All practice questions

Question 7181Question →

An optical communication sensor operates using an electromagnetic wave with a frequency of $1.5 \times 10^{14}\text{ Hz}$ in a vacuum. Given that the speed of light in vacuum is $3.0 \times 10^8\text{ m/s}$ , what is the wavelength of this electromagnetic radiation?

2.0 \times 10^{-6}\text{ m}

5.0 \times 10^5\text{ m}

4.5 \times 10^{22}\text{ m}

2.0 \times 10^{-7}\text{ m}

Show answer & explanation

Answer: $2.0 \times 10^{-6}\text{ m}$

Answer

The wavelength of the electromagnetic wave is

2.0 \times 10^{-6}\text{ m}

.

Using the electromagnetic wave relation

c = f\lambda

, dividing the speed of light (

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

) by the given frequency (

1.5 \times 10^{14}\text{ Hz}

) correctly gives

2.0 \times 10^{-6}\text{ m}

.

Step-by-Step Solution

1

Identify given quantities and formula

Speed of light

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

, frequency

f = 1.5 \times 10^{14}\text{ Hz}

, wave equation

c = f \lambda

The fundamental wave equation relates wave speed, frequency, and wavelength for all electromagnetic waves.

2

Rearrange formula to solve for wavelength

\lambda

\lambda = \frac{c}{f}

Isolating the target unknown variable allows direct calculation.

3

Substitute values and compute

$\lambda = \frac{3.0 \times 10^8}{1.5 \times 10^{14}} = 2.0 \times 10^{-6}\text{ m}$

Dividing the coefficients (

3.0 / 1.5 = 2.0

) and subtracting the powers of ten (

8 - 14 = -6

) yields the accurate wavelength.

Key Concept

Wave Equation for Electromagnetic Waves

Question 7182Question →

What is the bond angle between adjacent single covalent bonds formed by a tetrahedral $sp^3$ hybridized carbon atom in a saturated organic compound?

109.5^\circ

120^\circ

180^\circ

90^\circ

Show answer & explanation

Answer: $109.5^\circ$

Answer

The bond angle of a tetrahedral sp³ hybridized carbon atom is 109.5°.

In saturated organic compounds such as alkanes, the central carbon atom forms four single sigma bonds using four equivalent

sp^3

hybrid orbitals. According to VSEPR theory, four bonding electron pairs surrounding a central atom arrange themselves in a regular tetrahedral shape to minimize electrostatic repulsion, yielding a standard bond angle of

109.5^\circ

.

Step-by-Step Solution

1

Determine the hybridization state of a carbon atom forming four single sigma bonds.

Mixing one 2s orbital and three 2p orbitals gives four equivalent

sp^3

hybrid orbitals.

Saturated carbon forms four single bonds by directed valence orbital mixing.

2

Apply VSEPR theory to determine spatial orientation for four electron pairs around the central carbon.

To minimize electron pair repulsion, the four orbitals point toward the vertices of a regular tetrahedron.

Symmetrical four-coordinate electron pair repulsion produces tetrahedral spatial orientation.

3

Identify the characteristic inter-bond angle of a regular tetrahedron.

The angle between any two adjacent bonds is

109.5^\circ

(or

109^\circ 28'

).

This angle maximizes the distance between the four bonding electron pairs in three-dimensional space.

Key Concept

Tetrahedral Geometry and sp³ Hybridization Bond Angle

Estimated Time:45s

Question 7183Question →

A pure chemical compound analyzed using thin-layer or paper chromatography under standard conditions produces a single distinct spot on the developed chromatogram.

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because pure substances contain only one component and yield a single spot during chromatographic separation.

A fundamental criterion of chemical purity is that a pure substance gives a single spot on a chromatogram, as it contains no additional chemical species to separate out.

Step-by-Step Solution

1

Identify the core principle of chromatography as a test of purity.

Chromatography separates components of a sample based on their relative affinities for the stationary and mobile phases.

Understanding how chromatography separates substances helps evaluate purity criteria.

2

Relate component count to the number of spots observed.

A pure compound contains only one chemical species, so it does not separate into multiple components and produces a single spot.

The appearance of more than one spot signifies a mixture containing impurities.

Key Concept

Single spot on a chromatogram as a criterion of chemical purity

Question 7184Question →

Arrange the following types of electromagnetic radiation in order of increasing frequency (from lowest frequency to highest frequency).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence of electromagnetic radiations in order of increasing frequency is: Microwaves, Infrared radiation, Ultraviolet radiation, and Gamma rays.

Electromagnetic waves propagate at the constant speed

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

in a vacuum. Frequency increases as wavelength decreases across the spectrum. Microwaves have the longest wavelength and lowest frequency among the choices, followed by infrared radiation, ultraviolet radiation, and finally gamma rays, which possess the shortest wavelength and highest frequency.

Step-by-Step Solution

1

Recall the arrangement of the electromagnetic spectrum in terms of frequency and wavelength.

In the electromagnetic spectrum, frequency increases in the order: Radio waves

\rightarrow

Microwaves

\rightarrow

Infrared

\rightarrow

Visible light

\rightarrow

Ultraviolet

\rightarrow

X-rays

\rightarrow

Gamma rays.

Electromagnetic wave energy

E = hf

and frequency

f = \frac{c}{\lambda}

increase as wavelength decreases.

2

Identify the relative position of each given radiation type along the frequency scale.

Microwaves (

10^9 - 10^{11}\text{ Hz}

) < Infrared (

10^{11} - 10^{14}\text{ Hz}

) < Ultraviolet (

10^{15} - 10^{16}\text{ Hz}

) < Gamma rays (

>10^{19}\text{ Hz}

).

Comparing their characteristic frequency ranges determines their exact position in the sequence.

3

Order the items from lowest to highest frequency.

1st: Microwaves, 2nd: Infrared radiation, 3rd: Ultraviolet radiation, 4th: Gamma rays.

This sequence satisfies the requirement of strictly increasing frequency.

Key Concept

Electromagnetic Spectrum Ordering by Frequency and Wavelength

Question 7185Question →

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 7186Question →

If the matrix $A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & k & 1 \\ 3 & 1 & 4 \end{pmatrix}$ is singular, what is the value of $k$ ?

\frac{3}{2}

-\frac{3}{2}

\frac{7}{2}

-\frac{1}{2}

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

\frac{3}{2}

A matrix is singular if its determinant equals zero. Expanding the determinant of matrix

A

along its first row gives

1(4k - 1) - 0 + 2(2 - 3k) = -2k + 3

. Setting

-2k + 3 = 0

yields

k = \frac{3}{2}

.

Step-by-Step Solution

1

Apply the condition for a singular matrix

A matrix is singular when its determinant equals zero, so

\det(A) = 0

.

By definition, square matrices with zero determinant are singular.

2

Expand

\det(A)

along the first row

\det(A) = 1(4k - 1) - 0(8 - 3) + 2(2 - 3k) = 4k - 1 + 4 - 6k = -2k + 3

.

Laplace expansion along the first row simplifies computation due to the zero entry.

3

Solve for

k

-2k + 3 = 0 \implies 2k = 3 \implies k = \frac{3}{2}

.

Isolating the variable gives the required value of

k

.

Key Concept

Singular Matrix Condition and 3x3 Determinant Expansion

Question 7187Question →

Which of the following physical properties best confirms that a solid chemical sample is pure?

It melts completely at a sharp, fixed temperature.

It melts gradually over a wide range of temperatures.

It boils over a broad temperature interval above its normal boiling point.

It forms two separate spots when developed on a paper chromatogram.

Show answer & explanation

Answer: It melts completely at a sharp, fixed temperature.

Answer

It melts completely at a sharp, fixed temperature.

Pure solid compounds melt at a fixed, sharp temperature characteristic of that substance. A sharp melting point is a definitive physical criterion of purity for crystalline solids.

Step-by-Step Solution

1

Identify the standard criteria of purity for a solid substance.

A pure solid possesses a sharp and definite melting point.

Soluble impurities disrupt the uniform crystal lattice of a solid, depressing its melting point and causing it to melt over a broad temperature range.

Key Concept

Criteria of purity for solid substances

Question 7188Question →

A series alternating current circuit comprises a resistor with resistance $30\ \Omega$ , an inductor with inductive reactance $80\ \Omega$ , and a capacitor with capacitive reactance $40\ \Omega$ . What is the total impedance of the circuit?

50\ \Omega

150\ \Omega

70\ \Omega

40\ \Omega

Show answer & explanation

Answer: $50\ \Omega$

Answer

The impedance of the circuit is

50\ \Omega

.

The impedance

Z

of a series RLC circuit is calculated using the formula

Z = \sqrt{R^2 + (X_L - X_C)^2}

. Substituting the given values

R = 30\ \Omega

,

X_L = 80\ \Omega

, and

X_C = 40\ \Omega

yields

Z = \sqrt{30^2 + (80 - 40)^2} = \sqrt{900 + 1600} = 50\ \Omega

.

Step-by-Step Solution

1

Calculate the net reactance (

X_{net}

)

X_{net} = X_L - X_C = 80\ \Omega - 40\ \Omega = 40\ \Omega

Inductive and capacitive reactances are

180^\circ

out of phase in a series AC circuit.

2

Apply the series impedance formula

Z = \sqrt{R^2 + X_{net}^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50\ \Omega

Resistance and net reactance act at

90^\circ

phase to each other, requiring the Pythagorean relation.

Key Concept

Impedance of a Series RLC Circuit

Question 7189Question →

Match each redox description on the left with its corresponding classical or modern definition concept on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Addition of oxygen matches Classical oxidation; Loss of electrons matches Modern oxidation (electron transfer); Decrease in oxidation state matches Modern reduction (oxidation state change); Removal of oxygen matches Classical reduction.

Classical oxidation involves gaining oxygen, while classical reduction involves losing oxygen. In modern electronic terms, oxidation is the loss of electrons (OIL), and reduction is a decrease in oxidation state (reduction of oxidation number).

Step-by-Step Solution

1

Identify classical redox concepts based on oxygen transfer.

Addition of oxygen corresponds to classical oxidation, whereas removal of oxygen corresponds to classical reduction.

Classical definitions focused on the transfer of oxygen and hydrogen atoms.

2

Identify modern redox concepts based on electron transfer and oxidation numbers.

Loss of electrons corresponds to modern oxidation, and a decrease in oxidation state corresponds to modern reduction.

Modern concepts expand redox beyond oxygen/hydrogen to include electron movement and formal charge changes.

Key Concept

Distinguishing between classical (oxygen/hydrogen transfer) and modern (electron transfer and oxidation number) definitions of oxidation and reduction.

Estimated Time:1m 0s

Question 7190Question →

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 7191Question →

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 7192Question →

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 7193Question →

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 7194Question →

According to the kinetic molecular theory, collisions between gas particles and the walls of their container are completely inelastic, resulting in a continuous loss of kinetic energy.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False. Collisions of gas particles with container walls are assumed to be perfectly elastic, meaning no kinetic energy is lost.

The statement is incorrect because kinetic molecular theory assumes perfectly elastic collisions, preserving total kinetic energy and sustaining constant pressure.

Step-by-Step Solution

1

Identify the kinetic molecular theory postulate concerning molecular collisions.

The kinetic molecular theory specifies that all collisions between gas molecules or between molecules and the container walls are perfectly elastic.

Elasticity ensures that the average kinetic energy of the gas remains constant at a given temperature.

2

Compare the statement with the established postulate.

The statement claims collisions are 'completely inelastic' with a 'continuous loss of kinetic energy', which directly contradicts the elastic collision postulate.

An inelastic collision process would violate energy conservation for ideal gas systems at thermal equilibrium.

Key Concept

Elasticity of Collisions in Kinetic Molecular Theory

Question 7195Question →

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 7196Question →

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 7197Question →

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 7198Question →

Match each feature observed during electrical discharge through a gas with its corresponding pressure stage or physical characteristic.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Cathode Glow matches 'Luminous glow appearing right next to the cathode at around 10 mmHg pressure', Striations match 'Alternating bright and dark bands in the positive column at around 1 mmHg pressure', and Crookes Dark Space matches 'Dark region extending to fill most of the tube at very low pressure around 0.01 mmHg'.

Each feature of gas discharge corresponds to a specific pressure regime inside the discharge tube: Cathode Glow occurs at ~10 mmHg, Striations form at ~1 mmHg in the positive column, and Crookes Dark Space expands to cover most of the tube at ~0.01 mmHg where cathode rays are freely emitted.

Step-by-Step Solution

1

Identify the discharge stage for Cathode Glow.

Cathode glow occurs at moderate pressure (~10 mmHg) directly adjacent to the negative electrode.

Initial gas ionization near the cathode causes luminescence at this pressure stage.

2

Identify the discharge stage for Striations.

Striations represent light and dark divisions of the positive column at ~1 mmHg.

Periodic ionization and recombination along the tube produce alternating luminous discs.

3

Identify the discharge stage for Crookes Dark Space.

Crookes dark space expands as pressure drops to ~0.01 mmHg.

At very low pressures, electrons travel longer distances without colliding, causing the dark space to fill most of the discharge tube.

Key Concept

Stages of electric conduction through gases at reduced pressure.

Question 7199Question →

Using differentiation from first principles, what is the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the function $f(x) = 3x^2 - 2x$ ?

6x - 2

6x + 2

3x - 2

6x^2 - 2

Show answer & explanation

Answer: $6x - 2$

Answer

The derivative of

f(x) = 3x^2 - 2x

with respect to

x

is

6x - 2

.

Applying the first principles definition

\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

to

f(x) = 3x^2 - 2x

yields

\frac{3x^2 + 6xh + 3h^2 - 2x - 2h - (3x^2 - 2x)}{h} = \frac{6xh + 3h^2 - 2h}{h} = 6x + 3h - 2

. As

h \to 0

, this simplifies directly to

6x - 2

.

Step-by-Step Solution

1

Set up the definition of the derivative from first principles.

\frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The definition of the derivative is the limit of the difference quotient as

h

approaches zero.

2

Substitute

f(x+h)

and

f(x)

into the difference quotient.

f(x+h) = 3(x+h)^2 - 2(x+h) = 3(x^2 + 2xh + h^2) - 2x - 2h = 3x^2 + 6xh + 3h^2 - 2x - 2h

Expand

(x+h)^2

completely using algebraic identity.

3

Subtract

f(x)

from

f(x+h)

to find the numerator.

f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 2x - 2h) - (3x^2 - 2x) = 6xh + 3h^2 - 2h

Cancel out identical terms

3x^2

and

-2x

.

4

Divide by

h

and evaluate the limit as

h \to 0

.

\frac{6xh + 3h^2 - 2h}{h} = 6x + 3h - 2; \quad \lim_{h \to 0} (6x + 3h - 2) = 6x - 2

Factor out

h

to cancel the denominator, then set

h = 0

.

Key Concept

Differentiation from First Principles

Estimated Time:1m 30s

Question 7200Question →

Calculate the numerical values of the following sequence and progression quantities, then arrange them in ascending order (from smallest to largest):

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order of the numerical values is: the common ratio of the GP (value = 3), the common difference of the AP (value = 4), the 5th term of the AP (value = 8), and the sum to infinity of the GP (value = 10).

Evaluating each sequence property yields numerical values: the GP common ratio equals 3, the AP common difference equals 4, the AP 5th term equals 8, and the GP sum to infinity equals 10. Ordering these values from least to greatest produces the sequence 3, 4, 8, 10.

Step-by-Step Solution

1

Calculate the value for the first quantity (AP common difference)

d = 4

From

a + 6d = 27

and

a + 2d = 11

, subtract to find

4d = 16 \implies d = 4

.

2

Calculate the value for the second quantity (GP common ratio)

r = 3

From

ar^4 = 162

and

ar = 6

, divide to obtain

r^3 = 27 \implies r = 3

.

3

Calculate the value for the third quantity (GP sum to infinity)

S_{\infty} = 10

Apply the sum to infinity formula

S_{\infty} = \frac{a}{1-r} = \frac{5}{1 - 0.5} = 10

.

4

Calculate the value for the fourth quantity (AP 5th term)

T_5 = 8

Apply the nth term formula

T_5 = a + 4d = 2 + 4(1.5) = 8

.

5

Sort the calculated values in ascending order

3 < 4 < 8 < 10

Comparing the values gives

3

(GP common ratio),

4

(AP common difference),

8

(AP 5th term), and

10

(GP sum to infinity).

Key Concept

Nth term, common difference, common ratio, and sum to infinity calculations for AP and GP sequences.

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept