All practice questions

Question 761Question →

During a quality control inspection at a manufacturing plant, two independent automated scanners, $S_1$ and $S_2$ , inspect items along a production line. The probability that scanner $S_1$ detects a surface flaw on a defective item is $0.90$ , and the probability that scanner $S_2$ detects the flaw is $0.80$ . What is the probability that a surface flaw is detected by exactly one of the two scanners?

Show answer & explanation

Answer: 0.26

Answer

The probability that a surface flaw is detected by exactly one of the two scanners is 0.26.

The probability of a surface flaw being detected by exactly one scanner is calculated by combining two mutually exclusive compound events: scanner

S_1

detects it while scanner

S_2

fails (

0.90 \times 0.20 = 0.18

), and scanner

S_1

fails while scanner

S_2

detects it (

0.10 \times 0.80 = 0.08

). Adding these two values yields

0.18 + 0.08 = 0.26

.

Step-by-Step Solution

1

Determine the probabilities of non-detection for each scanner.

P(S_1') = 0.10, P(S_2') = 0.20

The probability of an event's complement is 1 minus the probability of the event.

2

Calculate the probability that scanner S_1 detects the flaw while scanner S_2 misses it.

P(S_1 \cap S_2') = 0.90 \times 0.20 = 0.18

Because the scanners operate independently, joint probability is the product of their individual probabilities.

3

Calculate the probability that scanner S_1 misses the flaw while scanner S_2 detects it.

P(S_1' \cap S_2) = 0.10 \times 0.80 = 0.08

Independence allows multiplication of individual probabilities.

4

Sum the probabilities of the two mutually exclusive outcomes.

P(\text{exactly one}) = 0.18 + 0.08 = 0.26

The addition law applies because detecting the flaw with only S_1 and only S_2 are mutually exclusive events.

Key Concept

Compound probability laws for independent and mutually exclusive events

Question 762Question →

The interior angles of a convex polygon form an arithmetic progression. If the smallest interior angle is $120^\circ$ and the common difference between consecutive interior angles is $5^\circ$ , how many sides does the polygon have?

Show answer & explanation

Answer: 9

Answer

The polygon has 9 sides.

Equating the interior angle sum formula

(n-2) \times 180^\circ

with the sum formula for an arithmetic progression

\frac{n}{2}[2(120^\circ) + (n-1)5^\circ]

yields the quadratic equation

n^2 - 25n + 144 = 0

. Solving gives

n = 9

and

n = 16

. For

n = 16

, the largest angle would be

195^\circ

, which is impossible for a convex polygon. Thus, the only valid number of sides is 9.

Step-by-Step Solution

1

Formulate expressions for the sum of the interior angles.

The interior angle sum of an

n

-sided polygon is

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

. As an arithmetic sequence with

a = 120^\circ

and

d = 5^\circ

, the sum is

S = \frac{n}{2}[2(120^\circ) + (n-1)5^\circ]

.

Both methods express the total sum of all interior angles of the polygon.

2

Equate the two sum formulas and simplify into a quadratic equation.

(n-2) \times 180 = \frac{n}{2}(235 + 5n) \implies n^2 - 25n + 144 = 0

.

Expanding and dividing by 5 reduces the equality to standard quadratic form.

3

Solve for

n

and test the validity of the roots based on polygon convexity.

(n - 9)(n - 16) = 0 \implies n = 9

or

n = 16

. Testing

n = 16

gives a largest angle of

120^\circ + 15(5^\circ) = 195^\circ

(invalid as it exceeds

180^\circ

). Testing

n = 9

gives a largest angle of

120^\circ + 8(5^\circ) = 160^\circ

(valid).

A convex polygon cannot have any interior angle greater than or equal to

180^\circ

.

Key Concept

Polygon interior angle sum and arithmetic progressions

Question 763Question →

Given the matrix $P = \begin{pmatrix} x & 1 & 2 \\ 3 & 0 & x \\ 1 & 2 & 1 \end{pmatrix}$ , for what positive value of $x$ is the determinant of $P$ equal to $3$ ?

Show answer & explanation

Answer: 2

Answer

The positive value of

x

is

2

.

Expanding the determinant of

P

along the first row yields

\det(P) = -2x^2 + x + 9

. Setting

\det(P) = 3

leads to

-2x^2 + x + 9 = 3

, which rearranges to

2x^2 - x - 6 = 0

. Factoring gives

(2x + 3)(x - 2) = 0

, leading to roots

x = -1.5

and

x = 2

. The positive value is

2

.

Step-by-Step Solution

1

Calculate the determinant of matrix

P

in terms of

x

\det(P) = -2x^2 + x + 9

Expanding along the first row gives

x(0 \cdot 1 - 2x) - 1(3 \cdot 1 - x \cdot 1) + 2(3 \cdot 2 - 0 \cdot 1) = -2x^2 + x + 9

.

2

Set up the quadratic equation using

\det(P) = 3

2x^2 - x - 6 = 0

Subtracting

3

from both sides gives

-2x^2 + x + 6 = 0

, which simplifies to

2x^2 - x - 6 = 0

.

3

Solve the quadratic equation for

x

x = 2 \text{ or } x = -1.5

Factoring

(2x + 3)(x - 2) = 0

yields

x = -\frac{3}{2}

and

x = 2

.

4

Select the positive solution as required by the problem statement

x = 2

The problem specifically requests the positive real value of

x

.

Key Concept

3x3 Determinant Evaluation and Quadratic Solving

Question 764Question →

Given that $x$ is a positive real number and the determinant of the matrix $A = \begin{pmatrix} x & 1 & 0 \\ 2 & x & 3 \\ 0 & 1 & 4 \end{pmatrix}$ is $2$ , find the value of $x$ .

Show answer & explanation

Answer: 2

Answer

The positive value of

x

is

2

.

Expanding the determinant along the first row yields

\det(A) = x(4x - 3) - 1(8 - 0) = 4x^2 - 3x - 8

. Equating this to

2

gives

4x^2 - 3x - 10 = 0

. Factoring the quadratic gives

(4x + 5)(x - 2) = 0

, which yields

x = 2

or

x = -1.25

. Because

x

must be positive, the correct value is

2

.

Step-by-Step Solution

1

Expand the

3 \times 3

determinant along the first row

\det(A) = x(4x - 3) - 1(8 - 0) + 0 = 4x^2 - 3x - 8

Expanding along the first row leverages the zero entry to simplify calculation of the determinant.

2

Equate the determinant expression to the given determinant value

4x^2 - 3x - 10 = 0

Setting the calculated determinant equal to

2

creates a quadratic equation in terms of

x

.

3

Factor the quadratic equation to find the candidate values for

x

(4x + 5)(x - 2) = 0 \implies x = 2 \text{ or } x = -1.25

Factoring determines all algebraic solutions that satisfy the determinant equation.

4

Apply the positivity constraint given in the problem statement

x = 2

The question restricts

x

to positive real numbers, discarding the negative root.

Key Concept

Determinant of a 3x3 matrix and quadratic equations

Estimated Time:2m 0s

Question 765Question →

What is the value of the definite integral $\int_{0}^{2} (3x^2 + 2) \, dx$ ?

Show answer & explanation

Answer: 12

Answer

12

To evaluate

\int_{0}^{2} (3x^2 + 2) \, dx

, integrate

3x^2 + 2

to get

x^3 + 2x

. Substituting the limits gives

(2^3 + 2(2)) - (0^3 + 2(0)) = (8 + 4) - 0 = 12

.

Step-by-Step Solution

1

Integrate the function term by term

\int (3x^2 + 2) \, dx = x^3 + 2x

By the power rule of integration, \int 3x^2 \, dx = x^3 and \int 2 \, dx = 2x.

2

Apply the limits of integration from 0 to 2

[x^3 + 2x]_0^2 = (2^3 + 2(2)) - (0^3 + 2(0)) = 12 - 0 = 12

Substitute the upper limit 2 into the antiderivative and subtract the value obtained by substituting the lower limit 0.

Key Concept

Evaluation of Definite Integrals

Question 766Question →

A fruit juice mixture is prepared by combining orange extract and water in the ratio $3 : 5$ . If a bottle contains $40\text{ litres}$ of this mixture, how many litres of water does it contain?

Show answer & explanation

Answer: 25

Answer

The volume of water in the mixture is 25 litres.

The mixture ratio of orange extract to water is

3 : 5

, giving a total of

3 + 5 = 8

equal parts. Dividing the total volume of

40\text{ litres}

by

8

parts gives

5\text{ litres}

per part. Since water makes up

5

parts of the ratio, its volume is

5 \times 5 = 25\text{ litres}

.

Step-by-Step Solution

1

Determine the total number of parts in the ratio.

3 + 5 = 8 parts

The full volume of the mixture is divided into equal portions represented by the sum of the ratio parts.

2

Calculate the volume per ratio part.

40 / 8 = 5 litres per part

Dividing the total volume by the total number of parts gives the size of a single part.

3

Find the quantity of water.

5 * 5 = 25 litres

Water accounts for 5 parts of the ratio, so multiply 5 parts by 5 litres per part.

Key Concept

Ratio Sharing and Proportional Division

Question 767Question →

Given the permutation equation ${^{n}P_4} = 42 \cdot {^{n}P_2}$ , what is the positive integer value of $n$ ?

Show answer & explanation

Answer: 9

Answer

The positive integer value of

n

is

9

.

Expanding

{^{n}P_4}

as

n(n-1)(n-2)(n-3)

and

{^{n}P_2}

as

n(n-1)

transforms the equation to

n(n-1)(n-2)(n-3) = 42n(n-1)

. Dividing both sides by

n(n-1)

gives

(n-2)(n-3) = 42

, which simplifies to

n^2 - 5n - 36 = 0

. Factoring this quadratic gives

(n-9)(n+4) = 0

. Since

n

must be a positive integer greater than or equal to

4

, the correct answer is

9

.

Step-by-Step Solution

1

Write out the expanded expressions for

{^{n}P_4}

and

{^{n}P_2}

.

{^{n}P_4} = n(n-1)(n-2)(n-3)

and

{^{n}P_2} = n(n-1)

.

By definition,

{^{n}P_r} = \frac{n!}{(n-r)!} = n(n-1)\dots(n-r+1)

.

2

Set up the algebraic equation based on the given problem statement.

n(n-1)(n-2)(n-3) = 42n(n-1)

.

Substitute the expanded permutation formulas into the given identity.

3

Simplify the equation by dividing both sides by the non-zero common product

n(n-1)

.

(n-2)(n-3) = 42

.

Since

n \ge 4

,

n(n-1) \neq 0

, so we can cancel these terms from both sides.

4

Expand and rearrange the resulting expression into a standard quadratic equation.

n^2 - 5n - 36 = 0

.

Expanding

(n-2)(n-3)

gives

n^2 - 5n + 6

; subtracting

42

yields

n^2 - 5n - 36 = 0

.

5

Solve the quadratic equation for

n

and discard non-physical roots.

n = 9

.

Factoring yields

(n-9)(n+4) = 0

, so

n = 9

or

n = -4

. A permutation requires

n

to be a positive integer

\ge 4

, so

n = 9

.

Key Concept

Solving algebraic equations involving permutations

{^{n}P_r} = \frac{n!}{(n-r)!}

Question 768Question →

A trader bought a quantity of palm oil. He sold $\frac{1}{3}$ of the total quantity at a profit of $20\%$ , $\frac{1}{2}$ of the remaining palm oil at a loss of $10\%$ , and the rest of the palm oil at a profit of $x\%$ . If his overall profit on the entire transaction was $15\%$ , what is the value of $x$ ?

Show answer & explanation

Answer: 35

Answer

The value of

x

is

35

.

The palm oil is divided into three equal portions of

\frac{1}{3}

each. The first portion contributes

\frac{20}{3}\%

profit, the second portion contributes a loss of

-\frac{10}{3}\%

, and the final portion contributes

\frac{x}{3}\%

profit. Summing these contributions yields

\frac{10 + x}{3}\%

. Equating this to the target overall profit of

15\%

gives

10 + x = 45

, which solves to

x = 35

.

Step-by-Step Solution

1

Determine the fractional portion of palm oil sold at each stage.

First portion =

\frac{1}{3}

. Remaining after first sale =

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

. Second portion =

\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}

. Final portion =

1 - \left(\frac{1}{3} + \frac{1}{3}\right) = \frac{1}{3}

.

Finding the exact fraction of the total quantity sold at each step is necessary to weight their respective profit/loss rates correctly.

2

Formulate the net percentage profit equation.

\text{Net Profit } \% = \left(\frac{1}{3} \times 20\%\right) + \left(\frac{1}{3} \times (-10\%)\right) + \left(\frac{1}{3} \times x\%\right) = \frac{20 - 10 + x}{3}\% = \frac{10 + x}{3}\%

The overall profit percentage is the sum of individual fractional contributions, treating loss as a negative percentage.

3

Set the net profit equal to

15\%

and solve for

x

.

\frac{10 + x}{3} = 15 \implies 10 + x = 45 \implies x = 35

Solving this linear equation gives the required profit percentage for the final portion.

Key Concept

Weighted Average Profit and Loss across fractional components of an asset

Estimated Time:2m 0s

Question 769Question →

What is the canonical non-negative remainder when $3^{45} - 11$ is divided by $7$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Using modular exponentiation,

3^3 = 27 \equiv -1 \pmod{7}

, so

3^{45} = (3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

. Subtracting 11 yields

-1 - 11 = -12 \pmod{7}

. Adding 14 (a multiple of 7) gives

-12 + 14 = 2

, which is the canonical non-negative remainder.

Step-by-Step Solution

1

Evaluate

3^{45} \pmod{7}

using power rules

3^{45} \equiv -1 \pmod{7}

Since

3^3 = 27 \equiv -1 \pmod{7}

, raising both sides to the power of 15 gives

(3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

.

2

Subtract 11 from the modular result

-12 \pmod{7}

Replacing

3^{45}

with

-1

gives

-1 - 11 = -12

.

3

Reduce -12 to its canonical non-negative remainder in

[0, 6]

2

Adding the smallest multiple of 7 that makes the result non-negative gives

-12 + 14 = 2

.

Key Concept

Modular Exponentiation and Negative Remainder Reduction

Question 770Question →

In a survey of $120$ tech entrepreneurs at an innovation hub, $65$ secured Angel Investment ( $A$ ), $55$ received Venture Capital ( $V$ ), and $50$ obtained Government Grants ( $G$ ). Furthermore, $25$ received both $A$ and $V$ , $20$ received both $V$ and $G$ , $22$ received both $A$ and $G$ , while $12$ received no funding from any of these three sources. How many entrepreneurs secured funding from exactly two of these sources?

Show answer & explanation

Answer: 52

Answer

The number of entrepreneurs who secured funding from exactly two sources is 52.

Subtracting the

12

unfunded entrepreneurs from the total

120

yields

108

funded entrepreneurs. Applying inclusion-exclusion gives

108 = 65 + 55 + 50 - (25 + 20 + 22) + n(A \cap V \cap G)

, which gives

n(A \cap V \cap G) = 5

. Subtracting

5

from each pairwise intersection yields

20

,

15

, and

17

for entrepreneurs receiving funding from exactly two sources. Summing these gives

52

.

Step-by-Step Solution

1

Determine the cardinality of the union of all three funding sets

n(A \cup V \cup G) = 108

Subtracting the 12 unfunded entrepreneurs from the universal set of 120 gives the total number of entrepreneurs who received at least one form of funding.

2

Solve for the number of entrepreneurs who received funding from all three sources using inclusion-exclusion

n(A \cap V \cap G) = 5

Using

n(A \cup V \cup G) = n(A) + n(V) + n(G) - n(A \cap V) - n(V \cap G) - n(A \cap G) + n(A \cap V \cap G)

, we get

108 = 65 + 55 + 50 - 25 - 20 - 22 + x

, which simplifies to

108 = 103 + x

, giving

x = 5

.

3

Calculate the count for each region representing exactly two funding sources

Only

A \cap V = 20

, Only

V \cap G = 15

, Only

A \cap G = 17

Subtracting the 3-set intersection (

x = 5

) from each pairwise intersection isolates the elements belonging to strictly two sets.

4

Sum the three strictly two-set regions

20 + 15 + 17 = 52

Adding the individual counts for the three disjoint regions gives the total number of entrepreneurs who received funding from exactly two sources.

Key Concept

Principle of Inclusion-Exclusion for three sets and cardinal partitioning of Venn diagrams

Question 771Question →

The frequency table below shows the fuel efficiency, measured in $\text{km/L}$ , for a fleet of $50$ delivery vans operated by a logistics company:

Fuel Efficiency (km/L)	Frequency ( $f$ )
10 – 14	8
15 – 19	12
20 – 24	20
25 – 29	10

What is the mean fuel efficiency of the fleet, in $\text{km/L}$ ?

Show answer & explanation

Answer: 20.2

Answer

The mean fuel efficiency of the fleet is 20.2 km/L.

The estimated mean is found by calculating the midpoint of each class interval, multiplying each midpoint by its class frequency, summing these products to get 1010, and dividing by the total frequency of 50, resulting in 20.2 km/L.

Step-by-Step Solution

1

Determine the midpoint (x) for each class interval.

Midpoints are 12 for 10–14, 17 for 15–19, 22 for 20–24, and 27 for 25��29.

For grouped frequency data, each class interval is represented by its midpoint x = (lower limit + upper limit) / 2.

2

Compute the product of frequency and midpoint (f * x) for each interval.

Products: (8 * 12) = 96, (12 * 17) = 204, (20 * 22) = 440, (10 * 27) = 270.

To calculate the total contribution of values from each class interval.

3

Sum all products (sum of f * x) and divide by the total number of delivery vans (sum of f).

sum of f * x = 96 + 204 + 440 + 270 = 1010. sum of f = 50. Mean = 1010 / 50 = 20.2.

The formula for the estimated mean of grouped data is mean = (sum of f * x) / (sum of f).

Key Concept

Measures of Central Tendency for Grouped Data (Grouped Mean)

Question 772Question →

A laboratory technician recorded the temperature changes (in °C) of a chemical reaction across five trials as $3$ , $6$ , $7$ , $9$ , and $15$ . What is the variance of this set of data?

Show answer & explanation

Answer: 16

Answer

The variance of the temperature changes is 16.

The mean of the given numbers is

\bar{x} = \frac{3 + 6 + 7 + 9 + 15}{5} = 8

. The squared deviations from the mean are

(3-8)^2 = 25

,

(6-8)^2 = 4

,

(7-8)^2 = 1

,

(9-8)^2 = 1

, and

(15-8)^2 = 49

. Summing these squared deviations gives

25 + 4 + 1 + 1 + 49 = 80

. Dividing by the number of observations

N = 5

yields the variance:

\frac{80}{5} = 16

.

Step-by-Step Solution

1

Calculate the arithmetic mean (\bar{x}) of the dataset.

\bar{x} = \frac{3 + 6 + 7 + 9 + 15}{5} = 8

The mean is required to determine the deviation of each individual value from the central value.

2

Compute the square of the deviation for each data point from the mean.

(3-8)^2 = 25, (6-8)^2 = 4, (7-8)^2 = 1, (9-8)^2 = 1, (15-8)^2 = 49

Squaring deviations ensures all values are positive and emphasizes larger departures from the mean.

3

Sum the squared deviations and divide by the total number of observations (N = 5).

\text{Variance} = \frac{25 + 4 + 1 + 1 + 49}{5} = \frac{80}{5} = 16

Variance measures the average of the squared deviations from the mean.

Key Concept

Population Variance for Ungrouped Data

Question 773Question →

In a survey of $100$ book club members regarding their reading preferences among Science Fiction ( $S$ ), Mystery ( $M$ ), and Historical Fiction ( $H$ ), it was found that $48$ read Science Fiction, $42$ read Mystery, and $38$ read Historical Fiction. Furthermore, $18$ read both Science Fiction and Mystery, $15$ read both Science Fiction and Historical Fiction, $14$ read both Mystery and Historical Fiction, and $8$ read all three genres. How many of the members read exactly one of these three genres?

Show answer & explanation

Answer: 58

Answer

The number of members who read exactly one of the three genres is

58

.

By isolating the disjoint regions of the 3-set Venn diagram, the number of members reading only Science Fiction is

23

, only Mystery is

18

, and only Historical Fiction is

17

. Adding these disjoint sets yields

58

.

Step-by-Step Solution

1

Determine exclusive two-set intersection regions

n(S \cap M \text{ only}) = 10

,

n(S \cap H \text{ only}) = 7

,

n(M \cap H \text{ only}) = 6

The given pairwise totals include members who read all three genres, so subtracting

n(S \cap M \cap H) = 8

isolates those in exactly two sets.

2

Determine exclusive single-set regions

n(S \text{ only}) = 23

,

n(M \text{ only}) = 18

,

n(H \text{ only}) = 17

Subtracting all overlapping regions within each set's boundary gives the number of members reading only that specific genre.

3

Sum the single-set regions

23 + 18 + 17 = 58

The set of members reading exactly one genre is the disjoint union of the three exclusive single-set regions.

Key Concept

3-Set Venn Diagram Cardinality and Disjoint Region Analysis

Question 774Question →

Find the smallest positive value of $\theta$ , in degrees, that satisfies the trigonometric equation $2\sin(3\theta - 30^\circ) = \sqrt{3}$ .

Show answer & explanation

Answer: 30

Answer

The smallest positive angle

\theta

is

30^\circ

.

To find the smallest positive value of

\theta

, first isolate the sine function by dividing both sides by 2 to obtain

\sin(3\theta - 30^\circ) = \frac{\sqrt{3}}{2}

. The smallest positive angle with a sine of

\frac{\sqrt{3}}{2}

is

60^\circ

. Setting

3\theta - 30^\circ = 60^\circ

yields

3\theta = 90^\circ

, which gives

\theta = 30^\circ

.

Step-by-Step Solution

1

Isolate the trigonometric ratio

\sin(3\theta - 30^\circ) = \frac{\sqrt{3}}{2}

Dividing both sides of

2\sin(3\theta - 30^\circ) = \sqrt{3}

by 2 simplifies the equation into standard form.

2

Determine the primary angle solution

3\theta - 30^\circ = 60^\circ

The smallest positive angle whose sine equals

\frac{\sqrt{3}}{2}

is

60^\circ

.

3

Solve the linear equation for

\theta

\theta = 30^\circ

Adding

30^\circ

to both sides gives

3\theta = 90^\circ

, and dividing by 3 yields

\theta = 30^\circ

.

Key Concept

Solving Trigonometric Equations with Linear Argument Transformations

Question 775Question →

Three point charges $q_1 = +2.0 \times 10^{-6}\text{ C}$ , $q_2 = +2.0 \times 10^{-6}\text{ C}$ , and $q_3 = -4.0 \times 10^{-6}\text{ C}$ are placed along a straight line at positions $x = 0\text{ m}$ , $x = 0.30\text{ m}$ , and $x = 0.60\text{ m}$ , respectively. Taking Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ , calculate the magnitude of the net electrostatic force acting on charge $q_2$ in newtons.

Show answer & explanation

Answer: 1.2

Answer

The magnitude of the net electrostatic force acting on charge

q_2

is

1.20\text{ N}

.

The force exerted on

q_2

by

q_1

is repulsive (

0.40\text{ N}

directed to the right) because both charges are positive. The force exerted on

q_2

by

q_3

is attractive (

0.80\text{ N}

directed to the right) because

q_2

is positive and

q_3

is negative. Since both component forces act in the same direction, the total net force magnitude is

0.40\text{ N} + 0.80\text{ N} = 1.20\text{ N}

.

Step-by-Step Solution

1

Calculate the repulsive force exerted by

q_1

on

q_2

F_{12} = 0.40\text{ N}

pointing to the right

Like charges repel each other, so

q_1

pushes

q_2

away along the

+x

-axis.

2

Calculate the attractive force exerted by

q_3

on

q_2

F_{32} = 0.80\text{ N}

pointing to the right

Unlike charges attract each other, so

q_3

pulls

q_2

towards itself along the

+x

-axis.

3

Sum the component electrostatic forces acting on

q_2

F_{\text{net}} = 0.40\text{ N} + 0.80\text{ N} = 1.20\text{ N}

Because both forces act in the exact same direction along the line, their magnitudes add directly.

Key Concept

Coulomb's Law and Principle of Superposition for Electrostatic Forces

Question 776Question →

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 777Question →

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 778Question →

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 779Question →

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 780Question →

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

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