All practice questions

Question 4101Question →

What is the exact numerical value of the trigonometric expression $\frac{4\cos^2 30^\circ + 2\sin^2 45^\circ}{\tan^2 60^\circ - \csc 30^\circ}$ ?

Show answer & explanation

Answer: 4

Answer

4

By evaluating the special angles directly:

4\cos^2 30^\circ = 3

,

2\sin^2 45^\circ = 1

,

\tan^2 60^\circ = 3

, and

\csc 30^\circ = 2

. Substituting these into the expression yields

\frac{3 + 1}{3 - 2} = \frac{4}{1} = 4

.

Step-by-Step Solution

1

Substitute the exact value of

\cos 30^\circ

4\cos^2 30^\circ = 4 \left(\frac{\sqrt{3}}{2}\right)^2 = 4 \left(\frac{3}{4}\right) = 3

The exact value of

\cos 30^\circ

is

\frac{\sqrt{3}}{2}

.

2

Substitute the exact value of

\sin 45^\circ

2\sin^2 45^\circ = 2 \left(\frac{1}{\sqrt{2}}\right)^2 = 2 \left(\frac{1}{2}\right) = 1

The exact value of

\sin 45^\circ

is

\frac{1}{\sqrt{2}}

.

3

Substitute the exact values of

\tan 60^\circ

and

\csc 30^\circ

\tan^2 60^\circ = (\sqrt{3})^2 = 3

and

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

The exact value of

\tan 60^\circ

is

\sqrt{3}

and

\csc 30^\circ

is the reciprocal of

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

.

4

Simplify the entire fraction

\frac{3 + 1}{3 - 2} = \frac{4}{1} = 4

Dividing the simplified numerator (4) by the simplified denominator (1) gives 4.

Key Concept

Evaluation of Special Angle Trigonometric Ratios and Reciprocal Functions

Question 4102Question →

Complete the sentence below by supplying a word nearest in meaning to 'clandestine' or 'stealthy' that correctly fits the context.

Fill in the blanks below

The security operative executed a entry into the fortified warehouse to retrieve the stolen documents without alerting the guards.

Show answer & explanation

Answer

surreptitious (or covert, furtive, stealthy)

The word 'surreptitious' (as well as 'covert' or 'furtive') means kept secret or done by stealth, which perfectly aligns with the prompt requirement 'clandestine' and fits the sentence context of entering a guarded facility undetected.

Step-by-Step Solution

1

Analyze the context of the sentence and the target word prompt.

The action described involves entering a guarded location without alerting security, which demands a word meaning done secretly, stealthily, or under cover.

Contextual clues like 'without alerting the guards' and the synonym prompt 'clandestine' establish the required semantic meaning.

2

Identify exact synonyms matching the required tone and grammatical class.

The adjective 'surreptitious' (or 'covert' / 'furtive') means obtained, done, or made by stealth, exactly matching 'clandestine'.

An adjective modifying 'entry' is required to express secret or stealthy movement.

Key Concept

Identifying synonyms and words nearest in meaning in contextual sentences

Estimated Time:1m 30s

Question 4103Question →

Which word nearest in meaning to 'overly servile' or 'fawning' correctly completes the blank in the sentence below?

Fill in the blanks below

The new governor preferred surrounding himself with candid advisors rather than aides who praised his every decision unconditionally.

Show answer & explanation

Answer

The word 'obsequious' (or synonyms such as 'sycophantic', 'servile', 'unctuous') correctly completes the sentence.

The target context requires a word denoting insincere flattery and servile behavior to contrast with 'candid advisors'. 'Obsequious' (along with 'sycophantic' or 'servile') means obedient or attentive to a servile degree, making it the precise synonym required.

Step-by-Step Solution

1

Analyze the sentence context for contrast and semantic clues.

The phrase 'preferred candid advisors rather than...' establishes a contrast between honest advisors and aides who offer unconditional praise.

Understanding the context helps determine that the missing word must describe individuals who flatter excessively and show submissive eagerness to please.

2

Identify the precise term matching 'overly servile' or 'fawning'.

'Obsequious' (and its close synonyms 'sycophantic' and 'servile') denotes excessive submissiveness and insincere flattery.

This precisely matches the required meaning and fits the formal register of the sentence.

Key Concept

Contextual Synonyms and High-Level Lexical Precision

Estimated Time:1m 30s

Question 4104Question →

A variable $y$ is partly constant and partly varies directly as $\sqrt{x}$ . Given that $y = 26$ when $x = 16$ , and $y = 38$ when $x = 49$ , what is the value of $y$ when $x = 64$ ?

Show answer & explanation

Answer: 42

Answer

The value of

y

when

x = 64

is

42

.

The relationship is given by the partial variation equation

y = a + b\sqrt{x}

. Substituting the pairs

(16, 26)

and

(49, 38)

yields the linear system

a + 4b = 26

and

a + 7b = 38

. Solving this system gives the constants

a = 10

and

b = 4

. Substituting

x = 64

into

y = 10 + 4\sqrt{64}

results in

y = 10 + 4(8) = 42

.

Step-by-Step Solution

1

Formulate the partial variation equation.

y = a + b\sqrt{x}

, where

a

and

b

are constants of variation.

Partial variation implies

y

is the sum of a constant term

a

and a term directly proportional to

\sqrt{x}

.

2

Set up simultaneous equations using the given pairs of

(x, y)

.

Equation 1:

a + 4b = 26

Equation 2:

a + 7b = 38

Evaluating

\sqrt{16} = 4

and

\sqrt{49} = 7

simplifies the relationship into two linear equations in two unknowns.

3

Solve for constants

a

and

b

.

b = 4

and

a = 10

Subtracting Equation 1 from Equation 2 yields

3b = 12 \implies b = 4

, and substituting

b = 4

back into Equation 1 gives

a = 10

.

4

Calculate

y

when

x = 64

.

y = 10 + 4(8) = 42

Using the specific formula

y = 10 + 4\sqrt{x}

for

x = 64

gives

y = 10 + 32 = 42

.

Key Concept

Partial Variation with Simultaneous Equations

Question 4105Question →

Provide the word nearest in meaning to 'cursory' that appropriately fills the blank in the sentence below.

Fill in the blanks below

The auditor performed a inspection of the company's financial statements, missing several obvious discrepancies because of his haste.

Show answer & explanation

Answer

perfunctory (or superficial)

The word 'perfunctory' (or 'superficial') is nearest in meaning to 'cursory', as both describe an action done routinely, hastily, and with minimal thoroughness, directly aligning with the context of missing discrepancies due to haste.

Step-by-Step Solution

1

Analyze the sentence context for clues regarding the intended meaning.

The phrase 'missing several obvious discrepancies because of his haste' indicates that the inspection was done quickly, routinely, and without thorough attention to detail.

Contextual indicators guide the selection of the correct synonym.

2

Determine the term nearest in meaning to 'cursory'.

The adjective 'perfunctory' means carried out with minimal effort or care, serving as a direct synonym for 'cursory' in this formal administrative context.

Matching the dictionary definition with contextual intent establishes the exact synonym.

Key Concept

Synonyms and Words Nearest in Meaning

Question 4106Question →

Match each data representation concept on the left with its corresponding mathematical formula or definition on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Sector angle of a category in a pie chart matches

\frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360^\circ

; Height of a histogram bar for equal class intervals matches Frequency of the class interval; Height of a histogram bar for unequal class intervals matches

\frac{\text{Class Frequency}}{\text{Class Width}}

; Upper class boundary of an interval

a - b

matches

b + 0.5

.

Each chart concept matches its standard definition: pie chart sector angles are proportional parts of

360^\circ

; histogram heights equal class frequencies when widths are equal, but equal frequency density when widths are unequal; and class boundaries adjust discrete limits by half the gap width.

Step-by-Step Solution

1

Identify the formula for calculating pie chart sector angles.

Sector angle =

\frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360^\circ

.

Pie charts distribute

360^\circ

proportionally according to the frequency of each category.

2

Determine histogram bar height representation under uniform class widths.

Bar height corresponds directly to the class frequency.

With uniform widths, the area of each rectangle is directly proportional to its height.

3

Determine histogram bar height representation under varying class widths.

Bar height corresponds to frequency density, calculated as

\frac{\text{Class Frequency}}{\text{Class Width}}

.

To maintain area proportional to frequency across varying widths, height must equal frequency divided by width.

4

Determine upper class boundary for discrete class intervals.

Upper boundary =

b + 0.5

.

Class boundaries eliminate gaps between discrete intervals by extending limits by half of the unit gap.

Key Concept

Data Representation Principles in Charts and Frequency Tables

Question 4107Question →

What is the indefinite integral $\int \left( 6x^2 - \frac{4}{x^2} + 4\sin(2x)\cos(2x) \right) dx$ ?

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

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

2x^3 - \frac{4}{x} + \frac{1}{2}\cos(4x) + C

12x + \frac{8}{x^3} + 8\cos(4x) + C

Show answer & explanation

Answer: $2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C$

Answer

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

Simplifying

4\sin(2x)\cos(2x)

to

2\sin(4x)

using double-angle identities and integrating

6x^2 - 4x^{-2} + 2\sin(4x)

term-by-term yields

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

.

Step-by-Step Solution

1

Use the double-angle trigonometric identity to simplify the product term.

4\sin(2x)\cos(2x) = 2(2\sin(2x)\cos(2x)) = 2\sin(4x)

Applying the double-angle identity

\sin(2\theta) = 2\sin\theta\cos\theta

transforms the product into a standard single sine term.

2

Express the reciprocal power with a negative exponent.

-\frac{4}{x^2} = -4x^{-2}

Rewriting the fraction in index form allows the power rule of integration to be applied directly.

3

Integrate each component term using standard integration formulas.

\int 6x^2 dx = 2x^3

,

\int -4x^{-2} dx = \frac{-4x^{-1}}{-1} = \frac{4}{x}

, and

\int 2\sin(4x) dx = 2\left(-\frac{1}{4}\cos(4x)\right) = -\frac{1}{2}\cos(4x)

Integrating

x^n

gives

\frac{x^{n+1}}{n+1}

and integrating

\sin(ax)

yields

-\frac{1}{a}\cos(ax)

.

4

Combine the integrated terms and append the constant of integration.

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

Indefinite integrals require an arbitrary constant

C

to represent the entire family of antiderivatives.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

Question 4108Question →

The second, fourth, and eighth terms of an arithmetic progression (AP) with a non-zero common difference form three consecutive terms of a geometric progression (GP). If the sum of the first $5$ terms of the AP is $45$ , what is the first term of the AP?

3

5

9

15

Show answer & explanation

Answer: 3

Answer

The first term of the AP is 3.

By expressing the 2nd, 4th, and 8th terms as

a+d

,

a+3d

, and

a+7d

, the geometric mean condition

(a+3d)^2 = (a+d)(a+7d)

reduces to

d = a

. Substituting

d = a

into the sum formula

S_5 = 5(a+2d) = 45

gives

15a = 45

, yielding a first term of 3.

Step-by-Step Solution

1

Express the 2nd, 4th, and 8th terms of the AP in terms of first term

a

and common difference

d

T_2 = a + d

,

T_4 = a + 3d

, and

T_8 = a + 7d

The

n

-th term of an AP is given by

T_n = a + (n-1)d

.

2

Apply the condition for consecutive terms of a GP

(a + 3d)^2 = (a + d)(a + 7d) \implies a^2 + 6ad + 9d^2 = a^2 + 8ad + 7d^2 \implies 2d^2 = 2ad \implies d = a

If three terms

x, y, z

form a GP, then

y^2 = xz

. Since

d \neq 0

, dividing by

2d

gives

d = a

.

3

Use the sum of the first 5 terms of the AP to set up an equation for

a

and

d

S_5 = \frac{5}{2}[2a + 4d] = 45 \implies 5(a + 2d) = 45 \implies a + 2d = 9

The sum of the first

n

terms of an AP is

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

.

4

Substitute

d = a

into the sum equation to solve for

a

a + 2(a) = 9 \implies 3a = 9 \implies a = 3

Substituting

d = a

simplifies the linear equation to solve directly for

a

.

Key Concept

Combining Arithmetic and Geometric Progression properties to set up and solve simultaneous equations.

Question 4109Question →

In $\Delta PQR$ , side $p = 3\text{ cm}$ , side $q = 8\text{ cm}$ , and the included angle $\angle R = 60^\circ$ . What is the length of side $r$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

r

is

7\text{ cm}

.

Using the Cosine Rule

r^2 = p^2 + q^2 - 2pq \cos R

with

p=3

,

q=8

, and

\angle R=60^\circ

yields

r^2 = 9 + 64 - 48(0.5) = 49

, which gives

r = 7\text{ cm}

.

Step-by-Step Solution

1

State the Cosine Rule formula for side

r

r^2 = p^2 + q^2 - 2pq \cos R

The Cosine Rule allows calculating the third side of a triangle when two sides and the included angle are given.

2

Substitute the given values into the formula

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

We are given

p = 3

,

q = 8

, and

\angle R = 60^\circ

.

3

Evaluate the trigonometric expression and simplify

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

Since

\cos 60^\circ = 0.5

, multiplying

2 \times 3 \times 8 \times 0.5

yields

24

.

4

Solve for

r

by taking the positive square root

r = 7\text{ cm}

Length must be a positive value, and

\sqrt{49} = 7

.

Key Concept

Cosine Rule for finding an unknown side given two sides and the included angle (SAS).

Question 4110Question →

A trader estimated the mass of a bag of rice to be $25\text{ kg}$ , but the actual mass of the bag was $20\text{ kg}$ . Calculate the percentage error in the trader's estimate.

Show answer & explanation

Answer: 25

Answer

The percentage error in the trader's estimate is

25\%

.

The absolute error is

25\text{ kg} - 20\text{ kg} = 5\text{ kg}

. Evaluating the error relative to the actual value gives

\frac{5\text{ kg}}{20\text{ kg}} = 0.25

, which equals

25\%

.

Step-by-Step Solution

1

Calculate the absolute error in measurement

Error =

|25 - 20| = 5\text{ kg}

Absolute error is the absolute difference between the estimated value and the true value.

2

Calculate the percentage error relative to the actual value

\text{Percentage Error} = \frac{5}{20} \times 100\% = 25\%

Percentage error must always be calculated using the actual (true) value as the denominator.

Key Concept

Percentage Error Calculation

Estimated Time:45s

Question 4111Question →

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

The board asked the financial director to curtail unnecessary expenses to maintain the company's stability.

Which word is nearest in meaning to curtail?

reduce

increase

detail

audit

Show answer & explanation

Answer: reduce

Answer

The word nearest in meaning to 'curtail' in this context is 'reduce'.

In the sentence, 'curtail' means to restrict, lessen, or decrease expenses. 'Reduce' directly aligns with this contextual meaning.

Step-by-Step Solution

1

Analyze the context of the sentence

The company needs stability, so spending must be cut back or limited.

Contextual clues indicate a need to minimize spending.

2

Identify the definition of 'curtail'

'Curtail' means to reduce in extent or quantity, or to impose a restriction on.

Matching the vocabulary term with its direct definition in context.

3

Select the correct synonym

'Reduce' accurately reflects the meaning of cutting back spending.

'Reduce' is the exact contextual synonym among the options.

Key Concept

Identifying contextual synonyms

Estimated Time:45s

Question 4112Question →

What is the exact simplified value of the trigonometric expression $\frac{\sin 60^\circ + \cos 30^\circ}{\tan 45^\circ + \tan 30^\circ}$ ?

\frac{3(\sqrt{3} - 1)}{2}

\frac{3 - \sqrt{3}}{2}

\frac{3\sqrt{6} - 3\sqrt{2}}{4}

\frac{\sqrt{3} + 1}{3}

Show answer & explanation

Answer: $\frac{3(\sqrt{3} - 1)}{2}$

Answer

\frac{3(\sqrt{3} - 1)}{2}

Substituting the exact values gives a numerator of

\sqrt{3}

and a denominator of

\frac{\sqrt{3}+1}{\sqrt{3}}

. Multiplying by the reciprocal of the denominator gives

\frac{3}{\sqrt{3}+1}

. Rationalizing the denominator by multiplying top and bottom by

(\sqrt{3}-1)

simplifies to

\frac{3(\sqrt{3}-1)}{2}

.

Step-by-Step Solution

1

Substitute the exact standard values for each trigonometric function

\sin 60^\circ = \frac{\sqrt{3}}{2}

,

\cos 30^\circ = \frac{\sqrt{3}}{2}

,

\tan 45^\circ = 1

,

\tan 30^\circ = \frac{1}{\sqrt{3}}

Special angles have precise surd representations that must be used in non-calculator assessments.

2

Simplify the numerator and denominator separately

Numerator:

\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3}

. Denominator:

1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}}

.

Combine like terms in the numerator and find a common denominator for the terms in the denominator.

3

Divide the numerator by the denominator

\frac{\sqrt{3}}{\frac{\sqrt{3}+1}{\sqrt{3}}} = \frac{3}{\sqrt{3}+1}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

4

Rationalize the denominator

\frac{3(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{3(\sqrt{3}-1)}{3-1} = \frac{3(\sqrt{3}-1)}{2}

Multiply the numerator and denominator by the conjugate

(\sqrt{3}-1)

to remove the radical from the denominator.

Key Concept

Evaluation and surd simplification of special angle trigonometric expressions

Question 4113Question →

The table below shows the score distribution of a group of students in a quiz:

Score ( $x$ )	1	2	3	4	5
Frequency ( $f$ )	4	3	$k$	2	1

If the mean score of the distribution is $2.5$ , what is the median score of the dataset?

2.5

3.0

2.0

4.0

Show answer & explanation

Answer: $2.5$

Answer

The median score of the dataset is

2.5

.

To find the median, we first find the missing frequency

k

using the mean formula

\text{Mean} = \frac{\sum fx}{\sum f}

. Setting

\frac{23 + 3k}{10 + k} = 2.5

gives

k = 4

. With

k = 4

, the total number of observations is

N = 14

. The median is the average of the 7th and 8th scores in the ordered dataset. From cumulative frequencies, the 7th score is

2

and the 8th score is

3

. Thus, the median is

\frac{2 + 3}{2} = 2.5

.

Step-by-Step Solution

1

Set up the equation for the mean to find the missing frequency

k

\text{Mean} = \frac{\sum fx}{\sum f} = \frac{1(4) + 2(3) + 3(k) + 4(2) + 5(1)}{4 + 3 + k + 2 + 1} = \frac{23 + 3k}{10 + k} = 2.5

The mean of an ungrouped frequency table is the sum of all values divided by total frequency.

2

Solve the linear equation for

k

23 + 3k = 2.5(10 + k) \implies 23 + 3k = 25 + 2.5k \implies 0.5k = 2 \implies k = 4

Cross-multiplying and grouping like terms isolates the variable

k

.

3

Determine the total frequency

N

and median positions

N = 4 + 3 + 4 + 2 + 1 = 14

. Median position = average of

\frac{14}{2}\text{th}

(

7\text{th}

) and

(\frac{14}{2} + 1)\text{th}

(

8\text{th}

) values.

For an even total number of observations

N

, the median is the arithmetic mean of the two central numbers.

4

Find the

7\text{th}

and

8\text{th}

scores using cumulative frequency and calculate median

Cumulative frequencies: Score 1 (1–4), Score 2 (5–7), Score 3 (8–11).

7\text{th}\text{ term} = 2

,

8\text{th}\text{ term} = 3

.

\text{Median} = \frac{2 + 3}{2} = 2.5

.

The 7th score is 2 and the 8th score is 3, making their average 2.5.

Key Concept

Calculating the median of ungrouped frequency data after finding a missing frequency using the mean

Estimated Time:2m 0s

Question 4114Question →

An arithmetic progression (A.P.) has a first term of $5$ and a common difference of $3$ . What is the $8^{\text{th}}$ term of this progression?

26

29

32

23

Show answer & explanation

Answer: $26$

Answer

The

8^{\text{th}}

term of the arithmetic progression is

26

.

The

8^{\text{th}}

term is calculated using the standard formula

T_n = a + (n - 1)d

. Substituting

a = 5

,

d = 3

, and

n = 8

gives

T_8 = 5 + 7(3) = 26

.

Step-by-Step Solution

1

Identify the given parameters from the problem

First term

a = 5

, common difference

d = 3

, and term position

n = 8

.

These are the values required for substitution into the

n^{\text{th}}

term formula of an A.P.

2

Write down the general formula for the

n^{\text{th}}

term of an arithmetic progression

T_n = a + (n - 1)d

This formula defines any term in an arithmetic progression based on its position.

3

Substitute the values into the formula and simplify

T_8 = 5 + (8 - 1) \times 3 = 5 + 7 \times 3 = 5 + 21 = 26

Performing the multiplication before addition yields the value of the

8^{\text{th}}

term.

Key Concept

Arithmetic Progression (A.P.)

n^{\text{th}}

term formula

Estimated Time:45s

Question 4115Question →

Two ships, $P$ and $Q$ , leave a port $O$ at the same time. Ship $P$ sails on a bearing of $040^\circ$ at a constant speed of $25\text{ km/h}$ , while Ship $Q$ sails on a bearing of $100^\circ$ at a constant speed of $40\text{ km/h}$ . What is the distance in kilometers between the two ships after $2$ hours?

Show answer & explanation

Answer: 70

Answer

The distance between the two ships after 2 hours is 70 km.

The distance traveled by Ship P in 2 hours is

50\text{ km}

and by Ship Q is

80\text{ km}

. The angle between their paths is

100^\circ - 040^\circ = 60^\circ

. Applying the Cosine Rule yields

PQ^2 = 50^2 + 80^2 - 2(50)(80)\cos(60^\circ) = 2500 + 6400 - 4000 = 4900

, giving a distance of

\sqrt{4900} = 70\text{ km}

.

Step-by-Step Solution

1

Calculate the distances traveled by Ship P and Ship Q after 2 hours

OP = 50\text{ km}

and

OQ = 80\text{ km}

Distance equals speed multiplied by time.

2

Find the angle between the lines of travel from port O

\angle POQ = 100^\circ - 40^\circ = 60^\circ

The angle between two bearings from a common origin is the difference between their bearing angles.

3

Use the Cosine Rule to calculate the side length PQ

PQ^2 = 50^2 + 80^2 - 2(50)(80)\cos(60^\circ) = 4900

The Cosine Rule

c^2 = a^2 + b^2 - 2ab\cos C

calculates the unknown opposite side given two side lengths and their included angle.

4

Take the square root to find PQ

PQ = 70\text{ km}

Taking the principal square root gives the final linear distance.

Key Concept

Applying the Cosine Rule to solve bearing and distance non-right triangle problems

Estimated Time:2m 0s

Question 4116Question →

A forest ranger at a control post $X$ observes a lookout tower at point $Y$ on a bearing of $072^\circ$ . What is the bearing of the control post $X$ from the lookout tower $Y$ ?

252^\circ

108^\circ

288^\circ

162^\circ

Show answer & explanation

Answer: $252^\circ$

Answer

The bearing of the control post

X

from the lookout tower

Y

is

252^\circ

.

The bearing of

Y

from

X

is

072^\circ

. To find the bearing of

X

from

Y

(the back bearing), we add

180^\circ

to the forward bearing because

072^\circ

is less than

180^\circ

. Calculation:

072^\circ + 180^\circ = 252^\circ

.

Step-by-Step Solution

1

Identify the forward bearing of point

Y

from point

X

.

Forward bearing =

072^\circ

.

The question states that

Y

is observed from

X

on a bearing of

072^\circ

.

2

Apply the rule for finding a back bearing when the forward bearing is less than

180^\circ

.

Back bearing =

\text{Forward bearing} + 180^\circ = 072^\circ + 180^\circ = 252^\circ

.

Since

072^\circ < 180^\circ

, we add

180^\circ

to determine the reverse direction from North at point

Y

.

Key Concept

Back Bearing / Reverse Bearing

Estimated Time:45s

Question 4117Question →

What is the value of $(-38) \pmod{7}$ expressed in standard non-negative remainder form?

4

-3

3

5

Show answer & explanation

Answer: 4

Answer

4

The value

4

is correct because

-38 = 7 \times (-6) + 4

. The remainder

4

lies in the standard non-negative range

0 \le r < 7

. Alternatively, adding multiples of

7

to

-38

gives

-38 + 35 = -3

, and adding

7

once more gives

-3 + 7 = 4

.

Step-by-Step Solution

1

Find the largest multiple of the modulus

7

that is less than or equal to

-38

.

The multiple is

7 \times (-6) = -42

.

Modular arithmetic requires the remainder

r

to satisfy

0 \le r < 7

in standard form.

2

Calculate the remainder by subtracting the multiple from the dividend.

-38 - (-42) = -38 + 42 = 4

.

The remainder is the non-negative difference between the number and the multiple of the modulus.

Key Concept

Modular Arithmetic with Negative Numbers

Estimated Time:1m 0s

Question 4118Question →

The 3rd term of an arithmetic progression (AP) is $14$ and its 7th term is $34$ . If the $n$ -th term of this AP is equal to the 4th term of a geometric progression (GP) whose first term is $2$ and common ratio is $3$ , what is the value of $n$ ?

10

11

12

9

Show answer & explanation

Answer: 11

Answer

The value of

n

is

11

.

The AP has first term

a = 4

and common difference

d = 5

, giving

T_n = 4 + 5(n-1) = 5n - 1

. The 4th term of the GP is

2 \times 3^{3} = 54

. Setting

5n - 1 = 54

gives

5n = 55

, so

n = 11

.

Step-by-Step Solution

1

Find the first term

a

and common difference

d

of the arithmetic progression.

a = 4

and

d = 5

The

n

-th term of an AP is given by

T_n = a + (n-1)d

. Using given terms:

T_3 = a + 2d = 14

and

T_7 = a + 6d = 34

. Subtracting the first equation from the second yields

4d = 20 \Rightarrow d = 5

. Substituting

d = 5

into

a + 2(5) = 14

gives

a = 4

.

2

Calculate the 4th term of the geometric progression (

G_4

).

G_4 = 54

The

m

-th term of a GP is given by

G_m = a_{gp} \cdot r^{m-1}

. With first term

a_{gp} = 2

and ratio

r = 3

,

G_4 = 2 \cdot 3^{4-1} = 2 \cdot 3^3 = 2 \cdot 27 = 54

.

3

Equate

T_n

to

G_4

and solve for

n

.

n = 11

Set

T_n = G_4 \Rightarrow 4 + (n-1)5 = 54

. Simplifying gives

(n-1)5 = 50 \Rightarrow n-1 = 10 \Rightarrow n = 11

.

Key Concept

Solving simultaneous AP/GP equations using the

n

-th term formulas

T_n = a + (n-1)d

and

G_n = a r^{n-1}

.

Estimated Time:2m 0s

Question 4119Question →

Which word nearest in meaning to 'clear and leaving no doubt' correctly completes the sentence below?

Fill in the blanks below

The committee chairman gave his support to the new environmental policy during the senate meeting.

Show answer & explanation

Answer

unequivocal

The word 'unequivocal' means expressed in a clear and definite way, leaving no doubt, which perfectly matches the required meaning in the sentence context.

Step-by-Step Solution

1

Analyze the contextual clues in the sentence.

The sentence describes a strong, direct endorsement given to a policy without room for uncertainty.

The context requires an adjective meaning absolute, unambiguous, or clear.

2

Identify the word nearest in meaning to 'clear and leaving no doubt'.

'Unequivocal' (or 'explicit' / 'unambiguous') accurately fits the blank.

'Unequivocal' means leaving no doubt, clear, and unambiguous.

Key Concept

Contextual Synonyms in Administrative and Formal Contexts

Estimated Time:1m 0s

Question 4120Question →

What is the larger real value of $x$ that satisfies the logarithmic equation $\log_x 8 + \log_4 x = \frac{5}{2}$ ?

8

4

6

16

Show answer & explanation

Answer: 8

Answer

The larger value of

x

that satisfies the equation is 8.

Using the change of base rule

\log_a b = \frac{\log_c b}{\log_c a}

, we express both logarithmic terms in base 2:

\log_x 8 = \frac{3}{\log_2 x}

and

\log_4 x = \frac{\log_2 x}{2}

. Setting

y = \log_2 x

gives

\frac{3}{y} + \frac{y}{2} = \frac{5}{2}

. Multiplying by

2y

yields

y^2 - 5y + 6 = 0

, which factors as

(y-2)(y-3) = 0

. Thus,

y = 2

or

y = 3

, giving solutions

x = 2^2 = 4

and

x = 2^3 = 8

. The larger solution is 8.

Step-by-Step Solution

1

Apply the change of base formula to express all logarithmic terms in base 2.

\log_x 8 = \frac{\log_2 8}{\log_2 x} = \frac{3}{\log_2 x} \quad \text{and} \quad \log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{1}{2}\log_2 x

Converting all terms to a common base (base 2) allows algebraic simplification.

2

Substitute

y = \log_2 x

into the given equation.

\frac{3}{y} + \frac{y}{2} = \frac{5}{2}

Using substitution converts the logarithmic expression into a rational algebraic equation.

3

Multiply the entire equation by

2y

to clear denominators and form a quadratic equation.

6 + y^2 = 5y \implies y^2 - 5y + 6 = 0

Clearing denominators transforms the relation into standard quadratic form.

4

Factor the quadratic equation to find the values of

y

.

(y - 2)(y - 3) = 0 \implies y = 2 \text{ or } y = 3

Factoring determines the possible powers of 2 for

x

.

5

Solve for

x

using

y = \log_2 x

and select the larger value.

x = 2^2 = 4 \quad \text{or} \quad x = 2^3 = 8. \text{ The larger value is } 8.

Converting back from

y

to

x

yields the final solutions for

x

.

Key Concept

Change of Base Theorem for Logarithms and Reduction to Quadratic Equations

Answer

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