All practice questions

Find the largest integer value of $x$ that satisfies the linear inequality $\frac{2x - 5}{3} \le \frac{x + 1}{4}$ .

Show answer & explanation

Answer: 4

Answer

The largest integer value of

x

satisfying the inequality is 4.

Multiplying the entire inequality by 12 yields 4(2x - 5) <= 3(x + 1). Expanding both sides produces 8x - 20 <= 3x + 3. Subtracting 3x and adding 20 gives 5x <= 23, which simplifies to x <= 4.6. The largest integer less than or equal to 4.6 is 4.

Step-by-Step Solution

1

Clear the denominators by multiplying both sides by 12.

4(2x - 5) \le 3(x + 1)

Multiplying by a positive number preserves the inequality direction while clearing fractions.

2

Expand both sides of the inequality using the distributive property.

8x - 20 \le 3x + 3

Multiply 4 through (2x - 5) and 3 through (x + 1).

3

Isolate the variable terms on one side and constant terms on the other.

5x \le 23 \implies x \le 4.6

Subtract 3x from both sides and add 20 to both sides, then divide by 5.

4

Determine the maximum integer value satisfying the inequality boundary.

4

Since x must be less than or equal to 4.6, the greatest whole integer satisfying this condition is 4.

Key Concept

Solving linear inequalities with fractions and finding integer bounds

Question 7382Question →

The table below records the daily water consumption, in liters, of $40$ households in a residential community:

Daily Water Consumption (liters)	Frequency ( $f$ )
$10 - 19$	$6$
$20 - 29$	$10$
$30 - 39$	$14$
$40 - 49$	$7$
$50 - 59$	$3$

Calculate the mean daily water consumption for this community in liters.

Show answer & explanation

Answer: 32.25

Answer

The mean daily water consumption is

32.25

liters.

The mean of a grouped frequency distribution is computed using the formula

\bar{x} = \frac{\sum f x}{\sum f}

, where

x

represents the midpoint of each class interval and

f

is the class frequency. Calculating the midpoints yields

14.5, 24.5, 34.5, 44.5, 54.5

. Multiplying each midpoint by its frequency gives products of

87, 245, 483, 311.5, 163.5

, which sum to

1290

. Dividing

1290

by the total frequency of

40

yields

32.25

liters.

Step-by-Step Solution

1

Find the class midpoint (

x

) for each interval by taking the average of the upper and lower limits of each class.

Midpoints are

14.5

,

24.5

,

34.5

,

44.5

, and

54.5

.

For grouped data, the midpoint serves as the representative value for all data within that class interval.

2

Compute the product of frequency and midpoint (

f \cdot x

) for each class interval.

Products are

87

,

245

,

483

,

311.5

, and

163.5

.

This accounts for the total sum contributed by each group.

3

Sum all products

\sum f x

and divide by the total number of households

\sum f = 40

.

\text{Mean} = \frac{1290}{40} = 32.25.

The formula for the estimated mean of grouped data is

\bar{x} = \frac{\sum f x}{\sum f}

.

Key Concept

Grouped Mean Estimation using Class Midpoints

Question 7383Question →

Three farmers, Audu, Ngozi, and Tayo, shared a bag of fertilizer in the ratio $2 : 3 : 5$ . If Ngozi received $45\text{ kg}$ , what was the total mass of the fertilizer shared?

75\text{ kg}

135\text{ kg}

150\text{ kg}

225\text{ kg}

Show answer & explanation

Answer: $150\text{ kg}$

Answer

The total mass of the fertilizer shared was

150\text{ kg}

.

The ratio

2 : 3 : 5

indicates that the fertilizer is divided into

2 + 3 + 5 = 10

equal parts. Ngozi receives 3 of these parts, which equals

45\text{ kg}

. Dividing

45\text{ kg}

by 3 gives

15\text{ kg}

per part. Multiplying

15\text{ kg}

by the total of 10 parts gives

150\text{ kg}

.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

Total parts =

2 + 3 + 5 = 10

parts.

The total quantity is divided into parts according to the given ratio

2 : 3 : 5

.

2

Determine the value of 1 part using Ngozi's share.

Value of 1 part =

\frac{45\text{ kg}}{3} = 15\text{ kg}

.

Ngozi corresponds to the middle term of the ratio, which is 3 parts out of 10.

3

Calculate the total mass by multiplying the value of 1 part by the total number of parts.

Total mass =

10 \times 15\text{ kg} = 150\text{ kg}

.

The total mass consists of all 10 parts combined.

Key Concept

Sharing quantities in a given ratio

Alternative Method

Use proportional fractions: Ngozi's fraction of the total is

\frac{3}{10}

. Let total mass be

M

. Then

\frac{3}{10} M = 45 \implies M = 45 \times \frac{10}{3} = 150\text{ kg}

.

Estimated Time:45s

Question 7384Question →

A rhombus has an area of $120\text{ cm}^2$ . If the length of one of its diagonals exceeds the length of the other diagonal by $14\text{ cm}$ , what is the perimeter of the rhombus in centimeters?

Show answer & explanation

Answer: 52

Answer

The perimeter of the rhombus is 52 cm.

The area of a rhombus is given by A = \frac{1}{2} d_1 d_2. Setting \frac{1}{2} d_1(d_1 + 14) = 120$ yields the quadratic equation d_1^2 + 14d_1 - 240 = 0, which factors to (d_1 - 10)(d_1 + 24) = 0. Taking the positive solution d_1 = 10\text{ cm} gives d_2 = 24\text{ cm}. The diagonals intersect at right angles, dividing the rhombus into four congruent right triangles with legs of 5 cm and 12 cm. The hypotenuse (side length s) is \sqrt{5^2 + 12^2} = 13\text{ cm}. Therefore, the perimeter is 4 \times 13\text{ cm} = 52\text{ cm}.

Step-by-Step Solution

1

Set up the area formula for a rhombus in terms of its diagonals

d_1 \cdot d_2 = 240

The area of a rhombus is given by A = \frac{1}{2} d_1 d_2, so \frac{1}{2} d_1 d_2 = 120.

2

Substitute d_2 = d_1 + 14 into the area equation

d_1^2 + 14d_1 - 240 = 0

The difference between the diagonal lengths is 14 cm.

3

Solve the quadratic equation for d_1

d_1 = 10\text{ cm} \text{ and } d_2 = 24\text{ cm}

Factoring gives (d_1 - 10)(d_1 + 24) = 0. Discarding the negative root yields d_1 = 10 cm.

4

Calculate the side length s using the right-angled triangle formed by the perpendicular bisecting diagonals

s = 13\text{ cm}

s = \sqrt{(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2} = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}.

5

Multiply side length by 4 to get the total perimeter

P = 52\text{ cm}

All four sides of a rhombus are equal, so Perimeter = 4 \times s = 4 \times 13 = 52 cm.

Key Concept

Perimeter and Area of a Rhombus using Diagonals and Pythagorean Theorem

Question 7385Question →

A solid metal trophy is formed by mounting a right circular cone of slant height $5\text{ cm}$ on top of a right circular cylinder of height $8\text{ cm}$ . Both the cone and the cylinder share a common base radius of $3\text{ cm}$ . If the trophy is completely melted down and recast to form a solid pyramid with a square base of side length $6\text{ cm}$ , what is the vertical height of the pyramid?

9\pi\text{ cm}

7\pi\text{ cm}

\frac{7\pi}{3}\text{ cm}

3\pi\text{ cm}

Show answer & explanation

Answer: $7\pi\text{ cm}$

Answer

7\pi\text{ cm}

First, find the height of the cone using the Pythagorean theorem:

h_{\text{cone}} = \sqrt{5^2 - 3^2} = 4\text{ cm}

. The volume of the conical section is

\frac{1}{3}\pi (3^2)(4) = 12\pi\text{ cm}^3

, and the volume of the cylindrical section is

\pi (3^2)(8) = 72\pi\text{ cm}^3

. The total volume melted is

12\pi + 72\pi = 84\pi\text{ cm}^3

. For the recast pyramid with square base area

6^2 = 36\text{ cm}^2

, its volume is

\frac{1}{3}(36)h = 12h

. Equating the volumes (

12h = 84\pi

) yields a vertical height of

7\pi\text{ cm}

.

Step-by-Step Solution

1

Calculate the vertical height of the conical top

h_{\text{cone}} = \sqrt{5^2 - 3^2} = 4\text{ cm}

The slant height

l

, radius

r

, and vertical height

h_{\text{cone}}

form a right-angled triangle where

h_{\text{cone}} = \sqrt{l^2 - r^2}

.

2

Compute the total volume of the composite metal solid

V_{\text{total}} = 12\pi + 72\pi = 84\pi\text{ cm}^3

The total volume is the sum of the cone volume

V_{\text{cone}} = \frac{1}{3}\pi (3^2)(4) = 12\pi\text{ cm}^3

and the cylinder volume

V_{\text{cylinder}} = \pi (3^2)(8) = 72\pi\text{ cm}^3

.

3

Equate the total volume to the volume of the square pyramid and solve for its height

h = 7\pi\text{ cm}

The base area of the square pyramid is

A = 6^2 = 36\text{ cm}^2

. Its volume is

V_{\text{pyramid}} = \frac{1}{3}(36)h = 12h

. Setting

12h = 84\pi

gives

h = \frac{84\pi}{12} = 7\pi\text{ cm}

.

Key Concept

Conservation of volume during recasting of 3D solids and combined solid geometry

Estimated Time:2m 0s

Question 7386Question →

If $y = \frac{e^{2x}}{\cos x}$ , calculate the value of the derivative $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 2

Answer

The value of the derivative

\frac{dy}{dx}

at

x = 0

is 2.

Applying the quotient rule to

y = \frac{e^{2x}}{\cos x}

yields

\frac{dy}{dx} = \frac{2e^{2x}\cos x + e^{2x}\sin x}{\cos^2 x}

. Substituting

x = 0

gives

\frac{2(1)(1) + (1)(0)}{1^2} = 2

.

Step-by-Step Solution

1

Set up the quotient rule components for

y = \frac{u(x)}{v(x)}

.

Let

u(x) = e^{2x}

and

v(x) = \cos x

.

The given function is a ratio of exponential and trigonometric functions.

2

Compute individual derivatives

\frac{du}{dx}

and

\frac{dv}{dx}

.

\frac{du}{dx} = 2e^{2x}

and

\frac{dv}{dx} = -\sin x

.

Using the chain rule for exponential functions and standard derivative rules for trigonometric functions.

3

Apply the quotient rule formula

\frac{dy}{dx} = \frac{v u' - u v'}{v^2}

.

\frac{dy}{dx} = \frac{\cos x (2e^{2x}) - e^{2x}(-\sin x)}{\cos^2 x} = \frac{e^{2x}(2\cos x + \sin x)}{\cos^2 x}

.

Combining terms gives the exact derivative function.

4

Evaluate the derivative at

x = 0

.

\frac{dy}{dx}\Big|_{x=0} = \frac{e^{0}(2\cos 0 + \sin 0)}{\cos^2 0} = \frac{1 \cdot (2 + 0)}{1} = 2

.

Using trigonometric and exponential values at zero:

e^0 = 1

,

\cos 0 = 1

, and

\sin 0 = 0

.

Key Concept

Differentiation of Exponential and Trigonometric Functions using Quotient Rule

Question 7387Question →

In a circle of radius $13\text{ cm}$ , two parallel chords $AB$ and $CD$ are drawn on the same side of the center $O$ . If $AB = 24\text{ cm}$ and $CD = 10\text{ cm}$ , calculate the perpendicular distance between the two chords in centimeters.

Show answer & explanation

Answer: 7

Answer

The perpendicular distance between the two chords is

7\text{ cm}

.

The perpendicular line from the center

O

to a chord bisects the chord. Applying the Pythagorean theorem to the right triangles formed by the radius (

13\text{ cm}

) and half-chords (

12\text{ cm}

and

5\text{ cm}

) yields distances of

5\text{ cm}

and

12\text{ cm}

from the center, respectively. Since both parallel chords are on the same side of the center, the distance between them is

12\text{ cm} - 5\text{ cm} = 7\text{ cm}

.

Step-by-Step Solution

1

Find the perpendicular distance from center

O

to chord

AB

d_1 = 5\text{ cm}

A line drawn from the center of a circle perpendicular to a chord bisects the chord. Using the right triangle formed by the radius, half-chord (

12\text{ cm}

), and perpendicular distance:

d_1 = \sqrt{13^2 - 12^2} = 5\text{ cm}

.

2

Find the perpendicular distance from center

O

to chord

CD

d_2 = 12\text{ cm}

Using the perpendicular bisector property for chord

CD

(half-chord is

5\text{ cm}

):

d_2 = \sqrt{13^2 - 5^2} = 12\text{ cm}

.

3

Calculate the distance between the parallel chords

7\text{ cm}

Because both chords are on the same side of the center

O

, the distance between them is the difference of their individual distances from the center:

12\text{ cm} - 5\text{ cm} = 7\text{ cm}

.

Key Concept

Perpendicular from the center of a circle to a chord bisects the chord

Question 7388Question →

The first term of an arithmetic progression (A.P.) is $7$ and its common difference is $4$ . What is the value of the $12^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 51

Answer

The

12^{\text{th}}

term of the arithmetic progression is

51

.

Applying the arithmetic progression

n^{\text{th}}

term formula

T_n = a + (n - 1)d

with first term

a = 7

, common difference

d = 4

, and term index

n = 12

gives

T_{12} = 7 + (12 - 1) \times 4 = 7 + 44 = 51

.

Step-by-Step Solution

1

Identify given variables

a = 7

,

d = 4

, and

n = 12

These are the necessary components to calculate the required term position in an arithmetic sequence.

2

Apply the

n^{\text{th}}

term formula for an AP

T_{12} = 7 + (12 - 1) \times 4

The standard formula

T_n = a + (n - 1)d

determines the value of any term in an AP.

3

Compute the numerical result

T_{12} = 7 + 44 = 51

Multiplying the common difference by 11 and adding the first term yields the correct value.

Key Concept

Calculating the nth term of an Arithmetic Progression

Question 7389Question →

The frequency distribution table below shows the scores obtained by $50$ candidates in a competitive assessment:

Score Interval	Frequency
$10 - 19$	$5$
$20 - 29$	$12$
$30 - 39$	$18$
$40 - 49$	$10$
$50 - 59$	$5$

When constructing a cumulative frequency curve (ogive) for this distribution, which of the following ordered pairs $(x, y)$ represents the point plotted for the class interval $30 - 39$ ?

(29.5, 35)

(39.5, 35)

(34.5, 35)

(39.5, 18)

Show answer & explanation

Answer: $(39.5, 35)$

Answer

The point plotted on the ogive for the class interval

30 - 39

is

(39.5, 35)

.

A cumulative frequency curve (ogive) is constructed by plotting the cumulative frequency of each class against its upper class boundary. For the interval

30 - 39

, the upper boundary is

39.5

and the cumulative frequency is

5 + 12 + 18 = 35

, resulting in the point

(39.5, 35)

.

Step-by-Step Solution

1

Determine the upper class boundary for the class interval

30 - 39

Upper class boundary =

\frac{39 + 40}{2} = 39.5

Cumulative frequencies in an ogive are plotted against the upper boundaries of each class interval.

2

Calculate the cumulative frequency up to the class interval

30 - 39

Cumulative frequency =

5 + 12 + 18 = 35

The cumulative frequency is the running sum of frequencies up to and including the given interval.

3

Combine the upper class boundary (

x

-axis) and cumulative frequency (

y

-axis) into an ordered pair

(x, y)

Point =

(39.5, 35)

In an ogive, points are plotted as (Upper Class Boundary, Cumulative Frequency).

Key Concept

Plotting Points on a Cumulative Frequency Curve (Ogive)

Question 7390Question →

Two boats depart simultaneously from a port $P$ . Boat $A$ travels along a straight path for $8\text{ km}$ , while Boat $B$ travels along another straight path for $15\text{ km}$ . If the angle between their paths at the port is $60^\circ$ , what is the distance between the two boats in kilometers?

Show answer & explanation

Answer: 13

Answer

The distance between the two boats is 13 km.

The distance between the two boats forms the third side of a triangle where two side lengths (

8\text{ km}

and

15\text{ km}

) and the included angle (

60^\circ

) are known. By the Cosine Rule,

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 120 = 169

, so

d = \sqrt{169} = 13\text{ km}

.

Step-by-Step Solution

1

Formulate the geometric model

A triangle with two sides of length

8\text{ km}

and

15\text{ km}

, and an included angle of

60^\circ

.

The paths of the two boats and the distance between them form a triangle with two given side lengths and the included angle.

2

Set up the Cosine Rule formula

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ)

The Cosine Rule (

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

) is used when two sides and the included angle are known (SAS configuration).

3

Evaluate the trigonometric term and simplify

d^2 = 64 + 225 - 240(0.5) = 289 - 120 = 169

Since

\cos(60^\circ) = 0.5

, the subtraction term reduces to

120

.

4

Calculate the principal square root

d = 13

Taking the positive square root gives the distance in kilometers.

Key Concept

Applying the Cosine Rule to calculate the unknown side of a triangle given two sides and the included angle (SAS).

Question 7391Question →

A meteorologist recorded the rainfall (in millimeters) in a town over five days as $2$ , $5$ , $6$ , $7$ , and $10$ . What is the mean deviation of the rainfall data?

2

0

10

6

Show answer & explanation

Answer: $2$

Answer

The mean deviation of the rainfall data is

2

.

To find the mean deviation, first calculate the mean of the data:

(2 + 5 + 6 + 7 + 10) / 5 = 6

. Next, calculate the absolute difference of each value from 6, which yields 4, 1, 0, 1, and 4. The average of these absolute differences is

(4 + 1 + 0 + 1 + 4) / 5 = 10 / 5 = 2

.

Step-by-Step Solution

1

Calculate the arithmetic mean (

\bar{x}

) of the dataset.

\bar{x} = \frac{2 + 5 + 6 + 7 + 10}{5} = \frac{30}{5} = 6

The mean is required to compute the deviation of each observation from the central value.

2

Find the absolute deviation

|x - \bar{x}|

for each data point.

|2 - 6| = 4, \quad |5 - 6| = 1, \quad |6 - 6| = 0, \quad |7 - 6| = 1, \quad |10 - 6| = 4

Mean deviation measures the average absolute distance of data values from the mean.

3

Sum the absolute deviations and divide by the total number of observations (

N = 5

).

\text{Mean Deviation} = \frac{4 + 1 + 0 + 1 + 4}{5} = \frac{10}{5} = 2

Dividing the sum of absolute deviations by

N

gives the mean deviation.

Key Concept

Mean Deviation

Question 7392Question →

Find the product matrix $AB$ given that $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix}$ .

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

\begin{pmatrix} 4 & 10 \\ 0 & 6 \end{pmatrix}

\begin{pmatrix} 4 & 23 \\ 1 & 8 \end{pmatrix}

\begin{pmatrix} 5 & 7 \\ 1 & 5 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}$

Answer

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

The matrix product

AB

is found by performing row-by-column multiplication. Computing row 1 gives

(1)(4)+(2)(1) = 6

and

(1)(5)+(2)(2) = 9

. Computing row 2 gives

(0)(4)+(3)(1) = 3

and

(0)(5)+(3)(2) = 6

. Thus, the resulting matrix is

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

.

Step-by-Step Solution

1

Set up row-by-column matrix multiplication formula for

2 \times 2

matrices.

The entry at row

i

, column

j

of product

AB

is the dot product of row

i

of

A

and column

j

of

B

.

Matrix multiplication requires combining rows of the left matrix with columns of the right matrix.

2

Calculate row 1 entries of the product matrix.

First element:

(1)(4) + (2)(1) = 4 + 2 = 6

. Second element:

(1)(5) + (2)(2) = 5 + 4 = 9

.

Multiply elements of row 1 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

3

Calculate row 2 entries of the product matrix.

Third element:

(0)(4) + (3)(1) = 0 + 3 = 3

. Fourth element:

(0)(5) + (3)(2) = 0 + 6 = 6

.

Multiply elements of row 2 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

4

Assemble the product matrix.

AB = \begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

.

Combine all computed row and column entries into a single

2 \times 2

matrix.

Key Concept

Matrix Multiplication (

2 \times 2

Matrix Product)

Question 7393Question →

A trapezium has parallel sides of length $12\text{ cm}$ and $18\text{ cm}$ , and a perpendicular height of $7\text{ cm}$ . What is the area of the trapezium?

105\text{ cm}^2

210\text{ cm}^2

216\text{ cm}^2

37\text{ cm}^2

Show answer & explanation

Answer: $105\text{ cm}^2$

Answer

The area of the trapezium is

105\text{ cm}^2

.

The area of a trapezium is calculated using the formula

\text{Area} = \frac{1}{2}(a + b)h

, where

a

and

b

are the lengths of the parallel sides, and

h

is the perpendicular height. Substituting

a = 12\text{ cm}

,

b = 18\text{ cm}

, and

h = 7\text{ cm}

yields

\text{Area} = \frac{1}{2}(12 + 18) \times 7 = \frac{1}{2}(30) \times 7 = 105\text{ cm}^2

.

Step-by-Step Solution

1

Identify the given dimensions of the trapezium

Parallel sides

a = 12\text{ cm}

,

b = 18\text{ cm}

, and height

h = 7\text{ cm}

.

These are the necessary parameters for the area formula of a trapezium.

2

Sum the lengths of the parallel sides

a + b = 12 + 18 = 30\text{ cm}

.

The area formula requires the sum of the bases.

3

Multiply by the perpendicular height and divide by 2

\text{Area} = \frac{1}{2} \times 30 \times 7 = 15 \times 7 = 105\text{ cm}^2.

Applying the standard formula

\text{Area} = \frac{1}{2}(a + b)h

gives the plane region's total area.

Key Concept

Area of a Trapezium

Question 7394Question →

In how many distinct ways can a president, a vice-president, and a secretary be chosen from a group of $7$ candidates, assuming no candidate can hold more than one position?

Show answer & explanation

Answer: 210

Answer

The total number of distinct ways to choose the three officers from 7 candidates is 210.

Selecting 3 distinct officers from a group of 7 candidates requires ordering 3 individuals out of 7, which equals

7 \times 6 \times 5 = 210

ways.

Step-by-Step Solution

1

Identify whether the problem involves permutations or combinations

Order is important because the positions (President, Vice-President, Secretary) are distinct.

Selecting person X as President and person Y as Secretary is different from selecting person Y as President and person X as Secretary.

2

Calculate the permutation

^7P_3

^7P_3 = 7 \times 6 \times 5 = 210

There are 7 choices for President, 6 remaining choices for Vice-President, and 5 remaining choices for Secretary.

Key Concept

Permutations of n items taken r at a time

Estimated Time:45s

Question 7395Question →

If $\log_2 (x + 3) + \log_2 4 = 5$ , what is the value of $x$ ?

25

5

18

3

Show answer & explanation

Answer: $5$

Answer

The value of

x

is

5

.

Using the logarithmic addition property

\log_b (M) + \log_b (N) = \log_b (M \cdot N)

, the equation

\log_2(x+3) + \log_2 4 = 5

becomes

\log_2[4(x+3)] = 5

. Converting to index form gives

4(x+3) = 2^5 = 32

. Dividing by

4

yields

x + 3 = 8

, so

x = 5

.

Step-by-Step Solution

1

Apply the product rule of logarithms:

\log_b A + \log_b B = \log_b (A \cdot B)

\log_2 [4(x + 3)] = 5

Logarithms with the same base that are added together can be combined by multiplying their arguments.

2

Convert the logarithmic equation into its equivalent exponential form

4(x + 3) = 2^5 = 32

If

\log_b Y = c

, then

Y = b^c

.

3

Divide both sides by 4 and solve for

x

x + 3 = 8 \implies x = 5

Linear algebraic simplification.

Key Concept

Product Law of Logarithms and Log-to-Exponential Conversion

Question 7396Question →

What is the simplified form of the expression $\frac{12}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ ?

3\sqrt{2} + 2\sqrt{3} - \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

2\sqrt{2} + 3\sqrt{3} + \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{10}

Show answer & explanation

Answer: $3\sqrt{2} + 2\sqrt{3} + \sqrt{30}$

Answer

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

By grouping the denominator as

(\sqrt{2}+\sqrt{3}) - \sqrt{5}

and multiplying by its conjugate

(\sqrt{2}+\sqrt{3}) + \sqrt{5}

, the denominator reduces to

2\sqrt{6}

. Multiplying the resulting fraction by

\sqrt{6}/\sqrt{6}

yields

\sqrt{12} + \sqrt{18} + \sqrt{30}

, which simplifies directly to

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

.

Step-by-Step Solution

1

Group the terms in the denominator as

((\sqrt{2} + \sqrt{3}) - \sqrt{5})

and multiply the numerator and denominator by its conjugate

((\sqrt{2} + \sqrt{3}) + \sqrt{5})

.

The fraction becomes

\frac{12((\sqrt{2} + \sqrt{3}) + \sqrt{5})}{((\sqrt{2} + \sqrt{3}) - \sqrt{5})((\sqrt{2} + \sqrt{3}) + \sqrt{5})}

.

Applying the difference of two squares to eliminate the outer radical.

2

Expand the denominator using

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

.

Denominator =

(\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2 = (2 + 2\sqrt{6} + 3) - 5 = 2\sqrt{6}

.

Simplifying the algebraic square of a binomial surd.

3

Divide the numerator by the constant factor of the denominator.

\frac{12(\sqrt{2} + \sqrt{3} + \sqrt{5})}{2\sqrt{6}} = \frac{6(\sqrt{2} + \sqrt{3} + \sqrt{5})}{\sqrt{6}}.

Simplifying numerical coefficients before further rationalization.

4

Rationalize the remaining monomial radical in the denominator by multiplying numerator and denominator by

\sqrt{6}

.

\frac{6(\sqrt{12} + \sqrt{18} + \sqrt{30})}{6} = \sqrt{12} + \sqrt{18} + \sqrt{30}.

Eliminating

\sqrt{6}

from the denominator.

5

Simplify each radical to its simplest surd form.

\sqrt{12} = 2\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2}, \quad \text{so } \sqrt{12} + \sqrt{18} + \sqrt{30} = 3\sqrt{2} + 2\sqrt{3} + \sqrt{30}.

Factoring out perfect square components from radical terms.

Key Concept

Rationalisation of trinomial surd denominators using repeated conjugate multiplication

Estimated Time:2m 0s

Question 7397Question →

What is the sum of all real values of $x$ that satisfy the logarithmic equation $\log_3(9^x + 81) - x = \log_3 30$ ?

4

10

30

6

Show answer & explanation

Answer: 4

Answer

The sum of all real values of

x

satisfying the equation is 4.

Using logarithmic properties,

\log_3(9^x + 81) - x = \log_3 30

simplifies to

\frac{9^x + 81}{3^x} = 30

. Substituting

u = 3^x

produces the quadratic equation

u^2 - 30u + 81 = 0

, which yields

u = 3

and

u = 27

. Solving

3^x = 3

and

3^x = 27

gives

x = 1

and

x = 3

. Their sum is

1 + 3 = 4

.

Step-by-Step Solution

1

Express

x

as

\log_3(3^x)

and apply the logarithm quotient rule.

\log_3\left(\frac{9^x + 81}{3^x}\right) = \log_3 30

Using

\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right)

combines terms into a single logarithm.

2

Equate the logarithmic arguments and split the fraction.

\frac{9^x + 81}{3^x} = 30 \implies 3^x + \frac{81}{3^x} = 30

Logarithmic equality with identical bases implies equality of their arguments.

3

Substitute

u = 3^x

to form and solve a quadratic equation.

u + \frac{81}{u} = 30 \implies u^2 - 30u + 81 = 0 \implies (u - 3)(u - 27) = 0 \implies u = 3 \text{ or } u = 27

Since

9^x = (3^x)^2 = u^2

, substituting

u = 3^x

converts the equation into a standard quadratic equation.

4

Solve for

x

from

3^x = u

and calculate the sum of the solutions.

3^x = 3 \implies x = 1; \quad 3^x = 27 \implies x = 3. \quad \text{Sum} = 1 + 3 = 4

Matching exponential bases gives

x = 1

and

x = 3

.

Key Concept

Logarithmic quotient identity and quadratic substitution for exponential equations

Estimated Time:2m 0s

Question 7398Question →

A closed cylindrical metal container has a total surface area of $54\pi\text{ cm}^2$ . What radius, in centimeters, of the circular base will yield the maximum volume for the container?

Show answer & explanation

Answer: 3

Answer

The radius of the circular base that maximizes the volume is 3 cm.

To find the radius that yields maximum volume, we first express height

h

in terms of radius

r

using the total surface area formula

2\pi r^2 + 2\pi rh = 54\pi

, giving

h = \frac{27 - r^2}{r}

. Substituting this into the volume equation

V = \pi r^2 h

gives

V(r) = 27\pi r - \pi r^3

. Setting the first derivative

\frac{dV}{dr} = 27\pi - 3\pi r^2

to zero yields

3\pi r^2 = 27\pi

, so

r^2 = 9

and

r = 3\text{ cm}

. The second derivative

\frac{d^2V}{dr^2} = -6\pi r

evaluated at

r = 3

is

-18\pi

, which is strictly negative, confirming that

r = 3\text{ cm}

maximizes volume.

Step-by-Step Solution

1

Set up the surface area equation for a closed cylinder with the given value.

2\pi r^2 + 2\pi r h = 54\pi

A closed cylinder consists of two circular bases (

2\pi r^2

) and a curved lateral surface (

2\pi r h

).

2

Express

h

in terms of

r

.

h = \frac{27 - r^2}{r}

Dividing the surface area equation by

2\pi

yields

r^2 + rh = 27

, allowing

h

to be isolated.

3

Substitute

h

into the volume formula

V = \pi r^2 h

to write volume as a function of

r

only.

V(r) = 27\pi r - \pi r^3

To maximize volume using calculus, the volume equation must be expressed in terms of a single variable.

4

Differentiate

V(r)

with respect to

r

and set the derivative equal to zero to find stationary points.

\frac{dV}{dr} = 27\pi - 3\pi r^2 = 0 \implies r = 3

Maximum volume occurs at a stationary point where the first derivative is zero.

5

Verify that

r = 3

produces a maximum using the second derivative test.

\frac{d^2V}{dr^2} = -6\pi r

; at

r = 3

,

\frac{d^2V}{dr^2} = -18\pi < 0

A negative second derivative indicates a local maximum.

Key Concept

Optimization and Stationary Points in Mensuration

Estimated Time:2m 0s

Question 7399Question →

An infinite geometric progression of positive terms has a sum to infinity of $16$ , and the sum of its first two terms is $12$ . An arithmetic progression has its first term equal to the first term of this geometric progression, and its $5^{\text{th}}$ term equal to the sum to infinity of the geometric progression. Calculate the $10^{\text{th}}$ term of the arithmetic progression.

Show answer & explanation

Answer: 26

Answer

The 10th term of the arithmetic progression is 26.

By solving the geometric progression system, we find the common ratio

r = \frac{1}{2}

and first term

a = 8

. Using

a = 8

as the first term of the arithmetic progression and setting its

5^{\text{th}}

term

A_5 = 16

, we determine the common difference

d = 2

. Calculating

A_{10} = 8 + 9(2)

yields

26

.

Step-by-Step Solution

1

Formulate equations for the geometric progression using the sum to infinity and sum of the first two terms

a = 16(1 - r)

and

a(1 + r) = 12

The standard formula for the sum to infinity of a GP is

S_\infty = \frac{a}{1-r}

and the sum of the first two terms is

S_2 = a + ar = a(1+r)

.

2

Solve for the common ratio

r

and first term

a

of the geometric progression

r = 0.5

and

a = 8

Substituting

a = 16(1-r)

yields

16(1-r^2) = 12 \implies r^2 = \frac{1}{4} \implies r = \frac{1}{2}

. Then

a = 16(1 - 0.5) = 8

.

3

Determine the common difference

d

of the arithmetic progression

d = 2

The first term of the AP is

A_1 = a = 8

and the 5th term is

A_5 = S_\infty = 16

. Using

A_5 = A_1 + 4d \implies 8 + 4d = 16 \implies d = 2

.

4

Calculate the 10th term of the arithmetic progression

A_{10} = 26

Using the AP

n^{\text{th}}

term formula

A_n = A_1 + (n-1)d

:

A_{10} = 8 + 9(2) = 26

.

Key Concept

Combining geometric progression parameters (sum to infinity and sum of terms) with arithmetic progression term formulas

Question 7400Question →

In a right-angled triangle $PQR$ , where $\angle Q = 90^\circ$ and $\tan P = \frac{5}{12}$ , what is the exact value of $\sin P + \cos P$ ?

\frac{17}{13}

\frac{7}{13}

\frac{17}{12}

\frac{12}{5}

Show answer & explanation

Answer: $\frac{17}{13}$

Answer

\frac{17}{13}

Using

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

, the triangle has opposite side

5

, adjacent side

12

, and hypotenuse

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

. Therefore,

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

and

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

, giving a sum of

\frac{17}{13}

.

Step-by-Step Solution

1

Identify the side lengths of the right triangle

PQR

using the given tangent ratio.

Opposite side to

P = 5

, adjacent side to

P = 12

.

By definition,

\tan P = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}

.

2

Calculate the length of the hypotenuse

PR

using the Pythagorean theorem.

Hypotenuse

PR = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

.

In a right triangle,

\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2

.

3

Determine

\sin P

and

\cos P

and compute their sum.

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

,

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

, so

\sin P + \cos P = \frac{5}{13} + \frac{12}{13} = \frac{17}{13}

.

\sin P = \frac{\text{Opposite}}{\text{Hypotenuse}}

and

\cos P = \frac{\text{Adjacent}}{\text{Hypotenuse}}

.

Key Concept

Basic Trigonometric Ratios and Pythagorean Triples

Estimated Time:1m 30s

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