All practice questions

Question 7681Question →

A straight line $L_1$ passes through the points $(k, 2)$ and $(4, 8)$ . If $L_1$ is perpendicular to the line $2x - 3y + 6 = 0$ , what is the value of $k$ ?

-5

0

8

13

Show answer & explanation

Answer: 8

Answer

The value of

k

is

8

.

The line

2x - 3y + 6 = 0

has a gradient of

\frac{2}{3}

. A line perpendicular to it must have a gradient of

-\frac{3}{2}

. Calculating the gradient of

L_1

using the points

(k, 2)

and

(4, 8)

gives

\frac{8-2}{4-k} = \frac{6}{4-k}

. Equating

\frac{6}{4-k} = -\frac{3}{2}

and solving for

k

gives

k = 8

.

Step-by-Step Solution

1

Find the gradient of the given line

2x - 3y + 6 = 0

Rearranging into

y = mx + c

form gives

3y = 2x + 6 \implies y = \frac{2}{3}x + 2

. Therefore, the gradient

m_2 = \frac{2}{3}

.

To find the perpendicular gradient, we first need the gradient of the given line.

2

Determine the gradient of line

L_1

Since

L_1

is perpendicular to the given line, its gradient

m_1

satisfies

m_1 \cdot m_2 = -1

. Thus,

m_1 = -\frac{1}{2/3} = -\frac{3}{2}

.

Perpendicular lines have gradients that are negative reciprocals of each other.

3

Express the gradient of

L_1

using the coordinates

(k, 2)

and

(4, 8)

and solve for

k

Gradient formula:

m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{4 - k} = \frac{6}{4 - k}

. Setting this equal to

-\frac{3}{2}

:

\frac{6}{4 - k} = -\frac{3}{2}

6 \cdot 2 = -3(4 - k)

12 = -12 + 3k

3k = 24 \implies k = 8

.

Equating the slope calculated from points to the perpendicular slope allows solving for the unknown coordinate

k

.

Key Concept

Perpendicular gradients and line slope formula

Question 7682Question →

A progressive wave traveling along a medium is described by the equation $y = 0.05 \sin(20\pi t - 4\pi x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. What is the speed of the wave in $\text{m/s}$ ?

Show answer & explanation

Answer: 5

Answer

The speed of the wave is

5.0\text{ m/s}

.

Comparing

y = 0.05 \sin(20\pi t - 4\pi x)

to the standard wave equation

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

, we find

\omega = 20\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

. The wave speed

v

is calculated as

v = \frac{\omega}{k} = \frac{20\pi}{4\pi} = 5.0\text{ m/s}

.

Step-by-Step Solution

1

Identify the wave parameters from the standard equation form

\omega = 20\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

Matching the given equation

y = 0.05 \sin(20\pi t - 4\pi x)

to

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

gives the values for angular frequency

\omega

and wave number

k

.

2

Compute wave speed using the relationship

v = \frac{\omega}{k}

v = \frac{20\pi}{4\pi} = 5.0\text{ m/s}

Wave speed is defined as the ratio of angular frequency to wave number.

Key Concept

Wave Speed from Wave Equation

Question 7683Question →

Five of the interior angles of a convex polygon are each equal to $140^\circ$ , while the remaining interior angles are each equal to $160^\circ$ . Calculate the number of sides of the polygon.

Show answer & explanation

Answer: 13

Answer

The number of sides of the polygon is 13.

Each

140^\circ

interior angle has an exterior angle of

40^\circ

, contributing

5 \times 40^\circ = 200^\circ

to the exterior angle sum. Each

160^\circ

interior angle has an exterior angle of

20^\circ

, contributing

(n - 5) \times 20^\circ

. Since the sum of exterior angles of any convex polygon is

360^\circ

, setting

200 + 20(n - 5) = 360

yields

20n = 260

, giving

n = 13

.

Step-by-Step Solution

1

Calculate the exterior angle measures

The exterior angles are

180^\circ - 140^\circ = 40^\circ

(for 5 vertices) and

180^\circ - 160^\circ = 20^\circ

(for the remaining

n - 5

vertices).

Interior and exterior angles at each vertex of a polygon form a linear pair and sum to

180^\circ

.

2

Apply the sum of exterior angles property

5(40^\circ) + (n - 5)(20^\circ) = 360^\circ

.

The sum of exterior angles of any convex polygon is always constant and equal to

360^\circ

.

3

Solve the linear equation for

n

200 + 20n - 100 = 360 \implies 20n = 260 \implies n = 13

.

Expanding terms and isolating

n

gives the exact number of sides.

Key Concept

Exterior angle sum property of convex polygons

Question 7684Question →

A circle of radius $7\text{ cm}$ is inscribed inside a square. What is the area of the region inside the square but outside the circle? (Take $\pi = \frac{22}{7}$ )

42\text{ cm}^2

154\text{ cm}^2

196\text{ cm}^2

12\text{ cm}^2

Show answer & explanation

Answer: $42\text{ cm}^2$

Answer

The area of the region inside the square but outside the circle is

42\text{ cm}^2

.

Because the circle is inscribed inside the square, the diameter of the circle is equal to the side length of the square:

s = 2r = 2(7) = 14\text{ cm}

. The area of the square is

14^2 = 196\text{ cm}^2

. The area of the circle is

\pi r^2 = \frac{22}{7} \times 7^2 = 154\text{ cm}^2

. Subtracting the area of the circle from the area of the square gives the region inside the square but outside the circle:

196 - 154 = 42\text{ cm}^2

.

Step-by-Step Solution

1

Determine the side length of the square.

Side length

s = 2 \times 7\text{ cm} = 14\text{ cm}

.

An inscribed circle touches all four sides of the square, so its diameter equals the side length of the square.

2

Calculate the area of the square.

Area of square

= s^2 = 14^2 = 196\text{ cm}^2

.

The area of a square with side length

s

is given by

s^2

.

3

Calculate the area of the inscribed circle.

Area of circle

= \pi r^2 = \frac{22}{7} \times 7^2 = 22 \times 7 = 154\text{ cm}^2

.

The area of a circle with radius

r

is given by

\pi r^2

.

4

Subtract the circle's area from the square's area to find the area of the shaded outer region.

Area

= 196\text{ cm}^2 - 154\text{ cm}^2 = 42\text{ cm}^2

.

The remaining area consists of the four corner regions between the circle and the bounding square.

Key Concept

Area of composite plane figures (shaded area between inscribed shape and container)

Question 7685Question →

Find the positive value of $x$ that satisfies the simultaneous equations $y - 2x = 3$ and $y = x^2 + 3$ .

2

-2

3

5

Show answer & explanation

Answer: $2$

Answer

The positive value of

x

is 2.

Rearranging the linear equation gives

y = 2x + 3

. Setting this equal to the second expression for

y

gives

2x + 3 = x^2 + 3

, which simplifies to

x^2 - 2x = 0

. Factoring out

x

yields

x(x - 2) = 0

. The solutions for

x

are

0

and

2

. The positive solution is

2

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the linear equation

y = 2x + 3

Isolating

y

prepares for substitution into the quadratic equation.

2

Substitute

y = 2x + 3

into the second equation

y = x^2 + 3

2x + 3 = x^2 + 3

This creates a single quadratic equation in terms of

x

.

3

Rearrange and factor the quadratic equation

x^2 - 2x = 0 \implies x(x - 2) = 0

Subtracting

3

and

2x

from both sides simplifies the equation to factorable form.

4

Solve for the non-zero (positive) value of

x

x = 2

Setting

x - 2 = 0

yields

x = 2

.

Key Concept

Solving simultaneous linear and quadratic equations by substitution

Question 7686Question →

Two identical positive point charges, each of magnitude $q = +2.5 \times 10^{-6}\text{ C}$ , are fixed in a vacuum at a distance of $0.60\text{ m}$ apart. A third point charge $q_0 = +1.0 \times 10^{-6}\text{ C}$ is placed on the perpendicular bisector of the line joining the two fixed charges, at a distance of $0.40\text{ m}$ from their midpoint. Taking Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ , what is the magnitude of the net electrostatic force acting on the third charge, in newtons?

Show answer & explanation

Answer: 0.144

Answer

The magnitude of the net electrostatic force acting on the third charge is

0.144\text{ N}

.

Each fixed charge exerts an equal repulsive electrostatic force of

0.09\text{ N}

on the third charge. Due to the symmetrical arrangement, the force components perpendicular to the bisector cancel each other out, while the parallel components add together, giving a net force of

2 \times 0.09 \times 0.8 = 0.144\text{ N}

.

Step-by-Step Solution

1

Determine the distance from each fixed charge to the third charge

r = 0.50\text{ m}

The charges form a right-angled triangle with base

0.30\text{ m}

(half of

0.60\text{ m}

) and height

0.40\text{ m}

, yielding a hypotenuse of

\sqrt{0.30^2 + 0.40^2} = 0.50\text{ m}

.

2

Calculate the magnitude of the individual repulsive force from one charge

F = 0.09\text{ N}

Applying Coulomb's law:

F = \frac{k q q_0}{r^2} = \frac{9.0 \times 10^9 \times 2.5 \times 10^{-6} \times 1.0 \times 10^{-6}}{0.25} = 0.09\text{ N}

.

3

Determine directional component of forces along the perpendicular bisector

\cos\theta = 0.8

The directional cosine along the axis of symmetry is the ratio of the adjacent side (

0.40\text{ m}

) to the hypotenuse (

0.50\text{ m}

).

4

Compute the net electrostatic force using vector addition

F_{\text{net}} = 0.144\text{ N}

Horizontal components cancel by symmetry, so

F_{\text{net}} = 2 F \cos\theta = 2 \times 0.09 \times 0.8 = 0.144\text{ N}

.

Key Concept

Vector superposition of Coulombic forces along an axis of symmetry

Question 7687Question →

A student library committee is selecting $5$ distinct books from a shelf containing $7$ novel titles and $5$ biography titles. If $2$ specific novel titles are mutually exclusive (they cannot both be selected together in the same combination), in how many ways can the selection of $5$ books be made such that at least $3$ novel titles are included?

426

436

516

546

Show answer & explanation

Answer: 436

Answer

436

To find the number of valid ways, we first find the total number of ways to select at least

3

novels from

7

novels and

5

biographies, which equals

546

. Next, we determine how many of these combinations contain both of the restricted novels while still having at least

3

novels overall. There are

110

such invalid selections. Subtracting

110

from

546

gives

436

valid ways.

Step-by-Step Solution

1

Calculate the total combinations with at least 3 novels without any restriction.

Case 1 (

3

Novels,

2

Biographies):

\binom{7}{3} \times \binom{5}{2} = 35 \times 10 = 350

.
Case 2 (

4

Novels,

1

Biography):

\binom{7}{4} \times \binom{5}{1} = 35 \times 5 = 175

.
Case 3 (

5

Novels,

0

Biographies):

\binom{7}{5} \times \binom{5}{0} = 21 \times 1 = 21

.
Total unrestricted combinations =

350 + 175 + 21 = 546

.

Establishing the total pool of choices that satisfy the constraint of selecting at least 3 novels.

2

Calculate the invalid combinations where both restricted novels are selected together AND at least 3 novels are included.

If both restricted novels are selected, we have already chosen

2

novels. To reach a total of

5

books with at least

3

novels, we must pick

3

additional books from the remaining

10

books (

5

remaining novels and

5

biographies), EXCLUDING the scenario where all

3

additional books are biographies (which would leave us with only

2

novels in total).
Combinations of

3

books from

10

remaining books =

\binom{10}{3} = 120

.
Combinations of

3

biographies from

5

biographies =

\binom{5}{3} = 10

.
Invalid combinations =

120 - 10 = 110

.

To apply the mutual exclusion constraint correctly, we must subtract only those invalid selections that also satisfy the condition of having at least 3 novels.

3

Subtract invalid combinations from total unrestricted combinations.

Valid combinations =

546 - 110 = 436

.

Subtracting the forbidden overlapping outcomes yields the net valid choices.

Key Concept

Combinations with multiple constraints and mutual exclusion

Question 7688Question →

A plane mechanical wave propagating through a fluid is represented by the equation $y = 0.02 \sin \left(250\pi t - \frac{5\pi}{2} x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. When the wave passes into a secondary fluid medium, its speed decreases to $60\text{ m/s}$ . Assuming the frequency remains constant, what is the wavelength of the wave in the secondary medium?

0.48\text{ m}

0.80\text{ m}

0.24\text{ m}

1.33\text{ m}

Show answer & explanation

Answer: $0.48\text{ m}$

Answer

The wavelength of the wave in the secondary medium is

0.48\text{ m}

.

Comparing the given equation to the general progressive wave equation

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

, we find the angular frequency

\omega = 250\pi\text{ rad/s}

. The wave frequency is

f = \frac{\omega}{2\pi} = 125\text{ Hz}

. Because wave frequency is invariant across media boundaries,

f

remains

125\text{ Hz}

in the secondary medium. Using the wave equation

v = f \lambda

, the wavelength in the secondary fluid is

\lambda = \frac{v}{f} = \frac{60\text{ m/s}}{125\text{ Hz}} = 0.48\text{ m}

.

Step-by-Step Solution

1

Extract angular frequency and wave number from the progressive wave equation

From

y = 0.02 \sin \left(250\pi t - \frac{5\pi}{2} x\right)

,

\omega = 250\pi\text{ rad/s}

and

k = \frac{5\pi}{2}\text{ rad/m}

.

Standard progressive wave equations follow the format

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

.

2

Determine the frequency of the wave

f = \frac{\omega}{2\pi} = \frac{250\pi}{2\pi} = 125\text{ Hz}

.

Frequency is related to angular frequency by

\omega = 2\pi f

.

3

Apply boundary transition rules to determine the new wavelength

Frequency

f

remains constant at

125\text{ Hz}

across media. Therefore,

\lambda_2 = \frac{v_2}{f} = \frac{60\text{ m/s}}{125\text{ Hz}} = 0.48\text{ m}

.

When a wave travels across different media boundaries, its frequency depends only on the source and remains constant, whereas speed and wavelength adjust accordingly.

Key Concept

Wave equation parameter extraction and frequency invariance during refraction

Estimated Time:1m 30s

Question 7689Question →

A resistance thermometer registers a resistance of $5.0\,\Omega$ at the ice point ( $0^\circ\text{C}$ ) and $25.0\,\Omega$ at the steam point ( $100^\circ\text{C}$ ). When placed in a heated liquid bath, the resistance measured is $30.0\,\Omega$ . What is the temperature of the bath on the Kelvin scale?

125\,\text{K}

398\,\text{K}

423\,\text{K}

373\,\text{K}

Show answer & explanation

Answer: $398\,\text{K}$

Answer

The temperature of the bath on the Kelvin scale is

398\,\text{K}

.

Using the linear interpolation formula for a thermometric property

\theta = \frac{R_\theta - R_0}{R_{100} - R_0} \times 100^\circ\text{C}

, substituting

R_0 = 5.0\,\Omega

,

R_{100} = 25.0\,\Omega

, and

R_\theta = 30.0\,\Omega

yields

\theta = \frac{25.0}{20.0} \times 100 = 125^\circ\text{C}

. Converting to absolute thermodynamic temperature gives

T = 125 + 273 = 398\,\text{K}

.

Step-by-Step Solution

1

Calculate the temperature on the Celsius scale using linear interpolation of thermometric property.

\theta = \frac{R_\theta - R_0}{R_{100} - R_0} \times 100^\circ\text{C} = \frac{30.0 - 5.0}{25.0 - 5.0} \times 100^\circ\text{C} = \frac{25.0}{20.0} \times 100^\circ\text{C} = 125^\circ\text{C}

The change in resistance is directly proportional to the temperature change between fixed points.

2

Convert the temperature from degrees Celsius to Kelvins.

T = \theta + 273 = 125 + 273 = 398\,\text{K}

Absolute temperature in Kelvin is obtained by adding 273 to the temperature in degrees Celsius.

Key Concept

Temperature Scale Interpolation and Kelvin Conversion

Estimated Time:1m 30s

Question 7690Question →

A music festival coordinator needs to select $4$ bands to perform from a pool of $8$ available bands. If $2$ specific bands insist on either both being selected or neither being selected, in how many different ways can the $4$ bands be chosen?

30

60

70

390

Show answer & explanation

Answer: $30$

Answer

The total number of ways to choose the bands under the given condition is

30

.

To satisfy the condition that the two specific bands are either both chosen or neither chosen, we evaluate two distinct cases. Case 1 (both chosen) requires choosing

2

additional bands from the remaining

6

, yielding

{}^6C_2 = 15

ways. Case 2 (neither chosen) requires choosing all

4

bands from the remaining

6

, yielding

{}^6C_4 = 15

ways. Adding both cases gives

15 + 15 = 30

ways.

Step-by-Step Solution

1

Analyze Case 1: Both specific bands are selected.

If both specific bands are included, we only need to select

2

more bands from the remaining

6

bands. The number of ways is

{}^6C_2 = \frac{6 \times 5}{2 \times 1} = 15

.

Since order does not matter in forming a group of performers, we use combinations.

2

Analyze Case 2: Neither of the specific bands is selected.

If neither of the

2

specific bands is chosen, all

4

bands must be selected from the remaining

6

bands. The number of ways is

{}^6C_4 = {}^6C_2 = 15

.

Excluding the

2

specific bands leaves

6

candidate bands to choose

4

from.

3

Sum the mutually exclusive cases.

\text{Total ways} = 15 + 15 = 30.

The two scenarios are disjoint, so the addition principle applies.

Key Concept

Combinations with conditional restrictions

Estimated Time:1m 30s

Question 7691Question →

Find the sum of all integer values of $x$ that satisfy both the linear inequality $2x - 1 \ge 3$ and the quadratic inequality $x^2 - 5x - 14 \le 0$ .

Show answer & explanation

Answer: 27

Answer

The sum of all integer values of

x

satisfying both inequalities is

27

.

Solving the linear inequality

2x - 1 \ge 3

yields

x \ge 2

. Solving the quadratic inequality

x^2 - 5x - 14 \le 0

by factoring gives

(x - 7)(x + 2) \le 0

, which defines the interval

-2 \le x \le 7

. Taking the intersection of

x \ge 2

and

-2 \le x \le 7

gives

2 \le x \le 7

. The integer values satisfying this range are

2, 3, 4, 5, 6,

and

7

, and their sum is

27

.

Step-by-Step Solution

1

Solve the linear inequality

2x \ge 4 \implies x \ge 2

Adding 1 to both sides and dividing by 2 isolates the variable

x

.

2

Factor and solve the quadratic inequality

(x - 7)(x + 2) \le 0 \implies -2 \le x \le 7

The roots of the quadratic equation are

x = 7

and

x = -2

. The parabola opens upward, so the expression is non-positive between the roots.

3

Determine the intersection of both solution sets

2 \le x \le 7

The values of

x

must simultaneously satisfy

x \ge 2

and

-2 \le x \le 7

.

4

List all integer solutions within the valid interval

x \in \{2, 3, 4, 5, 6, 7\}

These are all the whole numbers contained in the closed interval

[2, 7]

.

5

Sum the integer solutions

2 + 3 + 4 + 5 + 6 + 7 = 27

Summing the identified integer values yields the final required numerical answer.

Key Concept

Linear and Quadratic Inequalities

Estimated Time:1m 30s

Question 7692Question →

A physical quantity $Z$ is defined by the expression $Z = \frac{P \cdot V}{I \cdot t}$ , where $P$ represents pressure, $V$ represents volume, $I$ represents electric current, and $t$ represents time. Which of the following statements correctly classifies quantity $Z$ and expresses it in fundamental SI base units?

Quantity

Z

is a derived quantity with fundamental SI base units of

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Quantity

Z

is a fundamental quantity with fundamental SI base units of

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Quantity

Z

is a derived quantity with fundamental SI base units of

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\cdot\text{A}^{-1}

.

Quantity

Z

is a fundamental quantity with fundamental SI base units of

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-1}

.

Show answer & explanation

Answer: Quantity $Z$ is a derived quantity with fundamental SI base units of $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}$ .

Answer

Quantity

Z

is a derived quantity with fundamental SI base units of

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

The quantity

Z

represents electric potential (voltage), which is defined as work done per unit electric charge (

W/Q

). Since electric potential is derived from mass, length, time, and electric current, it is a derived physical quantity. Expanding

P \cdot V

yields energy units (

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

), and dividing by

I \cdot t

(

\text{A}\cdot\text{s}

) gives the correct base units of

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Step-by-Step Solution

1

Determine the SI base unit decomposition for pressure (

P

) and volume (

V

).

Pressure

P = \frac{\text{Force}}{\text{Area}} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

. Volume

V = \text{m}^3

. Therefore,

P \cdot V = (\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}) \cdot \text{m}^3 = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

.

Decomposing composite physical quantities into base units of mass, length, and time.

2

Determine the SI base unit decomposition for the denominator

I \cdot t

.

I \cdot t = \text{Electric current} \times \text{Time} = \text{A}\cdot\text{s}

.

Electric current (amperes, A) and time (seconds, s) are fundamental SI quantities.

3

Evaluate the full expression for

Z = \frac{P \cdot V}{I \cdot t}

in SI base units.

Base units of

Z = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Apply laws of indices to simplify base unit ratios.

4

Classify physical quantity

Z

.

Because

Z

(electric potential) is defined in terms of fundamental quantities (kg, m, s, A) rather than existing independently, it is a derived quantity.

The seven fundamental SI quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

Key Concept

Classification of physical quantities into fundamental and derived types and resolving complex derived quantities into base SI units.

Question 7693Question →

A trigonometric function is defined as $f(x) = a \sin(b x) + c$ , where $a > 0$ and $b > 0$ . The graph of $y = f(x)$ has a maximum value of $8$ , a minimum value of $-2$ , and a period of $120^\circ$ . What is the value of $a + b + c$ ?

Show answer & explanation

Answer: 11

Answer

The value of

a + b + c

is 11.

Using the maximum (

a + c = 8

) and minimum (

-a + c = -2

), solving simultaneously gives

a = 5

and

c = 3

. Using the period formula

T = \frac{360^\circ}{b} = 120^\circ

, we obtain

b = 3

. Adding these values together yields

5 + 3 + 3 = 11

.

Step-by-Step Solution

1

Set up a system of linear equations for the amplitude and vertical shift from the maximum and minimum bounds

a = 5

and

c = 3

Since

\sin(bx)

ranges from

-1

to

1

, the maximum is

a(1) + c = 8

and minimum is

a(-1) + c = -2

. Solving these simultaneous equations gives

c = 3

and

a = 5

.

2

Calculate the frequency coefficient

b

using the period formula

b = 3

For a sine curve specified in degrees, the period is

T = \frac{360^\circ}{b}

. Substituting

T = 120^\circ

yields

b = 3

.

3

Compute the requested sum

a + b + c

a + b + c = 11

Summing the calculated parameters

5 + 3 + 3 = 11

.

Key Concept

Determining parameters of a trigonometric graph from amplitude, vertical shift, and period.

Question 7694Question →

A parallel plate air capacitor of capacitance $8.0\text{ }\mu\text{F}$ is fully charged using a $40.0\text{ V}$ d.c. power supply and then disconnected from the source. A dielectric slab with a relative permittivity of $4.0$ is subsequently inserted to completely fill the region between the plates. Calculate the magnitude of the decrease in electrostatic energy stored in the capacitor, in microjoules ( $\mu\text{J}$ ).

Show answer & explanation

Answer: 4800

Answer

The magnitude of the decrease in stored electrostatic energy is 4800 μJ.

Disconnecting the battery ensures that the charge

Q = C_0 V = 320\text{ }\mu\text{C}

on the plates stays fixed. Inserting the dielectric increases capacitance fourfold to

32.0\text{ }\mu\text{F}

. The energy decreases from

U_i = 6400\text{ }\mu\text{J}

to

U_f = \frac{Q^2}{2C_f} = 1600\text{ }\mu\text{J}

, giving a total decrease of

4800\text{ }\mu\text{J}

.

Step-by-Step Solution

1

Calculate the initial energy stored in the air capacitor before disconnection.

Initial energy

U_i = \frac{1}{2} C_0 V^2 = \frac{1}{2} \times 8.0 \times 10^{-6} \text{ F} \times (40.0 \text{ V})^2 = 6.4 \times 10^{-3} \text{ J} = 6400 \text{ } \mu\text{J}

.

The initial state has known capacitance and potential difference.

2

Calculate the new capacitance with the dielectric present.

Final capacitance

C_f = K C_0 = 4.0 \times 8.0 \text{ } \mu\text{F} = 32.0 \text{ } \mu\text{F}

.

Inserting a dielectric of constant

K

scales the capacitance by

K

.

3

Calculate the final stored energy using charge conservation.

Final energy

U_f = \frac{U_i}{K} = \frac{6400 \text{ } \mu\text{J}}{4.0} = 1600 \text{ } \mu\text{J}

.

Disconnection forces the charge

Q

to remain fixed, so energy scales inversely with capacitance (

U = \frac{Q^2}{2C}

).

4

Find the difference between initial and final energy.

\Delta U = 6400 \text{ } \mu\text{J} - 1600 \text{ } \mu\text{J} = 4800 \text{ } \mu\text{J}.

The loss in electrostatic energy represents the work done by the field pulling the dielectric slab into the plates.

Key Concept

Effect of dielectric insertion on stored electrostatic energy under isolated (constant charge) conditions

Estimated Time:2m 0s

Question 7695Question →

In atomic physics, the energy $E$ of a photon is related to its frequency $f$ by the formula $E = h f$ , where $h$ is Planck's constant. What are the fundamental dimensions of Planck's constant $h$ ?

M L^2 T^{-1}

M L^2 T^{-2}

M L T^{-1}

M L^2 T^{-3}

Show answer & explanation

Answer: $M L^2 T^{-1}$

Answer

The fundamental dimensions of Planck's constant are

M L^2 T^{-1}

.

Planck's constant

h

is given by

h = E / f

. Since energy

E

has fundamental dimensions of

M L^2 T^{-2}

and frequency

f

has dimensions of

T^{-1}

, dividing energy by frequency yields

\frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1}

.

Step-by-Step Solution

1

Express Planck's constant in terms of energy and frequency

h = \frac{E}{f}

Rearranging the equation

E = h f

isolates

h

.

2

Determine the fundamental dimensions of energy (

E

) and frequency (

f

)

[E] = M L^2 T^{-2}

and

[f] = T^{-1}

Energy has dimensions of work (

F \times d = M L T^{-2} \cdot L = M L^2 T^{-2}

) and frequency is inverse time (

T^{-1}

).

3

Divide the dimensions of energy by the dimensions of frequency

[h] = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-2 - (-1)} = M L^2 T^{-1}

Applying the rules of indices simplifies the exponent of time.

Key Concept

Dimensional Analysis of Physical Constants

Estimated Time:1m 0s

Question 7696Question →

An electric charge of $+3.2 \times 10^{-19}\text{ C}$ is situated in a uniform electric field of strength $5.0 \times 10^4\text{ N C}^{-1}$ . What is the magnitude of the electrostatic force exerted on the charge?

6.4 \times 10^{-24}\text{ N}

1.6 \times 10^{-14}\text{ N}

1.6 \times 10^{-23}\text{ N}

5.0 \times 10^4\text{ N}

Show answer & explanation

Answer: $1.6 \times 10^{-14}\text{ N}$

Answer

The magnitude of the electrostatic force exerted on the charge is

1.6 \times 10^{-14}\text{ N}

.

The relationship between electric field intensity

E

, charge

q

, and electrostatic force

F

is given by

F = qE

. Multiplying

+3.2 \times 10^{-19}\text{ C}

by

5.0 \times 10^4\text{ N C}^{-1}

gives

1.6 \times 10^{-14}\text{ N}

, which correctly expresses the force magnitude in standard notation.

Step-by-Step Solution

1

Identify the given values and formula.

Charge

q = 3.2 \times 10^{-19}\text{ C}

, Electric field strength

E = 5.0 \times 10^4\text{ N C}^{-1}

. Formula:

F = qE

.

The force experienced by a charge in an electric field is the product of the charge magnitude and the field strength.

2

Substitute the values into the formula and calculate.

F = (3.2 \times 10^{-19}) \times (5.0 \times 10^4) = 16.0 \times 10^{-15}\text{ N} = 1.6 \times 10^{-14}\text{ N}

.

Multiplying the numerical coefficients

(3.2 \times 5.0 = 16.0)

and combining the powers of ten

(10^{-19} \times 10^4 = 10^{-15})

gives

1.6 \times 10^{-14}\text{ N}

in standard scientific notation.

Key Concept

Electric Field Intensity and Electrostatic Force (

F = qE

)

Question 7697Question →

A fixed line segment $AB$ has a length of $8\text{ cm}$ . A point $P$ moves in the plane such that the area of $\triangle PAB$ is always $20\text{ cm}^2$ . Which of the following best describes the locus of $P$ ?

A pair of parallel lines at a perpendicular distance of

5\text{ cm}

on opposite sides of

AB

A circle of radius

5\text{ cm}

centered at the midpoint of

AB

A pair of parallel lines at a perpendicular distance of

2.5\text{ cm}

on opposite sides of

AB

The perpendicular bisector of the line segment

AB

Show answer & explanation

Answer: A pair of parallel lines at a perpendicular distance of $5\text{ cm}$ on opposite sides of $AB$

Answer

A pair of parallel lines at a perpendicular distance of

5\text{ cm}

on opposite sides of

AB

The area of

\triangle PAB

is given by

\frac{1}{2} \times \text{base} \times \text{height}

. With a base

AB = 8\text{ cm}

, an area of

20\text{ cm}^2

requires a constant height

h = 5\text{ cm}

. The geometric locus of all points at a constant distance from a given straight line consists of two parallel lines situated at that distance on either side of the line.

Step-by-Step Solution

1

Express the area formula of the triangle in terms of base and height.

\text{Area} = \frac{1}{2} \times \text{base} \times h = \frac{1}{2} \times 8 \times h = 4h

The base of

\triangle PAB

is fixed as the length of segment

AB

, which is

8\text{ cm}

.

2

Calculate the constant perpendicular height

h

.

4h = 20 \implies h = 5\text{ cm}

Setting the calculated area equal to the given constant area of

20\text{ cm}^2

gives the required height.

3

Determine the geometric locus corresponding to a constant perpendicular height.

The locus of points at a fixed distance

h = 5\text{ cm}

from line

AB

is a pair of parallel lines running on either side of

AB

at a distance of

5\text{ cm}

.

Any point

P

lying on either of these two parallel lines maintains a perpendicular distance of

5\text{ cm}

from

AB

, ensuring

\text{Area}(\triangle PAB) = 20\text{ cm}^2

.

Key Concept

Locus at a constant distance from a straight line

Question 7698Question →

Find the equation of the normal to the curve $y = 2\sin x - \cos x$ at the point where $x = 0$ .

x + 2y + 2 = 0

2x - y - 1 = 0

x + 2y - 2 = 0

x - 2y - 2 = 0

Show answer & explanation

Answer: $x + 2y + 2 = 0$

Answer

x + 2y + 2 = 0

At

x = 0

, the

y

-coordinate is

2\sin(0) - \cos(0) = -1

. Evaluating the derivative

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

at

x = 0

yields a tangent slope of

2

. Since the normal is perpendicular to the tangent, its gradient is

-\frac{1}{2}

. Substituting the point

(0, -1)

and slope

-\frac{1}{2}

into the line formula yields

x + 2y + 2 = 0

.

Step-by-Step Solution

1

Find the

y

-coordinate of the point of contact

At

x = 0

,

y = 2\sin(0) - \cos(0) = 0 - 1 = -1

. The point is

(0, -1)

.

The line equation requires a point

(x_1, y_1)

on the curve.

2

Differentiate the curve to find

\frac{dy}{dx}

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

.

The derivative gives the gradient function of the curve.

3

Calculate the gradient of the tangent and normal at

x = 0

Tangent gradient

m_t = 2\cos(0) + \sin(0) = 2(1) + 0 = 2

. Normal gradient

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

.

The normal line is perpendicular to the tangent line.

4

Form the equation of the normal line

y - (-1) = -\frac{1}{2}(x - 0) \implies y + 1 = -\frac{1}{2}x \implies 2y + 2 = -x \implies x + 2y + 2 = 0

.

Apply the point-slope form

y - y_1 = m(x - x_1)

.

Key Concept

Equation of Normal to a Curve

Estimated Time:1m 30s

Question 7699Question →

If $132_x - 54_x = 45_x$ , where $x$ is a positive integer base, what is the value of $x$ ?

6

7

8

9

Show answer & explanation

Answer: 7

Answer

The value of the base

x

is 7.

Expanding all numbers in terms of powers of

x

yields

132_x = x^2 + 3x + 2

,

54_x = 5x + 4

, and

45_x = 4x + 5

. Substituting these into

132_x - 54_x = 45_x

gives

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

. Simplifying this equation results in

x^2 - 6x - 7 = 0

. Factoring

(x - 7)(x + 1) = 0

gives solutions

x = 7

and

x = -1

. Since a base must be a positive integer greater than 5, the base

x

is 7.

Step-by-Step Solution

1

Convert each term from base

x

to base 10 using place-value expansion.

132_x = 1 \cdot x^2 + 3 \cdot x^1 + 2 \cdot x^0 = x^2 + 3x + 2

,

54_x = 5x + 4

, and

45_x = 4x + 5

.

Converting all terms to a common base (base 10) allows standard algebraic solving.

2

Substitute the expanded expressions into the given equation.

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

.

Set up the algebraic equation in terms of

x

.

3

Simplify the equation into standard quadratic form.

x^2 - 2x - 2 = 4x + 5 \implies x^2 - 6x - 7 = 0

.

Combine like terms and move all terms to one side.

4

Solve the quadratic equation for

x

.

(x - 7)(x + 1) = 0 \implies x = 7

or

x = -1

.

Factor the quadratic expression.

5

Select the valid base.

x = 7

.

A number base must be a positive integer greater than the largest digit present in the equation (which is 5).

Key Concept

Solving equations involving unknown number bases by expanding in powers of the base.

Question 7700Question →

The derived SI unit of electrical resistance is the ohm ( $\Omega$ ). Which of the following correctly expresses the ohm strictly in terms of fundamental SI base units?

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\cdot\text{A}^{-2}

\text{V}\cdot\text{A}^{-1}

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

Show answer & explanation

Answer: $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}$

Answer

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}

Resistance is defined by Ohm's Law as voltage divided by current (

R = V / I

). Voltage is work done per unit charge (

V = W / Q

), and charge is current multiplied by time (

Q = I \cdot t

). Substituting the unit of work (

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

) gives

R = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}^2\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}

.

Step-by-Step Solution

1

Relate resistance to potential difference and current using Ohm's Law.

R = \frac{V}{I}

Resistance is defined as the ratio of potential difference to electric current.

2

Express electric potential difference (

V

) in terms of work (

W

) and charge (

Q

), where

Q = I \cdot t

.

V = \frac{W}{I \cdot t} \implies R = \frac{W}{I^2 \cdot t}

Electric potential difference is work done per unit charge, and electric charge is current multiplied by time.

3

Break down work (

W = \text{Force} \times \text{distance}

) into base SI units.

Unit of Work =

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

Force is mass times acceleration (

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}

), so work is

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

.

4

Substitute all base units into the resistance formula.

Unit of

R = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}^2 \cdot \text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}

Combining powers of base units yields the complete fundamental SI base unit expression for the ohm.

Key Concept

Derivation of SI base units from defining formulas of physical quantities

Answer

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