All practice questions

What is the result of the subtraction $524_8 - 267_8$ expressed in base 8?

235_8

257_8

345_8

343_8

Show answer & explanation

Answer: $235_8$

Answer

The correct answer is

235_8

.

Subtracting

267_8

from

524_8

using direct base-8 subtraction requires borrowing 8 when a digit is smaller than the digit subtracted from it. In the units column, borrowing 1 (which equals 8) gives

(4 + 8) - 7 = 5

. In the eights column, the borrowed-from digit 2 becomes 1; borrowing 1 (which equals 8) gives

(1 + 8) - 6 = 3

. In the sixty-fours column, the borrowed-from digit 5 becomes 4;

4 - 2 = 2

. Thus, the difference is

235_8

.

Step-by-Step Solution

1

Convert both octal numbers to base 10.

524_8 = (5 \times 8^2) + (2 \times 8^1) + (4 \times 8^0) = 320 + 16 + 4 = 340_{10}

, and

267_8 = (2 \times 8^2) + (6 \times 8^1) + (7 \times 8^0) = 128 + 48 + 7 = 183_{10}

.

Converting to decimal simplifies standard arithmetic operations.

2

Subtract the decimal values.

340_{10} - 183_{10} = 157_{10}

.

Obtain the difference in base 10.

3

Convert the decimal difference

157_{10}

back to base 8.

157 \div 8 = 19

remainder

5

;

19 \div 8 = 2

remainder

3

;

2 \div 8 = 0

remainder

2

. Reading remainders from bottom to top gives

235_8

.

Successive division by 8 yields the octal representation.

Key Concept

Non-decimal base arithmetic and borrowing rules

Question 7522Question →

The $5^{\text{th}}$ term of an arithmetic progression (AP) is $17$ and its common difference is $3$ . What is the first term of the progression?

5

2

29

-1

Show answer & explanation

Answer: $5$

Answer

The first term of the progression is

5

.

The

n^{\text{th}}

term formula of an arithmetic progression is

T_n = a + (n - 1)d

. Substituting

T_5 = 17

,

n = 5

, and

d = 3

gives

17 = a + 4(3) = a + 12

. Isolating

a

yields

a = 17 - 12 = 5

.

Step-by-Step Solution

1

Identify the given values and formula for the nth term of an AP.

The formula is

T_n = a + (n - 1)d

, with

T_5 = 17

,

n = 5

, and

d = 3

.

This formula connects the nth term, the first term, the number of terms, and the common difference.

2

Substitute the known values into the equation.

17 = a + (5 - 1) \times 3 \implies 17 = a + 12

.

Subtracting 1 from the term index 5 gives 4, and multiplying by 3 gives 12.

3

Solve for the first term

a

.

a = 17 - 12 = 5

.

Subtracting 12 from both sides isolates

a

.

Key Concept

n-th term of an Arithmetic Progression

Estimated Time:1m 0s

Question 7523Question →

A straight line $L_1$ passes through the points $A(-2, 1)$ and $B(4, 5)$ . A second line $L_2$ is perpendicular to $L_1$ and passes through the midpoint of the line segment $AB$ . If $L_2$ intersects the $x$ -axis at point $R$ , and $R$ divides the line segment joining $P(1, -2)$ and $Q(k, 4)$ internally in the ratio $1 : 2$ , what is the value of $k$ ?

7

4

-5

5

Show answer & explanation

Answer: $7$

Answer

The value of

k

is

7

.

The midpoint of

AB

is

(1, 3)

and the gradient of

L_1

is

\frac{2}{3}

. The gradient of the perpendicular line

L_2

is

-\frac{3}{2}

, yielding the equation

3x + 2y - 9 = 0

. Setting

y = 0

gives the

x

-intercept

R(3, 0)

. Applying the internal section formula for ratio

1 : 2

on the

x

-coordinates yields

\frac{k + 2}{3} = 3

, which solves directly to

k = 7

.

Step-by-Step Solution

1

Find the midpoint

M

of line segment

AB

and the gradient of line

L_1

.

Midpoint

M = \left(\frac{-2 + 4}{2}, \frac{1 + 5}{2}\right) = (1, 3)

. Gradient

m_1 = \frac{5 - 1}{4 - (-2)} = \frac{4}{6} = \frac{2}{3}

.

Line

L_2

passes through the midpoint

M

and its orientation depends on the gradient of

L_1

.

2

Determine the gradient

m_2

of

L_2

and its equation.

Since

L_2 \perp L_1

,

m_2 = -\frac{1}{m_1} = -\frac{3}{2}

. Equation of

L_2

:

y - 3 = -\frac{3}{2}(x - 1) \implies 3x + 2y - 9 = 0

.

Perpendicular lines have gradients whose product is

-1

(

m_1 \cdot m_2 = -1

).

3

Find the coordinates of point

R

, the

x

-intercept of

L_2

.

Set

y = 0

in

3x + 2y - 9 = 0 \implies 3x - 9 = 0 \implies x = 3

. Thus,

R = (3, 0)

.

The

x

-intercept occurs where

y = 0

on the coordinate plane.

4

Apply the section formula to find

k

.

Point

R(3, 0)

divides

P(1, -2)

and

Q(k, 4)

in ratio

1 : 2

. The

x

-coordinate is given by

x_R = \frac{1(k) + 2(1)}{1 + 2} = \frac{k + 2}{3}

. Setting

\frac{k + 2}{3} = 3

gives

k + 2 = 9 \implies k = 7

.

The internal section formula states that a point dividing

(x_1, y_1)

and

(x_2, y_2)

in ratio

m : n

has coordinates

\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)

.

Key Concept

Perpendicular lines, midpoints, intercepts, and section formula in coordinate geometry

Question 7524Question →

A fair spinner is divided into $8$ equal sectors numbered $1$ to $8$ . In an experiment, the spinner is spun $120$ times, and it lands on a multiple of $3$ a total of $36$ times. What is the positive difference between the experimental probability and the theoretical probability of landing on a multiple of $3$ ?

\frac{1}{20}

\frac{3}{10}

\frac{1}{4}

\frac{1}{10}

Show answer & explanation

Answer: $\frac{1}{20}$

Answer

The positive difference between the experimental probability and theoretical probability is

\frac{1}{20}

.

The correct option is

\frac{1}{20}

. The theoretical probability of hitting a multiple of

3

on an

8

-sector spinner is

\frac{2}{8} = \frac{1}{4}

. The experimental probability over

120

spins with

36

successes is

\frac{36}{120} = \frac{3}{10}

. Subtracting theoretical probability from experimental probability gives

\frac{3}{10} - \frac{1}{4} = \frac{6-5}{20} = \frac{1}{20}

.

Step-by-Step Solution

1

Calculate the theoretical probability of landing on a multiple of

3

.

The sample space is

\{1, 2, 3, 4, 5, 6, 7, 8\}

(total

8

outcomes). Multiples of

3

in this set are

\{3, 6\}

(

2

outcomes). Thus,

P(\text{Theoretical}) = \frac{2}{8} = \frac{1}{4}

.

Theoretical probability is the ratio of favorable outcomes to total possible outcomes in a fair trial.

2

Calculate the experimental probability (relative frequency) from the trial results.

P(\text{Experimental}) = \frac{36}{120} = \frac{3}{10}

.

Experimental probability is defined as the number of observed successful events divided by the total number of trials conducted.

3

Find the positive difference between the experimental and theoretical probabilities.

\text{Difference} = \frac{3}{10} - \frac{1}{4} = \frac{6}{20} - \frac{5}{20} = \frac{1}{20}

.

Subtracting the smaller theoretical probability from the larger experimental probability yields the positive difference.

Key Concept

Experimental versus Theoretical Probability

Estimated Time:1m 30s

Question 7525Question →

Two straight paths diverge from a junction $J$ at an angle of $120^\circ$ . A person walks $5\text{ km}$ along the first path to point $A$ , and another person walks $16\text{ km}$ along the second path to point $B$ . What is the direct distance between $A$ and $B$ in kilometres?

Show answer & explanation

Answer: 19

Answer

The direct distance between points

A

and

B

is

19\text{ km}

.

Using the Cosine Rule

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

with sides

5\text{ km}

and

16\text{ km}

and included angle

120^\circ

yields

c^2 = 5^2 + 16^2 - 2(5)(16)(-0.5) = 25 + 256 + 80 = 361

. Taking the square root gives

19\text{ km}

.

Step-by-Step Solution

1

Identify given values and setup the Cosine Rule formula

Side

a = 5

, side

b = 16

, and included angle

\theta = 120^\circ

The scenario provides two sides and the included angle (SAS configuration), which requires the Cosine Rule to find the third side.

2

Substitute the values into

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

c^2 = 5^2 + 16^2 - 2(5)(16) \cos(120^\circ)

Populating the formula allows evaluation of the unknown distance squared.

3

Evaluate the trigonometric term and simplify

c^2 = 25 + 256 - 160(-0.5) = 281 + 80 = 361

The cosine of an obtuse angle in the second quadrant (

120^\circ

) is negative:

\cos(120^\circ) = -0.5

.

4

Take the square root to find the distance

c

c = \sqrt{361} = 19\text{ km}

Taking the positive square root gives the physical distance between the two points.

Key Concept

Applying the Cosine Rule to find the length of an unknown side in a non-right triangle given two sides and the included angle (SAS).

Question 7526Question →

Find the indefinite integral $\int \left( 3x(x - 2) + 8\cos(4x) \right) dx$ .

x^3 - 3x^2 + 2\sin(4x) + C

x^3 - 3x^2 + 2\sin(4x)

x^3 - 3x^2 - 2\sin(4x) + C

x^3 - 3x^2 + 32\sin(4x) + C

Show answer & explanation

Answer: $x^3 - 3x^2 + 2\sin(4x) + C$

Answer

x^3 - 3x^2 + 2\sin(4x) + C

Expanding

3x(x - 2)

yields

3x^2 - 6x

. Integrating

3x^2

gives

x^3

, integrating

-6x

yields

-3x^2

, and integrating

8\cos(4x)

yields

2\sin(4x)

. Adding the arbitrary constant

C

produces

x^3 - 3x^2 + 2\sin(4x) + C

.

Step-by-Step Solution

1

Expand the algebraic term inside the integrand.

3x(x - 2) = 3x^2 - 6x

, expressing the integral as

\int (3x^2 - 6x + 8\cos(4x)) dx

.

Expanding the product converts the term into standard polynomial form suitable for integration.

2

Integrate each component term using basic integration rules.

\int 3x^2 dx = x^3

,

\int -6x dx = -3x^2

, and

\int 8\cos(4x) dx = 8 \cdot \frac{\sin(4x)}{4} = 2\sin(4x)

.

Apply the power rule

\int x^n dx = \frac{x^{n+1}}{n+1}

and trigonometric rule

\int \cos(kx) dx = \frac{\sin(kx)}{k}

.

3

Combine terms and attach the constant of integration.

x^3 - 3x^2 + 2\sin(4x) + C

An indefinite integral requires an arbitrary constant

C

to represent the full set of antiderivatives.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

Question 7527Question →

Evaluate the definite integral $\int_{0}^{3} (x^2 + 2) \, dx$ .

Show answer & explanation

Answer: 15

Answer

The value of the definite integral is 15.

Integrating

x^2 + 2

with respect to

x

yields

\left[\frac{x^3}{3} + 2x\right]_{0}^{3}

. Substituting the upper bound

x = 3

gives

\frac{27}{3} + 6 = 15

. Substituting the lower bound

x = 0

gives

0

. The net value is

15 - 0 = 15

.

Step-by-Step Solution

1

Find the antiderivative of

x^2 + 2

\frac{x^3}{3} + 2x

Apply the power rule of integration to each term.

2

Substitute the upper limit

x = 3

15

\frac{3^3}{3} + 2(3) = 9 + 6 = 15

3

Substitute the lower limit

x = 0

0

\frac{0^3}{3} + 2(0) = 0

4

Compute the difference between upper and lower limit values

15

15 - 0 = 15

Key Concept

Definite Integration of Polynomial Functions

Estimated Time:45s

Question 7528Question →

A bag contains $5$ red balls, $3$ blue balls, and $n$ green balls. The theoretical probability of drawing a blue ball at random from the bag is $\frac{3}{20}$ . In a probability experiment, a ball is drawn at random from the bag and replaced $500$ times. If a green ball is observed $340$ times, calculate the absolute difference between the experimental probability and the theoretical probability of drawing a green ball.

Show answer & explanation

Answer: 0.08

Answer

The absolute difference between the experimental probability and the theoretical probability of drawing a green ball is

0.08

(or

\frac{2}{25}

).

The theoretical probability of drawing a blue ball establishes that the bag contains

20

total balls, meaning there are

12

green balls. The theoretical probability of picking a green ball is therefore

\frac{12}{20} = 0.60

. The experimental probability is

\frac{340}{500} = 0.68

. Taking the positive difference gives

|0.68 - 0.60| = 0.08

.

Step-by-Step Solution

1

Find the total number of balls and the value of

n

n = 12

, Total balls =

20

The theoretical probability of picking a blue ball is

\frac{3}{8+n} = \frac{3}{20}

, giving

8 + n = 20

.

2

Find the theoretical probability of drawing a green ball

P(\text{Green})_{\text{theo}} = 0.60

There are

12

green balls out of

20

total balls, so

\frac{12}{20} = 0.60

.

3

Find the experimental probability of drawing a green ball

P(\text{Green})_{\text{exp}} = 0.68

In

500

trials, a green ball was observed

340

times, so

\frac{340}{500} = 0.68

.

4

Calculate the absolute difference

|0.68 - 0.60| = 0.08

Subtract the theoretical probability from the experimental probability and take the absolute value.

Key Concept

Comparison of theoretical probability based on outcome sample space and experimental probability based on observed trial relative frequencies.

Question 7529Question →

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

Show answer & explanation

Answer: 3

Answer

The value of

\frac{dy}{dx}

at

x = 0

is 3.

Applying the product rule to

e^{-x}\tan x

gives

-e^{-x}\tan x + e^{-x}\sec^2 x

, and applying the chain rule to

\ln(2x+1)

gives

\frac{2}{2x+1}

. Evaluating

\frac{dy}{dx} = -e^{-x}\tan x + e^{-x}\sec^2 x + \frac{2}{2x+1}

at

x = 0

yields

-e^0(0) + e^0(1) + \frac{2}{1} = 3

.

Step-by-Step Solution

1

Differentiate

u(x) = e^{-x} \tan x

using the product rule.

\frac{du}{dx} = -e^{-x} \tan x + e^{-x} \sec^2 x

The derivative of

e^{-x}

is

-e^{-x}

and the derivative of

\tan x

is \sec^2 x.

2

Differentiate

v(x) = \ln(2x + 1)

using the chain rule.

\frac{dv}{dx} = \frac{2}{2x + 1}

The derivative of

\ln(g(x))

is

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

, where

g(x) = 2x + 1

and

g'(x) = 2

.

3

Sum the derivatives to find the complete expression for

\frac{dy}{dx}

.

\frac{dy}{dx} = -e^{-x} \tan x + e^{-x} \sec^2 x + \frac{2}{2x + 1}

The derivative of a sum is equal to the sum of the derivatives.

4

Evaluate the derivative at

x = 0

.

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

Since

\tan(0) = 0

,

e^0 = 1

, and

\sec(0) = 1

, substituting

x = 0

simplifies the derivative to

0 + 1 + 2 = 3

.

Key Concept

Differentiation of Transcendental Functions (Product and Chain Rules)

Question 7530Question →

A constant-volume gas thermometer registers a pressure of $50\,\text{kPa}$ at the ice point ( $0^\circ\text{C}$ ) and $70\,\text{kPa}$ at the steam point ( $100^\circ\text{C}$ ). What is the temperature when the gas pressure measured by the thermometer is $62\,\text{kPa}$ ?

12^\circ\text{C}

60^\circ\text{C}

89^\circ\text{C}

310^\circ\text{C}

Show answer & explanation

Answer: $60^\circ\text{C}$

Answer

60^\circ\text{C}

The temperature

\theta

on the Celsius scale is given by the ratio of the change in thermometric property from the ice point to the total fundamental interval, scaled by 100. Substituting

P_0 = 50\,\text{kPa}

,

P_{100} = 70\,\text{kPa}

, and

P_\theta = 62\,\text{kPa}

gives

\theta = \frac{62 - 50}{70 - 50} \times 100 = \frac{12}{20} \times 100 = 60^\circ\text{C}

.

Step-by-Step Solution

1

Identify the given thermometric property values at the fixed points and target state

P_0 = 50\,\text{kPa}

,

P_{100} = 70\,\text{kPa}

, and

P_\theta = 62\,\text{kPa}

These represent the lower fixed point, upper fixed point, and unknown temperature reading respectively.

2

Apply the general thermometric scale conversion formula

\theta = \frac{P_\theta - P_0}{P_{100} - P_0} \times 100^\circ\text{C}

Temperature on the Celsius scale varies linearly with the thermometric property relative to fixed points.

3

Substitute the values into the formula and solve for

\theta

\theta = \frac{62 - 50}{70 - 50} \times 100 = \frac{12}{20} \times 100 = 60^\circ\text{C}

Carrying out the arithmetic yields the exact temperature of

60^\circ\text{C}

.

Key Concept

Linear interpolation on temperature scales using thermometric properties

Estimated Time:1m 0s

Question 7531Question →

A surveyor at station $A$ observes two landmarks, $B$ and $C$ . Landmark $B$ is located at a distance of $14\text{ km}$ from $A$ on a bearing of $025^\circ$ . Landmark $C$ is located at a distance of $6\sqrt{2}\text{ km}$ from $A$ on a bearing of $070^\circ$ . What is the direct distance between landmark $B$ and landmark $C$ ?

10\text{ km}

7\sqrt{2}\text{ km}

8\text{ km}

2\sqrt{46}\text{ km}

Show answer & explanation

Answer: $10\text{ km}$

Answer

The direct distance between landmark

B

and landmark

C

is

10\text{ km}

.

The included angle

\angle BAC

between the two bearings is

070^\circ - 025^\circ = 45^\circ

. Applying the Cosine Rule

a^2 = b^2 + c^2 - 2bc \cos A

gives

a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ = 72 + 196 - 168 = 100

. Taking the square root gives

10\text{ km}

.

Step-by-Step Solution

1

Determine the interior angle

\angle BAC

from the given bearings.

\angle BAC = 070^\circ - 025^\circ = 45^\circ

The difference between two bearings measured clockwise from North from the same point gives the included angle between the lines of sight.

2

Identify the side lengths adjacent to angle

A

in

\Delta ABC

.

c = AB = 14\text{ km}

,

b = AC = 6\sqrt{2}\text{ km}

, and included angle

A = 45^\circ

We have a Side-Angle-Side (SAS) triangle configuration, requiring the Cosine Rule to find the opposite side

a = BC

.

3

Apply the Cosine Rule

a^2 = b^2 + c^2 - 2bc \cos A

.

a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ

Substituting known values into the Cosine Rule formula.

4

Simplify the terms and solve for

a

.

a^2 = 72 + 196 - 168\sqrt{2}\left(\frac{1}{\sqrt{2}}\right) = 268 - 168 = 100 \implies a = \sqrt{100} = 10\text{ km}

Squaring

6\sqrt{2}

gives

36 \times 2 = 72

,

14^2 = 196

, and simplifying the cosine term yields

168

.

Key Concept

Cosine Rule for SAS non-right triangles in bearing contexts

Estimated Time:2m 0s

Question 7532Question →

Find the value of $x$ , in degrees, for $0^\circ \le x \le 90^\circ$ that satisfies the trigonometric equation $\sin 2x = \cos(x + 30^\circ)$ .

Show answer & explanation

Answer: 20

Answer

The value of

x

in the interval

0^\circ \le x \le 90^\circ

satisfying the equation is

20^\circ

.

Using the co-function identity

\cos \alpha = \sin(90^\circ - \alpha)

, we convert the right-hand side to

\sin(90^\circ - (x + 30^\circ)) = \sin(60^\circ - x)

. Equating the arguments gives

2x = 60^\circ - x

, which simplifies to

3x = 60^\circ

, yielding

x = 20^\circ

.

Step-by-Step Solution

1

Apply the co-function trigonometric identity

\cos(x + 30^\circ) = \sin(90^\circ - (x + 30^\circ)) = \sin(60^\circ - x)

Converting cosine to sine allows direct comparison of sine functions on both sides of the equation.

2

Set up the equation equating the angle expressions

2x = 60^\circ - x

Since

\sin(2x) = \sin(60^\circ - x)

and

x \in [0^\circ, 90^\circ]

, equating the principal angle arguments gives the primary solution.

3

Solve the linear equation for

x

3x = 60^\circ \implies x = 20^\circ

Adding

x

to both sides gives

3x = 60^\circ

, and dividing by 3 yields

x = 20^\circ

.

Key Concept

Co-function identities and simple trigonometric equations

Question 7533Question →

The mean of four positive integers is $9$ . If the mode of the set of numbers is $12$ and their median is $10$ , what is the value of the smallest integer?

4

2

6

5

Show answer & explanation

Answer: $4$

Answer

The smallest integer is 4.

Let the four positive integers in ascending order be

a \le b \le c \le d

. The sum of the four numbers is

4 \times 9 = 36

. For

12

to be the mode, it must appear at least twice, so

c = 12

and

d = 12

. The median of four numbers is

\frac{b + c}{2} = 10

, which gives

\frac{b + 12}{2} = 10

, so

b = 8

. Substituting these into the sum equation

a + 8 + 12 + 12 = 36

gives

a = 4

.

Step-by-Step Solution

1

Calculate the sum of all four integers from the given mean.

Sum

= 4 \times 9 = 36

.

The mean of

n

numbers is the sum divided by

n

.

2

Determine the two largest numbers using the mode.

Let the ordered numbers be

a \le b \le c \le d

. Since the mode is

12

and the set contains

4

numbers with a single mode,

c = 12

and

d = 12

.

For

12

to be the mode in a 4-element set without tie, it must appear at least twice.

3

Use the median to find the second integer

b

.

Median

= \frac{b + c}{2} = \frac{b + 12}{2} = 10 \implies b + 12 = 20 \implies b = 8

.

The median of an even number of ordered elements is the arithmetic mean of the two middle elements.

4

Solve for the smallest integer

a

.

a + b + c + d = 36 \implies a + 8 + 12 + 12 = 36 \implies a + 32 = 36 \implies a = 4

.

Subtracting the sum of the known three numbers from the total sum gives the smallest integer.

Key Concept

Using mean, median, and mode definitions simultaneously to deduce unknown values in an ungrouped dataset

Question 7534Question →

Two independent weather forecasting stations, $A$ and $B$ , operate in a region. The probability that station $A$ makes an accurate forecast on any given day is $0.80$ , and the probability that station $B$ makes an accurate forecast is $0.75$ . What is the probability that at least one of the two stations makes an accurate forecast on a given day?

Show answer & explanation

Answer: 0.95

Answer

The probability that at least one of the two stations makes an accurate forecast is

0.95

.

The probability of at least one event occurring is given by

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Because events

A

and

B

are independent,

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

. Substituting into the addition rule yields

0.80 + 0.75 - 0.60 = 0.95

. Alternatively, using the complement rule:

1 - P(A')P(B') = 1 - (1 - 0.80)(1 - 0.75) = 1 - (0.20 \times 0.25) = 1 - 0.05 = 0.95

.

Step-by-Step Solution

1

Calculate the probability of both events occurring simultaneously using the multiplication law for independent events.

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

Since the two forecasting stations operate independently, the joint probability is the product of their individual probabilities.

2

Apply the general addition law of probability to calculate the probability of at least one station making an accurate forecast.

P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.80 + 0.75 - 0.60 = 0.95

The probability of compound event 'at least one' corresponds to the union of the two events.

Key Concept

Addition and Multiplication Laws of Probability for Independent Events

Question 7535Question →

A transverse mechanical wave propagates along a stretched string. If the maximum speed of an oscillating particle of the string is equal to the speed of wave propagation, what is the ratio of the wave's amplitude to its wavelength?

2\pi

\frac{1}{2\pi}

\frac{1}{\pi}

1

Show answer & explanation

Answer: $\frac{1}{2\pi}$

Answer

The ratio of the wave's amplitude to its wavelength is

\frac{1}{2\pi}

.

For a transverse wave, the maximum speed of particle oscillation is

v_{p,\text{max}} = A\omega

, while the propagation speed of the wave pattern through the medium is

v_w = \frac{\omega}{k}

. Setting these two speeds equal gives

A\omega = \frac{\omega}{k}

, which simplifies to

A k = 1

. Substituting

k = \frac{2\pi}{\lambda}

yields

A \left(\frac{2\pi}{\lambda}\right) = 1

, giving the ratio

\frac{A}{\lambda} = \frac{1}{2\pi}

.

Step-by-Step Solution

1

Express the maximum speed of a particle in simple harmonic oscillation along a transverse wave.

The maximum particle speed is given by

v_{p,\text{max}} = A\omega

, where

A

is the amplitude and

\omega

is the angular frequency.

Particles in a transverse mechanical wave oscillate perpendicularly to wave propagation in simple harmonic motion.

2

Express the wave propagation speed in terms of angular frequency and wave number.

The wave propagation speed is

v_w = \frac{\omega}{k}

, where

k = \frac{2\pi}{\lambda}

is the angular wave number.

Wave propagation speed reflects the rate at which the wave energy and profile move through the medium.

3

Equate the maximum particle speed to the wave propagation speed and solve for

\frac{A}{\lambda}

.

A\omega = \frac{\omega}{k} \implies A k = 1 \implies A \left(\frac{2\pi}{\lambda}\right) = 1 \implies \frac{A}{\lambda} = \frac{1}{2\pi}

.

Setting

v_{p,\text{max}} = v_w

allows the angular frequency

\omega

to cancel out, directly linking amplitude and wavelength.

Key Concept

Transverse Wave Particle Oscillation vs. Wave Propagation Speed

Estimated Time:2m 0s

Question 7536Question →

What is the value of the definite integral $\int_{0}^{4} \sqrt{x} \, dx$ ?

\frac{16}{3}

8

12

\frac{8}{3}

Show answer & explanation

Answer: $\frac{16}{3}$

Answer

\frac{16}{3}

Applying the power rule to

x^{1/2}

gives

\frac{2}{3}x^{3/2}

. Substituting the upper limit

x = 4

yields

\frac{2}{3}(4)^{3/2} = \frac{2}{3}(8) = \frac{16}{3}

, and substituting the lower limit

x = 0

yields 0. Subtracting the lower limit result from the upper limit result gives

\frac{16}{3}

.

Step-by-Step Solution

1

Express the integrand with a fractional exponent

\sqrt{x} = x^{1/2}

Rewriting the square root as a fractional exponent allows the application of the power rule of integration.

2

Find the antiderivative using the power rule of integration

\int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}

The power rule states that

\int x^n \, dx = \frac{x^{n+1}}{n+1}

for

n \neq -1

.

3

Evaluate the antiderivative at the limits 4 and 0

\left[ \frac{2}{3}x^{3/2} \right]_{0}^{4} = \frac{2}{3}(4)^{3/2} - \frac{2}{3}(0)^{3/2} = \frac{2}{3}(8) - 0 = \frac{16}{3}

By the Fundamental Theorem of Calculus, evaluate

F(b) - F(a)

.

Key Concept

Definite Integration using the Power Rule

Question 7537Question →

For the domain $0^\circ \le x \le 180^\circ$ , what is the complete solution set to the trigonometric equation $4\sin^2 x - 3 = 0$ ?

\{60^\circ\}

\{60^\circ, 120^\circ\}

\{30^\circ, 150^\circ\}

\{60^\circ, 240^\circ\}

Show answer & explanation

Answer: $\{60^\circ, 120^\circ\}$

Answer

\{60^\circ, 120^\circ\}

Solving the equation

4\sin^2 x - 3 = 0

gives

\sin^2 x = \frac{3}{4}

, which simplifies to

\sin x = \frac{\sqrt{3}}{2}

within the interval

0^\circ \le x \le 180^\circ

. The principal angle is

60^\circ

. Since sine is positive in both the first and second quadrants, the second valid angle in the domain is

180^\circ - 60^\circ = 120^\circ

, giving the solution set

\{60^\circ, 120^\circ\}

.

Step-by-Step Solution

1

Isolate the trigonometric term

\sin^2 x = \frac{3}{4}

Rearrange

4\sin^2 x - 3 = 0

by adding 3 to both sides and dividing by 4.

2

Take the square root of both sides

\sin x = \pm \frac{\sqrt{3}}{2}

Taking the square root yields both positive and negative ratios.

3

Apply the domain restriction

0^\circ \le x \le 180^\circ

\sin x = \frac{\sqrt{3}}{2}

The sine function is non-negative in the first and second quadrants (

0^\circ \le x \le 180^\circ

), so the negative root has no solutions in this interval.

4

Determine the angles for

x

x = 60^\circ \text{ and } x = 180^\circ - 60^\circ = 120^\circ

The reference angle is

60^\circ

because

\sin 60^\circ = \frac{\sqrt{3}}{2}

. In Quadrant II, the corresponding angle is

180^\circ - 60^\circ = 120^\circ

.

Key Concept

Solving Quadratic Trigonometric Equations

Question 7538Question →

What is the simplified numerical value of $\frac{\sqrt{12}}{\sqrt{3} + 1} + \frac{\sqrt{12}}{\sqrt{3} - 1}$ ?

Show answer & explanation

Answer: 6

Answer

The simplified numerical value of the given expression is 6.

Combining the fractions using their common conjugate denominator

(\sqrt{3}+1)(\sqrt{3}-1) = 2

leads to a numerator of

2\sqrt{3}[(\sqrt{3}-1)+(\sqrt{3}+1)] = 2\sqrt{3}(2\sqrt{3}) = 12

. Dividing 12 by 2 yields the final answer of 6.

Step-by-Step Solution

1

Simplify the surd in the numerator

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Simplifying surds into basic form makes subsequent calculations simpler.

2

Combine the fractions by finding a common denominator

\frac{2\sqrt{3}(\sqrt{3}-1) + 2\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}+1)(\sqrt{3}-1)}

Multiplying denominators forms a conjugate pair, which rationalizes the combined denominator.

3

Simplify the numerator and the denominator

\text{Numerator} = 2\sqrt{3}(\sqrt{3}-1 + \sqrt{3}+1) = 2\sqrt{3}(2\sqrt{3}) = 12; \quad \text{Denominator} = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2

Expanding the numerator combines like surd terms, and using the difference of squares simplifies the denominator to a rational integer.

4

Perform the final division

\frac{12}{2} = 6

Dividing the simplified numerator by the rationalized denominator gives the final integer value.

Key Concept

Rationalization of binomial denominators using conjugate surds

Question 7539Question →

The sum of the first $n$ terms of a sequence is given by $S_n = 2n^2 + 3n$ . Find the $5^{\text{th}}$ term of a geometric progression whose first term is the $3^{\text{rd}}$ term of this sequence, and whose common ratio is equal to the common difference of this sequence.

3328

13312

6912

1280

Show answer & explanation

Answer: $3328$

Answer

The 5th term of the geometric progression is 3328.

Evaluating

S_1

,

S_2

, and

S_3

gives sequence terms

T_1 = 5

,

T_2 = 9

, and

T_3 = 13

. The common difference is

d = 9 - 5 = 4

. Using

T_3 = 13

as the first term of the GP and

r = 4

as the common ratio, the 5th term of the GP is

13 \times 4^{5-1} = 13 \times 256 = 3328

.

Step-by-Step Solution

1

Calculate the first few terms of the sequence using the sum formula

S_n = 2n^2 + 3n

.

S_1 = 2(1)^2 + 3(1) = 5

,

S_2 = 2(2)^2 + 3(2) = 14

,

S_3 = 2(3)^2 + 3(3) = 27

.

The sum formula gives cumulative sums, from which individual terms can be derived.

2

Find the 3rd term (

T_3

) and the common difference (

d

) of the arithmetic progression.

T_1 = 5

,

T_2 = S_2 - S_1 = 9

,

T_3 = S_3 - S_2 = 13

. Common difference

d = T_2 - T_1 = 4

.

The difference between consecutive cumulative sums gives the individual sequence terms, and their constant difference gives the common difference.

3

Define the parameters of the geometric progression (GP).

First term of GP

a = T_3 = 13

, common ratio

r = d = 4

.

The problem specifies that the first term of the GP is the 3rd term of the sequence and the common ratio equals the common difference.

4

Compute the 5th term of the geometric progression using

G_n = a \cdot r^{n-1}

.

G_5 = 13 \cdot 4^{5-1} = 13 \cdot 4^4 = 13 \cdot 256 = 3328

.

Applying the standard nth term formula for a GP with

n = 5

.

Key Concept

Arithmetic Progression sum formula to term conversion and Geometric Progression nth term evaluation

Estimated Time:2m 0s

Question 7540Question →

An arithmetic progression (A.P.) has a first term of $8$ and a common difference of $6$ . What is the $9^{\text{th}}$ term of the progression?

56

62

50

68

Show answer & explanation

Answer: $56$

Answer

The

9^{\text{th}}

term of the arithmetic progression is

56

.

The

n^{\text{th}}

term of an arithmetic progression is given by

T_n = a + (n - 1)d

. Substituting

a = 8

,

d = 6

, and

n = 9

yields

T_9 = 8 + (9 - 1) \times 6 = 8 + 48 = 56

.

Step-by-Step Solution

1

Identify the given terms from the problem

First term

a = 8

, common difference

d = 6

, and term position

n = 9

.

These are the parameters required for the

n^{\text{th}}

term formula of an A.P.

2

Apply the

n^{\text{th}}

term formula

T_n = a + (n - 1)d

T_9 = 8 + (9 - 1) \times 6

The

n^{\text{th}}

term requires multiplying the common difference by

(n - 1)

.

3

Simplify the expression

T_9 = 8 + 8 \times 6 = 8 + 48 = 56

Perform multiplication before addition according to standard order of operations.

Key Concept

General term of an Arithmetic Progression

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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