All practice questions

Question 7201Question →

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 7202Question →

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 7203Question →

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 7204Question →

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 7205Question →

A progressive wave propagating through an initial elastic medium is represented by the equation $y = 0.08 \sin(120\pi t - 4\pi x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. Upon crossing a boundary into a second medium, the wave speed decreases by $25\%$ . What is the wavelength of the wave in the second medium?

0.375\text{ m}

0.667\text{ m}

0.125\text{ m}

0.625\text{ m}

Show answer & explanation

Answer: $0.375\text{ m}$

Answer

0.375\text{ m}

Comparing

y = 0.08 \sin(120\pi t - 4\pi x)

with the standard wave equation

y = A \sin(\omega t - k x)

yields an angular frequency

\omega = 120\pi\text{ rad/s}

and wave number

k = 4\pi\text{ rad/m}

. The frequency is

f = \frac{120\pi}{2\pi} = 60\text{ Hz}

, and the initial speed is

v_1 = \frac{120\pi}{4\pi} = 30\text{ m/s}

. When passing into a new medium, the frequency remains constant at

60\text{ Hz}

. A

25\%

reduction in speed gives

v_2 = 0.75 \times 30\text{ m/s} = 22.5\text{ m/s}

. The new wavelength is therefore

\lambda_2 = \frac{22.5\text{ m/s}}{60\text{ Hz}} = 0.375\text{ m}

.

Step-by-Step Solution

1

Extract angular frequency and wave number from the wave equation

\omega = 120\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

Comparing the given equation

y = 0.08 \sin(120\pi t - 4\pi x)

with standard form

y = A \sin(\omega t - k x)

gives the values for

\omega

and

k

.

2

Calculate the frequency and initial wave speed in medium 1

f = \frac{\omega}{2\pi} = 60\text{ Hz}

and

v_1 = \frac{\omega}{k} = \frac{120\pi}{4\pi} = 30\text{ m/s}

Frequency is related to angular frequency by

f = \frac{\omega}{2\pi}

, and wave velocity is

v = \frac{\omega}{k}

.

3

Determine the speed in medium 2 and apply the principle of constant frequency

v_2 = 30 \times 0.75 = 22.5\text{ m/s}

and

f_2 = f_1 = 60\text{ Hz}

Wave frequency depends solely on the source and remains unchanged across boundary refraction, whereas speed decreases by

25\%

.

4

Calculate the new wavelength in medium 2

\lambda_2 = \frac{v_2}{f} = \frac{22.5}{60} = 0.375\text{ m}

Using the wave equation relationship

\lambda = \frac{v}{f}

.

Key Concept

Wave refraction across media boundaries and mathematical wave equation parameter matching

Estimated Time:2m 0s

Question 7206Question →

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

Question 7207Question →

A student was asked to approximate the fraction $\frac{5}{16}$ as a decimal rounded to $2$ significant figures. What is the percentage error introduced by this approximation?

$0.8\%

$0.81\%

$0.16\%

$0.008\%

Show answer & explanation

Answer: $0.8\%

Answer

The percentage error introduced by the approximation is

0.8\%

.

Converting

\frac{5}{16}

gives

0.3125

. Rounding to

2

significant figures gives an estimated value of

0.31

. The absolute error is

|0.31 - 0.3125| = 0.0025

. Dividing by the true value

0.3125

yields a relative error of

0.008

, which equals

0.8\%

when multiplied by

100\%

.

Step-by-Step Solution

1

Convert the fraction to an exact decimal

\frac{5}{16} = 0.3125

To find the true value of the number before approximation.

2

Round the exact decimal to 2 significant figures

0.3125 \approx 0.31

The first non-zero digits are 3 and 1; the next digit (2) is less than 5, so we round down.

3

Calculate the absolute error

|0.31 - 0.3125| = 0.0025

Absolute error is the magnitude of the difference between actual and estimated values.

4

Calculate the percentage error

\frac{0.0025}{0.3125} \times 100\% = 0.8\%

Percentage error is given by

\frac{\text{Absolute Error}}{\text{True Value}} \times 100\%

.

Key Concept

Fractions, Decimals, Percentages, and Approximations

Estimated Time:1m 15s

Question 7208Question →

A single viral particle contains both DNA and RNA simultaneously within its protective capsid.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False. A virus contains either DNA or RNA as its genetic material, never both concurrently within the same viral particle.

The statement is false because one of the fundamental structural rules of virology is that a virus possesses either DNA or RNA as its genetic material, never both simultaneously.

Step-by-Step Solution

1

Identify the genetic composition of viruses.

Viruses possess a core of nucleic acid surrounded by a protein coat (capsid).

Understanding the fundamental biochemical architecture of viruses is required to evaluate their structural characteristics.

2

Evaluate the nucleic acid rule for viral genomes.

Unlike cellular organisms (which contain both DNA and RNA), a specific virus particle contains exclusively DNA or RNA.

This is a key taxonomic feature distinguishing viruses from cellular life forms such as bacteria, plants, and animals.

Key Concept

Viral Nucleic Acid Composition

Question 7209Question →

A car is traveling along a straight horizontal road at a constant speed of $20\text{ m/s}$ . The driver suddenly spots an obstacle ahead and takes $0.5\text{ s}$ to react before applying the brakes. Once the brakes are applied, the car decelerates uniformly at a rate of $4\text{ m/s}^2$ until coming to a complete stop. What is the total distance traveled by the car from the moment the driver spots the obstacle until the car stops completely?

50\text{ m}

60\text{ m}

110\text{ m}

40\text{ m}

Show answer & explanation

Answer: $60\text{ m}$

Answer

The total distance traveled by the car is

60\text{ m}

.

The motion consists of two distinct stages: first, constant speed motion during the

0.5\text{ s}

reaction time giving

s_1 = 20 \times 0.5 = 10\text{ m}

; second, uniform deceleration from

20\text{ m/s}

to rest over

s_2 = \frac{u^2}{2a} = \frac{400}{8} = 50\text{ m}

. Adding both stages yields a total distance of

60\text{ m}

.

Step-by-Step Solution

1

Calculate the reaction distance traveled at constant speed before braking.

s_1 = v \times t = 20\text{ m/s} \times 0.5\text{ s} = 10\text{ m}

.

During the reaction time, acceleration is zero, so distance equals speed multiplied by time.

2

Calculate the braking distance using the third equation of motion.

Using

v^2 = u^2 + 2as

:

0 = (20)^2 + 2(-4)s_2 \implies 8s_2 = 400 \implies s_2 = 50\text{ m}

.

The car decelerates from

u = 20\text{ m/s}

to

v = 0\text{ m/s}

at

a = -4\text{ m/s}^2

.

3

Sum the reaction distance and the braking distance to obtain the total stopping distance.

s_{\text{total}} = s_1 + s_2 = 10\text{ m} + 50\text{ m} = 60\text{ m}

.

Total distance is the sum of distances covered in both stages of motion.

Key Concept

Multi-stage linear motion combining constant velocity reaction distance and uniform deceleration braking distance.

Estimated Time:1m 30s

Question 7210Question →

Match each physical wave propagation scenario on the left with its correct classification and particle vibration characteristics on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Oscillating electric charge in a vacuum matches non-mechanical wave with perpendicular field oscillations; pressure pulse in argon gas matches mechanical longitudinal wave with parallel oscillations; shear pulse on a stretched string matches mechanical transverse wave with perpendicular particle oscillations; water ripple on deep water matches mechanical surface wave with combined circular motion.

The correct pairings align each physical wave scenario with its underlying propagation mechanics. The oscillating charge producing field variations in a vacuum represents a non-mechanical transverse electromagnetic wave. The pressure disturbance in argon represents a longitudinal mechanical wave because fluids transmit energy through density compressions. The pulse on a stretched string is a transverse mechanical wave enabled by string tension. Surface water ripples are two-dimensional interface waves exhibiting combined longitudinal and transverse particle movement.

Step-by-Step Solution

1

Classify the disturbance in a vacuum (oscillating charge).

Identified as electromagnetic radiation, which is non-mechanical and transverse.

Electromagnetic waves do not require a material medium and involve mutually perpendicular field vectors perpendicular to propagation.

2

Analyze wave propagation through a gaseous medium (argon gas).

Identified as a purely longitudinal mechanical wave.

Gases lack shear strength and can only transmit mechanical energy through volume variations (compressions and rarefactions) parallel to the propagation vector.

3

Evaluate wave propagation along a stretched 1D solid string.

Identified as a transverse mechanical wave.

Tension in the solid string restores lateral shear displacements, causing medium elements to vibrate perpendicular to the string length.

4

Determine the nature of surface waves on liquid interfaces.

Identified as a mechanical surface wave with orbital motion.

Interface boundaries support gravity and surface tension restoring forces, producing elliptical/circular particle trajectories combining transverse and longitudinal components.

Key Concept

Classification of waves by medium requirement (mechanical vs. electromagnetic) and directional displacement mode (transverse, longitudinal, and surface orbital motion).

Estimated Time:2m 0s

Question 7211Question →

What are the values of $y$ that satisfy the simultaneous equations $3x + y = 10$ and $x^2 + y = 14$ ?

-1

or

4

-13

or

2

-2

or

13

7

or

22

Show answer & explanation

Answer: $-2$ or $13$

Answer

The values of

y

that satisfy the simultaneous equations are

-2

or

13

.

Rearranging the linear equation gives

y = 10 - 3x

. Substituting this into

x^2 + y = 14

yields

x^2 + 10 - 3x = 14

, which simplifies to

x^2 - 3x - 4 = 0

. Factoring gives

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

, so

x = 4

or

x = -1

. Substituting

x = 4

into

y = 10 - 3x

gives

y = -2

, and substituting

x = -1

gives

y = 13

. Thus, the required values of

y

are

-2

or

13

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation

y = 10 - 3x

Isolation of variable

y

allows straightforward substitution into the quadratic equation.

2

Substitute

y = 10 - 3x

into the quadratic equation

x^2 + y = 14

x^2 + (10 - 3x) = 14 \implies x^2 - 3x - 4 = 0

Creates a quadratic equation in terms of a single variable

x

.

3

Solve the quadratic equation

x^2 - 3x - 4 = 0

for

x

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

Factoring determines the

x

-coordinates of the solution points.

4

Substitute each

x

-value back into

y = 10 - 3x

to find the corresponding

y

-values

For

x = 4

:

y = 10 - 3(4) = -2

; for

x = -1

:

y = 10 - 3(-1) = 13

Evaluates the exact values of

y

requested by the question.

Key Concept

Simultaneous Linear and Quadratic Equations

Question 7212Question →

A fixed mass of gas in a rigid container exerts a pressure of $3.0\text{ atm}$ at a temperature of $27^\circ\text{C}$ . If the container is heated at constant volume until the temperature reaches $177^\circ\text{C}$ , what is the final pressure exerted by the gas?

2.0\text{ atm}

4.5\text{ atm}

19.7\text{ atm}

1.77\text{ atm}

Show answer & explanation

Answer: $4.5\text{ atm}$

Answer

The final pressure exerted by the gas is

4.5\text{ atm}

.

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

P_1/T_1 = P_2/T_2

). Converting temperatures to Kelvin gives

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

and

T_2 = 177 + 273 = 450\text{ K}

. Solving for

P_2 = 3.0 \times (450 / 300) = 4.5\text{ atm}

.

Step-by-Step Solution

1

Convert all given temperatures from Celsius to Kelvin

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

,

T_2 = 177 + 273 = 450\text{ K}

Gas laws require absolute temperature measured in Kelvin.

2

State the Pressure Law equation relating pressure and absolute temperature at constant volume

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

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

3

Substitute the known values into the equation and solve for final pressure (

P_2

)

P_2 = P_1 \times \frac{T_2}{T_1} = 3.0 \times \frac{450}{300} = 4.5\text{ atm}

Multiplying initial pressure by the ratio of absolute temperatures gives the final pressure.

Key Concept

Pressure Law (Gay-Lussac's Law)

Estimated Time:50s

Question 7213Question →

A laboratory technician records four physical quantities during an experiment: electric current, thermodynamic temperature, electric charge, and luminous intensity. Which of these recorded quantities is a derived physical quantity?

Electric charge

Electric current

Thermodynamic temperature

Luminous intensity

Show answer & explanation

Answer: Electric charge

Answer

Electric charge is the derived physical quantity.

Electric charge is a derived physical quantity because it is defined mathematically in terms of fundamental quantities: electric charge equals electric current multiplied by time (

Q = I \cdot t

). Its SI unit, the coulomb (C), is defined as an ampere-second (

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

).

Step-by-Step Solution

1

Identify the seven fundamental physical quantities defined in the SI system.

The seven base quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

Fundamental quantities are independent base quantities from which all other physical quantities are derived.

2

Analyze each quantity given in the scenario against the list of fundamental quantities.

Electric current, thermodynamic temperature, and luminous intensity are fundamental quantities. Electric charge is not on the list of base quantities.

Electric charge (

Q

) is derived from the base quantities electric current (

I

) and time (

t

) through the relation

Q = I \cdot t

.

Key Concept

Fundamental quantities are basic physical quantities that do not depend on other quantities. Derived quantities are defined in terms of fundamental quantities.

Question 7214Question →

Although non-alignment served as a foundational doctrine of Nigerian foreign policy, Nigeria's membership in the Organization of the Petroleum Exporting Countries (OPEC) introduced strategic trade-offs between anti-imperial political diplomacy and global market realities. Which of the following statements best analyzes how OPEC's economic imperatives modified Nigeria's application of Non-Aligned Movement (NAM) principles during international energy crises?

Nigeria subordinated its economic interests within OPEC by enforcing absolute oil boycotts against all Western nations whenever NAM issued anti-colonial resolutions.

Nigeria balanced NAM political non-alignment with pragmatic economic nationalism, utilizing OPEC price mechanisms to maximize domestic revenue while preserving trade relationships with major Western consumers.

Nigeria leveraged OPEC's collective decision-making structure to legally transform NAM from a non-binding diplomatic conference into a supranational military command.

Nigeria's commitment to NAM's principle of peaceful coexistence legally stripped its national government of sovereign authority to regulate multinational oil company production quotas.

Show answer & explanation

Answer: Nigeria balanced NAM political non-alignment with pragmatic economic nationalism, utilizing OPEC price mechanisms to maximize domestic revenue while preserving trade relationships with major Western consumers.

Answer

Nigeria balanced Non-Aligned Movement (NAM) political non-alignment with pragmatic economic nationalism, utilizing OPEC price mechanisms to maximize domestic revenue while preserving trade relationships with major Western consumers.

Nigeria's foreign policy under successive administrations navigated the Cold War era by adopting political non-alignment while maintaining an essential economic relationship with Western trade partners. Joining OPEC in 1971 enabled Nigeria to assert national control over its oil resources and secure market revenues through collective pricing without compromising its non-aligned diplomatic status.

Step-by-Step Solution

1

Analyze the core foreign policy objectives of Nigeria within the Non-Aligned Movement (NAM)

NAM promoted political independence, non-adherence to Cold War power blocs, anti-colonialism, and sovereign equality among developing nations.

Establishing the political baseline of Nigerian foreign policy principles is necessary to evaluate external trade-offs.

2

Evaluate the economic dynamics of Nigeria's membership in OPEC starting in 1971

OPEC functioned as an economic cartel aimed at coordinating production quotas and securing fair petroleum pricing for revenue-dependent developing countries.

Identifies the primary motive of OPEC participation as national fiscal interest and resource stewardship.

3

Synthesize the interaction between NAM political non-alignment and OPEC economic pragmatism

Nigeria maintained diplomatic independence from superpower blocs while engaging in commercial oil trade with Western markets, demonstrating that economic national interest took precedence over rigid ideological confrontation.

Examines how foreign policy formulation resolves competing political and economic priorities.

Key Concept

Economic Nationalism and Diplomatic Pragmatism in Foreign Policy

Question 7215Question →

Which of the following expressions represents the correct dimensions of pressure?

M L^{-1} T^{-2}

M L T^{-2}

M L^2 T^{-2}

M L^{-2} T^{-2}

Show answer & explanation

Answer: $M L^{-1} T^{-2}$

Answer

M L^{-1} T^{-2}

Pressure is defined as Force divided by Area. Substituting the fundamental dimensions gives Force as

M L T^{-2}

and Area as

L^2

. Dividing Force by Area results in

\frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2}

. Therefore, the expression

M L^{-1} T^{-2}

is correct.

Step-by-Step Solution

1

State the physical definition formula for pressure

Pressure

P = \frac{\text{Force}}{\text{Area}}

Pressure is defined as force applied perpendicular to a surface per unit area.

2

Substitute fundamental dimensions for force and area

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

Force has dimensions

M L T^{-2}

and area has dimensions

L^2

.

3

Simplify the index of length

L

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

Applying the rules of indices for length gives

L^1 \cdot L^{-2} = L^{-1}

.

Key Concept

Dimensions of Pressure

Question 7216Question →

When the polynomial $P(x) = x^3 - 2x^2 + ax + 8$ is divided by $(x - 3)$ , the remainder is $14$ . What is the value of the constant $a$ ?

Show answer & explanation

Answer: -1

Answer

The value of the constant

a

is

-1

.

According to the Remainder Theorem, dividing

P(x)

by

(x - 3)

means the remainder is

P(3)

. Evaluating

P(3) = 3^3 - 2(3)^2 + 3a + 8 = 27 - 18 + 3a + 8 = 17 + 3a

. Setting this equal to the remainder

14

gives

17 + 3a = 14

, which simplifies to

3a = -3

and yields

a = -1

.

Step-by-Step Solution

1

Apply the Remainder Theorem

The remainder when

P(x)

is divided by

(x - 3)

is equal to

P(3)

.

The Remainder Theorem states that dividing a polynomial

P(x)

by

(x - c)

yields a remainder equal to

P(c)

.

2

Substitute

x = 3

into the polynomial and set equal to the given remainder

3^3 - 2(3)^2 + a(3) + 8 = 14

Setting the value of

P(3)

equal to

14

allows us to form a linear equation for the unknown constant

a

.

3

Simplify numerical terms in the equation

27 - 18 + 3a + 8 = 14 \implies 17 + 3a = 14

Evaluate exponents and multiplication to isolate the term containing

a

.

4

Solve for

a

3a = -3 \implies a = -1

Subtract

17

from both sides and divide by

3

.

Key Concept

Polynomial Remainder Theorem

Question 7217Question →

In triangle $XYZ$ , side length $x = 3\text{ cm}$ , side length $y = 8\text{ cm}$ , and the included angle $\angle Z = 60^\circ$ . What is the length of side $z$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

z

is

7\text{ cm}

.

Using the Cosine Rule

z^2 = x^2 + y^2 - 2xy \cos Z

, substituting

x = 3

,

y = 8

, and

\angle Z = 60^\circ

yields

z^2 = 3^2 + 8^2 - 2(3)(8)(0.5) = 9 + 64 - 24 = 49

. Taking the positive square root gives

z = 7\text{ cm}

.

Step-by-Step Solution

1

Identify known triangle components and select the appropriate rule

Two sides and the included angle (SAS) are given:

x = 3

,

y = 8

,

\angle Z = 60^\circ

, requiring the Cosine Rule.

When given two sides and the included angle (SAS), the Cosine Rule is used to find the third side.

2

Substitute values into the Cosine Rule formula

z^2 = x^2 + y^2 - 2xy \cos Z

z^2 = 3^2 + 8^2 - 2(3)(8)\cos 60^\circ

Direct algebraic substitution of side lengths and angle measure into the Cosine Rule.

3

Evaluate the trigonometric term and simplify the arithmetic expression

z^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49

The exact value of

\cos 60^\circ

is

0.5

.

4

Solve for side length

z

by taking the principal square root

z = \sqrt{49} = 7\text{ cm}

Side length must be positive.

Key Concept

Applying the Cosine Rule to find the third side of a non-right-angled triangle given two sides and an included angle (SAS).

Estimated Time:1m 30s

Question 7218Question →

Match each ecological pyramid phenomenon or energy flow concept on the left with its correct metabolic or thermodynamic explanation on the right. Which pairing correctly matches each concept to its underlying biological cause?

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match: (1) Inverted biomass pyramid in aquatic systems with producer turnover rate and rapid reproduction; (2) Invariably upright energy pyramid with Second Law of Thermodynamics heat loss; (3) Stepwise reduction in available energy with metabolic and waste loss of approximately 90% per level; (4) Biomagnification with non-biodegradable fat-soluble toxins concentrating in smaller higher-level biomasses.

Each ecological concept matches its precise biological mechanism: aquatic biomass inversion is caused by rapid turnover of primary producers; energy pyramids are strictly upright due to metabolic heat loss (Second Law of Thermodynamics); energy reduction across trophic levels stems from respiration and excretion losses (~90%); and biomagnification occurs because persistent toxins accumulate in fat tissue as biomass decreases at higher levels.

Step-by-Step Solution

1

Analyze standing crop vs. productivity in aquatic habitats

Recognize that phytoplankton productivity is high despite low standing biomass due to rapid population turnover, creating an inverted biomass pyramid.

Explains why biomass pyramids can be inverted while energy production rates remain normal.

2

Apply thermodynamic laws to ecological energy transfer

Establish that energy cannot be recycled and entropy increases via metabolic heat release during each conversion.

Demonstrates why energy pyramids are strictly upright in every natural ecosystem.

3

Examine trophic efficiency calculations

Relate energy loss across trophic levels to physiological processes like movement, excretion, and cellular respiration (~90% lost).

Calculates net energy available to secondary and tertiary consumers.

4

Trace toxic substance dynamics in trophic chains

Correlate fat-solubility and biological persistence with increased toxin concentration at apex trophic levels.

Defines the biological mechanism behind biomagnification.

Key Concept

Thermodynamic laws in energy flow, trophic efficiencies, ecological pyramid structures, and biological magnification.

Question 7219Question →

An online retail customer completes a legitimate electronic purchase using their valid debit card, successfully receives the ordered goods at their home address, but later files a false dispute with their card-issuing bank claiming the transaction was unauthorized in order to obtain a full refund while keeping the product. Which form of electronic business fraud risk is demonstrated in this scenario?

Chargeback fraud

Phishing attack

Business-to-Consumer credential spoofing

Indemnity insurance claim failure

Show answer & explanation

Answer: Chargeback fraud

Answer

Chargeback fraud

The scenario describes chargeback fraud, which occurs when a buyer makes a legal online purchase, receives the merchandise, and then dishonestly contacts their financial institution to claim the charge was fraudulent or unauthorized in order to obtain a refund without returning the merchandise.

Step-by-Step Solution

1

Analyze the customer action described in the scenario

The buyer completed an authentic purchase and accepted delivery of the goods.

Establishing that the transaction itself was legitimate rules out third-party account takeover or identity theft.

2

Examine the nature of the fraud committed post-delivery

The buyer falsely disputed the payment with the bank to force a financial reversal while retaining the item.

Forcing an unwarranted credit reversal through payment gateways constitutes chargeback fraud (friendly fraud).

Key Concept

Chargeback Fraud and Financial Risks in E-Commerce

Question 7220Question →

Read the dramatic extract below:

KING EZE: (Pacing the throne room after learning of his army's unexpected defeat)
"My pride has been my undoing! I ignored the council's wisdom and trusted solely in my own grand illusion of invincibility. It is not the gods who brought this ruin, but my own arrogance."

In the excerpt above, King Eze's self-acknowledgment primarily highlights which dramatic element?

Tragic flaw (Hamartia)

Dramatic irony

An aside

Comic relief

Show answer & explanation

Answer: Tragic flaw (Hamartia)

Answer

The dramatic element highlighted in the excerpt is the tragic flaw (Hamartia), specifically hubris (excessive pride).

The correct answer is tragic flaw because King Eze directly attributes his downfall to his internal personality defect—excessive pride and arrogance—which is the classic definition of hamartia in drama.

Step-by-Step Solution

1

Analyze the character's speech in the excerpt

King Eze admits that his own pride, arrogance, and refusal to heed wisdom led to his army's defeat.

Identifying internal character traits versus external events determines the specific dramatic device at play.

2

Match the character's admission to dramatic terminology

An internal defect or error in judgment within a protagonist that leads to their demise is known in tragic drama as hamartia or a tragic flaw.

Connecting literary definition with textual evidence confirms the correct dramatic element.

Key Concept

Tragic Flaw (Hamartia)

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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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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