All practice questions

Question 641Question →

A student calculated the value of $\frac{0.048}{0.006}$ as $8.5$ . Calculate the percentage error in the student's calculation.

Show answer & explanation

Answer: 6.25

Answer

The percentage error is

6.25\%

.

Evaluating

0.048 \div 0.006

yields a true value of

8

. The error in estimation is

|8.5 - 8| = 0.5

. Dividing this error by the true value of

8

and multiplying by

100\%

gives a percentage error of

6.25\%

.

Step-by-Step Solution

1

Calculate the true value of the expression

True value =

\frac{0.048}{0.006} = 8

The exact result is required to establish the baseline for percentage error calculation.

2

Determine the magnitude of the error

Error =

|8.5 - 8| = 0.5

Error is defined as the absolute difference between the estimated value and the true value.

3

Compute the percentage error

Percentage error =

\frac{0.5}{8} \times 100\% = 6.25\%

Percentage error expresses the error as a percentage of the true value.

Key Concept

Percentage Error

Estimated Time:45s

Question 642Question →

In a survey of 120 final year secondary school students in Lagos, 47 offer Physics ( $P$ ), 57 offer Chemistry ( $C$ ), and 48 offer Biology ( $B$ ). Records show that 17 students offer both Physics and Chemistry, 20 offer both Chemistry and Biology, and 15 offer both Physics and Biology. If the number of students who offer none of these three subjects is three times the number of students who offer all three subjects, find the total number of students who offer exactly one of the three subjects.

Show answer & explanation

Answer: 63

Answer

63 students offer exactly one of the three subjects.

By applying the principle of inclusion-exclusion for three sets, the total union is

|P \cup C \cup B| = 47 + 57 + 48 - (17 + 20 + 15) + x = 100 + x

, where

x

is the number of students taking all three subjects. Setting the universal set total to 120 gives

(100 + x) + 3x = 120

, solving to

x = 5

. Subtracting the relevant intersections yields 20 students for Physics only, 25 for Chemistry only, and 18 for Biology only. Summing these gives 63.

Step-by-Step Solution

1

Define unknown variables and set up the inclusion-exclusion formula for the union of three sets.

|P \cup C \cup B| = 100 + x

, where

x = |P \cap C \cap B|

.

The Principle of Inclusion-Exclusion states that

|P \cup C \cup B| = |P| + |C| + |B| - (|P \cap C| + |C \cap B| + |P \cap B|) + |P \cap C \cap B|

.

2

Formulate and solve an equation for the total number of students in the universal set.

120 = (100 + x) + 3x \implies 4x = 20 \implies x = 5

.

The total number of students equals those taking at least one subject plus those taking none (

3x

).

3

Determine the number of students taking exactly two subjects.

Physics and Chemistry only =

17 - 5 = 12

; Chemistry and Biology only =

20 - 5 = 15

; Physics and Biology only =

15 - 5 = 10

.

Subtract the triple intersection count (

x = 5

) from each pairwise intersection count.

4

Calculate the number of students offering only Physics, only Chemistry, and only Biology.

Physics only =

47 - (12 + 10 + 5) = 20

; Chemistry only =

57 - (12 + 15 + 5) = 25

; Biology only =

48 - (10 + 15 + 5) = 18

.

Subtract the sum of the two-subject-only regions and the three-subject region from each set's total cardinality.

5

Sum the counts of students offering exactly one subject.

20 + 25 + 18 = 63

.

The total offering exactly one subject is the sum of the three disjoint single-subject regions.

Key Concept

3-Set Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition

Question 643Question →

The electrical resistance $R$ of a wire varies directly as its length $L$ and inversely as the square of its diameter $d$ . A wire of length $50\text{ m}$ and diameter $2\text{ mm}$ has a resistance of $5\text{ ohms}$ . Calculate the resistance (in ohms) of a wire made of the same material with a length of $80\text{ m}$ and a diameter of $4\text{ mm}$ .

Show answer & explanation

Answer: 2

Answer

The resistance of the wire is

2\text{ ohms}

.

The variation model is

R = \frac{k L}{d^2}

. Substituting

R = 5\text{ ohms}

,

L = 50\text{ m}

, and

d = 2\text{ mm}

gives

5 = \frac{50k}{4}

, so

k = 0.4

. Substituting

k = 0.4

,

L = 80\text{ m}

, and

d = 4\text{ mm}

yields

R = \frac{0.4 \times 80}{4^2} = \frac{32}{16} = 2\text{ ohms}

.

Step-by-Step Solution

1

Set up the variation formula

R = \frac{k L}{d^2}

Resistance varies directly as length

L

and inversely as the square of diameter

d

.

2

Calculate the constant of variation

k

k = 0.4

Substitute the initial values

R = 5

,

L = 50

, and

d = 2

into the variation equation.

3

Compute the new resistance

R

R = 2\text{ ohms}

Substitute

k = 0.4

,

L = 80

, and

d = 4

into the formula.

Key Concept

Direct and Inverse Joint Variation

Estimated Time:1m 30s

Question 644Question →

Given that $\tan \theta + \cot \theta = 4$ for an acute angle $\theta$ , what is the exact decimal value of $\sin^4 \theta + \cos^4 \theta$ ?

Show answer & explanation

Answer: 0.875

Answer

The exact decimal value of

\sin^4 \theta + \cos^4 \theta

is 0.875.

By writing

\tan \theta + \cot \theta

as

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = 4

, we find that

\sin \theta \cos \theta = \frac{1}{4}

. Squaring the fundamental identity

\sin^2 \theta + \cos^2 \theta = 1

gives

\sin^4 \theta + 2\sin^2 \theta \cos^2 \theta + \cos^4 \theta = 1

. Isolating

\sin^4 \theta + \cos^4 \theta

yields

1 - 2(\sin \theta \cos \theta)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

.

Step-by-Step Solution

1

Rewrite the expression

\tan \theta + \cot \theta = 4

using sine and cosine ratios

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 4

Applies the fundamental quotient identities for tangent and cotangent.

2

Combine fractions over a common denominator and apply the Pythagorean identity

\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = 4 \implies \sin \theta \cos \theta = \frac{1}{4}

Utilizes the identity

\sin^2 \theta + \cos^2 \theta = 1

.

3

Relate

\sin^4 \theta + \cos^4 \theta

to

(\sin^2 \theta + \cos^2 \theta)^2

\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2(\sin \theta \cos \theta)^2

Uses the algebraic identity

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

where

a = \sin^2 \theta

and

b = \cos^2 \theta

.

4

Substitute

\sin \theta \cos \theta = \frac{1}{4}

into the algebraic relation

1 - 2\left(\frac{1}{4}\right)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

Evaluates the expression to obtain the final decimal result.

Key Concept

Trigonometric Identities and Algebraic Polynomial Expansion

Estimated Time:2m 0s

Question 645Question →

What is the $y$ -intercept of the tangent line to the curve $y = x^3 - 2x + 4$ at the point where $x = 1$ ?

Show answer & explanation

Answer: 2

Answer

The y-intercept of the tangent line is 2.

The curve evaluated at

x=1

gives point

(1,3)

. The derivative

y'=3x^2-2

gives slope

m=1

at

x=1

. The tangent line equation is

y-3=1(x-1)

, which simplifies to

y=x+2

. The

y

-intercept occurs at

x=0

, giving

y=2

.

Step-by-Step Solution

1

Find the y-coordinate of the point on the curve at

x = 1

.

At

x = 1

,

y = (1)^3 - 2(1) + 4 = 3

. The point of tangency is

(1, 3)

.

The point of tangency lies on the curve.

2

Find the gradient function of the curve by differentiation.

\frac{dy}{dx} = 3x^2 - 2

.

The first derivative represents the slope of the tangent line.

3

Calculate the slope of the tangent line at

x = 1

.

m = 3(1)^2 - 2 = 1

.

Substituting

x = 1

into the derivative yields the slope at that specific point.

4

Formulate the equation of the tangent line.

y - 3 = 1(x - 1) \implies y = x + 2

.

Use point-slope form

y - y_1 = m(x - x_1)

with point

(1,3)

and slope

m=1

.

5

Find the

y

-intercept of the tangent line.

Setting

x = 0

gives

y = 2

.

The

y

-intercept is the value of

y

where the line crosses the vertical axis.

Key Concept

Tangents and Normals to Curves

Question 646Question →

Determine the least non-negative integer congruent to $68 \pmod{9}$ .

Show answer & explanation

Answer: 5

Answer

The least non-negative integer congruent to

68 \pmod{9}

is 5.

Dividing 68 by 9 gives a quotient of 7 and a remainder of 5. Since

0 \le 5 < 9

, the value 5 is the standard non-negative remainder.

Step-by-Step Solution

1

Divide 68 by the modulus 9 using the division algorithm.

68 = 9 \times 7 + 5

Every integer

a

can be written uniquely as

a = nq + r

where

q

is the quotient and

0 \le r < n

.

2

Extract the non-negative remainder

r

.

r = 5

In modular arithmetic, the least non-negative integer congruent to

a \pmod{n}

is the remainder

r

when

a

is divided by

n

.

Key Concept

Modular Arithmetic Remainder

Question 647Question →

A solid right circular cylinder of radius $5\text{ cm}$ and height $12\text{ cm}$ has a conical cavity of the same radius and height hollowed out from its top face. What is the total surface area of the remaining solid in $\text{cm}^2$ , expressed as a multiple of $\pi$ (that is, find the value of $K$ where the total surface area is $K\pi\text{ cm}^2$ )?

Show answer & explanation

Answer: 210

Answer

The total surface area of the remaining solid is

210\pi\text{ cm}^2

, so the required numerical coefficient

K

is

210

.

The total surface area consists of three parts: the flat circular base at the bottom (

25\pi\text{ cm}^2

), the outer curved surface of the cylinder (

120\pi\text{ cm}^2

), and the newly created inner curved surface of the conical cavity (

65\pi\text{ cm}^2

). Adding these together yields

25\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2

, giving the coefficient

K = 210

.

Step-by-Step Solution

1

Calculate the slant height (

l

) of the conical cavity.

l = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}

.

The slant height forms the hypotenuse of the right triangle whose leg lengths are the base radius and the vertical height of the cone.

2

Determine the area of all exposed boundary surfaces of the remaining solid.

Flat bottom base area

= \pi(5)^2 = 25\pi\text{ cm}^2

; Outer curved cylindrical surface area

= 2\pi(5)(12) = 120\pi\text{ cm}^2

; Inner curved conical surface area

= \pi(5)(13) = 65\pi\text{ cm}^2

.

Hollowing out the cone creates an internal curved boundary while leaving the outer cylindrical boundary and the flat bottom base exposed.

3

Sum the areas of all exposed surfaces to calculate the total surface area.

Total Surface Area

= 25\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2

.

The total surface area is the sum of all external and internal exposed surface areas.

Key Concept

Surface Area of Hollowed and Composite 3D Solids

Question 648Question →

If $y = 5e^{2x} + \cos(4x)$ , find the value of $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 10

Answer

10

Differentiating

y = 5e^{2x} + \cos(4x)

with respect to

x

yields

\frac{dy}{dx} = 10e^{2x} - 4\sin(4x)

. Substituting

x = 0

gives

10e^0 - 4\sin(0) = 10(1) - 0 = 10

.

Step-by-Step Solution

1

Differentiate

y = 5e^{2x} + \cos(4x)

with respect to

x

.

\frac{dy}{dx} = 10e^{2x} - 4\sin(4x)

The derivative of

e^{ax}

is

a e^{ax}

and the derivative of

\cos(ax)

is

-a \sin(ax)

.

2

Evaluate the derivative at

x = 0

.

10

Substitute

x = 0

into

10e^{2x} - 4\sin(4x)

to obtain

10(1) - 4(0) = 10

.

Key Concept

Differentiation of exponential and trigonometric functions

Question 649Question →

The sum of the interior angles of a convex polygon is three times the sum of its exterior angles. Calculate the number of sides of the polygon.

Show answer & explanation

Answer: 8

Answer

The polygon has 8 sides.

The sum of the exterior angles of any convex polygon is always

360^\circ

. Since the sum of the interior angles is three times this value, the interior angle sum is

3 \times 360^\circ = 1080^\circ

. Setting

(n - 2) \times 180^\circ = 1080^\circ

gives

n - 2 = 6

, which yields

n = 8

sides.

Step-by-Step Solution

1

Determine the exterior angle sum

The sum of exterior angles for any convex polygon is

360^\circ

.

The exterior angles of any convex polygon sum to a full turn (

360^\circ

).

2

Compute the sum of the interior angles

Sum of interior angles =

3 \times 360^\circ = 1080^\circ

.

The question states that the interior angle sum is three times the exterior angle sum.

3

Apply the interior angle sum formula

(n - 2) \times 180^\circ = 1080^\circ

.

The interior angle sum of an

n

-sided convex polygon is given by

(n - 2) \times 180^\circ

.

4

Solve for

n

n - 2 = \frac{1080^\circ}{180^\circ} = 6 \implies n = 8

.

Dividing

1080^\circ

by

180^\circ

gives

6

, and adding

2

gives

n = 8

.

Key Concept

Relationship between the sum of interior and exterior angles of a convex polygon

Estimated Time:1m 0s

Question 650Question →

What is the simplified numerical value of $27^{\frac{2}{3}} \times 9^{-\frac{1}{2}}$ ?

Show answer & explanation

Answer: 3

Answer

The simplified numerical value of the expression is 3.

Evaluating

27^{\frac{2}{3}}

gives

(\sqrt[3]{27})^2 = 3^2 = 9

. Evaluating

9^{-\frac{1}{2}}

gives

\frac{1}{\sqrt{9}} = \frac{1}{3}

. Multiplying

9

by

\frac{1}{3}

yields

3

.

Step-by-Step Solution

1

Rewrite bases in terms of prime factors

27 = 3^3

and

9 = 3^2

Expressing bases in power-of-3 form allows direct application of index laws.

2

Simplify each indexed term using

(a^m)^n = a^{m \cdot n}

and

a^{-n} = \frac{1}{a^n}

27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9

and

9^{-\frac{1}{2}} = (3^2)^{-\frac{1}{2}} = 3^{-1} = \frac{1}{3}

Fractional indices denote roots and negative indices denote reciprocals.

3

Multiply the resulting numbers

9 \times \frac{1}{3} = 3

Simplifying the final product yields the single numeric answer.

Key Concept

Fractional and Negative Index Laws

Question 651Question →

In a technology conference of $60$ software developers, $35$ write code in Python, $28$ write in Java, and $12$ write in both Python and Java. How many developers write in neither of these two languages?

Show answer & explanation

Answer: 9

Answer

The number of developers who write in neither Python nor Java is

9

.

The total number of developers writing in at least one language is given by

35 + 28 - 12 = 51

. Subtracting this from the universal set size of

60

gives

60 - 51 = 9

developers who write in neither language.

Step-by-Step Solution

1

Calculate the number of developers who write in at least one of the languages.

n(P \cup J) = 35 + 28 - 12 = 51

By the principle of inclusion-exclusion for two sets,

n(P \cup J) = n(P) + n(J) - n(P \cap J)

.

2

Subtract from the universal set total to get the complement.

n((P \cup J)') = 60 - 51 = 9

The number of elements outside the union is the universal set total minus the union cardinality.

Key Concept

Two-set inclusion-exclusion and complement cardinality

Estimated Time:1m 0s

Question 652Question →

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 653Question →

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 654Question →

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 655Question →

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 656Question →

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 657Question →

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 658Question →

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 659Question →

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 660Question →

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

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