All practice questions

Determine the area under the curve $y = 3x^2 - 3$ above the $x$ -axis between $x = 1$ and $x = 3$ .

Show answer & explanation

Answer: 20

Answer

The area under the curve between

x = 1

and

x = 3

is 20 square units.

The area under the curve

y = 3x^2 - 3

from

x = 1

to

x = 3

is obtained by calculating the definite integral

\int_{1}^{3} (3x^2 - 3) \, dx = [x^3 - 3x]_{1}^{3} = (3^3 - 3(3)) - (1^3 - 3(1)) = 18 - (-2) = 20

.

Step-by-Step Solution

1

Set up the definite integral representing the area under the curve.

A = \int_{1}^{3} (3x^2 - 3) \, dx

The area bounded by a non-negative curve

y = f(x)

, the

x

-axis, and vertical lines

x = a

and

x = b

is given by

\int_{a}^{b} f(x) \, dx

.

2

Integrate the polynomial function term by term.

\int (3x^2 - 3) \, dx = x^3 - 3x + C

Applying the power rule of integration:

\int 3x^2 dx = x^3

and

\int 3 dx = 3x

.

3

Apply the Fundamental Theorem of Calculus by substituting the limits of integration.

[x^3 - 3x]_{1}^{3} = (3^3 - 3(3)) - (1^3 - 3(1)) = 18 - (-2) = 20

Evaluating

F(b) - F(a)

gives

(27 - 9) - (1 - 3) = 18 - (-2) = 20

.

Key Concept

Definite Integrals and Area Under Curves

Question 542Question →

A Nigerian exporter sells cocoa valued at $2,500,000$ Naira ( $\text{NGN}$ ) to an importer in the United States. If the prevailing foreign exchange rate is $1\text{ USD} = 1,250\text{ NGN}$ , how much will the importer pay in US Dollars ( $\text{USD}$ )?

Show answer & explanation

Answer: 2000

Answer

The importer will pay 2,000 USD.

To convert an amount expressed in domestic currency (Naira) to a foreign currency (US Dollars), divide the total domestic value by the exchange rate. Dividing

2,500,000\text{ NGN}

by

1,250\text{ NGN per USD}

gives

2,000\text{ USD}

.

Step-by-Step Solution

1

Identify the values given in the problem statement.

Total export value =

2,500,000\text{ NGN}

; Exchange rate =

1,250\text{ NGN}

per

\text{USD}

.

Establishing the target foreign currency and domestic currency values is the essential first step in exchange rate conversion.

2

Divide the amount in domestic currency by the exchange rate per US Dollar.

\frac{2,500,000}{1,250} = 2,000\text{ USD}

Converting from domestic currency to foreign currency requires dividing the domestic currency value by the units of domestic currency per unit of foreign currency.

Key Concept

Foreign Exchange Rate Conversion

Question 543Question →

An X-ray tube used in medical diagnostic imaging produces continuous X-rays with a minimum cutoff wavelength of $0.04125\text{ nm}$ . Given that Planck's constant $h = 6.6 \times 10^{-34}\text{ J}\cdot\text{s}$ , the speed of light $c = 3.0 \times 10^{8}\text{ m/s}$ , and the elementary charge $e = 1.6 \times 10^{-19}\text{ C}$ , what is the accelerating potential difference across the tube in kilovolts (kV)?

Show answer & explanation

Answer: 30

Answer

The accelerating potential difference across the X-ray tube is 30 kV.

By Duane-Hunt's law, the maximum energy of an emitted X-ray photon corresponds to the complete conversion of an electron's kinetic energy acquired across potential difference

V

:

e V = \frac{h c}{\lambda_{\text{min}}}

. Substituting

h = 6.6 \times 10^{-34}\text{ J}\cdot\text{s}

,

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

,

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

, and

\lambda_{\text{min}} = 4.125 \times 10^{-11}\text{ m}

yields

V = 30,000\text{ V} = 30\text{ kV}

.

Step-by-Step Solution

1

Convert the given wavelength into SI units of meters

\lambda_{\text{min}} = 4.125 \times 10^{-11}\text{ m}

Standard SI units are required for calculations involving physical constants.

2

State the Duane-Hunt relation for maximum photon energy

e V = E_{\text{max}} = \frac{h c}{\lambda_{\text{min}}}

The maximum photon energy occurs when all kinetic energy of an accelerating electron is converted into a single X-ray photon.

3

Rearrange the equation and substitute the numerical values to solve for

V

V = \frac{(6.6 \times 10^{-34})(3.0 \times 10^{8})}{(1.6 \times 10^{-19})(4.125 \times 10^{-11})} = 30,000\text{ V}

Evaluates the required accelerating potential difference in volts.

4

Express the final voltage in kilovolts (kV)

V = 30\text{ kV}

The question asks for the answer specifically in kilovolts.

Key Concept

Duane-Hunt Law and Cutoff Wavelength in X-ray Production

Question 544Question →

A beam balance measures the mass of a metal block as $15\text{ kg}$ on Earth. The block is then transported to a planet where the acceleration due to gravity is $3.8\text{ m s}^{-2}$ and suspended from a spring balance calibrated in newtons. If the spring balance has a positive zero error of $+2.0\text{ N}$ (reading $+2.0\text{ N}$ when unloaded), what is the displayed reading on the spring balance in newtons?

Show answer & explanation

Answer: 59

Answer

The displayed reading on the spring balance is

59\text{ N}

.

The beam balance establishes that the block has a constant mass of

15\text{ kg}

. On the planet, the gravitational pull on this mass is

W = 15\text{ kg} \times 3.8\text{ m s}^{-2} = 57\text{ N}

. Because the spring balance reads

+2.0\text{ N}

when unloaded, suspending the block causes the pointer to register

57\text{ N} + 2.0\text{ N} = 59\text{ N}

.

Step-by-Step Solution

1

Determine the true mass using beam balance principles

Mass

m = 15\text{ kg}

An equal-arm beam balance measures invariant scalar mass independently of local gravitational field strength.

2

Calculate the true weight on the planet

Weight

W = 57\text{ N}

Weight is the force of gravity acting on mass, calculated as

W = mg = 15\text{ kg} \times 3.8\text{ m s}^{-2} = 57\text{ N}

.

3

Incorporate the positive zero error to find the scale pointer reading

Displayed reading =

59\text{ N}

A positive zero error means the scale indicates

+2.0\text{ N}

when no load is attached. Therefore, Scale Reading = Actual Weight + Zero Offset =

57\text{ N} + 2.0\text{ N} = 59\text{ N}

.

Key Concept

Mass is an intrinsic property measured by a beam balance, whereas weight is a force measured by a spring balance and affected by zero errors.

Estimated Time:1m 0s

Question 545Question →

A fixed mass of hydrogen gas occupies a volume of $600\text{ cm}^3$ at $27^\circ\text{C}$ . What is its volume in $\text{cm}^3$ when the temperature is raised to $127^\circ\text{C}$ at constant pressure?

Show answer & explanation

Answer: 800

Answer

800

Converting initial and final temperatures to Kelvin gives

300\text{ K}

and

400\text{ K}

respectively. Applying Charles's Law (

V_1 / T_1 = V_2 / T_2

) gives

V_2 = 600 \times (400 / 300) = 800\text{ cm}^3

.

Step-by-Step Solution

1

Convert given temperatures from Celsius to Kelvin

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

and

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

Charles's Law requires temperature values to be expressed on the absolute (Kelvin) temperature scale.

2

Set up Charles's Law formula relating volume and absolute temperature

\frac{V_1}{T_1} = \frac{V_2}{T_2} \implies V_2 = V_1 \times \frac{T_2}{T_1}

At constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature.

3

Substitute the initial volume and absolute temperatures into the formula

V_2 = 600 \times \frac{400}{300} = 800\text{ cm}^3

Simplifying the expression yields the final gas volume.

Key Concept

Charles's Law and Absolute Temperature Scale

Question 546Question →

A rigid steel cylinder contains methane gas at a pressure of $150\text{ kPa}$ and a temperature of $47^\circ\text{C}$ . If the cylinder is heated to $127^\circ\text{C}$ while maintaining a constant volume, what is the final pressure of the gas in $\text{kPa}$ ?

Show answer & explanation

Answer: 187.5

Answer

The final pressure of the methane gas is

187.5\text{ kPa}

.

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

P \propto T

). Converting temperatures to Kelvin gives

T_1 = 320\text{ K}

and

T_2 = 400\text{ K}

. Applying

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

yields

\frac{150}{320} = \frac{P_2}{400}

, which gives

P_2 = 187.5\text{ kPa}

.

Step-by-Step Solution

1

Convert given temperatures to the Kelvin absolute scale.

T_1 = 47 + 273 = 320\text{ K}

and

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

All gas law calculations require temperatures to be expressed in Kelvin.

2

Set up the Pressure Law equation relating pressure and absolute temperature at constant volume.

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

According to Gay-Lussac's Law, pressure is directly proportional to absolute temperature when volume is fixed.

3

Substitute the known values into the equation and solve for

P_2

.

P_2 = \frac{150 \times 400}{320} = 187.5\text{ kPa}

Cross-multiplication gives

P_2 = 187.5\text{ kPa}

.

Key Concept

Pressure Law (Gay-Lussac's Law)

Question 547Question →

During the thermal decomposition of dinitrogen pentoxide, $2\text{N}_2\text{O}_5(g) \rightarrow 4\text{NO}_2(g) + \text{O}_2(g)$ , the concentration of $\text{N}_2\text{O}_5$ decreases from $0.80\text{ mol dm}^{-3}$ to $0.20\text{ mol dm}^{-3}$ over a time interval of $5\text{ minutes}$ . What is the average rate of decomposition of $\text{N}_2\text{O}_5$ in $\text{mol dm}^{-3}\text{s}^{-1}$ ?

Show answer & explanation

Answer: 0.002

Answer

0.002 mol dm⁻³ s⁻¹

The average rate of decomposition of a reactant is calculated by dividing the decrease in its concentration by the elapsed time in seconds. Here, the concentration change is

\Delta C = 0.80 - 0.20 = 0.60\text{ mol dm}^{-3}

and the time in seconds is

5 \times 60 = 300\text{ s}

. Thus, the rate is

\frac{0.60}{300} = 0.002\text{ mol dm}^{-3}\text{s}^{-1}

.

Step-by-Step Solution

1

Calculate the change in concentration of the reactant

\Delta [\text{N}_2\text{O}_5] = 0.80\text{ mol dm}^{-3} - 0.20\text{ mol dm}^{-3} = 0.60\text{ mol dm}^{-3}

The rate depends on the amount of reactant consumed during the reaction period.

2

Convert time from minutes to seconds

\Delta t = 5\text{ min} \times 60\text{ s/min} = 300\text{ s}

The requested unit is per second (

\text{s}^{-1}

), so time must be converted to standard SI units.

3

Divide the concentration change by the total time in seconds

\text{Rate} = \frac{0.60\text{ mol dm}^{-3}}{300\text{ s}} = 0.002\text{ mol dm}^{-3}\text{s}^{-1}

Average rate of reaction is defined as the change in concentration per unit time.

Key Concept

Average Rate of Reaction from Concentration Change

Question 548Question →

A specific tax of $\text{₦}80$ per unit is imposed on a market commodity. The price elasticity of demand for the commodity is $0.6$ , while its price elasticity of supply is $1.4$ . What is the tax burden per unit borne by the consumer in Naira?

Show answer & explanation

Answer: 56

Answer

The tax burden per unit borne by the consumer is 56 Naira.

The incidence of tax on consumers depends on relative elasticity. The formula for the consumer's burden is

T \times \frac{E_s}{E_d + E_s}

. Substituting the given values gives

80 \times \frac{1.4}{0.6 + 1.4} = 80 \times 0.7 = 56

Naira.

Step-by-Step Solution

1

Identify given variables and elasticity values.

Tax per unit (

T

) =

\text{₦}80

, Price elasticity of demand (

E_d

) =

0.6

, Price elasticity of supply (

E_s

) =

1.4

.

These values determine the proportion of the tax burden shifted to buyers versus sellers.

2

Apply the tax incidence formula for the consumer's share.

\text{Consumer Share} = T \times \left( \frac{E_s}{E_d + E_s} \right)

Tax incidence on consumers is directly proportional to supply elasticity relative to the sum of demand and supply elasticities.

3

Compute the numerical value.

\text{Consumer Share} = 80 \times \left( \frac{1.4}{0.6 + 1.4} \right) = 80 \times 0.7 = 56

Multiplying the per-unit tax by the consumer incidence proportion yields the exact burden per unit in Naira.

Key Concept

Tax Incidence and Price Elasticity

Question 549Question →

In a given financial year, an economy records a Net National Product ( $\text{NNP}$ ) of $\text{N}6,200\text{ million}$ and a Capital Consumption Allowance (Depreciation) of $\text{N}450\text{ million}$ . The factor income earned by domestic citizens from abroad is $\text{N}320\text{ million}$ , while the factor income paid to foreign residents within the economy is $\text{N}510\text{ million}$ . What is the value of the Gross Domestic Product ( $\text{GDP}$ ) of this economy in million Naira?

Show answer & explanation

Answer: 6840

Answer

6840 million Naira

To calculate Gross Domestic Product (GDP), first derive Gross National Product (GNP) by adding Capital Consumption Allowance (Depreciation) to Net National Product (NNP):

\text{GNP} = 6,200 + 450 = 6,650\text{ million Naira}

. Next, compute Net Factor Income from Abroad (NFIA) as factor income received from abroad minus factor income paid to foreigners:

\text{NFIA} = 320 - 510 = -190\text{ million Naira}

. Finally, apply the national accounting identity

\text{GNP} = \text{GDP} + \text{NFIA}

, rearranged as

\text{GDP} = \text{GNP} - \text{NFIA} = 6,650 - (-190) = 6,840\text{ million Naira}

.

Step-by-Step Solution

1

Calculate Gross National Product (GNP) from Net National Product (NNP) and Depreciation

GNP = 6,200 + 450 = 6,650 million Naira

Gross aggregates include depreciation, whereas net aggregates exclude it:

\text{GNP} = \text{NNP} + \text{Depreciation}

.

2

Determine Net Factor Income from Abroad (NFIA)

NFIA = 320 - 510 = -190 million Naira

NFIA is defined as factor income earned from abroad by citizens minus factor income paid to foreign residents domestically.

3

Determine Gross Domestic Product (GDP) using GNP and NFIA

GDP = 6,650 - (-190) = 6,840 million Naira

Because

\text{GNP} = \text{GDP} + \text{NFIA}

, rearranging gives

\text{GDP} = \text{GNP} - \text{NFIA}

. Subtracting a negative value is equivalent to adding its positive magnitude.

Key Concept

Relationship between basic national income aggregates (GDP, GNP, NNP, NDP), Depreciation, and Net Factor Income from Abroad (NFIA).

Question 550Question →

A specific indirect tax of $\text{₦}50$ per unit is levied on a commodity. If the price elasticity of demand for the commodity is $1.5$ and the price elasticity of supply is $0.5$ , how much of the tax per unit (in Naira) is borne by the producer?

Show answer & explanation

Answer: 37.5

Answer

The producer bears 37.5 Naira per unit of the tax burden.

The economic incidence of a specific tax depends on the relative price elasticities of demand (

E_d

) and supply (

E_s

). The producer's share per unit is calculated as

T \times \frac{E_d}{E_d + E_s}

. Substituting

T = 50

,

E_d = 1.5

, and

E_s = 0.5

yields

50 \times \frac{1.5}{2.0} = 37.5

Naira.

Step-by-Step Solution

1

Identify the tax incidence formula for the producer's share

Formula:

\text{Producer's Burden} = T \times \left(\frac{E_d}{E_d + E_s}\right)

Tax burden distribution between buyers and sellers depends inversely on their relative price elasticities.

2

Substitute the values into the equation

\text{Producer's Burden} = 50 \times \left(\frac{1.5}{1.5 + 0.5}\right) = 50 \times 0.75 = 37.5

With elastic demand (

E_d = 1.5

) relative to inelastic supply (

E_s = 0.5

), the producer absorbs

75\%

of the tax.

Key Concept

Tax Incidence and Relative Elasticity of Demand and Supply

Question 551Question →

On a topographic map, index contours are drawn at elevations of $400\text{ m}$ and $700\text{ m}$ . There are $5$ equal contour intervals (spaces) between these two index contours. A radio transmission mast is located on a contour line situated $3$ contour intervals higher than the $700\text{ m}$ index contour. What is the elevation, in meters, of the contour line where the radio mast is located?

Show answer & explanation

Answer: 880

Answer

The elevation of the contour line where the radio mast stands is

880\text{ m}

.

The elevation difference between the

400\text{ m}

and

700\text{ m}

index lines is

300\text{ m}

. With

5

equal spaces separating them, the contour interval is

60\text{ m}

. Adding

3

contour intervals (

180\text{ m}

) to the

700\text{ m}

index line yields an elevation of

880\text{ m}

.

Step-by-Step Solution

1

Calculate the elevation difference between the given index contours.

Vertical difference =

700\text{ m} - 400\text{ m} = 300\text{ m}

.

Index contours serve as reference lines to determine the vertical rise across intervening contour spaces.

2

Determine the contour interval (the vertical distance between consecutive contour lines).

Contour Interval =

\frac{300\text{ m}}{5} = 60\text{ m}

.

Dividing the total height difference between index lines by the number of spaces gives the value of a single contour interval.

3

Calculate the target elevation at 3 contour intervals above the

700\text{ m}

index line.

Target elevation =

700\text{ m} + (3 \times 60\text{ m}) = 880\text{ m}

.

Adding three vertical intervals (

180\text{ m}

) to the

700\text{ m}

reference line gives the exact height of the target contour line.

Key Concept

Contour Interval Determination and Relief Elevation Calculation

Question 552Question →

A weather monitoring station situated at longitude $55^\circ\text{W}$ records a local solar time of $08:20\text{ AM}$ . At the exact same instant, a maritime research vessel logs a local solar time of $02:40\text{ PM}$ . What is the longitude of the maritime research vessel in degrees East ( $^\circ\text{E}$ )?

Show answer & explanation

Answer: 40

Answer

The longitude of the maritime research vessel is

40^\circ\text{E}

.

The time difference between

08:20\text{ AM}

(

55^\circ\text{W}

) and

02:40\text{ PM}

(

14:40

) is

6\text{ hours and } 20\text{ minutes}

, which equals

380\text{ minutes}

. At a rate of

4\text{ minutes per degree}

of longitude, this represents a longitudinal difference of

95^\circ

. Since the vessel's local solar time is ahead, it is situated to the East. Measuring

95^\circ

East from

55^\circ\text{W}

crosses the Greenwich Meridian (

0^\circ

) after

55^\circ

, placing the vessel at

40^\circ\text{E}

.

Step-by-Step Solution

1

Calculate the time difference between the weather monitoring station and the maritime research vessel.

Time difference =

14:40 - 08:20 = 6\text{ hours } 20\text{ minutes} = 380\text{ minutes}

.

Converting 02:40 PM to 24-hour solar time (14:40) allows direct subtraction of the times.

2

Convert the time difference into angular degrees of longitude.

Angular distance =

\frac{380\text{ minutes}}{4\text{ minutes per degree}} = 95^\circ

.

The Earth rotates

360^\circ

in 24 hours, which corresponds to

1^\circ

of longitude every 4 minutes.

3

Determine the direction and calculate the vessel's longitude.

95^\circ\text{ East} - 55^\circ\text{ West to Prime Meridian} = 40^\circ\text{E}

.

Because the local solar time at the vessel is ahead (later in the day), the vessel is located to the East. Traveling

95^\circ

East starting from

55^\circ\text{W}

uses

55^\circ

to reach

0^\circ

(Prime Meridian), leaving

40^\circ

in the Eastern Hemisphere.

Key Concept

Longitude calculation from local solar time difference across the Prime Meridian

Estimated Time:1m 30s

Question 553Question →

A commercial bakery acquires a heavy-duty industrial generator under a hire purchase agreement. The cash price of the generator is $₦3,200,000$ . The agreement requires an initial deposit of $25\%$ of the cash price, with the remaining balance paid in $10$ equal monthly installments of $₦270,000$ each. What is the total hire purchase interest charged on this transaction in Naira?

Show answer & explanation

Answer: 300000

Answer

The total hire purchase interest charged on the transaction is 300,000 NGN.

The total financial outlay under the hire purchase agreement consists of the

25\%

deposit (

₦800,000

) plus the

10

monthly installments of

₦270,000

(

₦2,700,000

), totaling

₦3,500,000

. Subtracting the cash price of

₦3,200,000

yields a total hire purchase interest (carrying charge) of

₦300,000

.

Step-by-Step Solution

1

Calculate the initial deposit amount

Initial deposit = ₦800,000

The deposit is 25% of the cash price of ₦3,200,000 (0.25 × 3,200,000 = 800,000).

2

Calculate the cumulative total of all monthly installment payments

Total installments = ₦2,700,000

10 monthly installments of ₦270,000 equal 10 × 270,000 = 2,700,000.

3

Calculate the total hire purchase price

Total hire purchase price = ₦3,500,000

Total Hire Purchase Price = Initial Deposit + Total Installments = 800,000 + 2,700,000 = 3,500,000.

4

Deduct the cash price from the total hire purchase price to determine the total finance/interest charge

Total interest = ₦300,000

Interest = Total Hire Purchase Price - Cash Price = 3,500,000 - 3,200,000 = 300,000.

Key Concept

Calculation of total hire purchase price and carrying charges (interest)

Question 554Question →

A cubic curve defined by $y = ax^3 + bx^2 + cx + d$ has a local maximum at $(-1, 10)$ and a point of inflexion at $(1, 2)$ . What is the value of $y$ at the local minimum of the curve?

Show answer & explanation

Answer: -6

Answer

The local minimum value of

y

on the curve is

-6

.

By setting up a system of equations using the conditions for the local maximum at

(-1, 10)

and the point of inflexion at

(1, 2)

, the cubic curve is uniquely determined as

y = 0.5x^3 - 1.5x^2 - 4.5x + 7.5

. Setting the derivative

\frac{dy}{dx} = 1.5(x-3)(x+1) = 0

gives

x = 3

as the

x

-coordinate of the local minimum. Evaluating

y(3)

gives

-6

.

Step-by-Step Solution

1

Differentiate the general cubic equation to obtain expressions for the first and second derivatives.

\frac{dy}{dx} = 3ax^2 + 2bx + c

and

\frac{d^2y}{dx^2} = 6ax + 2b

.

Stationary points require

\frac{dy}{dx} = 0

and points of inflexion require

\frac{d^2y}{dx^2} = 0

.

2

Apply the point of inflexion conditions at

(1, 2)

.

6a(1) + 2b = 0 \implies b = -3a

, and

a(1)^3 + b(1)^2 + c(1) + d = 2 \implies -2a + c + d = 2

.

At a point of inflexion, the second derivative is zero, and the point lies on the curve.

3

Apply the stationary point and coordinate conditions at the local maximum

(-1, 10)

.

3a(-1)^2 + 2b(-1) + c = 0 \implies 9a + c = 0 \implies c = -9a

, and

-a + b - c + d = 10 \implies 5a + d = 10

.

At a local maximum, the first derivative is zero, and the point lies on the curve.

4

Solve the system of linear equations for coefficients

a, b, c, d

.

a = 0.5

,

b = -1.5

,

c = -4.5

,

d = 7.5

.

Combining

-11a + d = 2

and

5a + d = 10

yields

16a = 8 \implies a = 0.5

.

5

Find the

x

-coordinate of the local minimum by solving

\frac{dy}{dx} = 0

.

1.5x^2 - 3x - 4.5 = 0 \implies 1.5(x - 3)(x + 1) = 0 \implies x = 3

(since

x = -1

is the local maximum).

Evaluating

\frac{d^2y}{dx^2}(3) = 6(0.5)(3) + 2(-1.5) = 6 > 0

confirms a local minimum at

x = 3

.

6

Calculate the value of

y

at

x = 3

.

y = 0.5(3)^3 - 1.5(3)^2 - 4.5(3) + 7.5 = -6

.

Substituting

x = 3

into the curve equation gives the value of

y

at the local minimum.

Key Concept

Determining polynomial coefficients from stationary and inflexion point conditions to find extreme values.

Question 555Question →

If $24_x \times 13_x = 345_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 7

Answer

The value of the base

x

is 7.

Expanding

24_x

,

13_x

, and

345_x

into base 10 yields

(2x + 4)(x + 3) = 3x^2 + 4x + 5

. Expanding the left side gives

2x^2 + 10x + 12

. Equating and simplifying gives

x^2 - 6x - 7 = 0

, which factors as

(x - 7)(x + 1) = 0

. The positive integer solution greater than 5 is

x = 7

.

Step-by-Step Solution

1

Convert all base

x

numbers to decimal (base 10) expressions.

24_x = 2x + 4

,

13_x = x + 3

, and

345_x = 3x^2 + 4x + 5

.

Place-value expansion allows algebraic manipulation in standard base 10.

2

Multiply the expanded factors on the left-hand side.

(2x + 4)(x + 3) = 2x^2 + 10x + 12

.

Applying the distributive property of multiplication.

3

Equate the expanded left-hand side to the right-hand side and rearrange into standard quadratic form.

3x^2 + 4x + 5 - (2x^2 + 10x + 12) = 0 \implies x^2 - 6x - 7 = 0

.

Setting the quadratic expression equal to zero enables factoring.

4

Factor the quadratic equation and select the valid base.

(x - 7)(x + 1) = 0 \implies x = 7

(rejecting

x = -1

).

A number base must be a positive integer strictly greater than any individual digit in the given numbers (max digit is 5).

Key Concept

Solving polynomial equations derived from number base expansion.

Question 556Question →

If $\log_2 (x - 1) + \log_4 (x - 1) + \log_{16} (x - 1) = \frac{7}{2}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Converting all terms to base 2 yields

\log_2(x-1) + \frac{1}{2}\log_2(x-1) + \frac{1}{4}\log_2(x-1) = \frac{7}{4}\log_2(x-1)

. Setting

\frac{7}{4}\log_2(x-1) = \frac{7}{2}

gives

\log_2(x-1) = 2

. Exponentiating both sides in base 2 gives

x - 1 = 2^2 = 4

, which results in

x = 5

.

Step-by-Step Solution

1

Convert each logarithmic term to base 2 using the change of base property.

\log_4(x-1) = \frac{1}{2}\log_2(x-1) \quad \text{and} \quad \log_{16}(x-1) = \frac{1}{4}\log_2(x-1)

Bases 4 and 16 are powers of 2 (

4 = 2^2

and

16 = 2^4

), allowing transformation to a common base.

2

Substitute these equivalent base-2 terms into the original equation.

\log_2(x-1) + \frac{1}{2}\log_2(x-1) + \frac{1}{4}\log_2(x-1) = \frac{7}{2}

This consolidates the equation into a single logarithmic variable,

\log_2(x-1)

.

3

Factor out

\log_2(x-1)

and add the fractional coefficients.

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

Summing the coefficients

1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4}

.

4

Isolate

\log_2(x-1)

and solve for

x

.

\log_2(x-1) = 2 \implies x - 1 = 2^2 = 4 \implies x = 5

Multiplying both sides by

\frac{4}{7}

yields

\log_2(x-1) = 2

, and rewriting in exponential form gives

x = 5

.

Key Concept

Change of base rule for logarithms:

\log_{b^k} a = \frac{1}{k}\log_b a

Question 557Question →

Find the number of distinct arrangements of the letters of the word $\text{PARALLEL}$ such that no two letters $\text{'L'}$ are adjacent.

Show answer & explanation

Answer: 1200

Answer

1200

To ensure no two letters 'L' are adjacent, we use the gap method. First, arrange the 5 non-L letters (P, A, R, A, E). Because 'A' repeats twice, there are 5! / 2! = 60 distinct arrangements. These 5 letters form 6 available gaps (including the ends). Selecting 3 of these 6 gaps to insert the 3 identical 'L's can be done in C(6, 3) = 20 ways. Multiplying these gives 60 × 20 = 1200 valid arrangements.

Step-by-Step Solution

1

Count the frequency of each letter in the word PARALLEL.

The word has 8 letters: 1 P, 2 A's, 1 R, 3 L's, and 1 E.

Recognizing repeated elements is essential for permutations with duplicates.

2

Arrange the non-restricted letters (P, A, R, A, E).

Number of arrangements = 5! / 2! = 60.

The letter 'A' is repeated twice, so we divide 5! by 2!.

3

Calculate the number of available gaps for placing the 3 'L's so that no two are adjacent.

5 arranged letters create 6 gap positions. Choosing 3 gaps gives C(6, 3) = (6 × 5 × 4) / (3 × 2 × 1) = 20 ways.

Placing at most one 'L' per gap guarantees that no two 'L's will be adjacent.

4

Multiply the number of arrangements of non-L letters by the gap choices.

Total arrangements = 60 × 20 = 1200.

By the fundamental counting principle, total arrangements equal the product of independent choices.

Key Concept

Permutations with repeated elements and non-adjacency constraints using the Gap Method

Estimated Time:1m 30s

Question 558Question →

Find the value of

x

that satisfies the exponential equation

\sqrt{\frac{125^{x+1}}{5^{x-1}}} = 25^{x-1}

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Converting all terms to base 5 yields

5^{x+2}

on the left-hand side and

5^{2x-2}

on the right-hand side. Setting the exponents equal gives

x + 2 = 2x - 2

, which solves to

x = 4

.

Step-by-Step Solution

1

Express all terms with a common base of 5

125 = 5^3

and

25 = 5^2

Converting terms to prime base 5 allows the application of standard laws of indices.

2

Simplify the fraction inside the square root

\frac{5^{3x+3}}{5^{x-1}} = 5^{(3x+3) - (x-1)} = 5^{2x+4}

Subtract the denominator exponent from the numerator exponent when dividing like bases.

3

Apply the square root as a fractional exponent

\sqrt{5^{2x+4}} = (5^{2x+4})^{1/2} = 5^{x+2}

Taking the square root of a power is equivalent to multiplying the exponent by 1/2.

4

Equate the simplified exponents of both sides

x + 2 = 2x - 2

With identical bases of 5 on both sides, the exponents must be equal.

5

Solve the linear equation for x

x = 4

Rearranging terms gives

2x - x = 2 + 2

, which yields

x = 4

.

Key Concept

Indices and Laws of Indices

Question 559Question →

A function

f(x)

is defined by

f(x) = \begin{cases} \frac{x^2 + kx - 10}{x - 2}, & x \neq 2 \\ 7, & x = 2 \end{cases}

If

f(x)

is continuous at

x = 2

, what is the numerical value of the constant

k

?

Show answer & explanation

Answer: 3

Answer

The numerical value of the constant

k

is 3.

By definition of continuity,

f(x)

is continuous at

x = 2

if

\lim_{x \to 2} f(x) = f(2) = 7

. As

x \to 2

, the denominator

x - 2

approaches

0

. For the quotient to have a finite limit, the numerator

x^2 + kx - 10

must also evaluate to

0

at

x = 2

, yielding

2^2 + 2k - 10 = 0

. Solving this gives

2k = 6

, so

k = 3

. Substituting

k = 3

gives

\lim_{x \to 2} \frac{(x-2)(x+5)}{x-2} = 7

, confirming that

k = 3

is correct.

Step-by-Step Solution

1

Apply the definition of continuity at

x = 2

\lim_{x \to 2} f(x) = f(2) = 7

A function

f(x)

is continuous at

x = a

if and only if

\lim_{x \to a} f(x) = f(a)

.

2

Set the numerator to zero at the point of discontinuity

x = 2

2^2 + k(2) - 10 = 0

Because the denominator

(x - 2) \to 0

as

x \to 2

, the limit can only exist if the numerator also approaches

0

, forming an indeterminate form

\frac{0}{0}

that can be simplified.

3

Solve for the unknown parameter

k

4 + 2k - 10 = 0 \implies 2k - 6 = 0 \implies k = 3

Linear algebraic equation solving.

4

Verify that the simplified limit equals

f(2)

\lim_{x \to 2} \frac{x^2 + 3x - 10}{x - 2} = \lim_{x \to 2} \frac{(x - 2)(x + 5)}{x - 2} = \lim_{x \to 2} (x + 5) = 7

Canceling the common factor

(x - 2)

yields

7

, which matches

f(2) = 7

.

Key Concept

Continuity of a Piecewise Function and Limit Existence

Question 560Question →

Find the value of

x

that satisfies the exponential equation

\frac{100^{x+1}}{1000^{x-1}} = 10^{2x-1}

Show answer & explanation

Answer: 2

Answer

The value of

x

that satisfies the equation is 2.

Rewriting the terms in base 10 gives

\frac{10^{2x+2}}{10^{3x-3}} = 10^{2x-1}

. Using the division rule of indices yields

10^{-x+5} = 10^{2x-1}

. Equating exponents gives

-x+5 = 2x-1

, which simplifies to

3x = 6

, so

x = 2

.

Step-by-Step Solution

1

Convert each power to base 10

100^{x+1} = (10^2)^{x+1} = 10^{2x+2}

and

1000^{x-1} = (10^3)^{x-1} = 10^{3x-3}

Converting all non-prime composite bases to powers of a common fundamental base allows exponent comparison.

2

Apply the quotient rule of indices to the left side

\frac{10^{2x+2}}{10^{3x-3}} = 10^{(2x+2)-(3x-3)} = 10^{-x+5}

According to the index quotient law

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

, subtract the denominator's exponent from the numerator's exponent.

3

Equate exponents of equal bases

-x + 5 = 2x - 1

If

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

for

a > 0, a \neq 1

, then

f(x) = g(x)

.

4

Solve the linear equation for

x

3x = 6 \implies x = 2

Isolate the variable

x

to find its value.

Key Concept

Exponential equations solvable by converting to a common base

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept