All practice questions

Question 7641Question →

A vibrating tuning fork generates a longitudinal sound wave in air. The fork completes $120$ full oscillations in $0.40\text{ s}$ . If the wavelength of the sound wave in air is $1.15\text{ m}$ , what is the speed of propagation of the wave in $\text{m/s}$ ?

Show answer & explanation

Answer: 345

Answer

The speed of propagation of the sound wave in air is

345\text{ m/s}

.

The frequency of oscillation is determined by dividing the number of oscillations by the total time taken:

f = \frac{120}{0.40\text{ s}} = 300\text{ Hz}

. Substituting the frequency and given wavelength into the wave equation

v = f \lambda

yields

v = 300\text{ Hz} \times 1.15\text{ m} = 345\text{ m/s}

.

Step-by-Step Solution

1

Determine the frequency of the longitudinal wave.

f = 300\text{ Hz}

Frequency is the number of complete oscillations per unit time:

f = \frac{N}{t} = \frac{120}{0.40\text{ s}} = 300\text{ Hz}

.

2

Calculate the wave propagation speed using the wave equation.

v = 345\text{ m/s}

The speed of a progressive wave is given by

v = f \lambda = 300\text{ Hz} \times 1.15\text{ m} = 345\text{ m/s}

.

Key Concept

Relationship between frequency, wavelength, and wave propagation speed in a mechanical medium

Question 7642Question →

Given the $3 \times 3$ matrix $M = \begin{pmatrix} k & 2 & 1 \\ 3 & k-1 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ , where $k > 5$ . If $\det(M^2) = 100$ and $\det(M) < 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

Using the matrix determinant power identity

\det(M^2) = (\det(M))^2 = 100

and the condition

\det(M) < 0

, we find

\det(M) = -10

. Expanding

\det(M)

along the first row yields

\det(M) = k(k-9) - 2(1) + 1(13-k) = k^2 - 10k + 11

. Setting this equal to

-10

gives the quadratic equation

k^2 - 10k + 21 = 0

, which factors as

(k-3)(k-7) = 0

. Given

k > 5

, the unique solution is

k = 7

.

Step-by-Step Solution

1

Apply determinant properties for matrix powers

\det(M) = -10

Because

\det(M^2) = (\det(M))^2 = 100

and it is given that

\det(M) < 0

, taking the negative square root gives

\det(M) = -10

.

2

Evaluate the determinant of matrix

M

using first-row expansion

\det(M) = k^2 - 10k + 11

Expanding along row 1 gives

k((k-1)(1) - 8) - 2(3(1) - 2(1)) + 1(3(4) - (k-1)(1)) = k^2 - 9k - 2 + 13 - k = k^2 - 10k + 11

.

3

Set up and simplify the quadratic equation for

k

k^2 - 10k + 21 = 0

Equating

k^2 - 10k + 11

to

-10

yields

k^2 - 10k + 21 = 0

.

4

Solve the quadratic equation and enforce the inequality condition

k > 5

k = 7

Factoring

(k-3)(k-7) = 0

gives

k = 3

or

k = 7

. Applying the restriction

k > 5

selects

k = 7

.

Key Concept

Determinant Properties of Matrix Powers and 3x3 Matrix Expansion

Question 7643Question →

Complete the statement below regarding indicator selection and color transition during an acid-base titration.

Fill in the blanks below

In the volumetric titration of a strong acid against a weak base, is the most suitable indicator, and it changes color from to red at the acidic equivalence point when acid is added to the base.

Show answer & explanation

Answer

In the volumetric titration of a strong acid against a weak base, methyl orange is the most suitable indicator, and it changes color from yellow to red at the acidic equivalence point when acid is added to the base.

Titrations involving a strong acid and a weak base yield an acidic equivalence point (

\text{pH } 3 - 5

). Methyl orange is ideal because its transition range is

\text{pH } 3.1 - 4.4

. In the initial basic environment of the conical flask, methyl orange is yellow, and it shifts to red/pink as the acid endpoint is reached.

Step-by-Step Solution

1

Determine the pH of the equivalence point for the titration system.

Titrating a strong acid against a weak base produces a salt that undergoes hydrolysis, yielding an acidic equivalence point with

\text{pH} < 7

(typically between

\text{pH } 3

and

5

).

The conjugate acid of the weak base hydrolyzes in water to release hydrogen ions

\text{H}^+

.

2

Match the equivalence point pH range with the appropriate indicator.

Methyl orange has an effective indicator range of

\text{pH } 3.1\text{ to } 4.4

, which overlaps with the sharp pH drop of this titration curve.

An indicator is suitable only if its working range coincides with the steep vertical region of the titration curve.

3

Identify the initial and endpoint colors of methyl orange.

Before titration begins (in basic solution), methyl orange is yellow. At the endpoint after excess acid is introduced, it turns red.

Methyl orange is yellow in alkaline media (

\text{pH} > 4.4

) and turns pink/red in acidic media (

\text{pH} < 3.1

).

Key Concept

Indicator Selection and Endpoint Color Changes in Volumetric Analysis

Question 7644Question →

A binary operation $*$ defined on the set of real numbers $\mathbb{R}$ is given by $a * b = a^2 + 2b - 5$ . If $3 * x = 12$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Applying the binary operation definition

a * b = a^2 + 2b - 5

to

3 * x

gives

3^2 + 2x - 5 = 2x + 4

. Equating

2x + 4 = 12

yields

2x = 8

, so

x = 4

.

Step-by-Step Solution

1

Substitute

a = 3

and

b = x

into the operational rule

a * b = a^2 + 2b - 5

.

3 * x = 3^2 + 2x - 5 = 9 + 2x - 5 = 2x + 4

To express the operation

3 * x

as an algebraic expression in terms of

x

.

2

Set the simplified algebraic expression equal to the given value of 12.

2x + 4 = 12

The question states that

3 * x = 12

.

3

Solve the linear equation for

x

.

2x = 8 \implies x = 4

Subtract 4 from both sides and divide by 2.

Key Concept

Evaluating binary operations and solving algebraic equations involving defined operational rules.

Question 7645Question →

The acoustic intensity $S$ (defined as power per unit area) of a sound wave propagating through a medium of density $\rho$ at speed $v$ is given by the empirical relationship $S = k A^x \omega^2 \rho v^w$ , where $A$ is the wave displacement amplitude, $\omega$ is the angular frequency, and $k$ is a dimensionless constant. Using the principles of dimensional analysis, calculate the numerical value of the exponent $x$ .

Show answer & explanation

Answer: 2

Answer

The numerical value of the exponent

x

is 2.

By applying the principle of dimensional homogeneity, the dimensions of intensity

[S] = M T^{-3}

are equated to

[A]^x [\omega]^2 [\rho] [v]^w = M L^{x - 3 + w} T^{-2 - w}

. Equating time exponents yields

-3 = -2 - w \implies w = 1

. Equating length exponents yields

0 = x - 3 + w \implies x = 2

.

Step-by-Step Solution

1

Determine the fundamental dimensions of acoustic intensity

S

[S] = \frac{[\text{Power}]}{[\text{Area}]} = \frac{M L^2 T^{-3}}{L^2} = M L^0 T^{-3}

Intensity is defined as power delivered per unit surface area perpendicular to the direction of propagation.

2

Write the dimensional formulas for all variables in the given equation

S = k A^x \omega^2 \rho^1 v^w

[A] = L

,

[\omega] = T^{-1}

,

[\rho] = M L^{-3}

,

[v] = L T^{-1}

Each physical quantity must be resolved into fundamental SI dimensions of Mass (

M

), Length (

L

), and Time (

T

).

3

Formulate the dimensional balance equation

M^1 L^0 T^{-3} = L^x \cdot T^{-2} \cdot M^1 L^{-3} \cdot L^w T^{-w} = M^1 L^{x - 3 + w} T^{-2 - w}

For physical validity, the dimensions on both sides of an equation must be identical (principle of dimensional homogeneity).

4

Equate the exponents of Time (

T

) to solve for

w

-3 = -2 - w \implies w = 1

The power of

T

on the left side must equal the sum of powers of

T

on the right side.

5

Equate the exponents of Length (

L

) to find

x

0 = x - 3 + w \implies 0 = x - 3 + 1 \implies x = 2

Substituting

w = 1

into the length exponent balance yields the value of

x

.

Key Concept

Principle of Dimensional Homogeneity

Estimated Time:2m 0s

Question 7646Question →

Given the matrices $P = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 0 \\ -2 & 1 \end{pmatrix}$ , what is the product matrix $PQ$ ?

\begin{pmatrix} 10 & 0 \\ -2 & 4 \end{pmatrix}

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

\begin{pmatrix} 10 & 15 \\ -3 & -2 \end{pmatrix}

\begin{pmatrix} 16 & 3 \\ 13 & 4 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}$

Answer

The product matrix

PQ

is

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

To find

PQ

, multiply each row of matrix

P

by each column of matrix

Q

. The entry in row 1, column 1 is

2(5) + 3(-2) = 4

. The entry in row 1, column 2 is

2(0) + 3(1) = 3

. The entry in row 2, column 1 is

1(5) + 4(-2) = -3

. The entry in row 2, column 2 is

1(0) + 4(1) = 4

. Putting these together yields

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

Step-by-Step Solution

1

Compute the entries of the top row of

PQ

using the first row of

P

and both columns of

Q

.

First entry:

(2)(5) + (3)(-2) = 10 - 6 = 4

. Second entry:

(2)(0) + (3)(1) = 0 + 3 = 3

.

Matrix entry

(1,1)

is the dot product of Row 1 of

P

and Column 1 of

Q

; entry

(1,2)

is the dot product of Row 1 of

P

and Column 2 of

Q

.

2

Compute the entries of the bottom row of

PQ

using the second row of

P

and both columns of

Q

.

Third entry:

(1)(5) + (4)(-2) = 5 - 8 = -3

. Fourth entry:

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

.

Matrix entry

(2,1)

is the dot product of Row 2 of

P

and Column 1 of

Q

; entry

(2,2)

is the dot product of Row 2 of

P

and Column 2 of

Q

.

3

Assemble the computed entries into a

2 \times 2

matrix.

PQ = \begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

Combining row results gives the final product matrix.

Key Concept

Matrix Multiplication (Row-by-Column Dot Product)

Estimated Time:1m 15s

Question 7647Question →

A binary operation $\ast$ on the set of real numbers $\mathbb{R} \setminus \{2\}$ is defined by $a \ast b = 2a + 2b - ab - 2$ . If $y^{-1}$ represents the inverse of an element $y$ under the operation $\ast$ , find the value of $x$ such that $(x \ast 3) \ast 4^{-1} = 5$ .

Show answer & explanation

Answer: 8

Answer

The value of x is 8.

To solve for x, first find the identity element by solving a * e = a, yielding e = 1. Next, compute 4^{-1} from 4 * 4^{-1} = 1, which gives 4^{-1} = 2.5. Then simplify x * 3 to 4 - x. Finally, substitute into (4 - x) * 2.5 = 5 and solve for x to get 8.

Step-by-Step Solution

1

Find the identity element e of the operation

e = 1

By definition, a * e = a. Substituting into the operation gives 2a + 2e - ae - 2 = a, which simplifies to (a - 2)(1 - e) = 0. Since a != 2, e must equal 1.

2

Calculate the inverse element 4^{-1}

4^{-1} = 2.5

By definition of an inverse element, 4 * 4^{-1} = e = 1. Applying the operation formula yields 2(4) + 2(4^{-1}) - 4(4^{-1}) - 2 = 1, which simplifies to 6 - 2(4^{-1}) = 1, so 4^{-1} = 2.5.

3

Express x * 3 in terms of x

x * 3 = 4 - x

Evaluating x * 3 using the operational definition gives 2x + 2(3) - 3x - 2 = 4 - x.

4

Solve the main equation (x * 3) * 4^{-1} = 5 for x

x = 8

Substituting x * 3 = 4 - x and 4^{-1} = 2.5 into the equation yields (4 - x) * 2.5 = 5. Applying the operation gives 2(4 - x) + 2(2.5) - 2.5(4 - x) - 2 = 5, which simplifies to 1 + 0.5x = 5, giving x = 8.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 7648Question →

Two capacitors with capacitances of $10\text{ }\mu\text{F}$ and $15\text{ }\mu\text{F}$ are connected in parallel. What is the equivalent capacitance of the combination in microfarads ( $\mu\text{F}$ )?

Show answer & explanation

Answer: 25

Answer

The equivalent capacitance of the parallel combination is

25\text{ }\mu\text{F}

.

When capacitors are connected in parallel, the total equivalent capacitance is equal to the direct sum of the individual capacitances:

C_{\text{eq}} = C_1 + C_2 = 10\text{ }\mu\text{F} + 15\text{ }\mu\text{F} = 25\text{ }\mu\text{F}

.

Step-by-Step Solution

1

Identify the relationship for parallel capacitors

C_{\text{eq}} = C_1 + C_2

Capacitors connected in parallel store charge independently across the same potential difference, so their capacitances add directly.

2

Substitute the given values into the formula

C_{\text{eq}} = 10\text{ }\mu\text{F} + 15\text{ }\mu\text{F}

The circuit contains two capacitors of

10\text{ }\mu\text{F}

and

15\text{ }\mu\text{F}

in parallel.

3

Calculate the total capacitance

25\text{ }\mu\text{F}

Simple addition of the two values yields

25\text{ }\mu\text{F}

.

Key Concept

Equivalent Capacitance of Parallel Connected Capacitors

Question 7649Question →

A businesswoman invested $\text{₦}50,000$ in a financial fund. Part of the money was invested at a simple interest rate of $6\%$ per annum, and the remaining part was invested at $8\%$ simple interest per annum. If the total interest earned at the end of $1$ year was $\text{₦}3,600$ , what was the amount, in Naira, invested at the $8\%$ interest rate?

Show answer & explanation

Answer: 30000

Answer

The amount invested at the

8\%

interest rate is

\text{₦}30,000

.

Setting up the linear equation for annual simple interest gives

0.06(50,000 - x) + 0.08x = 3,600

. Simplifying this yields

3,000 + 0.02x = 3,600

, which solves to

x = 30,000

. Thus,

\text{₦}30,000

was invested at

8\%

.

Step-by-Step Solution

1

Define variables for the two investment amounts.

Let

x

be the amount in Naira invested at

8\%

, so

(50,000 - x)

is the amount invested at

6\%

.

The total capital of

\text{₦}50,000

is split into two distinct portions.

2

Formulate the total interest expression using the simple interest formula

I = \frac{P \times R \times T}{100}

.

\frac{6}{100}(50,000 - x) + \frac{8}{100}x = 3,600

The sum of annual interests from both parts equals the total interest earned of

\text{₦}3,600

.

3

Expand the terms and solve the linear equation for

x

.

3,000 + 0.02x = 3,600 \implies 0.02x = 600 \implies x = 30,000

Isolating

x

yields the exact principal amount allocated to the

8\%

interest rate.

Key Concept

Simple Interest and Allocation of Principal across Different Interest Rates

Estimated Time:1m 30s

Question 7650Question →

A farmer bought a motorcycle for $\text{₦}250,000$ and later sold it at a profit of $12\%$ . What is the selling price of the motorcycle in Naira?

Show answer & explanation

Answer: 280000

Answer

The selling price of the motorcycle is ₦280,000.

The cost price of the motorcycle is

\text{₦}250,000

. A profit of

12\%

means an additional

\frac{12}{100} \times 250,000 = \text{₦}30,000

. Adding this profit to the cost price gives a selling price of

\text{₦}250,000 + \text{₦}30,000 = \text{₦}280,000

.

Step-by-Step Solution

1

Calculate the profit amount in Naira

Profit = ₦30,000

Profit is calculated as 12% of the original cost price of ₦250,000.

2

Determine the final selling price

Selling Price = ₦280,000

Selling price equals cost price plus the profit made.

Key Concept

Percentage Profit and Selling Price

Question 7651Question →

A wave traveling through a first medium is described by the displacement equation $y = 0.02 \sin \left(100\pi t - \frac{4\pi}{3} x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. As the wave enters a second medium, its speed becomes $120\text{ m/s}$ . What is the wavelength of the wave in the second medium?

2.4\text{ m}

1.5\text{ m}

0.42\text{ m}

3.0\text{ m}

Show answer & explanation

Answer: $2.4\text{ m}$

Answer

The wavelength of the wave in the second medium is

2.4\text{ m}

.

Comparing the wave equation

y = 0.02 \sin\left(100\pi t - \frac{4\pi}{3} x\right)

with

y = A \sin(\omega t - kx)

gives

\omega = 100\pi\text{ rad/s}

. The frequency is

f = \frac{\omega}{2\pi} = 50\text{ Hz}

. Because frequency does not change when crossing boundaries, the wavelength in the second medium where speed is

120\text{ m/s}

is

\lambda = \frac{v}{f} = \frac{120}{50} = 2.4\text{ m}

.

Step-by-Step Solution

1

Extract the angular frequency

\omega

from the given wave equation.

The angular frequency

\omega = 100\pi\text{ rad/s}

.

The standard wave equation is

y = A \sin(\omega t - kx)

.

2

Calculate the frequency of the wave.

f = \frac{\omega}{2\pi} = \frac{100\pi}{2\pi} = 50\text{ Hz}

.

Frequency is determined by the source and remains constant regardless of the medium.

3

Determine the wavelength in the second medium using the new wave speed.

\lambda_2 = \frac{v_2}{f} = \frac{120\text{ m/s}}{50\text{ Hz}} = 2.4\text{ m}

.

Applying the wave equation

v = f\lambda

with the updated speed in the second medium.

Key Concept

Invariance of wave frequency across media boundaries and extraction of wave parameters from the mathematical wave equation.

Question 7652Question →

If $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = k\sqrt{15}$ , find the value of $k$ .

Show answer & explanation

Answer: 2

Answer

The value of

k

is

2

.

Combining the fractions gives a common denominator of

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

. Expanding the numerator gives

(8+2\sqrt{15}) - (8-2\sqrt{15}) = 4\sqrt{15}

. Dividing by 2 yields

2\sqrt{15}

, giving

k = 2

.

Step-by-Step Solution

1

Combine the fractions on the left-hand side over a common denominator.

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

Combining two fractions with conjugate denominators simplifies the expression.

2

Evaluate the denominator using the difference of two squares formula

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

.

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

Multiplying conjugate surds eliminates the radical signs in the denominator.

3

Expand both squared terms in the numerator and subtract them.

(8 + 2\sqrt{15}) - (8 - 2\sqrt{15}) = 4\sqrt{15}

Expanding

(a \pm b)^2 = a^2 \pm 2ab + b^2

gives

5 \pm 2\sqrt{15} + 3 = 8 \pm 2\sqrt{15}

.

4

Divide the resulting numerator by the denominator and solve for

k

.

\frac{4\sqrt{15}}{2} = 2\sqrt{15} \Rightarrow k = 2

Dividing

4\sqrt{15}

by

2

yields

2\sqrt{15}

, so matching the coefficients gives

k = 2

.

Key Concept

Rationalization of Denominators and Difference of Conjugate Surd Fractions

Estimated Time:1m 30s

Question 7653Question →

A sound wave of frequency $250\text{ Hz}$ propagates through a metal rod at a speed of $5000\text{ m s}^{-1}$ . What is the distance between a compression and the adjacent rarefaction in the rod?

20\text{ m}

0.05\text{ m}

10\text{ m}

1.25 \times 10^6\text{ m}

Show answer & explanation

Answer: $10\text{ m}$

Answer

The distance between a compression and the adjacent rarefaction is

10\text{ m}

.

First, find the wavelength using

\lambda = \frac{v}{f} = \frac{5000\text{ m s}^{-1}}{250\text{ Hz}} = 20\text{ m}

. In a longitudinal mechanical wave propagating through a medium, one complete wavelength is the distance between two successive compressions or two successive rarefactions. The distance from a compression to the adjacent rarefaction is half a wavelength, giving

\frac{20\text{ m}}{2} = 10\text{ m}

.

Step-by-Step Solution

1

Calculate the wavelength (\lambda) of the wave using the wave equation.

\lambda = \frac{v}{f} = \frac{5000\text{ m s}^{-1}}{250\text{ Hz}} = 20\text{ m}

The fundamental wave relationship is

v = f\lambda

, so wavelength is the ratio of wave speed to frequency.

2

Determine the distance between a compression and the adjacent rarefaction.

\text{Distance} = \frac{\lambda}{2} = \frac{20\text{ m}}{2} = 10\text{ m}

In a longitudinal wave, one full wavelength is the distance between two consecutive compressions. The distance between a compression and the immediate next rarefaction is half of one wavelength.

Key Concept

Distance between consecutive compression and rarefaction in a longitudinal wave

Question 7654Question →

In $\Delta ABC$ , side $a = 10\text{ cm}$ , side $b = 16\text{ cm}$ , and $\sin A = \frac{3}{8}$ . If $\angle B$ is an obtuse angle, what is the exact value of $\cos B$ ?

-\frac{4}{5}

\frac{4}{5}

\frac{3}{5}

-\frac{3}{5}

Show answer & explanation

Answer: $-\frac{4}{5}$

Answer

-\frac{4}{5}

Applying the Sine Rule gives

\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{3}{5}

. Since

\angle B

is an obtuse angle, it lies in the second quadrant where cosine is negative. Using

\cos B = -\sqrt{1 - \sin^2 B}

, we get

\cos B = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\frac{4}{5}

.

Step-by-Step Solution

1

Apply the Sine Rule to calculate

\sin B

.

\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{6}{10} = \frac{3}{5}

The Sine Rule states that

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

.

2

Determine the sign of

\cos B

based on the given angle type.

Since

\angle B

is an obtuse angle (

90^\circ < B < 180^\circ

),

\cos B < 0

.

Cosine is negative in the second quadrant.

3

Calculate

\cos B

using the Pythagorean trigonometric identity.

\cos B = -\sqrt{1 - \sin^2 B} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\sqrt{\frac{16}{25}} = -\frac{4}{5}

Substitute

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

into

\cos B = -\sqrt{1 - \sin^2 B}

.

Key Concept

Sine Rule and Trigonometric Ratios of Obtuse Angles

Question 7655Question →

By definition, the derivative of the function $f(x) = \frac{4}{x}$ at $x = 2$ is given by the limit of the difference quotient $\lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$ . What is the value of this limit?

-1

1

-2

-4

Show answer & explanation

Answer: $-1$

Answer

-1

Substituting

f(2+h) = \frac{4}{2+h}

and

f(2) = 2

into the difference quotient gives

\frac{\frac{4}{2+h} - 2}{h} = \frac{-2h}{h(2+h)} = \frac{-2}{2+h}

. Evaluating the limit as

h \to 0

yields

\frac{-2}{2} = -1

.

Step-by-Step Solution

1

Calculate

f(2)

and

f(2+h)

f(2) = \frac{4}{2} = 2

and

f(2+h) = \frac{4}{2+h}

These are the two values needed for the difference quotient numerator.

2

Subtract

f(2)

from

f(2+h)

and find a common denominator

f(2+h) - f(2) = \frac{4}{2+h} - 2 = \frac{4 - 2(2+h)}{2+h} = \frac{4 - 4 - 2h}{2+h} = \frac{-2h}{2+h}

Simplifying the numerator expression algebraically.

3

Divide the numerator by

h

to form the difference quotient

\frac{\frac{-2h}{2+h}}{h} = \frac{-2}{2+h}

Canceling the common factor

h

in the numerator and denominator.

4

Evaluate the limit as

h \to 0

\lim_{h \to 0} \frac{-2}{2+h} = \frac{-2}{2+0} = -1

Direct substitution of

h = 0

into the simplified expression gives the instantaneous rate of change.

Key Concept

Differentiation of reciprocal functions from first principles

Estimated Time:1m 30s

Question 7656Question →

If $y = e^{3x}\cos(2x) + \ln(x + 1)$ , calculate the value of $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 4

Answer

The derivative evaluated at

x = 0

is 4.

Differentiating

y = e^{3x}\cos(2x) + \ln(x + 1)

with respect to

x

yields

\frac{dy}{dx} = 3e^{3x}\cos(2x) - 2e^{3x}\sin(2x) + \frac{1}{x+1}

. Evaluating this derivative at

x = 0

gives

3(1)(1) - 2(1)(0) + 1 = 4

.

Step-by-Step Solution

1

Differentiate the product

u(x) = e^{3x}\cos(2x)

using the product rule and chain rule.

\frac{du}{dx} = 3e^{3x}\cos(2x) - 2e^{3x}\sin(2x)

By the product rule

\frac{d}{dx}(uv) = u'v + uv'

, where

\frac{d}{dx}(e^{3x}) = 3e^{3x}

and

\frac{d}{dx}(\cos(2x)) = -2\sin(2x)

.

2

Differentiate the logarithmic term

v(x) = \ln(x + 1)

.

\frac{dv}{dx} = \frac{1}{x + 1}

The derivative of

\ln(g(x))

is

\frac{g'(x)}{g(x)}

.

3

Combine the terms to write the complete derivative

\frac{dy}{dx}

.

\frac{dy}{dx} = 3e^{3x}\cos(2x) - 2e^{3x}\sin(2x) + \frac{1}{x + 1}

The derivative of a sum of functions is the sum of their individual derivatives.

4

Evaluate

\frac{dy}{dx}

at

x = 0

.

\frac{dy}{dx}\Big|_{x=0} = 3e^0\cos(0) - 2e^0\sin(0) + \frac{1}{0 + 1} = 3(1)(1) - 2(1)(0) + 1 = 4

Substitute

x = 0

using

e^0 = 1

,

\cos(0) = 1

, and

\sin(0) = 0

.

Key Concept

Differentiation of Trigonometric, Exponential, and Logarithmic Functions

Estimated Time:1m 30s

Question 7657Question →

A logistics firm purchased a commercial delivery van for $\text{₦}800,000$ . If the value of the van depreciates by $10\%$ in the first year and by $15\%$ in the second year, what is the value of the van at the end of the second year?

\text{₦}612,000

\text{₦}600,000

\text{₦}188,000

\text{₦}648,000

Show answer & explanation

Answer: $\text{₦}612,000$

Answer

\text{₦}612,000

To find the remaining value of an asset undergoing successive annual depreciation, multiply the initial cost by each annual retention factor

(1 - r)

. For Year 1 at

10\%

, the value retention factor is

0.90

, making the value

\text{₦}720,000

. For Year 2 at

15\%

, the retention factor is

0.85

, yielding a final value of

\text{₦}720,000 \times 0.85 = \text{₦}612,000

.

Step-by-Step Solution

1

Calculate the value of the van at the end of the first year after a 10% depreciation.

\text{Value after Year 1} = \text{₦}800,000 \times \left(1 - \frac{10}{100}\right) = \text{₦}800,000 \times 0.90 = \text{₦}720,000

Depreciation decreases the principal asset value by the specified percentage.

2

Calculate the value of the van at the end of the second year after a 15% depreciation on the year-one value.

\text{Value after Year 2} = \text{₦}720,000 \times \left(1 - \frac{15}{100}\right) = \text{₦}720,000 \times 0.85 = \text{₦}612,000

Successive depreciation is compounded on the reduced value at the start of each period.

Key Concept

Successive Asset Depreciation

Question 7658Question →

A convex polygon has $44$ diagonals. What is the total number of distinct triangles that can be formed by joining any three vertices of this polygon?

Show answer & explanation

Answer: 165

Answer

165

The number of diagonals of an

n

-sided convex polygon is given by

\binom{n}{2} - n = \frac{n(n-3)}{2}

. Setting this equal to

44

yields

n(n-3) = 88

, which simplifies to

n^2 - 3n - 88 = 0

. Factoring gives

(n-11)(n+8) = 0

, so

n = 11

. The total number of distinct triangles formed by choosing any 3 vertices from an 11-sided polygon is

\binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165

.

Step-by-Step Solution

1

Set up the equation for the number of diagonals in terms of the number of vertices

n

\frac{n(n-3)}{2} = 44

Choosing any 2 vertices from

n

vertices gives

\binom{n}{2}

total connecting line segments. Subtracting the

n

boundary sides leaves the diagonals.

2

Solve the quadratic equation for

n

n^2 - 3n - 88 = 0 \implies (n-11)(n+8) = 0 \implies n = 11

A polygon must have a positive integer number of vertices, so

n = 11

.

3

Compute the number of distinct triangles using combinations

\binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165

In a convex polygon, no three vertices are collinear, so every unique combination of 3 vertices forms a distinct triangle.

Key Concept

Application of combinations to geometry (polygon diagonals and triangle selection)

Estimated Time:2m 0s

Question 7659Question →

What is the magnitude of the electric field intensity, in $\text{N C}^{-1}$ , at a point $2.0\text{ m}$ away from a isolated point charge of $+4.0 \times 10^{-6}\text{ C}$ in a vacuum? (Take Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ )

Show answer & explanation

Answer: 9000

Answer

The magnitude of the electric field intensity is

9000\text{ N C}^{-1}

.

The electric field intensity

E

produced by a point charge

q

at distance

r

is given by

E = \frac{kq}{r^2}

. Substituting

k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}

,

q = 4.0 \times 10^{-6}\text{ C}

, and

r = 2.0\text{ m}

yields

E = \frac{9.0 \times 10^9 \times 4.0 \times 10^{-6}}{4.0} = 9000\text{ N C}^{-1}

.

Step-by-Step Solution

1

Identify the given physical quantities and formula

q = 4.0 \times 10^{-6}\text{ C}

,

r = 2.0\text{ m}

,

k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}

. Formula:

E = \frac{kq}{r^2}

The magnitude of electric field intensity due to a single point charge is given by Coulomb's field law.

2

Substitute the values and calculate the electric field strength

E = \frac{9.0 \times 10^9 \times 4.0 \times 10^{-6}}{2.0^2} = \frac{36000}{4} = 9000\text{ N C}^{-1}

Perform basic arithmetic simplification to determine the numerical result.

Key Concept

Electric Field Intensity due to a Point Charge

Question 7660Question →

Given the simultaneous equations $2x - y = 1$ and $2x^2 - xy + y^2 + x - y = 17$ , let $(x_1, y_1)$ and $(x_2, y_2)$ be the real solution pairs such that $x_1 > x_2$ . What is the value of $2x_1 + y_2$ ?

1

3

9

-3

Show answer & explanation

Answer: 1

Answer

1

Substituting

y = 2x - 1

into

2x^2 - xy + y^2 + x - y = 17

simplifies correctly to

4x^2 - 4x - 15 = 0

. Factorizing gives roots

x = \frac{5}{2}

and

x = -\frac{3}{2}

. Given

x_1 > x_2

, we set

x_1 = \frac{5}{2}

and

x_2 = -\frac{3}{2}

. Substituting

x_2

back into the linear equation gives

y_2 = -4

. Calculating

2x_1 + y_2 = 2\left(\frac{5}{2}\right) + (-4) = 5 - 4 = 1

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation.

y = 2x - 1

Substitution method requires expressing one variable in terms of the other.

2

Substitute

y = 2x - 1

into the quadratic equation

2x^2 - xy + y^2 + x - y = 17

.

2x^2 - x(2x - 1) + (2x - 1)^2 + x - (2x - 1) = 17

This converts the system into a single quadratic equation in terms of

x

.

3

Expand and simplify the algebraic expression.

2x^2 - 2x^2 + x + 4x^2 - 4x + 1 + x - 2x + 1 = 17 \implies 4x^2 - 4x + 2 = 17 \implies 4x^2 - 4x - 15 = 0

Combining like terms reveals the standard quadratic form

ax^2 + bx + c = 0

.

4

Solve the quadratic equation

4x^2 - 4x - 15 = 0

by factorization.

(2x - 5)(2x + 3) = 0 \implies x = \frac{5}{2}

or

x = -\frac{3}{2}

Finding the two real values for

x

.

5

Identify

x_1

and

x_2

according to

x_1 > x_2

and compute corresponding

y

-values.

x_1 = \frac{5}{2} \implies y_1 = 2\left(\frac{5}{2}\right) - 1 = 4

;

x_2 = -\frac{3}{2} \implies y_2 = 2\left(-\frac{3}{2}\right) - 1 = -4

Determining the complete coordinate solution pairs.

6

Evaluate the expression

2x_1 + y_2

.

2\left(\frac{5}{2}\right) + (-4) = 5 - 4 = 1

Answering the specific value requested in the problem statement.

Key Concept

Solving simultaneous linear and quadratic equations using substitution

Estimated Time:2m 30s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept