All practice questions

Question 101Question →

Tayo Traders had a trade debtors balance of $\text{₦}450,000$ before year-end adjustments for the financial year ended 31st December 2024. At the end of the year, additional unrecorded bad debts of $\text{₦}15,000$ were identified to be written off. During the same year, cash of $\text{₦}10,000$ was received from a customer whose debt of $\text{₦}10,000$ had been written off in 2022. What is the net trade debtors balance to be shown in the Statement of Financial Position as at 31st December 2024?

Show answer & explanation

Answer: 435000

Answer

The net trade debtors balance to be shown in the Statement of Financial Position at 31st December 2024 is

\text{₦}435,000

.

To calculate the net trade debtors figure for the Statement of Financial Position, only additional bad debts written off at year-end (

\text{₦}15,000

) are deducted from the unadjusted trade debtors balance of

\text{₦}450,000

, resulting in

\text{₦}435,000

. Bad debts recovered (

\text{₦}10,000

) are recognized as revenue in the Income Statement and received in cash, having no effect on closing trade debtors.

Step-by-Step Solution

1

Determine the unadjusted trade debtors balance

Unadjusted Trade Debtors =

\text{₦}450,000

This represents the recorded receivables prior to writing off year-end unrecorded bad debts.

2

Subtract additional bad debts written off at year-end

Adjusted Trade Debtors =

\text{₦}450,000 - \text{₦}15,000 = \text{₦}435,000

Bad debts written off represent irrecoverable amounts and must be deducted directly from trade debtors.

3

Evaluate the treatment of bad debts recovered

Effect on Trade Debtors =

\text{₦}0

Bad debts recovered during the year are recorded by debiting Cash/Bank and crediting Bad Debts Recovered (an income account). Because the original personal account of the debtor was already cleared when written off in a previous period, recovery does not alter the closing trade debtors balance.

Key Concept

Treatment of Bad Debts Written Off and Bad Debts Recovered on the Statement of Financial Position

Estimated Time:1m 30s

Question 102Question →

Given that $P$ varies directly as the square of $r$ , and $P = 48$ when $r = 4$ , calculate the value of $P$ when $r = 6$ .

Show answer & explanation

Answer: 108

Answer

The value of

P

when

r = 6

is

108

.

Since

P

varies directly as

r^2

, the relationship is expressed as

P = k r^2

. Substituting the given values

P = 48

and

r = 4

yields

48 = 16k

, so

k = 3

. Substituting

k = 3

and

r = 6

into the equation gives

P = 3 \times 6^2 = 3 \times 36 = 108

.

Step-by-Step Solution

1

Set up the variation equation using constant of variation

k

P = k r^2

Direct variation with the square of a variable means

P

is directly proportional to

r^2

.

2

Substitute the initial values

P = 48

and

r = 4

to determine

k

48 = k \times 4^2 \implies 48 = 16k \implies k = 3

Finding the variation constant

k

allows us to establish a specific relationship between

P

and

r

.

3

Calculate

P

for

r = 6

using the specific equation

P = 3 r^2

P = 3 \times 6^2 = 3 \times 36 = 108

Evaluating the formula with the new input

r = 6

yields the required value of

P

.

Key Concept

Direct variation involving square powers

Question 103Question →

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 104Question →

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 105Question →

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 106Question →

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 107Question →

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 108Question →

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 109Question →

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 110Question →

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 111Question →

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 112Question →

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 113Question →

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 114Question →

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 115Question →

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 116Question →

Find the total number of distinct solutions to the trigonometric equation $2\cos^2(2x) + \sin(2x) - 1 = 0$ in the interval $0^\circ \le x \le 360^\circ$ .

Show answer & explanation

Answer: 6

Answer

The total number of distinct solutions is 6.

Substituting

\cos^2(2x) = 1 - \sin^2(2x)

gives the quadratic

2\sin^2(2x) - \sin(2x) - 1 = 0

, which factors into

(2\sin(2x) + 1)(\sin(2x) - 1) = 0

. For

0^\circ \le x \le 360^\circ

, the angle argument

2x

covers

0^\circ \le 2x \le 720^\circ

. The equation

\sin(2x) = 1

provides 2 values for

x

(

45^\circ, 225^\circ

), while

\sin(2x) = -\frac{1}{2}

provides 4 values for

x

(

105^\circ, 165^\circ, 285^\circ, 345^\circ

). Summing these gives 6 distinct solutions in total.

Step-by-Step Solution

1

Use the Pythagorean trigonometric identity

\cos^2(2x) = 1 - \sin^2(2x)

to express the entire equation in terms of

\sin(2x)

.

2(1 - \sin^2(2x)) + \sin(2x) - 1 = 0 \implies 2 - 2\sin^2(2x) + \sin(2x) - 1 = 0

Converting all trigonometric terms to a single function allows the equation to be solved as a polynomial.

2

Rearrange and factorize the resulting quadratic equation in terms of

\sin(2x)

.

2\sin^2(2x) - \sin(2x) - 1 = 0 \implies (2\sin(2x) + 1)(\sin(2x) - 1) = 0

Factorization splits the quadratic trigonometric equation into two simple linear trigonometric equations.

3

Determine the expanded domain for

2x

given

0^\circ \le x \le 360^\circ

.

0^\circ \le 2x \le 720^\circ

Multiplying the bounds of

x

by 2 accounts for two full rotations in the unit circle.

4

Solve the first linear equation

\sin(2x) = 1

within

0^\circ \le 2x \le 720^\circ

.

2x = 90^\circ, 450^\circ \implies x = 45^\circ, 225^\circ

(2 distinct solutions)

The sine function equals 1 at

90^\circ

in the first revolution and at

90^\circ + 360^\circ = 450^\circ

in the second revolution.

5

Solve the second linear equation

\sin(2x) = -\frac{1}{2}

within

0^\circ \le 2x \le 720^\circ

.

2x = 210^\circ, 330^\circ, 570^\circ, 690^\circ \implies x = 105^\circ, 165^\circ, 285^\circ, 345^\circ

(4 distinct solutions)

The sine function is negative in the 3rd and 4th quadrants of both revolutions.

6

Combine the solution counts from both cases.

Total number of solutions =

2 + 4 = 6

.

Adding the valid solutions from both factor equations gives the complete set of roots.

Key Concept

Solving quadratic trigonometric equations across multiple revolutions

Estimated Time:3m 0s

Question 117Question →

The table below shows the distribution of masses (in kg) of cocoa bags harvested on a farm:

Mass (kg)	Frequency ( $f$ )
$10 - 19$	$6$
$20 - 29$	$k$
$30 - 39$	$15$
$40 - 49$	$12$
$50 - 59$	$7$

If the estimated mean mass of the distribution is $35.3\text{ kg}$ , calculate the value of the missing frequency $k$ .

Show answer & explanation

Answer: 10

Answer

The value of the missing frequency is 10.

To find the missing frequency, compute the midpoints (

x

) of each mass class interval: 14.5, 24.5, 34.5, 44.5, and 54.5. Next, express the sum of frequencies as

\sum f = 40 + k

and the sum of products of frequency and midpoint as

\sum fx = 6(14.5) + k(24.5) + 15(34.5) + 12(44.5) + 7(54.5) = 1520 + 24.5k

. Equating the mean expression

\frac{\sum fx}{\sum f}

to

35.3

gives

\frac{1520 + 24.5k}{40 + k} = 35.3

. Cross-multiplying and solving yields

1520 + 24.5k = 1412 + 35.3k

, which simplifies to

10.8k = 108

, giving

k = 10

.

Step-by-Step Solution

1

Determine the class midpoints (

x

) for all intervals.

Class midpoints are 14.5, 24.5, 34.5, 44.5, and 54.5.

Grouped mean calculations require representative midpoint values for each class interval.

2

Formulate expressions for total frequency

\sum f

and total weighted sum

\sum fx

.

\sum f = 40 + k and \sum fx = 1520 + 24.5k.

These algebraic expressions are necessary to substitute into the mean formula.

3

Set up and solve the linear equation using the given mean of 35.3.

\frac{1520 + 24.5k}{40 + k} = 35.3 \implies 10.8k = 108 \implies k = 10.

Equating the algebraic mean expression to the numerical mean allows solving for the unknown frequency k.

Key Concept

Measures of Central Tendency for Grouped Data

Question 118Question →

In triangle $LMN$ , the side lengths are given as $l = 7\text{ cm}$ , $m = 8\text{ cm}$ , and $n = 13\text{ cm}$ . What is the measure of angle $N$ in degrees?

Show answer & explanation

Answer: 120

Answer

The measure of angle

N

is

120^\circ

.

Using the Cosine Rule for angle

N

,

\cos N = \frac{l^2 + m^2 - n^2}{2lm} = \frac{7^2 + 8^2 - 13^2}{2(7)(8)} = \frac{-56}{112} = -0.5

. The angle whose cosine is

-0.5

within the interior angles of a triangle (

0^\circ < N < 180^\circ

) is

120^\circ

.

Step-by-Step Solution

1

Set up the Cosine Rule formula for angle

N

\cos N = \frac{l^2 + m^2 - n^2}{2lm}

The Cosine Rule relates the three sides of any triangle to the cosine of one of its interior angles.

2

Substitute the known side lengths into the formula

\cos N = \frac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \frac{49 + 64 - 169}{112}

Side

n = 13\text{ cm}

is opposite to angle

N

and must be subtracted in the numerator.

3

Simplify the fraction

\cos N = \frac{-56}{112} = -0.5

Evaluating the numerical expression gives a negative value, indicating that angle

N

is obtuse.

4

Calculate the principal inverse cosine angle for the triangle

N = 120^\circ

Since

\cos 60^\circ = 0.5

,

\cos(180^\circ - 60^\circ) = -0.5

, giving

N = 120^\circ

.

Key Concept

Applying the Cosine Rule to find an obtuse angle given three side lengths (SSS)

Question 119Question →

The fourth term of an arithmetic progression (A.P.) is $15$ and the ninth term is $35$ . What is the common difference of the progression?

Show answer & explanation

Answer: 4

Answer

The common difference of the arithmetic progression is

4

.

Using the A.P. term formula

T_n = a + (n-1)d

, the fourth term gives

a + 3d = 15

and the ninth term gives

a + 8d = 35

. Subtracting the two equations yields

5d = 20

, which simplifies directly to

d = 4

.

Step-by-Step Solution

1

Express the given terms using the n-th term formula

T_n = a + (n-1)d

a + 3d = 15

and

a + 8d = 35

The

n^{\text{th}}

term formula relates any term to the first term (

a

) and common difference (

d

).

2

Subtract the equation for the fourth term from the ninth term

5d = 20

Subtracting eliminates the first term

a

and leaves a simple equation in terms of

d

.

3

Divide by

5

to solve for

d

d = 4

Dividing both sides of

5d = 20

by

5

gives the common difference.

Key Concept

Finding the common difference of an Arithmetic Progression given two non-consecutive terms

Question 120Question →

What is the mean deviation of the data set $4, 7, 8, 11, 15$ ?

Show answer & explanation

Answer: 3.2

Answer

The mean deviation of the data set is 3.2.

To find the mean deviation, first calculate the mean of the dataset, which is 9. Then, compute the absolute difference of each number from 9, obtaining values of 5, 2, 1, 2, and 6. Finally, divide the sum of these absolute values (16) by the total number of items (5) to get 3.2.

Step-by-Step Solution

1

Calculate the arithmetic mean of the given data set.

The mean

\bar{x} = 9

.

The mean is needed to evaluate how far each data point deviates from the central value.

2

Find the absolute difference between each value and the mean.

The absolute deviations are 5, 2, 1, 2, and 6.

Mean deviation measures dispersion using absolute distances, ignoring negative signs.

3

Calculate the mean of the absolute deviations.

Mean deviation = 3.2.

Dividing the total sum of absolute deviations (16) by the number of observations (5) gives the mean deviation.

Key Concept

Mean Deviation for Ungrouped Data

Answer

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