All practice questions

Find the positive integer value of $x$ that satisfies the logarithmic equation $x^{\log_3 x} = 81x^3$ .

Show answer & explanation

Answer: 81

Answer

The positive integer value of

x

is 81.

Taking the logarithm base 3 on both sides transforms

x^{\log_3 x} = 81x^3

into

(\log_3 x)^2 = 4 + 3\log_3 x

. Substituting

u = \log_3 x

yields the quadratic equation

u^2 - 3u - 4 = 0

, which factors as

(u-4)(u+1) = 0

. The roots are

u = 4

(

x = 3^4 = 81

) and

u = -1

(

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

). Since

x

must be a positive integer, the correct value is 81.

Step-by-Step Solution

1

Take logarithm base 3 of both sides of the equation

x^{\log_3 x} = 81x^3

\log_3(x^{\log_3 x}) = \log_3(81x^3)

Taking the logarithm with base 3 allows us to simplify the exponent containing

\log_3 x

.

2

Apply logarithmic identities

\log_b(a^k) = k \log_b a

and

\log_b(mn) = \log_b m + \log_b n

(\log_3 x)^2 = \log_3 81 + 3\log_3 x = 4 + 3\log_3 x

Expanding the products and powers reduces the equation into a single logarithmic variable

\log_3 x

.

3

Substitute

u = \log_3 x

to create a quadratic equation

u^2 - 3u - 4 = 0

Setting

u = \log_3 x

converts the equation into standard quadratic form.

4

Factor the quadratic expression

(u - 4)(u + 1) = 0 \implies u = 4 \text{ or } u = -1

Factoring allows us to find all possible real values for

u

.

5

Convert back to

x

using

x = 3^u

and select the positive integer root

x = 3^4 = 81

or

x = 3^{-1} = \frac{1}{3}

. The positive integer solution is

x = 81

.

The question specifically requests the positive integer solution, eliminating

x = \frac{1}{3}

.

Key Concept

Solving equations with variable exponents by taking logarithms and reducing to a quadratic form.

Question 562Question →

A sample of nitrogen gas enclosed in a constant-volume container exerts a pressure of $2.50\text{ atm}$ at a temperature of $-23^\circ\text{C}$ . To what temperature, in degrees Celsius ( $^\circ\text{C}$ ), must the gas be heated so that its pressure increases to $4.00\text{ atm}$ ?

Show answer & explanation

Answer: 127

Answer

The gas must be heated to

127^\circ\text{C}

.

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

P \propto T

). Converting

-23^\circ\text{C}

to Kelvin gives

250\text{ K}

. Solving

\frac{2.50}{250} = \frac{4.00}{T_2}

gives

T_2 = 400\text{ K}

. Converting back to Celsius (

400 - 273

) yields the correct temperature of

127^\circ\text{C}

.

Step-by-Step Solution

1

Convert the initial temperature from Celsius to the thermodynamic temperature scale (Kelvin).

T_1 = -23 + 273 = 250\text{ K}

Gas laws strictly require temperature to be expressed in Kelvin.

2

Use Gay-Lussac's Pressure Law equation

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

to solve for

T_2

.

T_2 = \frac{P_2 \times T_1}{P_1} = \frac{4.00 \times 250}{2.50} = 400\text{ K}

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

3

Convert the calculated temperature

T_2

back to degrees Celsius.

t_2 = 400 - 273 = 127^\circ\text{C}

The question explicitly requests the final temperature in degrees Celsius.

Key Concept

Pressure Law (Gay-Lussac's Law)

Question 563Question →

The speed $v$ of a longitudinal wave propagating through a gas depends on the pressure $P$ of the gas and its density $\rho$ according to the dimensional relationship $v = C P^x \rho^y$ , where $C$ is a dimensionless constant. Using dimensional analysis, what is the numerical value of $x - y$ ?

Show answer & explanation

Answer: 1

Answer

The numerical value of

x - y

is 1.0.

By applying the principle of dimensional homogeneity, the exponents are determined as

x = 0.5

(for pressure) and

y = -0.5

(for density). Thus,

x - y = 0.5 - (-0.5) = 1.0

.

Step-by-Step Solution

1

Determine the dimensions of speed, pressure, and density.

[v] = [L T^{-1}]

,

[P] = [M L^{-1} T^{-2}]

, and

[\rho] = [M L^{-3}]

.

Dimensional homogeneity requires expressed physical quantities to be broken down into fundamental dimensions (

M

,

L

,

T

).

2

Substitute dimensions into the relationship

v = C P^x \rho^y

.

[L T^{-1}] = [M L^{-1} T^{-2}]^x \, [M L^{-3}]^y = M^{x+y} \, L^{-x-3y} \, T^{-2x}

.

This establishes a system of algebraic equations by equating exponents of corresponding fundamental dimensions.

3

Solve for exponents

x

and

y

.

From time

T

:

-2x = -1 \implies x = 0.5

. From mass

M

:

x + y = 0 \implies y = -0.5

.

Equating the powers of fundamental dimensions on both sides yields the values of

x

and

y

.

4

Calculate the required expression

(x - y)

.

x - y = 0.5 - (-0.5) = 1.0

.

Subtracting negative

0.5

from

0.5

results in

1.0

.

Key Concept

Dimensional Analysis and Homogeneity

Question 564Question →

A naturally occurring sample of boron consists of two stable isotopes, $^{10}\text{B}$ and $^{11}\text{B}$ . If the relative percentage abundance of $^{10}\text{B}$ is $20.0\%$ and that of $^{11}\text{B}$ is $80.0\%$ , what is the relative atomic mass of boron?

Show answer & explanation

Answer: 10.8

Answer

The relative atomic mass of boron is

10.8

.

The relative atomic mass of an element is defined as the weighted average mass of its naturally occurring isotopes relative to

\frac{1}{12}\text{th}

the mass of a carbon-12 atom. Applying the formula

\text{RAM} = \frac{\sum (\text{isotopic mass} \times \% \text{ abundance})}{100}

, we get

\frac{(10 \times 20) + (11 \times 80)}{100} = \frac{200 + 880}{100} = 10.8

.

Step-by-Step Solution

1

Determine the mass contribution of the

^{10}\text{B}

isotope

10 \times 0.20 = 2.0

The weighted contribution of an isotope is its mass multiplied by its fractional abundance.

2

Determine the mass contribution of the

^{11}\text{B}

isotope

11 \times 0.80 = 8.8

The weighted contribution of the second isotope is calculated using its percentage abundance.

3

Sum the weighted contributions to find the relative atomic mass

2.0 + 8.8 = 10.8

The relative atomic mass of an element is the weighted average mass of all naturally occurring isotopes relative to carbon-12.

Key Concept

Calculation of Relative Atomic Mass from Isotopic Abundances

Question 565Question →

Two sound waves of frequencies $440\text{ Hz}$ and $445\text{ Hz}$ travel through air and superpose to produce beats. What is the resulting beat frequency, in hertz?

Show answer & explanation

Answer: 5

Answer

The beat frequency produced by the superposition of the two sound waves is

5\text{ Hz}

.

When two waves of slightly different frequencies interfere, periodic variations in sound intensity called beats occur. The number of beats heard per second is equal to the absolute difference between the frequencies of the two superposing waves:

f_{\text{beat}} = |f_2 - f_1| = |445\text{ Hz} - 440\text{ Hz}| = 5\text{ Hz}

.

Step-by-Step Solution

1

Identify the frequencies of the interfering waves

f_1 = 440\text{ Hz}

and

f_2 = 445\text{ Hz}

Beat frequency is determined by the absolute difference between the individual wave frequencies.

2

Subtract the lower frequency from the higher frequency to find the beat frequency

f_{\text{beat}} = |445 - 440| = 5\text{ Hz}

The rate of periodic intensity variation (beat frequency) is governed by

f_{\text{beat}} = |f_2 - f_1|

.

Key Concept

Beat Frequency and Wave Superposition

Question 566Question →

A metal surface inside a vacuum cell is illuminated by incident photons each having an energy of $6.2\text{ eV}$ . If the maximum kinetic energy of the emitted photoelectrons is $2.4\text{ eV}$ , what is the work function of the metal in electron-volts ( $\text{eV}$ )?

Show answer & explanation

Answer: 3.8

Answer

The work function of the metal is

3.8\text{ eV}

.

According to Einstein's photoelectric equation, the total energy of an incident photon (

E = 6.2\text{ eV}

) equals the work function of the metal (

W_0

) plus the maximum kinetic energy of the ejected photoelectrons (

K_{\max} = 2.4\text{ eV}

). Subtracting the kinetic energy from the photon energy gives

W_0 = 6.2\text{ eV} - 2.4\text{ eV} = 3.8\text{ eV}

.

Step-by-Step Solution

1

State Einstein's photoelectric equation

E = W_0 + K_{\max}

Relates the incident photon energy to the metal work function and photoelectron kinetic energy.

2

Rearrange the equation to solve for the work function

W_0 = E - K_{\max}

Isolates the work function

W_0

on one side of the equation.

3

Substitute the given numerical values and compute

W_0 = 6.2\text{ eV} - 2.4\text{ eV} = 3.8\text{ eV}

Evaluates the difference to obtain the minimum energy needed to remove an electron.

Key Concept

Einstein's Photoelectric Equation and Work Function

Question 567Question →

A beaker containing water of density $1000\text{ kg/m}^3$ rests on a digital weighing scale, giving an initial reading of $1.50\text{ kg}$ . A solid aluminum block of mass $0.80\text{ kg}$ and density $2500\text{ kg/m}^3$ is suspended from a string and completely immersed in the water without touching the bottom or sides of the beaker. What is the new reading on the digital weighing scale, in kilograms? (Take $g = 10\text{ m/s}^2$ )

Show answer & explanation

Answer: 1.82

Answer

The new reading on the digital weighing scale is

1.82\text{ kg}

.

When the aluminum block is fully submerged in the water, it displaces a volume of water equal to its own volume (

V = \frac{0.80}{2500} = 3.2 \times 10^{-4}\text{ m}^3

). The mass of this displaced water is

m_{\text{water}} = 1000 \times 3.2 \times 10^{-4} = 0.32\text{ kg}

. The upthrust exerted by the water upward on the block is equal to the weight of the displaced water (

3.2\text{ N}

). By Newton's Third Law, the block exerts an equal and opposite downward reaction force (

3.2\text{ N}

) on the water. This extra downward force adds an equivalent mass of

0.32\text{ kg}

to the digital scale reading, making the new reading

1.50\text{ kg} + 0.32\text{ kg} = 1.82\text{ kg}

.

Step-by-Step Solution

1

Calculate the volume of the submerged block

Volume

V = 3.2 \times 10^{-4}\text{ m}^3

The volume of fluid displaced by a completely submerged body equals the volume of the body itself.

2

Find the mass of the displaced water

Mass of displaced water

m_{\text{water}} = 0.32\text{ kg}

According to Archimedes' principle, the upthrust equals the weight of the displaced fluid, which corresponds to a displaced mass of

\rho_{\text{water}} V

.

3

Apply Newton's Third Law to determine the change in scale reading

Scale reading increase

\Delta m = 0.32\text{ kg}

The fluid exerts an upward buoyant force on the block, so by Newton's Third Law, the block exerts an equal downward reaction force on the fluid, transferring an effective weight equal to the upthrust onto the scale.

4

Compute the total new scale reading

New scale reading

= 1.82\text{ kg}

Sum the initial mass reading of the beaker system (

1.50\text{ kg}

) and the mass of the displaced water (

0.32\text{ kg}

).

Key Concept

Apparent weight transfer, Archimedes' principle, and Newton's Third Law

Question 568Question →

A particle undergoing simple harmonic motion moves with an angular frequency of $4\text{ rad/s}$ and an amplitude of $0.5\text{ m}$ . What is the maximum speed of the particle in $\text{m/s}$ ?

Show answer & explanation

Answer: 2

Answer

The maximum speed of the particle is

2.0\text{ m/s}

.

The magnitude of velocity in simple harmonic motion varies with displacement

x

according to

v = \omega \sqrt{A^2 - x^2}

. The speed reaches its maximum value when the particle passes through the equilibrium position (

x = 0

), giving

v_{\text{max}} = \omega A

. Substituting

\omega = 4\text{ rad/s}

and

A = 0.5\text{ m}

gives

v_{\text{max}} = 4 \times 0.5 = 2.0\text{ m/s}

.

Step-by-Step Solution

1

Identify the given physical parameters.

\omega = 4\text{ rad/s}

and

A = 0.5\text{ m}

These values define the speed profile of the simple harmonic oscillator.

2

Apply the SHM formula for maximum speed.

v_{\text{max}} = \omega A

Peak speed occurs at the equilibrium position where displacement is zero.

3

Substitute the values to calculate the maximum speed.

v_{\text{max}} = 4 \times 0.5 = 2.0\text{ m/s}

Multiplying angular frequency by amplitude yields the maximum linear velocity.

Key Concept

Maximum speed in Simple Harmonic Motion

Question 569Question →

A diver is swimming at a depth of $3.5\text{ m}$ below the surface of a freshwater lake. If the density of water is $1000\text{ kg/m}^3$ and the acceleration due to gravity $g = 10\text{ m/s}^2$ , what is the gauge pressure exerted on the diver in pascals?

Show answer & explanation

Answer: 35000

Answer

35000 Pa

The gauge pressure exerted by a static column of fluid is given by

P = h \rho g

. Using the values

h = 3.5\text{ m}

,

\rho = 1000\text{ kg/m}^3

, and

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

, the pressure is

P = 3.5 \times 1000 \times 10 = 35000\text{ Pa}

.

Step-by-Step Solution

1

Identify the formula for liquid hydrostatic pressure

P = h \rho g

Gauge pressure at a depth

h

in a static fluid depends on depth, fluid density, and gravitational field strength.

2

Substitute the given numerical values

P = 3.5 \times 1000 \times 10

Substitute depth

h = 3.5\text{ m}

, density

\rho = 1000\text{ kg/m}^3

, and

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

.

3

Perform the multiplication to determine the pressure

35000\text{ Pa}

Complete the calculation to get the pressure in SI units (Pascals).

Key Concept

Hydrostatic Pressure in Static Fluids

Estimated Time:45s

Question 570Question →

A capacitor of capacitance $50\ \mu\text{F}$ is connected across an alternating current (AC) source operating at a frequency of $\frac{100}{\pi}\ \text{Hz}$ . What is the capacitive reactance of the capacitor?

Show answer & explanation

Answer: 100

Answer

The capacitive reactance of the capacitor is

100\ \Omega

.

Capacitive reactance

X_C

is given by the formula

X_C = \frac{1}{2\pi f C}

. Substituting

C = 50 \times 10^{-6}\ \text{F}

and

f = \frac{100}{\pi}\ \text{Hz}

into the formula yields

X_C = \frac{1}{2\pi (100/\pi) (50 \times 10^{-6})} = \frac{1}{10^{-2}} = 100\ \Omega

.

Step-by-Step Solution

1

Convert capacitance to farads and state all given values

C = 50 \times 10^{-6}\ \text{F}

and

f = \frac{100}{\pi}\ \text{Hz}

Calculations require standard SI base units.

2

Apply the formula for capacitive reactance

X_C = \frac{1}{2\pi f C}

Capacitive reactance measures the opposition offered by a capacitor to alternating current.

3

Substitute the values and calculate the result

X_C = \frac{1}{2\pi \cdot \frac{100}{\pi} \cdot (50 \times 10^{-6})} = \frac{1}{10,000 \times 10^{-6}} = 100\ \Omega

The factor

\pi

cancels out directly, making the arithmetic simple.

Key Concept

Capacitive Reactance in AC Circuits

Question 571Question →

A solid uniform cylinder of height $0.20\text{ m}$ and cross-sectional area $5.0 \times 10^{-3}\text{ m}^2$ floats vertically at the boundary between oil of density $800\text{ kg/m}^3$ and water of density $1000\text{ kg/m}^3$ . If a height of $0.08\text{ m}$ of the cylinder extends into the water layer while the remaining upper portion is completely covered by the oil layer, what is the mass of the cylinder in kilograms?

Show answer & explanation

Answer: 0.88

Answer

The mass of the cylinder is

0.88\text{ kg}

.

According to the Law of Flotation, a floating body displaces its own weight of fluid. When floating at the interface of two immiscible liquids, the total mass of the body equals the sum of the masses of the displaced liquids. Displaced water mass is

\rho_w A h_w = 0.40\text{ kg}

and displaced oil mass is

\rho_o A h_o = 0.48\text{ kg}

, giving a total cylinder mass of

0.88\text{ kg}

.

Step-by-Step Solution

1

Find the height of the cylinder submerged in the oil layer.

h_o = 0.20\text{ m} - 0.08\text{ m} = 0.12\text{ m}

The total cylinder height is

0.20\text{ m}

, and

0.08\text{ m}

is submerged in water.

2

Calculate the volumes of water and oil displaced by the cylinder.

V_w = 5.0 \times 10^{-3} \times 0.08 = 4.0 \times 10^{-4}\text{ m}^3

;

V_o = 5.0 \times 10^{-3} \times 0.12 = 6.0 \times 10^{-4}\text{ m}^3

Volume displaced in each fluid equals cross-sectional area multiplied by the submerged height in that fluid.

3

Calculate the mass of the floating cylinder using the Law of Flotation.

m = \rho_w V_w + \rho_o V_o = (1000 \times 4.0 \times 10^{-4}) + (800 \times 6.0 \times 10^{-4}) = 0.40\text{ kg} + 0.48\text{ kg} = 0.88\text{ kg}

For a floating object in static equilibrium, its mass equals the total mass of the fluids displaced by its submerged parts.

Key Concept

Law of Flotation in Layered Liquids

Estimated Time:1m 30s

Question 572Question →

A tripositive ion, $X^{3+}$ , has a mass number of $56$ and contains $23$ electrons. How many neutrons are present in the nucleus of an atom of element $X$ ?

Show answer & explanation

Answer: 30

Answer

30

To find the number of neutrons, first determine the atomic number (number of protons) of element

X

. The ion

X^{3+}

carries a

+3

charge because it lost 3 electrons. Since

X^{3+}

has 23 electrons, the neutral atom

X

has

23 + 3 = 26

electrons, which means it has 26 protons. The mass number (

A = 56

) is the sum of protons (

Z

) and neutrons (

N

). Thus,

N = 56 - 26 = 30

.

Step-by-Step Solution

1

Determine the atomic number (number of protons) of element

X

Protons (

Z

) = 26

The tripositive ion

X^{3+}

has lost 3 electrons. The neutral atom has

23 + 3 = 26

electrons, which equals its proton count.

2

Calculate the number of neutrons

Neutrons (

N

) = 30

Subtract the atomic number from the mass number:

N = A - Z = 56 - 26 = 30

.

Key Concept

Calculation of subatomic particles in ions using atomic number and mass number relationships

Question 573Question →

The saturated vapour pressure of water at the dew point of a mass of air is $12\text{ mmHg}$ , while the saturated vapour pressure at the actual air temperature is $24\text{ mmHg}$ . Calculate the relative humidity of the air.

Show answer & explanation

Answer: 50

Answer

The relative humidity of the air is 50%.

Relative humidity is the ratio of the saturated vapour pressure at the dew point to the saturated vapour pressure at the actual air temperature, expressed as a percentage: (12 mmHg / 24 mmHg) * 100% = 50%.

Step-by-Step Solution

1

Identify the saturated vapour pressure at the dew point and at the air temperature.

SVP at dew point = 12 mmHg; SVP at air temperature = 24 mmHg.

Relative humidity relies on the ratio of partial vapour pressure (SVP at dew point) to maximum vapour pressure at air temperature.

2

Apply the relative humidity formula.

Relative Humidity = (SVP at dew point / SVP at air temperature) * 100%

This formula defines the percentage saturation of the air.

3

Substitute the values and evaluate.

(12 / 24) * 100% = 50%

Dividing 12 by 24 gives 0.5, which equals 50% when multiplied by 100.

Key Concept

Relative Humidity Calculation

Estimated Time:45s

Question 574Question →

An FM radio station transmits electromagnetic waves at a frequency of $1.0 \times 10^8\text{ Hz}$ in a vacuum. Given that the speed of light in a vacuum is $3.0 \times 10^8\text{ m/s}$ , what is the wavelength of these radio waves in meters?

Show answer & explanation

Answer: 3

Answer

The wavelength of the radio waves is

3.0\text{ m}

.

Applying the wave equation

c = f\lambda

, rearranging to

\lambda = \frac{c}{f}

, and substituting

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

and

f = 1.0 \times 10^8\text{ Hz}

gives a wavelength of

3.0\text{ m}

.

Step-by-Step Solution

1

Identify given parameters and key formula

Speed of light

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

, frequency

f = 1.0 \times 10^8\text{ Hz}

, wave equation

c = f\lambda

The electromagnetic wave equation relates speed, frequency, and wavelength.

2

Rearrange for wavelength and substitute given values

\lambda = \frac{c}{f} = \frac{3.0 \times 10^8}{1.0 \times 10^8} = 3.0\text{ m}

Dividing the speed of propagation by the wave frequency yields the spatial wavelength.

Key Concept

Wave equation relating speed, frequency, and wavelength of electromagnetic waves

Question 575Question →

The line $y = 2x - 1$ intersects the curve $y = x^2 - 4x + 4$ at two distinct points, $(x_1, y_1)$ and $(x_2, y_2)$ . Calculate the sum of the $y$ -coordinates of these two points of intersection, $y_1 + y_2$ .

Show answer & explanation

Answer: 10

Answer

The sum of the

y

-coordinates of the points of intersection is 10.

Equating

y = 2x - 1

and

y = x^2 - 4x + 4

yields

x^2 - 6x + 5 = 0

, whose solutions are

x = 1

and

x = 5

. Substituting these values into

y = 2x - 1

gives

y_1 = 1

and

y_2 = 9

. The sum

y_1 + y_2 = 1 + 9 = 10

.

Step-by-Step Solution

1

Equate the linear and quadratic equations to eliminate

y

.

x^2 - 6x + 5 = 0

Setting

2x - 1 = x^2 - 4x + 4

allows finding the

x

-coordinates of the intersection points.

2

Solve the quadratic equation for

x

.

x_1 = 1

and

x_2 = 5

Factoring

(x - 1)(x - 5) = 0

yields the two

x

-values.

3

Determine the corresponding

y

-values using

y = 2x - 1

.

y_1 = 1

and

y_2 = 9

Substituting

x = 1

yields

y = 1

, and substituting

x = 5

yields

y = 9

.

4

Calculate the sum of the

y

-coordinates.

10

Adding

y_1 + y_2 = 1 + 9 = 10

.

Key Concept

Solving simultaneous linear and quadratic equations to find coordinates of intersection

Question 576Question →

The $4^{\text{th}}$ term of an arithmetic progression (A.P.) is $15$ and the $9^{\text{th}}$ term is $35$ . Calculate the sum of the first $10$ terms of the progression.

Show answer & explanation

Answer: 210

Answer

The sum of the first 10 terms of the progression is 210.

Using the

n^{\text{th}}

term formula

T_n = a + (n-1)d

, the equations

a + 3d = 15

and

a + 8d = 35

yield common difference

d = 4

and first term

a = 3

. Substituting these into the sum formula

S_n = \frac{n}{2}[2a + (n-1)d]

for

n = 10

yields

S_{10} = 5[2(3) + 9(4)] = 210

.

Step-by-Step Solution

1

Set up equations for the given terms

a + 3d = 15

and

a + 8d = 35

The

n^{\text{th}}

term of an A.P. is given by

T_n = a + (n-1)d

.

2

Solve for the common difference

d

d = 4

Subtracting

(a + 3d = 15)

from

(a + 8d = 35)

gives

5d = 20

, so

d = 4

.

3

Solve for the first term

a

a = 3

Substituting

d = 4

into

a + 3(4) = 15

yields

a = 15 - 12 = 3

.

4

Calculate the sum of the first 10 terms

S_{10} = 210

Applying

S_{10} = \frac{10}{2}[2(3) + 9(4)] = 5(6 + 36) = 210

.

Key Concept

Finding terms and sums of an Arithmetic Progression using simultaneous linear equations

Estimated Time:1m 30s

Question 577Question →

An alternating current (AC) series circuit contains a resistor, an inductor, and a capacitor. The root-mean-square (RMS) potential differences measured across the resistor, inductor, and capacitor are $80\text{ V}$ , $110\text{ V}$ , and $50\text{ V}$ , respectively. What is the total supply voltage across the circuit in volts?

Show answer & explanation

Answer: 100

Answer

The total supply voltage across the series AC circuit is

100\text{ V}

.

In a series alternating current circuit, the voltages across the resistor, inductor, and capacitor are not in phase. The resistor voltage is in phase with the current, whereas inductor voltage leads by

90^\circ

and capacitor voltage lags by

90^\circ

. The total voltage is calculated using vector addition:

V = \sqrt{V_R^2 + (V_L - V_C)^2}

. Substituting the given values gives

V = \sqrt{80^2 + (110 - 50)^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = 100\text{ V}

.

Step-by-Step Solution

1

Identify the RMS potential differences across each component.

V_R = 80\text{ V}

,

V_L = 110\text{ V}

, and

V_C = 50\text{ V}

.

In a series AC circuit, voltages across reactive components are out of phase with the resistor voltage.

2

Calculate the net reactive voltage difference between the inductor and capacitor.

V_L - V_C = 110\text{ V} - 50\text{ V} = 60\text{ V}

.

Inductive voltage leads current by

90^\circ

while capacitive voltage lags current by

90^\circ

, making them

180^\circ

out of phase with each other.

3

Determine total supply voltage using vector (phasor) addition.

V = \sqrt{V_R^2 + (V_L - V_C)^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100\text{ V}

.

The resistive voltage and net reactive voltage are perpendicular (

90^\circ

phase angle difference).

Key Concept

Phasor Addition of Voltages in a Series AC Circuit

Question 578Question →

Evaluate the definite integral $\int_{1}^{3} (4x^3 - 6x) \, dx$ .

Show answer & explanation

Answer: 56

Answer

The value of the definite integral is 56.

To evaluate the definite integral

\int_{1}^{3} (4x^3 - 6x) \, dx

, integrate the function term-by-term to obtain

F(x) = x^4 - 3x^2

. Evaluating

F(x)

at the upper bound

x = 3

yields

54

, and evaluating at the lower bound

x = 1

yields

-2

. Subtracting the lower bound value from the upper bound value gives

54 - (-2) = 56

.

Step-by-Step Solution

1

Find the indefinite integral of the function

\int (4x^3 - 6x) \, dx = x^4 - 3x^2

Apply the power rule of integration \int x^n \, dx = \frac{x^{n+1}}{n+1}

2

Substitute the upper limit x = 3 into the antiderivative F(x)

F(3) = 3^4 - 3(3)^2 = 81 - 27 = 54

Evaluate F(b) for b = 3

3

Substitute the lower limit x = 1 into the antiderivative F(x)

F(1) = 1^4 - 3(1)^2 = 1 - 3 = -2

Evaluate F(a) for a = 1

4

Subtract F(a) from F(b)

54 - (-2) = 56

Apply the Fundamental Theorem of Calculus: \int_{a}^{b} f(x) \, dx = F(b) - F(a)

Key Concept

Definite Integration using the Fundamental Theorem of Calculus

Question 579Question →

A monochromatic beam of light with a wavelength of $600\text{ nm}$ in air enters a glass block of refractive index $1.50$ . What is the wavelength of the light inside the glass block in nanometers ( $\text{nm}$ )?

Show answer & explanation

Answer: 400

Answer

The wavelength of the light inside the glass block is

400\text{ nm}

.

The speed and wavelength of light both decrease by a factor of the refractive index

n

when entering a medium from air, while the frequency stays constant (

v = f\lambda

). Thus,

\lambda = \frac{\lambda_0}{n} = \frac{600}{1.50} = 400\text{ nm}

.

Step-by-Step Solution

1

Identify the relevant formula connecting refractive index and wavelength

\lambda = \frac{\lambda_0}{n}

When light passes from air into a medium, its frequency remains unchanged while its speed and wavelength decrease proportionally by a factor of the refractive index

n

.

2

Substitute the given values into the equation

\lambda = \frac{600\text{ nm}}{1.50}

The wavelength in air is

600\text{ nm}

and the refractive index of glass is

1.50

.

3

Perform the calculation

\lambda = 400\text{ nm}

Dividing

600

by

1.50

yields

400

.

Key Concept

Refraction and Wavelength Change in a Medium

Question 580Question →

Find the $y$ -intercept of the tangent line to the curve $y = x^3 - 6x + 8$ at the point where $x = 1$ .

Show answer & explanation

Answer: 6

Answer

The

y

-intercept of the tangent line is 6.

Differentiating

y = x^3 - 6x + 8

yields

\frac{dy}{dx} = 3x^2 - 6

. At

x = 1

, the point on the curve is

(1, 3)

and the gradient of the tangent is

m = -3

. Substituting into the point-slope form

y - 3 = -3(x - 1)

gives

y = -3x + 6

. Setting

x = 0

identifies the

y

-intercept as 6.

Step-by-Step Solution

1

Find the point of tangency on the curve

For

x = 1

,

y = (1)^3 - 6(1) + 8 = 3

. The point is

(1, 3)

.

The tangent line touches the curve at the point corresponding to

x = 1

.

2

Find the gradient function of the curve

\frac{dy}{dx} = 3x^2 - 6

The derivative of the curve equation gives the gradient of the tangent at any point.

3

Calculate the gradient of the tangent line at

x = 1

m = 3(1)^2 - 6 = -3

Substitute

x = 1

into the derivative.

4

Formulate the equation of the tangent line

y - 3 = -3(x - 1) \implies y = -3x + 6

Apply the point-slope line equation

y - y_1 = m(x - x_1)

using

(1, 3)

and

m = -3

.

5

Determine the

y

-intercept

Setting

x = 0

yields

y = 6

.

The

y

-intercept is the value of

y

when

x = 0

.

Key Concept

Tangents to Curves and Axis Intercepts

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