari Game Reserve to estimate the population size of the variegated grasshopper (*Zonocerus variegatus*). A total of 180 grasshoppers were counted across all 15 quadrat throws. What is the estimated total population of *Zonocerus variegatus* in the entire $1.2\text{ hectare}$ study area?

Cevabı ve açıklamayı göster

Cevap: 36000

Cevap

The estimated total population of *Zonocerus variegatus* in the entire study area is 36,000 grasshoppers.

To estimate total population, first calculate total sampled area (

15 \times 4\text{ m}^2 = 60\text{ m}^2

). Dividing total individuals counted (

180

) by total sampled area (

60\text{ m}^2

) yields a mean density of

3\text{ grasshoppers/m}^2

. Extrapolating this density across the total study area (

12,000\text{ m}^2

) gives

3 \times 12,000 = 36,000

grasshoppers.

Adım Adım Çözüm

1

Calculate the surface area of one quadrat frame

2\text{ m} \times 2\text{ m} = 4\text{ m}^2

Establishing the individual quadrat area is required to find the total sampling footprint.

2

Determine the total area sampled across all 15 quadrat throws

15 \times 4\text{ m}^2 = 60\text{ m}^2

Multiplying single quadrat area by total throws gives the aggregate area sampled.

3

Compute the mean population density per square meter

\frac{180\text{ grasshoppers}}{60\text{ m}^2} = 3\text{ grasshoppers/m}^2

Population density is defined as the total number of organisms counted divided by total sampled area.

4

Extrapolate population density to the total study area

3\text{ grasshoppers/m}^2 \times 12,000\text{ m}^2 = 36,000\text{ grasshoppers}

Multiplying density by the full area of the ecosystem section yields the total estimated population.

Anahtar Kavram

Quadrat Sampling and Population Extrapolation

Soru 28Soru →

Apex Trading PLC has an authorized share capital of $500,000$ ordinary shares of $₦2$ each. The directors issued $60\%$ of these shares to the public. The company subsequently called up $₦1.50$ per issued share. However, shareholders holding $20,000$ shares defaulted on paying a final call of $₦0.50$ per share. Additionally, the company's financial records show $₦150,000$ in $10\%$ debentures, current assets of $₦320,000$ , and current liabilities of $₦140,000$ . What is the total paid-up share capital of the company in Naira?

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Cevap: 440000

Cevap

The total paid-up share capital of the company is ₦440,000.

Paid-up capital is calculated by determining total called-up capital (

300,000 \text{ issued shares} \times ₦1.50 = ₦450,000

) and subtracting unpaid calls (

20,000 \text{ shares} \times ₦0.50 = ₦10,000

), yielding a total paid-up share capital of ₦440,000.

Adım Adım Çözüm

1

Calculate the total number of issued ordinary shares

300,000 shares

Issued capital is the portion of authorized capital offered to subscribers (

500,000 \times 60\% = 300,000

shares).

2

Calculate total called-up capital

₦450,000

Called-up capital is the amount requested by the company from shareholders (

300,000 \text{ shares} \times ₦1.50 = ₦450,000

).

3

Calculate calls in arrears (unpaid called-up capital)

₦10,000

Calls in arrears represent the defaulted portion of called-up capital (

20,000 \text{ shares} \times ₦0.50 = ₦10,000

).

4

Subtract calls in arrears from total called-up capital

₦440,000

Paid-up share capital equals total called-up capital minus calls in arrears (

₦450,000 - ₦10,000 = ₦440,000

).

Anahtar Kavram

Distinction between Authorized, Issued, Called-Up, and Paid-Up Capital

Soru 29Soru →

Find the smallest positive value of $\theta$ (in degrees) satisfying the trigonometric equation $\sin(3\theta) + \sqrt{3}\cos(3\theta) = \sqrt{2}$ .

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The smallest positive value of

\theta

is

25^\circ

.

Dividing the equation by

2

reduces it to

\sin(3\theta + 60^\circ) = \frac{\sqrt{2}}{2}

. The principal acute angle is

45^\circ

. The second quadrant angle giving a positive sine is

180^\circ - 45^\circ = 135^\circ

. Setting

3\theta + 60^\circ = 135^\circ

gives

3\theta = 75^\circ

, which yields

\theta = 25^\circ

. This is smaller than any positive solution generated by the first quadrant branch.

Adım Adım Çözüm

1

Transform the left-hand side into a single harmonic function

R\sin(3\theta + \alpha)

.

Dividing the equation by

2

yields

\frac{1}{2}\sin(3\theta) + \frac{\sqrt{3}}{2}\cos(3\theta) = \frac{\sqrt{2}}{2}

, which simplifies to

\sin(3\theta + 60^\circ) = \frac{\sqrt{2}}{2}

.

The identity

\sin(A + B) = \sin A \cos B + \cos A \sin B

allows us to combine

\sin(3\theta)

and

\cos(3\theta)

using

\cos 60^\circ = \frac{1}{2}

and

\sin 60^\circ = \frac{\sqrt{3}}{2}

.

2

Find the quadrant solutions for the angle

3\theta + 60^\circ

.

First quadrant:

3\theta + 60^\circ = 45^\circ + 360^\circ k \implies 3\theta = -15^\circ + 360^\circ k

.
Second quadrant:

3\theta + 60^\circ = 135^\circ + 360^\circ k \implies 3\theta = 75^\circ + 360^\circ k

.

Since the sine of the angle is positive (

\frac{\sqrt{2}}{2}

), solutions exist in both the first (

45^\circ

) and second (

180^\circ - 45^\circ = 135^\circ

) quadrants.

3

Calculate the smallest positive angle

\theta

.

For

k = 0

in the second quadrant branch,

3\theta = 75^\circ \implies \theta = 25^\circ

. (The first quadrant branch yields

\theta = -5^\circ

for

k=0

and

\theta = 115^\circ

for

k=1

). Thus,

\theta = 25^\circ

is the smallest positive solution.

Comparing all non-negative resulting angles demonstrates that

25^\circ

is the smallest strictly positive solution.

Anahtar Kavram

Solving linear trigonometric equations of the form

A\sin x + B\cos x = C

using

R

-formula reduction.

Soru 30Soru →

Given that $245_x - 157_x = 66_x$ , find the value of the base $x$ .

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of the base

x

is 8.

Expanding all terms in powers of

x

gives

2x^2 + 4x + 5 - (x^2 + 5x + 7) = 6x + 6

. Simplifying gives

x^2 - 7x - 8 = 0

, which factors into

(x - 8)(x + 1) = 0

. Since a number base must be a positive integer greater than any digit present in the equation (the maximum digit here is 7), the only valid base is

x = 8

.

Adım Adım Çözüm

1

Convert each positional number into its base 10 polynomial expansion.

245_x = 2x^2 + 4x + 5

,

157_x = x^2 + 5x + 7

, and

66_x = 6x + 6

.

A number

d_2 d_1 d_0

in base

x

represents

d_2 x^2 + d_1 x^1 + d_0 x^0

in base 10.

2

Substitute the expanded expressions into the given subtraction equation.

(2x^2 + 4x + 5) - (x^2 + 5x + 7) = 6x + 6

This translates the base

x

relationship into a standard base 10 equation.

3

Simplify and rearrange into standard quadratic form.

x^2 - 7x - 8 = 0

Expanding the subtraction yields

x^2 - x - 2 = 6x + 6

. Subtracting

(6x + 6)

from both sides produces a quadratic set to zero.

4

Factorize the quadratic expression.

(x - 8)(x + 1) = 0 \implies x = 8 \text{ or } x = -1

Finding the roots provides potential mathematical values for

x

.

5

Apply number base constraints to select the valid root.

x = 8

A valid base must be a positive integer strictly greater than any individual digit in the expression. Since 7 appears in

157_x

,

x > 7

, ruling out

-1

and confirming

x = 8

.

Anahtar Kavram

Converting numbers from an unknown base

x

to base 10 polynomials to solve algebraic equations.

Soru 31Soru →

Find the number of integer values of $x$ that satisfy the inequality $3x^2 - 10x - 8 \leq 0$ .

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The number of integer values of

x

satisfying the inequality is 5.

Factoring

3x^2 - 10x - 8 \leq 0

gives

(3x + 2)(x - 4) \leq 0

. The region where the quadratic expression is non-positive lies between the roots

x = -\frac{2}{3}

and

x = 4

, yielding

-\frac{2}{3} \leq x \leq 4

. The integers falling within this closed interval are

0, 1, 2, 3,

and

4

, giving a total of 5 integer solutions.

Adım Adım Çözüm

1

Factor the quadratic expression

(3x + 2)(x - 4) \leq 0

Factoring allows us to find the critical boundary values of the inequality.

2

Find the critical values (roots)

x = -\frac{2}{3}

and

x = 4

Setting each linear factor to zero determines where the expression changes sign.

3

Determine the solution set interval

-\frac{2}{3} \leq x \leq 4

Since the coefficient of

x^2

is positive, the quadratic curve is convex (U-shaped), so the expression is less than or equal to zero between the roots.

4

List and count the integer solutions

Integers:

0, 1, 2, 3, 4

(Total = 5)

The smallest integer greater than or equal to

-\frac{2}{3}

is

0

, and the largest integer less than or equal to

4

is

4

.

Anahtar Kavram

Solving quadratic inequalities and identifying integer solutions within a continuous range.

Soru 32Soru →

The derived SI unit of force is the newton ( $\text{N}$ ), which can be expressed in terms of fundamental SI base units as $\text{kg}\cdot\text{m}\cdot\text{s}^{x}$ . What is the numerical value of the exponent $x$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The numerical value of the exponent

x

for time in the base SI expression of the newton is

-2

.

Force is defined by Newton's second law as mass times acceleration (

F = ma

). In SI fundamental units, mass is measured in kilograms (

\text{kg}

) and acceleration in meters per second squared (

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

). Combining these yields

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

. Therefore, the power of time in seconds (

x

) is

-2

.

Adım Adım Çözüm

1

Identify the relationship between force and fundamental quantities

Force is defined as mass multiplied by acceleration (

F = ma

)

This relates force to fundamental quantities of mass, length, and time.

2

Substitute the SI base units of mass and acceleration

Mass unit is

\text{kg}

; acceleration unit is

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

Kilogram, meter, and second are standard fundamental SI units.

3

Determine the exponent of seconds

The combined base unit expression is

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

, giving

x = -2

Equating the powers of seconds gives

x = -2

.

Anahtar Kavram

Expressing derived units in terms of fundamental SI base units

Soru 33Soru →

The derived SI unit of electric potential, the volt ( $\text{V}$ ), can be expressed in terms of fundamental SI base units as $\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d$ . What is the value of the exponent $c$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The exponent

c

corresponding to the base unit second (

\text{s}

) is

-3

.

Electric potential difference (

V

) is defined as work (

W

) per unit charge (

Q

). In terms of SI base units, work has units of

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

and charge has units of

\text{A} \cdot \text{s}

. Dividing work by charge gives

\frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{A} \cdot \text{s}} = \text{kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

. Equating this to

\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d

yields

c = -3

.

Adım Adım Çözüm

1

Relate electric potential to work and charge

\text{1 V} = \frac{\text{1 J}}{\text{1 C}}

Electric potential difference is defined as work done per unit electric charge.

2

Break down Joules into base SI units

\text{1 J} = \text{1 N} \cdot \text{m} = (\text{1 kg} \cdot \text{m} \cdot \text{s}^{-2}) \cdot \text{m} = \text{1 kg} \cdot \text{m}^2 \cdot \text{s}^{-2}

Work is force multiplied by displacement, and force is mass multiplied by acceleration.

3

Break down Coulombs into base SI units

\text{1 C} = \text{1 A} \cdot \text{s}

Electric charge is electric current multiplied by time.

4

Substitute base unit expressions into the ratio for electric potential

\text{1 V} = \frac{\text{1 kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{1 A} \cdot \text{s}} = \text{1 kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

Combining powers of base units yields the complete fundamental representation.

5

Determine the exponent value

c

c = -3

Comparing

\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d

to

\text{kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

shows that the exponent of

\text{s}

is

-3

.

Anahtar Kavram

Expressing derived units in terms of fundamental SI base units

Soru 34Soru →

If $\log_4(x + 2) - \log_{16}(x + 2) = 1$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The value of

x

is 14.

Applying the change of base formula converts

\log_{16}(x + 2)

into

\frac{1}{2}\log_4(x + 2)

. The given equation reduces to

\frac{1}{2}\log_4(x + 2) = 1

, which means

\log_4(x + 2) = 2

. Writing this in exponential form yields

x + 2 = 4^2 = 16

, giving

x = 14

.

Adım Adım Çözüm

1

Change the base of

\log_{16}(x + 2)

to base 4

\log_{16}(x + 2) = \frac{\log_4(x + 2)}{\log_4 16} = \frac{1}{2}\log_4(x + 2)

Using the change of base identity

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

allows all logarithmic terms to share base 4.

2

Substitute and simplify the expression on the left-hand side

\log_4(x + 2) - \frac{1}{2}\log_4(x + 2) = \frac{1}{2}\log_4(x + 2) = 1

Subtracting half of the term from the whole term leaves half of the term.

3

Isolate the logarithmic expression

\log_4(x + 2) = 2

Multiplying both sides of the equation by 2.

4

Convert from logarithmic to index form

x + 2 = 4^2 = 16

By definition,

\log_b a = c \iff b^c = a

.

5

Solve for the variable

x

x = 14

Subtracting 2 from both sides gives the final value of

x

.

Anahtar Kavram

Logarithms and Change of Base

Soru 35Soru →

The table below shows the production possibilities for an economy producing Good X and Good Y:

Combination	Good X (units)	Good Y (units)
A	0	50
B	10	45
C	20	35
D	30	20
E	40	0

What is the opportunity cost of increasing the production of Good X from 20 units to 30 units, expressed in units of Good Y?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The opportunity cost of increasing the production of Good X from 20 units to 30 units is 15 units of Good Y.

Moving from Combination C to Combination D on the Production Possibility Curve increases the output of Good X from 20 units to 30 units. To achieve this, the economy must reduce its output of Good Y from 35 units to 20 units. The opportunity cost is the amount of Good Y given up, which equals

35 - 20 = 15

units of Good Y.

Adım Adım Çözüm

1

Find the initial production level of Good Y at Combination C (Good X = 20 units).

35 units of Good Y

This establishes the baseline production of Good Y prior to increasing Good X.

2

Find the new production level of Good Y at Combination D (Good X = 30 units).

20 units of Good Y

This establishes the remaining production of Good Y after resources are reallocated to produce 10 additional units of Good X.

3

Calculate the difference in Good Y output between Combination C and Combination D.

35 - 20 = 15

units of Good Y

Opportunity cost is defined as the foregone quantity of Good Y required to obtain the additional units of Good X.

Anahtar Kavram

Calculating Opportunity Cost from a Production Possibility Schedule

Soru 36Soru →

On 1st April 2025, Horizon Ltd issued $\text{₦}800,000$ , $12\%$ debentures at a discount of $5\%$ . Interest is payable semi-annually on 30th September and 31st March. What is the total amount of debenture interest (in $\text{₦}$ ) to be charged to the Profit and Loss Account for the financial year ended 31st December 2025?

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Cevap: 72000

Cevap

The total debenture interest to be charged to the Profit and Loss Account for the financial year ended 31st December 2025 is ₦72,000.

Debenture interest is computed at the stated annual rate of

12\%

on the nominal value of

\text{₦}800,000

, resulting in an annual interest of

\text{₦}96,000

. Because the debentures were issued on 1st April 2025 and the financial year ends on 31st December 2025, interest accrued for only 9 months. The total interest charged to the Profit and Loss Account is

\text{₦}96,000 \times \frac{9}{12} = \text{₦}72,000

.

Adım Adım Çözüm

1

Calculate annual debenture interest on nominal value.

Annual interest = 12% of ₦800,000 = ₦96,000.

Debenture interest is calculated on nominal (par) value, regardless of whether debentures are issued at par, premium, or discount.

2

Determine the time period between debenture issue date and financial year-end.

From 1st April 2025 to 31st December 2025 is 9 months.

Interest expense must be recognized for the portion of the year the debentures were outstanding.

3

Calculate time-apportioned debenture interest expense.

₦96,000 × (9 / 12) = ₦72,000.

The Profit and Loss Account requires the exact accrued expense for the 9 active months of the accounting period.

Anahtar Kavram

Time Apportionment of Debenture Interest on Nominal Value

Soru 37Soru →

The following balances were extracted from the books of Zenith Nigeria Plc as at 31st December 2025:

Account Item	Amount (₦)
Issued Share Capital (Ordinary shares of ₦1 each)	500,000
Share Premium	50,000
General Reserve	30,000
10% Debentures	100,000
Non-Current Assets (Net Book Value)	620,000
Current Assets	180,000
Current Liabilities	70,000

What is the balance of the Retained Earnings (Profit and Loss Account) as at 31st December 2025?

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Cevap: 50000

Cevap

The balance of Retained Earnings as at 31st December 2025 is ₦50,000.

Under the company accounting framework, Net Assets equals total Shareholders' Equity. Total Assets of ₦800,000 less Current Liabilities of ₦70,000 and Non-Current Liabilities of ₦100,000 leaves Net Assets of ₦630,000. Subtracting the known equity items (Share Capital ₦500,000 + Share Premium ₦50,000 + General Reserve ₦30,000 = ₦580,000) yields the Retained Earnings balance of ₦50,000.

Adım Adım Çözüm

1

Calculate Total Assets

₦800,000

Total Assets is the sum of Non-Current Assets (₦620,000) and Current Assets (₦180,000).

2

Calculate Net Assets (Capital Employed by Equity)

₦630,000

Net Assets is determined by deducting Current Liabilities (₦70,000) and Non-Current Liabilities (₦100,000) from Total Assets (₦800,000).

3

Sum known Shareholders' Equity components

₦580,000

Add Share Capital (₦500,000), Share Premium (₦50,000), and General Reserve (₦30,000).

4

Deduct known equity components from Net Assets to solve for Retained Earnings

₦50,000

Retained Earnings = Net Assets (₦630,000) - Known Equity (₦580,000) = ₦50,000.

Anahtar Kavram

Accounting Equation for Company Statement of Financial Position

Soru 38Soru →

The following financial information was extracted from the single-entry records of Kalu, a sole trader, for the year ended 31 December 2025:

- Opening creditors balance (1 January 2025): $₦24,500$
- Payments made to creditors by cheque: $₦115,000$
- Discount received from suppliers: $₦3,200$
- Returns outwards: $₦4,800$
- Contra set-off with sales ledger: $₦2,500$
- Closing creditors balance (31 December 2025): $₦31,000$

What is the total value of credit purchases for the year in Naira?

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Cevap: 132000

Cevap

The total value of credit purchases for the year is

₦132,000

.

By reconstructing the Purchases Ledger Control Account, the debit side total is

₦156,500

(comprising cheque payments of

₦115,000

, discount received of

₦3,200

, returns outwards of

₦4,800

, contra set-off of

₦2,500

, and closing balance of

₦31,000

). Subtracting the opening balance of

₦24,500

gives the total credit purchases of

₦132,000

.

Adım Adım Çözüm

1

Determine the debit side total of the Purchases Ledger Control Account

Debit Total =

115,000 + 3,200 + 4,800 + 2,500 + 31,000 = ₦156,500

All transactions that reduce liabilities (payments, discounts received, returns, set-offs) and the closing balance are debited to the control account.

2

Calculate the missing credit purchases balancing figure

Credit Purchases =

156,500 - 24,500 = ₦132,000

Credit purchases increase the liability to creditors and represent the credit entry needed to balance the Purchases Ledger Control Account against the opening credit balance.

Anahtar Kavram

Reconstructing the Purchases Ledger Control Account to find missing credit purchases.

Soru 39Soru →

A sole trader extracted the following ledger balances and adjustment details at the end of the accounting period:

Account / Information	Amount ( $\text{N}$ )
Gross Profit	$120,000$
Discount Received	$4,500$
Rent Paid	$18,000$
Salaries and Wages Paid	$32,000$
General Expenses	$15,000$
Trade Debtors	$50,000$

Additional information at year end:
1. Rent paid includes $\text{N}3,000$ prepaid for the next financial period.
2. Accrued salaries and wages at the end of the year amount to $\text{N}4,000$ .
3. A provision for doubtful debts is to be created at $5\%$ of trade debtors.

What is the net profit for the year in Naira ( $\text{N}$ )?

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Cevap: 56000

Cevap

The net profit for the year is N56,000.

To determine the net profit, add revenues (Gross profit of N120,000 + Discount received of N4,500 = N124,500) and subtract all adjusted operating expenses (Rent of N15,000 + Salaries of N36,000 + General expenses of N15,000 + Doubtful debts provision of N2,500 = N68,500). Net profit equals N124,500 - N68,500 = N56,000.

Adım Adım Çözüm

1

Calculate Total Income

Total Income = N124,500

Discount received is revenue income and must be added to gross profit in the Profit and Loss Account.

2

Adjust Rent for prepayment

Adjusted Rent = N15,000

Prepayments relate to the next accounting period and must be deducted from rent paid during the current period.

3

Adjust Salaries and Wages for accrual

Adjusted Salaries = N36,000

Accrued expenses represent benefits consumed but not yet paid, so they must be added to wages paid under the matching concept.

4

Compute Provision for Doubtful Debts

Provision = N2,500

Creating a 5% provision on trade debtors of N50,000 creates an operating expense of 0.05 * 50,000 = N2,500.

5

Sum total adjusted operating expenses

Total Expenses = N68,500

Total operating expenses include rent (N15,000), salaries (N36,000), general expenses (N15,000), and provision for doubtful debts (N2,500).

6

Deduct total expenses from total income to find Net Profit

Net Profit = N56,000

Net Profit is computed as Total Income (N124,500) minus Total Operating Expenses (N68,500).

Anahtar Kavram

Net Profit Determination with Year-End Adjustments

Soru 40Soru →

The following financial records were extracted from the books of Bayo Craftsmanship Ltd for the year ended 31 December 2025:

Account Item	Amount ( $\text{₦}$ )
Stock of raw materials (1 Jan 2025)	$15,000$
Stock of raw materials (31 Dec 2025)	$12,000$
Purchases of raw materials	$85,000$
Carriage inwards on raw materials	$4,000$
Direct factory wages	$45,000$
Production royalties	$8,000$
Factory rent	$18,000$
Depreciation of factory plant	$6,000$

Calculate the total Prime Cost for the year in Naira ( $\text{₦}$ ).

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Cevap: 145000

Cevap

The Prime Cost for the year is ₦145,000.

Prime Cost is the sum total of all direct expenses incurred in production. It is calculated as Cost of Raw Materials Consumed (

15,000 + 85,000 + 4,000 - 12,000 = 92,000

) plus Direct Factory Wages (

45,000

) plus Production Royalties (

8,000

), yielding

\text{₦}145,000

. Indirect costs like factory rent and depreciation are overheads and must be excluded.

Adım Adım Çözüm

1

Calculate the Cost of Raw Materials Consumed

₦92,000

Cost of raw materials consumed is found by adding carriage inwards to purchases of raw materials, adding opening stock, and deducting closing stock:

15,000 + 85,000 + 4,000 - 12,000 = 92,000

.

2

Identify all direct manufacturing cost components

Raw Materials Consumed (₦92,000), Direct Factory Wages (₦45,000), and Production Royalties (₦8,000)

Prime Cost consists strictly of direct costs (Direct Materials + Direct Labour + Direct Expenses).

3

Sum the direct costs to arrive at Prime Cost

₦145,000

Adding raw materials consumed, direct labor, and production royalties:

92,000 + 45,000 + 8,000 = 145,000

.

Anahtar Kavram

Prime Cost consists of the sum of direct materials consumed, direct wages/labour, and direct expenses (such as royalties). Indirect expenses like factory rent and plant depreciation belong to factory overheads.

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