All practice questions

Which word nearest in meaning to 'obstinately unyielding' or 'stubborn' correctly completes the sentence below?

Fill in the blanks below

Despite prolonged mediation by the reconciliation committee, the faction leaders remained completely in their positions, frustrating all attempts at compromise.

Show answer & explanation

Answer

The word 'intransigent' (or 'obdurate') correctly completes the sentence, as it signifies an unyielding, refusal to compromise attitude that matches the context of failed mediation.

The term 'intransigent' means refusing to agree or compromise, perfectly matching the context where faction leaders frustrate mediation by holding obstinately to their positions.

Step-by-Step Solution

1

Analyze the context of the sentence

The clause 'frustrating all attempts at compromise' indicates that the faction leaders refused to alter their positions despite mediation.

Contextual clues demand an adjective denoting severe stubbornness or unyielding refusal to compromise.

2

Identify the precise lexical synonym required

Words such as 'intransigent', 'obdurate', or 'recalcitrant' precisely convey an uncompromising, obstinate attitude within formal socio-political register.

These terms function as exact formal synonyms for 'obstinately unyielding' in high-level English usage.

Key Concept

Contextual Vocabulary and Synonyms

Question 4082Question →

Given that $\sin \theta = \frac{5}{13}$ , where $\theta$ is an acute angle, evaluate the value of $13 \cos \theta - 12 \tan \theta$ . What is the numerical value?

Show answer & explanation

Answer: 7

Answer

The numerical value of the expression

13 \cos \theta - 12 \tan \theta

is 7.

For an acute angle

\theta

with

\sin \theta = \frac{5}{13}

, the corresponding right triangle has an opposite side of 5, a hypotenuse of 13, and an adjacent side of

\sqrt{13^2 - 5^2} = 12

. Therefore,

\cos \theta = \frac{12}{13}

and

\tan \theta = \frac{5}{12}

. Evaluating

13 \cos \theta - 12 \tan \theta

yields

13\left(\frac{12}{13}\right) - 12\left(\frac{5}{12}\right) = 12 - 5 = 7

.

Step-by-Step Solution

1

Determine

\cos \theta

using the right triangle ratio or Pythagorean identity.

\cos \theta = \frac{12}{13}

Since

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}

, the adjacent side is

\sqrt{13^2 - 5^2} = 12

. Because

\theta

is acute,

\cos \theta

is positive.

2

Determine

\tan \theta

using the ratio of opposite to adjacent sides.

\tan \tan \theta = \frac{5}{12}

\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}$.

3

Substitute the evaluated ratios into

13 \cos \theta - 12 \tan \theta

and simplify.

13\left(\frac{12}{13}\right) - 12\left(\frac{5}{12}\right) = 12 - 5 = 7

Multiplying clears the denominators, leaving

12 - 5 = 7

.

Key Concept

Basic Trigonometric Ratios and Pythagorean Triples

Estimated Time:1m 15s

Question 4083Question →

If $\sin \theta = \frac{3}{5}$ for an acute angle $\theta$ , what is the exact value of $5 \cos \theta + 4 \tan \theta$ ?

Show answer & explanation

Answer: 7

Answer

The exact value of

5 \cos \theta + 4 \tan \theta

is 7.

For an acute angle

\theta

with

\sin \theta = \frac{3}{5}

, the corresponding right triangle has opposite side = 3, hypotenuse = 5, and adjacent side = 4. Using trig definitions,

\cos \theta = \frac{4}{5}

and

\tan \theta = \frac{3}{4}

. Evaluating

5 \cos \theta + 4 \tan \theta

gives

5\left(\frac{4}{5}\right) + 4\left(\frac{3}{4}\right) = 4 + 3 = 7

.

Step-by-Step Solution

1

Find the adjacent side of the right-angled triangle.

Adjacent side =

\sqrt{5^2 - 3^2} = 4

.

By the Pythagorean theorem (

a^2 + b^2 = c^2

), where opposite = 3 and hypotenuse = 5.

2

Determine the values of

\cos \theta

and

\tan \theta

.

\cos \theta = \frac{4}{5}

and

\tan \theta = \frac{3}{4}

.

Using fundamental trigonometric definitions: cosine is adjacent/hypotenuse and tangent is opposite/adjacent.

3

Substitute these values into

5 \cos \theta + 4 \tan \theta

and simplify.

5\left(\frac{4}{5}\right) + 4\left(\frac{3}{4}\right) = 4 + 3 = 7

.

Performing simple multiplication and addition gives 7.

Key Concept

Basic Trigonometric Ratios in Right Triangles

Question 4084Question →

Given that $z$ varies directly as $x^2$ and inversely as $\sqrt{y}$ , and $z = 12$ when $x = 2$ and $y = 9$ , what is the value of $z$ when $x = 3$ and $y = 16$ ?

\frac{81}{4}

\frac{27}{2}

\frac{243}{16}

\frac{64}{9}

Show answer & explanation

Answer: $\frac{81}{4}$

Answer

\frac{81}{4}

The joint variation formula is

z = \frac{k x^2}{\sqrt{y}}

. Substituting the given values

x = 2, y = 9, z = 12

gives

12 = \frac{4k}{3}

, which yields

k = 9

. Evaluating

z

for

x = 3

and

y = 16

gives

z = \frac{9 \times 3^2}{\sqrt{16}} = \frac{81}{4}

.

Step-by-Step Solution

1

Set up the joint variation equation

z = \frac{k x^2}{\sqrt{y}}

Direct variation means

x^2

is in the numerator, and inverse variation means

\sqrt{y}

is in the denominator.

2

Substitute the initial values to solve for the constant of variation

k

12 = \frac{k (2)^2}{\sqrt{9}} \implies 12 = \frac{4k}{3} \implies 4k = 36 \implies k = 9

Using

x = 2

,

y = 9

, and

z = 12

allows us to find the constant

k

.

3

Calculate the new value of

z

using

x = 3

and

y = 16

z = \frac{9 (3)^2}{\sqrt{16}} = \frac{9 \times 9}{4} = \frac{81}{4}

Substitute

k = 9

,

x = 3

, and

y = 16

into the variation formula.

Key Concept

Joint Variation involving powers and roots

Estimated Time:1m 30s

Question 4085Question →

Convert the fractional binary number $0.1101_2$ to its equivalent base 10 (decimal) value. What is the decimal value?

Show answer & explanation

Answer: 0.8125

Answer

The base 10 value of

0.1101_2

is

0.8125

.

To convert a fractional binary number to decimal, expand each digit after the radix point using decreasing negative powers of 2 (

2^{-1}, 2^{-2}, 2^{-3}, 2^{-4}

). Evaluating

1(0.5) + 1(0.25) + 0(0.125) + 1(0.0625)

yields

0.8125

.

Step-by-Step Solution

1

Write the given binary fraction in place-value expansion using powers of 2

0.1101_2 = 1 \cdot 2^{-1} + 1 \cdot 2^{-2} + 0 \cdot 2^{-3} + 1 \cdot 2^{-4}

Positions after the binary point represent negative powers of 2 starting from

2^{-1}

.

2

Evaluate each fractional component

2^{-1} = 0.5

,

2^{-2} = 0.25

,

2^{-3} = 0.125

,

2^{-4} = 0.0625

Calculating standard decimal values for binary fractional places.

3

Add the non-zero fractional terms together

0.5 + 0.25 + 0.0625 = 0.8125

Summing the decimal values gives the complete converted decimal representation.

Key Concept

Conversion of fractional numbers from base 2 to base 10

Question 4086Question →

The electrical resistance $R$ of a wire varies directly as its length $L$ and inversely as the square of its diameter $d$ . If a wire of length $36\text{ m}$ and diameter $3\text{ mm}$ has a resistance of $16\ \Omega$ , what is the resistance, in ohms, of a wire of the same material with a length of $45\text{ m}$ and a diameter of $5\text{ mm}$ ?

Show answer & explanation

Answer: 7.2

Answer

The resistance of the wire is 7.2 ohms.

The equation governing the relation is

R = \frac{kL}{d^2}

. Substituting the initial parameters

R=16\ \Omega

,

L=36\text{ m}

, and

d=3\text{ mm}

gives

16 = \frac{36k}{9} = 4k

, which yields

k = 4

. Using

k = 4

with the new dimensions

L=45\text{ m}

and

d=5\text{ mm}

gives

R = \frac{4 \times 45}{5^2} = \frac{180}{25} = 7.2\ \Omega

.

Step-by-Step Solution

1

Formulate the joint variation equation

R = \frac{kL}{d^2}

Direct variation places length

L

in the numerator and inverse variation of the square of diameter

d

places

d^2

in the denominator.

2

Determine the variation constant

k

k = 4

Substituting

R = 16

,

L = 36

, and

d = 3

gives

16 = \frac{36k}{9} \implies 16 = 4k \implies k = 4

.

3

Calculate the new resistance

R = 7.2\ \Omega

Substituting

k = 4

,

L = 45

, and

d = 5

into

R = \frac{kL}{d^2}

yields

R = \frac{4 \times 45}{25} = 7.2

.

Key Concept

Joint Variation involving direct proportionality and inverse square law

Question 4087Question →

Read the passage below carefully:

Over the past century, standard ethnographic documentation of West African oral traditions relied predominantly on foreign institutions collecting recordings for distant archives, leaving local communities detached from their own intellectual heritage. Recently, however, a shift toward community-led digital archiving has redefined regional cultural preservation. The process begins with participatory field recording, wherein community elders and local youth collaboratively document oral narratives, songs, and proverbs using accessible digital technology. Following data collection, local archival teams undertake systematic transcription and contextual metadata annotation, ensuring that linguistic nuances and cultural background are accurately catalogued in indigenous dialects. Once digitized, these repositories undergo decentralized community verification, allowing local custodians to review, curate, and restrict sensitive sacred content before public distribution. Ultimately, this community-driven workflow culminates in open-access digital platforms that reintegrate oral traditions into local school curricula, restoring agency to indigenous communities while safeguarding fragile intangible heritage for future generations.

Based on the passage above, arrange the key stages of the community-led oral archiving workflow in their correct logical sequence from initial collection to final implementation.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence of stages in the community-led archiving workflow is: 1) Collaboratively capturing oral narratives and songs using accessible digital recording tools in the field, 2) Transcribing audio recordings and attaching detailed contextual metadata in indigenous dialects, 3) Conducting community-led verification to review and curate sensitive sacred material prior to public release, and 4) Reintegrating the verified digital repositories into school curricula to restore community cultural agency.

The author outlines a clear chronological progression for community-led oral history preservation: initial participatory field recording leads into transcription and cataloguing, followed by local custodian verification and curation, which ultimately culminates in educational curriculum integration.

Step-by-Step Solution

1

Identify the initial phase of the archiving process described in the passage.

Field recording using accessible digital tools is established as the starting point.

The text explicitly states that the process begins with participatory field recording by elders and youth.

2

Identify the subsequent analytical and cataloguing stage.

Systematic transcription and metadata annotation follow data collection.

The passage notes that following data collection, teams undertake transcription and annotation in local dialects.

3

Determine the content governance and curation phase.

Decentralized community verification and restriction of sensitive material occur next.

The text explains that once digitized, repositories undergo review by local custodians before public distribution.

4

Determine the final stage of educational integration.

Reintegration into school curricula marks the culmination of the workflow.

The passage concludes by highlighting that the process ultimately culminates in open-access educational platforms.

Key Concept

Sequential Summary and Logical Progression of Arguments

Question 4088Question →

Given that $\theta$ is an acute angle satisfying the relationship $\sec \theta + \tan \theta = 3$ , what is the exact value of $5\sin \theta$ ?

Show answer & explanation

Answer: 4

Answer

The exact value of

5\sin \theta

is 4.

Using the identity

\sec^2 \theta - \tan^2 \theta = 1

, we deduce

(\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1

. Given

\sec \theta + \tan \theta = 3

, it follows that

\sec \theta - \tan \theta = \frac{1}{3}

. Solving the system of equations yields

\sec \theta = \frac{5}{3}

and

\tan \theta = \frac{4}{3}

, which gives

\sin \theta = \frac{4}{5}

. Multiplying by 5 gives the final answer of 4.

Step-by-Step Solution

1

Apply the trigonometric Pythagorean identity

\sec^2 \theta - \tan^2 \theta = 1

This relates secant and tangent functions directly.

2

Factorize the identity and solve for

\sec \theta - \tan \theta

(\sec \theta - \tan \theta)(3) = 1 \implies \sec \theta - \tan \tan \theta = \frac{1}{3}

Using the algebraic identity

a^2 - b^2 = (a - b)(a + b)

.

3

Set up a linear system to solve for

\sec \theta

and

\tan \theta

\sec \theta = \frac{5}{3}, \quad \tan \theta = \frac{4}{3}

Adding and subtracting the equations

\sec \theta + \tan \theta = 3

and \sec \theta - \tan \theta = \frac{1}{3} gives the individual function values.

4

Calculate

\sin \theta

and evaluate

5\sin \theta

\sin \theta = \frac{\tan \theta}{\sec \theta} = \frac{4/3}{5/3} = \frac{4}{5} \implies 5\sin \theta = 4

The quotient of tangent and secant gives sine.

Key Concept

Pythagorean Trigonometric Identities

Alternative Method

Draw a right-angled triangle where hypotenuse over adjacent plus opposite over adjacent equals 3:

\frac{c + a}{b} = 3

. By Pythagorean theorem

c^2 - a^2 = b^2

, so

\frac{c - a}{b} = \frac{1}{3}

. Solving yields

a/c = 4/5

, hence

\sin \theta = 4/5

and

5\sin \theta = 4

.

Estimated Time:2m 0s

Question 4089Question →

The hourly operational cost, $C$ Naira, of an industrial water pump is partly constant and partly varies jointly as the flow rate, $r$ in litres per second, and the square of the pressure head, $h$ in metres. When $r = 10\text{ L/s}$ and $h = 4\text{ m}$ , the operational cost is $\text{N}620$ . When $r = 15\text{ L/s}$ and $h = 2\text{ m}$ , the operational cost is $\text{N}380$ . What is the operational cost in Naira when $r = 20\text{ L/s}$ and $h = 3\text{ m}$ ?

Show answer & explanation

Answer: 668

Answer

The operational cost when

r = 20

and

h = 3

is 668 Naira.

The partial and joint variation relationship is defined by

C = k_1 + k_2 r h^2

. Substituting the two given states gives the simultaneous equations

620 = k_1 + 160k_2

and

380 = k_1 + 60k_2

. Subtracting these equations gives

100k_2 = 240

, so

k_2 = 2.4

. Substituting

k_2 = 2.4

into the second equation yields

k_1 = 236

. Finally, evaluating

C

for

r = 20

and

h = 3

gives

C = 236 + 2.4(20)(3^2) = 236 + 432 = 668

.

Step-by-Step Solution

1

Set up the variation equation

C = k_1 + k_2 r h^2

, where

k_1

is the constant part and

k_2

is the constant of joint variation.

The problem states that

C

is partly constant (

k_1

) and partly varies jointly as

r

and

h^2

(

k_2 r h^2

).

2

Form simultaneous linear equations using the given data points

(1)

620 = k_1 + 160k_2

and (2)

380 = k_1 + 60k_2

Substituting

r = 10, h = 4, C = 620

gives

10 \times 4^2 = 160

. Substituting

r = 15, h = 2, C = 380

gives

15 \times 2^2 = 60

.

3

Solve for the constants

k_1

and

k_2

k_2 = 2.4

and

k_1 = 236

Subtracting equation (2) from (1) eliminates

k_1

, giving

100k_2 = 240 \implies k_2 = 2.4

. Substituting back into equation (2) gives

k_1 = 380 - 60(2.4) = 236

.

4

Calculate the operational cost for the target parameters

C = 236 + 2.4 \times 20 \times 3^2 = 668

Substitute

k_1 = 236

,

k_2 = 2.4

,

r = 20

, and

h = 3

into the variation formula.

Key Concept

Partial and Joint Variation

Question 4090Question →

A regular hexagon has a side length of $6\text{ cm}$ . At each vertex of the hexagon, a circular sector of radius $3\text{ cm}$ is formed inside the figure. Taking $\pi = \frac{22}{7}$ and $\sqrt{3} = 1.732$ , what is the area of the remaining region inside the hexagon not covered by the sectors, in $\text{cm}^2$ , correct to two decimal places?

Show answer & explanation

Answer: 36.96

Answer

The area of the remaining region inside the hexagon is

36.96\text{ cm}^2

.

The total area of the regular hexagon is computed by multiplying the area of one equilateral triangle of side

6\text{ cm}

by 6, yielding

54\sqrt{3} = 54 \times 1.732 = 93.528\text{ cm}^2

. Each interior angle of a regular hexagon is

120^\circ

, so each vertex sector has a central angle of

120^\circ

and radius

3\text{ cm}

. The area of one sector is

\frac{120^\circ}{360^\circ} \times \frac{22}{7} \times 3^2 = \frac{66}{7}\text{ cm}^2

. The total area for all six sectors is

6 \times \frac{66}{7} = \frac{396}{7} \approx 56.5714\text{ cm}^2

. Subtracting this from the total area gives

93.528 - 56.5714 = 36.9566\text{ cm}^2

, which rounds to

36.96\text{ cm}^2

.

Step-by-Step Solution

1

Determine the interior angle of the regular hexagon.

Each interior angle is

120^\circ

.

The formula for the interior angle of a regular polygon with

n

sides is

\frac{(n-2) \times 180^\circ}{n}

.

2

Calculate the total area of the 6 circular sectors at the vertices.

Total sector area is

\frac{396}{7} \approx 56.5714\text{ cm}^2

.

Each sector has a central angle of

120^\circ

and radius

3\text{ cm}

. With 6 sectors, the total area is

6 \times \frac{120^\circ}{360^\circ} \times \frac{22}{7} \times 3^2 = 18 \times \frac{22}{7} = \frac{396}{7}\text{ cm}^2

.

3

Calculate the total area of the regular hexagon.

Hexagon area is

93.528\text{ cm}^2

.

A regular hexagon consists of 6 equilateral triangles of side length

6\text{ cm}

. Area =

6 \times \left(\frac{\sqrt{3}}{4} \times 6^2\right) = 54\sqrt{3} = 54 \times 1.732 = 93.528\text{ cm}^2

.

4

Subtract the sector area from the total hexagon area.

93.528 - 56.5714 = 36.9566\text{ cm}^2 \approx 36.96\text{ cm}^2

.

The remaining area is the total area minus the area occupied by the six corner sectors.

Key Concept

Area of regular polygons and circular sectors

Estimated Time:2m 30s

Question 4091Question →

Supply the word nearest in meaning to 'haughty' or 'arrogantly disdainful' that correctly completes the sentence below.

Fill in the blanks below

The committee condemned the delegate's attitude toward his colleagues during the plenary session.

Show answer & explanation

Answer

The word that best completes the sentence with the meaning of haughty or arrogantly disdainful is 'supercilious' (or synonyms such as 'haughty', 'disdainful', or 'imperious').

The word 'supercilious' means behaving or looking as though one thinks one is superior to others, which directly matches the target meaning of haughty or arrogantly disdainful.

Step-by-Step Solution

1

Analyze the sentence context

The sentence describes a negative conduct by a delegate during a formal session that brought condemnation from a committee.

Establishing context helps determine the required register and sentiment of the missing word.

2

Identify the target meaning

The target concept requires a word expressing haughtiness or arrogant disdain.

The stem specifies 'haughty' or 'arrogantly disdainful' as the target meaning.

3

Select the appropriate synonym

'Supercilious' fits grammatically as an adjective modifying 'attitude' and precisely conveys disdainful arrogance.

'Supercilious' matches the formal literary register expected in standard exam assessments.

Key Concept

Synonyms and Words Nearest in Meaning

Estimated Time:1m 30s

Question 4092Question →

What is the range of values of $x$ that satisfies the inequality $3 - 2x > 9$ ?

x < -3

x > -3

x < 3

x > 3

Show answer & explanation

Answer: $x < -3$

Answer

x < -3

Subtracting 3 from both sides yields

-2x > 6

. Dividing both sides by

-2

and reversing the inequality symbol gives

x < -3

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality.

-2x > 6

To isolate the variable term

-2x

on the left-hand side.

2

Divide both sides by

-2

and reverse the inequality sign.

x < -3

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Reversing inequality signs upon multiplication or division by negative numbers

Estimated Time:45s

Question 4093Question →

In the sentence below, identify the word that is nearest in meaning to the underlined word.

'The managing director's perspicacious assessment of the volatile market conditions enabled the corporation to avert imminent financial collapse.'

Which word is nearest in meaning to the underlined word as used in the sentence?

astute

conspicuous

obtuse

meticulous

Show answer & explanation

Answer: astute

Answer

The word nearest in meaning to 'perspicacious' is 'astute'.

In this context, 'perspicacious' refers to having strong insight, keen judgment, and mental sharpness. The word 'astute' directly corresponds to this meaning, as an astute assessment demonstrates acute perception and shrewdness in navigating volatile situations.

Step-by-Step Solution

1

Analyze the context of the sentence

The target word describes an assessment of 'volatile market conditions' that successfully prevented 'imminent financial collapse'. This implies sharp insight, keen discernment, and shrewdness.

Contextual clues in vocabulary questions dictate the specific nuance of the target word.

2

Define the target word 'perspicacious'

Perspicacious means having keen mental perception, discernment, and understanding; insightful and shrewd.

Establishing the precise definition allows evaluation against option choices.

3

Evaluate options against contextual meaning

'Astute' means shrewd, discerning, and mentally sharp, making it the exact synonym required.

Selecting the option that preserves both the semantic meaning and contextual nuance.

Key Concept

Identifying contextual synonyms in formal vocabulary

Estimated Time:1m 0s

Question 4094Question →

In $\Delta ABC$ , side $a = 5\text{ cm}$ , side $b = 5\sqrt{2}\text{ cm}$ , and $\angle A = 30^\circ$ . What are all possible values for $\angle B$ ?

45^\circ

or

135^\circ

45^\circ

only

60^\circ

or

120^\circ

135^\circ

only

Show answer & explanation

Answer: $45^\circ$ or $135^\circ$

Answer

45^\circ

or

135^\circ

Applying the Sine Rule gives

\frac{5}{\sin 30^\circ} = \frac{5\sqrt{2}}{\sin B}

, so

\sin B = \frac{\sqrt{2}}{2}

. The angles whose sine is

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

between

0^\circ

and

180^\circ

are

45^\circ

and

135^\circ

. Checking angle sums:

30^\circ + 45^\circ = 75^\circ < 180^\circ

and

30^\circ + 135^\circ = 165^\circ < 180^\circ

, so both

45^\circ

and

135^\circ

yield valid triangles.

Step-by-Step Solution

1

Set up the Sine Rule formula relating sides

a

,

b

and angles

A

,

B

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule connects two sides and their opposite angles in any non-right triangle.

2

Substitute given values

a = 5

,

b = 5\sqrt{2}

, and

A = 30^\circ

\frac{5}{\sin 30^\circ} = \frac{5\sqrt{2}}{\sin B} \implies \frac{5}{0.5} = \frac{5\sqrt{2}}{\sin B} \implies 10 = \frac{5\sqrt{2}}{\sin B}

Since

\sin 30^\circ = \frac{1}{2}

, simplifying yields the ratio.

3

Solve for

\sin B

\sin B = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2}

Isolating

\sin B

gives the principal trigonometric ratio value.

4

Determine all valid values for angle

B

in the range

(0^\circ, 180^\circ)

Acute

B_1 = \arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

; Obtuse

B_2 = 180^\circ - 45^\circ = 135^\circ

. Both are valid since

30^\circ + 135^\circ = 165^\circ < 180^\circ

.

Since

b > a

, the ambiguous case (SSA) produces two valid distinct triangle solutions.

Key Concept

Sine Rule and the Ambiguous Case (SSA)

Estimated Time:1m 0s

Question 4095Question →

A student measured the length of a room as $4.00\text{ m}$ instead of the actual length of $5.00\text{ m}$ . What is the percentage error in the measurement?

20\%

25\%

80\%

2\%

Show answer & explanation

Answer: $20\%$

Answer

20\%

The absolute error is the difference between the measured value (

4.00\text{ m}

) and the actual value (

5.00\text{ m}

), which is

1.00\text{ m}

. Dividing the error by the actual value gives $\frac{1.00}{5.00} = 0.20$, which equals

20\%

when expressed as a percentage.

Step-by-Step Solution

1

Identify the actual value and the measured value.

Actual length =

5.00\text{ m}

, Measured length =

4.00\text{ m}

.

Percentage error calculations require establishing the true baseline quantity.

2

Calculate the absolute error.

Absolute Error =

|5.00 - 4.00| = 1.00\text{ m}

.

Error is defined as the magnitude of the difference between the true value and measured value.

3

Compute the percentage error.

Percentage Error = $\frac{1.00}{5.00} \times 100\% = 20\%$.

Percentage error is found by dividing the absolute error by the actual value and multiplying by

100\%

.

Key Concept

Percentage Error Calculation

Question 4096Question →

What is the result of the subtraction $523_8 - 267_8$ in base 8?

234_8

256_8

244_8

344_8

Show answer & explanation

Answer: $234_8$

Answer

234_8

Performing place-by-place subtraction in base 8 requires borrowing 8 whenever a top digit is smaller than the bottom digit. Borrowing 1 from the tens place adds 8 to the units place (

3 + 8 - 7 = 4

). The tens place becomes 1, borrowing 1 from the hundreds place adds 8 (

1 + 8 - 6 = 3

). The hundreds place becomes 4, giving

4 - 2 = 2

. Thus, the answer is

234_8

.

Step-by-Step Solution

1

Subtract the units column (

3 - 7

in base 8)

Borrow

1

from the middle column (which represents

8

). The units position becomes

3 + 8 = 11

. Then

11 - 7 = 4

.

Since

3 < 7

, borrowing from the next higher position (base 8) is required.

2

Subtract the middle column (

1 - 6

in base 8)

After borrowing, the middle digit

2

becomes

1

. Borrow

1

from the hundreds column (representing

8

). The middle position becomes

1 + 8 = 9

. Then

9 - 6 = 3

.

The middle digit was reduced by

1

due to the previous borrow, requiring another borrow from the left.

3

Subtract the hundreds column (

4 - 2

in base 8)

The left digit

5

was reduced to

4

. Then

4 - 2 = 2

.

Complete the subtraction for the leading column.

4

Combine the resulting digits

234_8

Concatenating the results from left to right gives the final answer in base 8.

Key Concept

Subtraction in non-decimal number bases

Question 4097Question →

Find the set of real values of $x$ that satisfies the inequality $\frac{3 - x}{x + 2} \geq 1$ .

-2 < x \leq \frac{1}{2}

-2 \leq x \leq \frac{1}{2}

x < -2 \text{ or } x \geq \frac{1}{2}

x \geq \frac{1}{2}

Show answer & explanation

Answer: $-2 < x \leq \frac{1}{2}$

Answer

-2 < x \leq \frac{1}{2}

Subtracting 1 from both sides yields

\frac{1 - 2x}{x + 2} \geq 0

. The critical points are

x = \frac{1}{2}

and

x = -2

. Testing values shows that the fraction is positive for

-2 < x < \frac{1}{2}

and equal to zero at

x = \frac{1}{2}

. Since

x = -2

causes division by zero, it is excluded from the interval, giving

-2 < x \leq \frac{1}{2}

.

Step-by-Step Solution

1

Subtract 1 from both sides of the inequality to set one side to zero.

\frac{3 - x}{x + 2} - 1 \geq 0

Direct cross-multiplication is invalid because the sign of

(x + 2)

depends on

x

.

2

Combine the terms over a common denominator.

\frac{(3 - x) - (x + 2)}{x + 2} \geq 0 \implies \frac{1 - 2x}{x + 2} \geq 0

Simplifying the numerator yields a clear rational inequality expression.

3

Identify the critical points and domain restrictions.

Numerator critical point:

x = \frac{1}{2}

; Denominator restriction:

x \neq -2

.

The quotient changes sign around

x = \frac{1}{2}

and

x = -2

, and division by zero is undefined.

4

Test the intervals

(-\infty, -2)

,

(-2, \frac{1}{2}]

, and

(\frac{1}{2}, \infty)

.

For

x \in (-2, \frac{1}{2}]

, the expression

\frac{1 - 2x}{x + 2}

is non-negative.

When

x = 0

,

\frac{1}{2} > 0

(positive). Outside this interval, the ratio is negative.

Key Concept

Solving Rational and Linear/Quadratic Inequalities

Estimated Time:2m 0s

Question 4098Question →

In $\Delta ABC$ , side lengths are given as $a = 6\text{ cm}$ and $b = 6\sqrt{3}\text{ cm}$ , and $\angle A = 30^\circ$ . If $\angle B$ is an obtuse angle, what is the length of side $c$ ?

6\text{ cm}

12\text{ cm}

6\sqrt{3}\text{ cm}

3\sqrt{3}\text{ cm}

Show answer & explanation

Answer: $6\text{ cm}$

Answer

The length of side

c

is

6\text{ cm}

.

Using the Sine Rule,

\frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B}

, which yields

\sin B = \frac{\sqrt{3}}{2}

. The two possible values for

\angle B

are

60^\circ

(acute) and

120^\circ

(obtuse). The problem explicitly specifies that

\angle B

is obtuse, so

\angle B = 120^\circ

. Subtracting from

180^\circ

gives

\angle C = 30^\circ

. Since

\angle A = \angle C = 30^\circ

, the triangle is isosceles, making side

c

equal to side

a

, which is

6\text{ cm}

.

Step-by-Step Solution

1

Apply the Sine Rule to find

\sin B

\frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B} \implies 12 = \frac{6\sqrt{3}}{\sin B} \implies \sin B = \frac{\sqrt{3}}{2}

The Sine Rule relates side lengths to the sines of their opposite angles.

2

Determine the value of

\angle B

using the given condition

\angle B = 60^\circ

or

\angle B = 180^\circ - 60^\circ = 120^\circ

. Since

\angle B

is obtuse,

\angle B = 120^\circ

.

The inverse sine function yields two possible angle solutions between

0^\circ

and

180^\circ

(the ambiguous case).

3

Calculate the third angle

\angle C

\angle C = 180^\circ - (\angle A + \angle B) = 180^\circ - (30^\circ + 120^\circ) = 30^\circ

The sum of interior angles in any triangle is

180^\circ

.

4

Find the length of side

c

Since

\angle A = 30^\circ

and

\angle C = 30^\circ

,

\Delta ABC

is isosceles with side

c = a = 6\text{ cm}

.

Sides opposite to equal angles in a triangle are equal in length.

Key Concept

Ambiguous Case of the Sine Rule (SSA Condition)

Alternative Method

Alternatively, apply the Cosine Rule for angle

A

:

a^2 = b^2 + c^2 - 2bc \cos A \implies 6^2 = (6\sqrt{3})^2 + c^2 - 2(6\sqrt{3})c \cos 30^\circ

. Simplifying gives

36 = 108 + c^2 - 18c \implies c^2 - 18c + 72 = 0

. Factoring yields

(c - 6)(c - 12) = 0

, giving

c = 6\text{ cm}

or

c = 12\text{ cm}

. For

c = 12\text{ cm}

,

b^2 + a^2 = 108 + 36 = 144 = c^2

, making

\angle C = 90^\circ

and

\angle B = 60^\circ

(acute). Thus,

c = 6\text{ cm}

corresponds to the obtuse angle

\angle B = 120^\circ

.

Estimated Time:2m 0s

Question 4099Question →

Find the number of integers that satisfy the compound inequality $-3 < 2x + 1 \le 9$ .

Show answer & explanation

Answer: 6

Answer

The number of integer solutions satisfying the inequality is 6.

Subtracting 1 across

-3 < 2x + 1 \le 9

gives

-4 < 2x \le 8

. Dividing by 2 yields

-2 < x \le 4

. The integers in this interval are

-1, 0, 1, 2, 3,

and

4

, giving a total of 6 integer solutions.

Step-by-Step Solution

1

Subtract 1 from all parts of the compound inequality.

-4 < 2x \le 8

Isolate the variable term

2x

in the middle.

2

Divide all parts by 2.

-2 < x \le 4

Solve for

x

by undoing the coefficient of 2.

3

Identify the set of integer solutions within the interval

(-2, 4]

.

x \in \{-1, 0, 1, 2, 3, 4\}

The endpoint

-2

is excluded due to the strict inequality (

<

), while the endpoint

4

is included due to the inclusive inequality (

\\le

).

4

Count the elements in the solution set.

6

There are 6 distinct integer values in the set.

Key Concept

Solving compound linear inequalities and identifying integer solution sets.

Question 4100Question →

A student evaluated the expression $\frac{0.0256 \times 1.5}{0.0064}$ by first rounding each number in the expression to $1$ significant figure before completing the computation. What is the percentage error in the student's result compared to the exact value?

66.67\%

40.00\%

166.67\%

33.33\%

Show answer & explanation

Answer: $66.67\%$

Answer

The percentage error in the student's result is

66.67\%

.

The exact evaluation yields

\frac{0.0384}{0.0064} = 6

. Rounding each quantity to

1

significant figure gives

0.03

,

2

, and

0.006

, which evaluates to

\frac{0.06}{0.006} = 10

. The absolute error is

|10 - 6| = 4

, and dividing by the true value

6

yields

\frac{4}{6} \times 100\% = 66.67\%

.

Step-by-Step Solution

1

Calculate the exact value of the expression

Numerator:

0.0256 \times 1.5 = 0.0384

. Division:

\frac{0.0384}{0.0064} = 6

.

Establishing the true reference value is required to calculate percentage error.

2

Round each number in the expression to

1

significant figure

0.0256 \rightarrow 0.03

,

1.5 \rightarrow 2

,

0.0064 \rightarrow 0.006

.

Following the approximation directive in the problem statement.

3

Calculate the estimated value using the rounded numbers

Estimated value =

\frac{0.03 \times 2}{0.006} = \frac{0.06}{0.006} = 10

.

Obtaining the student's evaluated result.

4

Determine the percentage error

Absolute Error =

|10 - 6| = 4

. Percentage Error =

\frac{4}{6} \times 100\% = 66.67\%

.

Percentage error is defined as

\frac{|\text{Estimated} - \text{True}|}{\text{True}} \times 100\%

.

Key Concept

Percentage error calculation with significant figure approximations

Estimated Time:2m 0s

Answer

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Common Mistakes

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