All practice questions

Question 821Question →

A glass flask of volume $1000\text{ cm}^3$ is filled completely with mercury at a temperature of $10^\circ\text{C}$ . The linear expansivity of the glass is $9.0 \times 10^{-6}\text{ K}^{-1}$ and the real cubic expansivity of mercury is $1.8 \times 10^{-4}\text{ K}^{-1}$ . What volume of mercury (in $\text{cm}^3$ ) will overflow when the system is heated to $110^\circ\text{C}$ ?

Show answer & explanation

Answer: 15.3

Answer

The volume of mercury that overflows is

15.3\text{ cm}^3

.

The apparent expansion of the liquid equals its real expansion minus the expansion of the container. Since

\gamma_v = 3\alpha = 2.7 \times 10^{-5}\text{ K}^{-1} = 0.27 \times 10^{-4}\text{ K}^{-1}

, the apparent cubic expansivity is

\gamma_a = 1.8 \times 10^{-4} - 0.27 \times 10^{-4} = 1.53 \times 10^{-4}\text{ K}^{-1}

. Multiplying by initial volume (

1000\text{ cm}^3

) and temperature change (

100\text{ K}

) yields an overflow volume of

15.3\text{ cm}^3

.

Step-by-Step Solution

1

Calculate the volume expansivity of the glass vessel (

\gamma_v

)

\gamma_v = 3 \times 9.0 \times 10^{-6}\text{ K}^{-1} = 2.7 \times 10^{-5}\text{ K}^{-1} = 0.27 \times 10^{-4}\text{ K}^{-1}

The volumetric (cubic) expansivity of a solid container is three times its linear expansivity.

2

Determine the apparent cubic expansivity of mercury (

\gamma_a

)

\gamma_a = \gamma_r - \gamma_v = 1.8 \times 10^{-4} - 0.27 \times 10^{-4} = 1.53 \times 10^{-4}\text{ K}^{-1}

The apparent expansion of a liquid accounts for both the expansion of the liquid itself and the expansion of the containing vessel.

3

Calculate the overflow volume (apparent expansion

\Delta V_a

)

\Delta V_a = V_0 \times \gamma_a \times \Delta T = 1000 \times 1.53 \times 10^{-4} \times 100 = 15.3\text{ cm}^3

The volume of liquid that overflows corresponds directly to its apparent volume increase.

Key Concept

Real and Apparent Cubical Expansivity of Liquids

Estimated Time:1m 30s

Question 822Question →

Given the matrices $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & k \end{pmatrix}$ , find the value of $k$ such that the determinant of the product matrix $AB$ is equal to 50.

Show answer & explanation

Answer: 3

Answer

The value of

k

is 3.

By applying the determinant product rule

\det(AB) = \det(A)\det(B)

, we find

\det(A) = (4)(3) - (2)(1) = 10

. Given

\det(AB) = 50

, it follows that

\det(B) = 50 / 10 = 5

. Since

\det(B) = (1)(k) - (-1)(2) = k + 2

, setting

k + 2 = 5

gives

k = 3

.

Step-by-Step Solution

1

Calculate the determinant of matrix A

\det(A) = (4 \times 3) - (2 \times 1) = 12 - 2 = 10

The determinant of a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} is ad - bc.

2

Apply the product property of determinants

\det(B) = \frac{\det(AB)}{\det(A)} = \frac{50}{10} = 5

For any two square matrices of the same dimension, \det(AB) = \det(A) \cdot \det(B).

3

Express the determinant of matrix B in terms of k and solve

\det(B) = (1)(k) - (-1)(2) = k + 2 = 5 \implies k = 3

Equating the calculated determinant formula for B to its numerical value of 5.

Key Concept

Determinant Product Property and 2x2 Matrix Determinant

Estimated Time:1m 30s

Question 823Question →

A point $P(x, y)$ moves in the Cartesian plane such that it is at all times equidistant from two parallel lines given by the equations $3x + 4y - 12 = 0$ and $3x + 4y + 4 = 0$ . The locus of $P$ intersects the straight line $x - 2y - 8 = 0$ at the point $(a, b)$ . What is the value of $a - b$ ?

Show answer & explanation

Answer: 6

Answer

The value of

a - b

is 6.

The locus of a point moving equidistant between two parallel lines is the parallel line lying midway between them. Combining the parallel line equations

3x + 4y - 12 = 0

and

3x + 4y + 4 = 0

yields the locus line

3x + 4y - 4 = 0

. Solving the system formed by this locus line and

x - 2y - 8 = 0

gives

x = 4

and

y = -2

. Therefore,

a = 4

and

b = -2

, so

a - b = 4 - (-2) = 6

.

Step-by-Step Solution

1

Find the equation of the locus of point P

Locus equation:

3x + 4y - 4 = 0

The locus of points equidistant from two parallel lines

ax + by + c_1 = 0

and

ax + by + c_2 = 0

is a parallel line midway between them, given by

ax + by + \frac{c_1 + c_2}{2} = 0

.

2

Solve the simultaneous equations to find the intersection point

(a, b)

a = 4

and

b = -2

Substitute

x = 2y + 8

into

3x + 4y - 4 = 0

to get

3(2y + 8) + 4y - 4 = 0

, which yields

10y = -20

, so

y = -2

and

x = 4

.

3

Calculate the difference

a - b

6

Subtract

b = -2

from

a = 4

to obtain

4 - (-2) = 6

.

Key Concept

Locus of points equidistant from two parallel lines and intersection of straight lines

Question 824Question →

A railway line is laid using steel rails, each of length $15\text{ m}$ , at a temperature of $20^\circ\text{C}$ . What minimum gap, in millimetres ( $\text{mm}$ ), must be left between consecutive rails so that they just touch without buckling when heated to $60^\circ\text{C}$ ? [Linear expansivity of steel = $1.2 \times 10^{-5}\text{ K}^{-1}$ ]

Show answer & explanation

Answer: 7.2

Answer

The minimum gap required between consecutive rails is

7.2\text{ mm}

.

The expansion in length

\Delta L

is given by

\Delta L = L_0 \alpha \Delta T

. Substituting the initial length

L_0 = 15\text{ m}

, linear expansivity

\alpha = 1.2 \times 10^{-5}\text{ K}^{-1}

, and temperature change

\Delta T = 40\text{ K}

gives

\Delta L = 7.2 \times 10^{-3}\text{ m}

, which corresponds to

7.2\text{ mm}

.

Step-by-Step Solution

1

Determine the change in temperature

\Delta T = 60^\circ\text{C} - 20^\circ\text{C} = 40\text{ K}

Thermal expansion is driven by the temperature difference between the final and initial states.

2

Set up the linear expansion formula

\Delta L = L_0 \alpha \Delta T

The linear expansion of a solid bar depends on its initial length, the material's linear expansivity, and the temperature change.

3

Calculate the expansion in metres and convert to millimetres

\Delta L = 15 \times (1.2 \times 10^{-5}) \times 40 = 7.2 \times 10^{-3}\text{ m} = 7.2\text{ mm}

Multiplying the value in metres by

1000

yields the required measurement in millimetres.

Key Concept

Linear Expansivity and Thermal Expansion of Solids

Estimated Time:1m 30s

Question 825Question →

The centripetal acceleration $a$ of a particle moving in a circular path depends on its linear speed $v$ and the radius $r$ of the path according to the formula $a = k v^x r^y$ , where $k$ is a dimensionless constant. Using dimensional analysis, what is the numerical value of the product $x \cdot y$ ?

Show answer & explanation

Answer: -2

Answer

The numerical value of the product

x \cdot y

is

-2

.

By dimensional analysis, centripetal acceleration has dimensions

[a] = \text{L T}^{-2}

, velocity

[v] = \text{L T}^{-1}

, and radius

[r] = \text{L}

. Substituting into

a = k v^x r^y

gives

\text{L T}^{-2} = \text{L}^{x+y} \text{T}^{-x}

. Equating exponents of time yields

-x = -2 \implies x = 2

. Equating exponents of length gives

x + y = 1 \implies 2 + y = 1 \implies y = -1

. Consequently,

x \cdot y = 2 \cdot (-1) = -2

.

Step-by-Step Solution

1

Identify the base dimensions for each physical quantity.

[a] = \text{M}^0 \text{L}^1 \text{T}^{-2}

,

[v] = \text{M}^0 \text{L}^1 \text{T}^{-1}

, and

[r] = \text{M}^0 \text{L}^1 \text{T}^0

.

Dimensional analysis requires substituting fundamental dimensions of mass, length, and time.

2

Set up the dimensional homogeneity equation.

\text{L}^1 \text{T}^{-2} = (\text{L T}^{-1})^x \cdot (\text{L})^y = \text{L}^{x+y} \text{T}^{-x}.

Since

k

is dimensionless, the net dimensions on both sides of the equation must be identical.

3

Solve for exponents

x

and

y

by equating powers of corresponding base units.

For \text{T}:

-x = -2 \implies x = 2

. For \text{L}:

x + y = 1 \implies 2 + y = 1 \implies y = -1

.

Equating coefficients of identical base dimensions gives a system of linear equations.

4

Multiply the derived values of

x

and

y

.

x \cdot y = 2 \cdot (-1) = -2

.

The question asks specifically for the product of exponents

x

and

y

.

Key Concept

Dimensional Homogeneity and Exponent Analysis

Question 826Question →

In a mathematics examination paper consisting of $8$ questions, a candidate is required to answer $5$ questions in total. If the candidate must answer the first $2$ questions, in how many ways can the candidate select the remaining questions?

Show answer & explanation

Answer: 20

Answer

The candidate can select the questions in 20 ways.

Because the first 2 questions are mandatory, the choice is reduced to picking 3 additional questions from the remaining 6 questions. Since selection order is irrelevant, the number of distinct ways is $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$.

Step-by-Step Solution

1

Determine the remaining number of questions needed

3 questions

Since 2 out of the required 5 questions are compulsory, the candidate must choose 3 more.

2

Determine the available pool of remaining questions

6 questions

Subtracting the 2 compulsory questions from the 8 total questions leaves 6 questions available.

3

Calculate the combinations using the nCr formula

20 ways

The order in which questions are selected does not matter, so we use combinations: $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$.

Key Concept

Combinations with restricted or fixed choices

Question 827Question →

The $2^{\text{nd}}$ , $5^{\text{th}}$ , and $14^{\text{th}}$ terms of a non-constant arithmetic progression (AP) form the first three consecutive terms of a geometric progression (GP). If the sum of the first $6$ terms of the AP is $72$ , calculate the common difference of the AP.

Show answer & explanation

Answer: 4

Answer

The common difference of the arithmetic progression is

4

.

Equating the square of the middle GP term

(a+4d)^2

to the product of the outer terms

(a+d)(a+13d)

yields

3d^2 = 6ad

, which simplifies to

d = 2a

. Substituting

2a = d

into the AP sum formula

S_6 = 3(2a+5d) = 72

gives

3(6d) = 72 \implies 18d = 72

, so

d = 4

.

Step-by-Step Solution

1

Express the

2^{\text{nd}}

,

5^{\text{th}}

, and

14^{\text{th}}

terms of the AP algebraically

T_2 = a + d

,

T_5 = a + 4d

,

T_{14} = a + 13d

The

n^{\text{th}}

term of an AP is given by

T_n = a + (n-1)d

.

2

Apply the consecutive terms property of a GP

(a + 4d)^2 = (a + d)(a + 13d)

For three consecutive terms of a GP, the square of the middle term equals the product of the first and third terms.

3

Simplify the quadratic equation to find the relationship between

a

and

d

a^2 + 8ad + 16d^2 = a^2 + 14ad + 13d^2 \implies 3d^2 = 6ad \implies d = 2a

Since the AP is non-constant,

d \neq 0

, allowing division by

3d

.

4

Formulate the sum of the first

6

terms of the AP

S_6 = 3(2a + 5d) = 72 \implies 2a + 5d = 24

The sum of the first

n

terms of an AP is

S_n = \frac{n}{2}[2a + (n-1)d]

.

5

Substitute

2a = d

into the linear equation and solve for

d

d + 5d = 24 \implies 6d = 24 \implies d = 4

Replacing

2a

with

d

reduces the equation to a single variable.

Key Concept

Relating non-consecutive terms of an Arithmetic Progression to form a Geometric Progression

Question 828Question →

Given the matrix $A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$ , if $A^2 - 7A + kI = \mathbf{0}$ , where $I$ is the $2 \times 2$ identity matrix and $\mathbf{0}$ is the $2 \times 2$ zero matrix, what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is 10.

By matrix multiplication and algebraic evaluation,

A^2 - 7A = -10I

. Substituting into

A^2 - 7A + kI = \mathbf{0}

yields

-10I + kI = \mathbf{0}

, which gives

k = 10

. Alternatively, by the Cayley-Hamilton Theorem, any

2 \times 2

matrix

A

satisfies

A^2 - \text{tr}(A)A + \det(A)I = \mathbf{0}

. Here

\text{tr}(A) = 3 + 4 = 7

and

\det(A) = (3)(4) - (2)(1) = 10

, directly giving

k = \det(A) = 10

.

Step-by-Step Solution

1

Calculate the matrix product

A^2

A^2 = \begin{pmatrix} (3\cdot 3 + 2\cdot 1) & (3\cdot 2 + 2\cdot 4) \\ (1\cdot 3 + 4\cdot 1) & (1\cdot 2 + 4\cdot 4) \end{pmatrix} = \begin{pmatrix} 11 & 14 \\ 7 & 18 \end{pmatrix}

Squaring matrix

A

involves multiplying rows of

A

by columns of

A

.

2

Perform scalar multiplication for

7A

7A = \begin{pmatrix} 21 & 14 \\ 7 & 28 \end{pmatrix}

Each entry of matrix

A

is multiplied by the scalar 7.

3

Subtract

7A

from

A^2

A^2 - 7A = \begin{pmatrix} 11 - 21 & 14 - 14 \\ 7 - 7 & 18 - 28 \end{pmatrix} = \begin{pmatrix} -10 & 0 \\ 0 & -10 \end{pmatrix} = -10I

Subtracting corresponding entries yields a scalar multiple of the identity matrix.

4

Solve for the unknown scalar

k

-10I + kI = \mathbf{0} \implies k = 10

Setting

(k - 10)I = \mathbf{0}

implies

k - 10 = 0

, so

k = 10

.

Key Concept

Matrix Polynomial Equations and Cayley-Hamilton Theorem

Question 829Question →

Find the value of $r$ such that ${^{8}P_r} = 6720$ .

Show answer & explanation

Answer: 5

Answer

The value of

r

is 5.

Expanding

{^{8}P_r}

into consecutive decreasing factors starting from 8 yields

8 \times 7 \times 6 \times 5 \times 4 = 6720

. Counting the number of factors multiplied (8, 7, 6, 5, 4) gives exactly 5 factors, so

r = 5

.

Step-by-Step Solution

1

Write the formula for permutation

{^{8}P_r}

as a product of descending integers.

{^{8}P_r} = 8 \times 7 \times 6 \times \dots \times (8 - r + 1)

By definition,

{^{n}P_r}

represents the product of

r

consecutive factors starting from

n

and decreasing by 1.

2

Perform sequential multiplication starting from 8 until reaching 6720.

Product of 5 factors:

8 \times 7 \times 6 \times 5 \times 4 = 6720

Multiplying factors gives:

8 \times 7 = 56

;

56 \times 6 = 336

;

336 \times 5 = 1680

;

1680 \times 4 = 6720

.

3

Count the number of terms in the product to find

r

.

r = 5

Since 5 consecutive integers were multiplied together to obtain 6720, the subset size

r

is 5.

Key Concept

Permutations of

n

distinct items taken

r

at a time

Estimated Time:1m 0s

Question 830Question →

In Bohr's atomic model of the hydrogen atom, the radius of the ground state orbit ( $n = 1$ ) is $0.053\text{ nm}$ . According to de Broglie's condition for stationary electron orbits, what is the de Broglie wavelength of the electron in its second excited state? Express your answer in nanometers ( $\text{nm}$ ).

Show answer & explanation

Answer: 1

Answer

The de Broglie wavelength of the electron in its second excited state is

1.00\text{ nm}

.

The second excited state corresponds to the quantum number

n = 3

. According to Bohr's atomic model, the radius of the

n

-th orbit is given by

r_n = n^2 r_1

, yielding

r_3 = 3^2 \times 0.053\text{ nm} = 0.477\text{ nm}

. De Broglie explained Bohr's angular momentum quantization by showing that an integral number of electron matter-waves must fit around the orbital circumference:

2\pi r_n = n \lambda_n

. Solving for

\lambda_3

gives

\lambda_3 = \frac{2\pi r_3}{3} = 2\pi \times 3 \times 0.053\text{ nm} \approx 1.00\text{ nm}

.

Step-by-Step Solution

1

Identify the principal quantum number for the specified energy state

n = 3

The ground state corresponds to

n = 1

, the first excited state to

n = 2

, and the second excited state to

n = 3

.

2

Calculate the radius of the third stationary orbit

r_3 = 0.477\text{ nm}

In Bohr's model, the radius of the

n

-th orbit is proportional to

n^2

, so

r_3 = 3^2 \times 0.053\text{ nm} = 9 \times 0.053\text{ nm} = 0.477\text{ nm}

.

3

Apply de Broglie's standing wave condition for stationary orbits

\lambda_3 = \frac{2\pi r_3}{3}

De Broglie postulated that a stationary orbit contains an integral number of electron de Broglie wavelengths around its circumference:

2\pi r_n = n \lambda_n

.

4

Substitute values to compute the wavelength

\lambda_3 = 1.00\text{ nm}

\lambda_3 = \frac{2 \times 3.1416 \times 0.477\text{ nm}}{3} = 2\pi \times 3 \times 0.053\text{ nm} \approx 0.9992\text{ nm} \approx 1.00\text{ nm}

.

Key Concept

De Broglie Standing Wave Quantization in Bohr Atomic Model

Question 831Question →

Let $A$ be a $3 \times 3$ square matrix such that $\det(A) > 0$ . If the matrix satisfies the property $\det(3A^{-1}) = \det(A^T A)$ , what is the value of $\det(A)$ ?

Show answer & explanation

Answer: 3

Answer

The value of

\det(A)

is 3.

Using fundamental determinant identities for a

3 \times 3

matrix (

n = 3

):

\det(3A^{-1}) = 3^3 \det(A^{-1}) = \frac{27}{\det(A)}

and

\det(A^T A) = \det(A^T)\det(A) = (\det(A))^2

. Equating them yields

\frac{27}{\det(A)} = (\det(A))^2 \implies (\det(A))^3 = 27

, which yields

\det(A) = 3

.

Step-by-Step Solution

1

Express

\det(3A^{-1})

in terms of

\det(A)

using determinant scaling and inverse properties.

\det(3A^{-1}) = 3^3 \det(A^{-1}) = \frac{27}{\det(A)}.

For an

n \times n

matrix

M

, scaling by constant

k

gives

\det(kM) = k^n \det(M)

, and

\det(M^{-1}) = \frac{1}{\det(M)}

.

2

Express

\det(A^T A)

in terms of

\det(A)

using transpose and multiplication properties.

\det(A^T A) = \det(A^T)\det(A) = (\det(A))^2.

The determinant of a product is the product of determinants, and

\det(A^T) = \det(A)

.

3

Set the two simplified expressions equal to each other and solve for

\det(A)

.

\frac{27}{\det(A)} = (\det(A))^2 \implies (\det(A))^3 = 27 \implies \det(A) = 3.

Taking the cube root of both sides gives the unique real value since

\det(A) > 0

.

Key Concept

Properties of Determinants (Scalar Multiplication, Transpose, Inverse, and Matrix Products)

Estimated Time:1m 30s

Question 832Question →

An electric scooter starts from rest and accelerates uniformly at a rate of $3\text{ m/s}^2$ for a time of $6\text{ s}$ . What is the total distance, in meters, traveled by the scooter during this period?

Show answer & explanation

Answer: 54

Answer

The total distance traveled by the scooter is

54\text{ m}

.

Using the second equation of linear motion

s = ut + \frac{1}{2}at^2

with

u = 0\text{ m/s}

,

a = 3\text{ m/s}^2

, and

t = 6\text{ s}

yields

s = 0 + \frac{1}{2}(3)(36) = 54\text{ m}

.

Step-by-Step Solution

1

Identify known variables from the stem

Initial velocity

u = 0\text{ m/s}

, acceleration

a = 3\text{ m/s}^2

, time interval

t = 6\text{ s}

Extracting given values provides the foundation for selecting the correct kinematic equation.

2

Select the appropriate equation of linear motion

s = ut + \frac{1}{2}at^2

This formula directly relates displacement

s

to initial velocity

u

, constant acceleration

a

, and time

t

.

3

Calculate the numerical displacement

s = (0\text{ m/s})(6\text{ s}) + \frac{1}{2}(3\text{ m/s}^2)(6\text{ s})^2 = 0 + \frac{1}{2}(3)(36) = 54\text{ m}

Evaluating the mathematical expression yields the final total distance.

Key Concept

Uniform Linear Acceleration

Question 833Question →

In an electric circuit, two capacitors $C_1 = 12\text{ }\mu\text{F}$ and $C_2 = 6\text{ }\mu\text{F}$ are connected in series. This series combination is then connected in parallel with a third capacitor $C_3$ of unknown value. When a direct-current potential difference of $100\text{ V}$ is applied across the entire network, the total electrostatic energy stored in the circuit is $100\text{ mJ}$ . What is the capacitance of $C_3$ in microfarads ( $\mu\text{F}$ )?

Show answer & explanation

Answer: 16

Answer

The capacitance of

C_3

is

16\text{ }\mu\text{F}

.

First, the series combination of

12\text{ }\mu\text{F}

and

6\text{ }\mu\text{F}

yields an equivalent branch capacitance of

4\text{ }\mu\text{F}

. Second, using

E = \frac{1}{2} C_{eq} V^2

with

E = 0.100\text{ J}

and

V = 100\text{ V}

gives a total circuit equivalent capacitance of

20\text{ }\mu\text{F}

. Finally, subtracting the branch capacitance from the total parallel equivalent capacitance gives

C_3 = 20\text{ }\mu\text{F} - 4\text{ }\mu\text{F} = 16\text{ }\mu\text{F}

.

Step-by-Step Solution

1

Calculate the effective capacitance of the series branch containing

C_1

and

C_2

C_{12} = 4\text{ }\mu\text{F}

Capacitors in series combine reciprocally:

\frac{1}{C_{12}} = \frac{1}{C_1} + \frac{1}{C_2} = \frac{1}{12} + \frac{1}{6} = \frac{3}{12} \implies C_{12} = 4\text{ }\mu\text{F}

.

2

Determine the total equivalent capacitance

C_{eq}

of the circuit using the given stored energy and voltage

C_{eq} = 20\text{ }\mu\text{F}

Energy stored in a capacitor network is

E = \frac{1}{2} C_{eq} V^2

. Rearranging gives

C_{eq} = \frac{2E}{V^2} = \frac{2 \times 0.100\text{ J}}{(100\text{ V})^2} = 20 \times 10^{-6}\text{ F} = 20\text{ }\mu\text{F}

.

3

Calculate the unknown capacitance

C_3

from the parallel combination formula

C_3 = 16\text{ }\mu\text{F}

Because the branch

C_{12}

and

C_3

are in parallel,

C_{eq} = C_{12} + C_3 \implies 20\text{ }\mu\text{F} = 4\text{ }\mu\text{F} + C_3 \implies C_3 = 16\text{ }\mu\text{F}

.

Key Concept

Series and parallel combinations of capacitors combined with electrostatic energy storage

Estimated Time:2m 0s

Question 834Question →

A student obtained a mean score of $62$ in $5$ subjects. After the score of a $6\text{th}$ subject was added, the overall mean score became $65$ . What is the score obtained in the $6\text{th}$ subject?

Show answer & explanation

Answer: 80

Answer

The score obtained in the 6th subject is 80.

The sum of scores for the first 5 subjects is

5 \times 62 = 310

. With the 6th subject included, the total sum of scores becomes

6 \times 65 = 390

. The score of the 6th subject is the difference between these two totals:

390 - 310 = 80

.

Step-by-Step Solution

1

Calculate the total score of the initial 5 subjects

310

The sum of scores for ungrouped data is equal to the number of data items multiplied by the mean: 5 × 62 = 310.

2

Calculate the total score of all 6 subjects after including the new score

390

The new total score is obtained by multiplying the new count of subjects by the new mean: 6 × 65 = 390.

3

Determine the score of the 6th subject

80

The difference between the total score of 6 subjects and the total score of 5 subjects gives the score of the 6th subject: 390 - 310 = 80.

Key Concept

Calculating a missing data value given the mean of ungrouped data before and after addition

Question 835Question →

A brass container with an initial capacity of $500\text{ cm}^3$ at $20^\circ\text{C}$ is filled completely with ethanol. When the container and its contents are uniformly heated to $70^\circ\text{C}$ , a volume of $12\text{ cm}^3$ of ethanol overflows from the container. Given that the linear expansivity of brass is $2.0 \times 10^{-5}\text{ K}^{-1}$ , calculate the real cubic expansivity of ethanol in units of $10^{-4}\text{ K}^{-1}$ .

Show answer & explanation

Answer: 5.4

Answer

The real cubic expansivity of ethanol is

5.4 \times 10^{-4}\text{ K}^{-1}

, which corresponds to a value of

5.4

in units of

10^{-4}\text{ K}^{-1}

.

The real volume increase of a liquid consists of the apparent expansion (observed overflow volume) plus the volume expansion of the container itself. By finding the apparent cubic expansivity

\gamma_a = \frac{12}{500 \times 50} = 4.8 \times 10^{-4}\text{ K}^{-1}

and adding the vessel's cubic expansivity

\gamma_v = 3 \times 2.0 \times 10^{-5} = 0.6 \times 10^{-4}\text{ K}^{-1}

, we obtain the real cubic expansivity

\gamma_r = 5.4 \times 10^{-4}\text{ K}^{-1}

, which yields

5.4

in the required units.

Step-by-Step Solution

1

Determine the temperature increase of the system

\Delta T = 70^\circ\text{C} - 20^\circ\text{C} = 50\text{ K}

The thermal expansion is driven by the change in temperature.

2

Calculate the apparent cubic expansivity of ethanol (\gamma_a)

\gamma_a = \frac{\Delta V_{overflow}}{V_0 \cdot \Delta T} = \frac{12\text{ cm}^3}{500\text{ cm}^3 \times 50\text{ K}} = 4.8 \times 10^{-4}\text{ K}^{-1}

The overflow volume represents the apparent volume increase of the liquid relative to the expanding vessel.

3

Calculate the cubic expansivity of the brass vessel (\gamma_v)

\gamma_v = 3 \times \alpha_{brass} = 3 \times 2.0 \times 10^{-5}\text{ K}^{-1} = 6.0 \times 10^{-5}\text{ K}^{-1} = 0.6 \times 10^{-4}\text{ K}^{-1}

Cubic expansivity of a solid vessel is three times its linear expansivity.

4

Compute the real cubic expansivity of ethanol (\gamma_r)

\gamma_r = \gamma_a + \gamma_v = 4.8 \times 10^{-4}\text{ K}^{-1} + 0.6 \times 10^{-4}\text{ K}^{-1} = 5.4 \times 10^{-4}\text{ K}^{-1}

The real expansion of a liquid is the sum of its apparent expansion and the expansion of the containing vessel.

Key Concept

Relationship between Real and Apparent Cubic Expansivity of Liquids

Question 836Question →

A neutral insulated conductor loses $5.0 \times 10^{13}$ electrons during an electrostatics experiment. What is the magnitude of the net electric charge, in microcoulombs ( $\mu\text{C}$ ), acquired by the conductor? (Take elementary charge $e = 1.6 \times 10^{-19}\text{ C}$ )

Show answer & explanation

Answer: 8

Answer

The magnitude of the net electric charge acquired by the conductor is

8\text{ }\mu\text{C}

.

According to the principle of charge quantization, the total electric charge

Q

is calculated using

Q = n e

. Multiplying

5.0 \times 10^{13}

electrons by

1.6 \times 10^{-19}\text{ C}

yields

8.0 \times 10^{-6}\text{ C}

, which converts to

8\text{ }\mu\text{C}

.

Step-by-Step Solution

1

Identify the relevant formula for quantization of charge.

The net charge acquired is given by

Q = n e

.

Electric charge is quantized, so the total charge magnitude equals the number of transferred electrons multiplied by the magnitude of charge on a single electron.

2

Calculate the magnitude of charge in Coulombs.

Q = (5.0 \times 10^{13}) \times (1.6 \times 10^{-19}\text{ C}) = 8.0 \times 10^{-6}\text{ C}

.

Multiplying the quantity of removed electrons by the elementary charge gives total charge in Coulombs.

3

Convert the calculated value from Coulombs to microcoulombs.

8.0 \times 10^{-6}\text{ C} = 8\text{ }\mu\text{C}

.

Since

1\text{ }\mu\text{C} = 10^{-6}\text{ C}

, dividing

8.0 \times 10^{-6}

by

10^{-6}

yields 8.

Key Concept

Quantization of Electric Charge

Question 837Question →

In a double-slit experiment, monochromatic light of wavelength $500\text{ nm}$ is incident normally on two narrow slits. If the third-order ( $m = 3$ ) bright fringe is observed at an angle of $30^\circ$ from the central maximum, what is the slit separation, $d$ , in micrometers ( $\mu\text{m}$ )?

Show answer & explanation

Answer: 3

Answer

The slit separation is

3.0\text{ }\mu\text{m}

.

For bright fringes in a double-slit setup, constructive interference occurs when

d \sin\theta = m\lambda

. Rearranging for slit separation yields

d = \frac{m\lambda}{\sin\theta}

. Substituting

m = 3

,

\lambda = 0.50\text{ }\mu\text{m}

, and

\theta = 30^\circ

gives

d = \frac{3 \times 0.50\text{ }\mu\text{m}}{\sin 30^\circ} = \frac{1.50\text{ }\mu\text{m}}{0.50} = 3.0\text{ }\mu\text{m}

.

Step-by-Step Solution

1

Identify the constructive interference equation for Young's double-slit experiment.

d \sin\theta = m\lambda

Bright fringes occur where waves from the two slits interfere constructively, which corresponds to path differences equal to integer multiples of the wavelength.

2

Convert the given wavelength to micrometers and substitute known values.

\lambda = 0.50\text{ }\mu\text{m}

,

m = 3

,

\sin(30^\circ) = 0.50

Converting units to micrometers early simplifies direct calculation of

d

in

\mu\text{m}

.

3

Rearrange the equation and evaluate for

d

.

d = \frac{3 \times 0.50\text{ }\mu\text{m}}{0.50} = 3.0\text{ }\mu\text{m}

Dividing the numerator by

\sin(30^\circ) = 0.50

doubles the value of

m\lambda

.

Key Concept

Angular position condition for constructive interference in double-slit diffraction

Question 838Question →

If $y = \frac{e^{3x}}{\cos x + \sin x}$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Show answer & explanation

Answer: 2

Answer

The value of

\frac{dy}{dx}

at

x = 0

is

2

.

Applying the quotient rule

\frac{dy}{dx} = \frac{v u' - u v'}{v^2}

to

y = \frac{e^{3x}}{\cos x + \sin x}

yields

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

. Evaluating this expression at

x = 0

gives

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

.

Step-by-Step Solution

1

Identify numerator

u(x)

and denominator

v(x)

for the quotient rule.

u(x) = e^{3x}

and

v(x) = \cos x + \sin x

.

The function

y

is expressed as a quotient of exponential and trigonometric functions.

2

Find the first derivatives of

u(x)

and

v(x)

.

u'(x) = 3e^{3x}

and

v'(x) = -\sin x + \cos x

.

Derivative of

e^{kx}

is

k e^{kx}

, derivative of

\cos x

is

-\sin x

, and derivative of

\sin x

is

\cos x

.

3

Substitute

u(x)

,

v(x)

,

u'(x)

, and

v'(x)

into the quotient rule formula

\frac{dy}{dx} = \frac{u'v - uv'}{v^2}

.

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

.

The quotient rule is required to differentiate

\frac{u(x)}{v(x)}

.

4

Evaluate the derivative expression at

x = 0

.

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

.

At

x = 0

,

e^0 = 1

,

\cos 0 = 1

, and

\sin 0 = 0

.

Key Concept

Differentiation of exponential and trigonometric functions using the quotient rule

Question 839Question →

A progressive wave traveling through a uniform medium is described by the displacement equation $y = 0.04 \sin\left(150\pi t - 6\pi x\right)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. What is the speed of propagation of the wave?

Show answer & explanation

Answer: 25

Answer

The speed of propagation of the wave is 25 m/s.

By matching

y = 0.04 \sin(150\pi t - 6\pi x)

to the standard wave equation

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

, we obtain

\omega = 150\pi\text{ rad/s}

and

k = 6\pi\text{ rad/m}

. The speed of the wave

v

is calculated as

v = \frac{\omega}{k} = \frac{150\pi}{6\pi} = 25\text{ m/s}

.

Step-by-Step Solution

1

Compare the given wave equation with the general progressive wave equation

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

.

Angular frequency

\omega = 150\pi\text{ rad/s}

and wave number

k = 6\pi\text{ rad/m}

.

Matching coefficients allows direct extraction of angular frequency and spatial wave number.

2

Calculate the wave speed using the relationship

v = \frac{\omega}{k}

.

v = \frac{150\pi}{6\pi} = 25\text{ m/s}

.

The velocity of a progressive wave is equal to the ratio of its angular frequency to its wave number.

Key Concept

Determining wave velocity from progressive wave equation parameters

Question 840Question →

According to Bohr's model of the hydrogen atom, the total energy of an electron in the first excited state ( $n = 2$ ) is $-3.40\text{ eV}$ . Calculate the electric potential energy of the electron in this state, in electron-volts ( $\text{eV}$ ).

Show answer & explanation

Answer: -6.8

Answer

The electric potential energy of the electron in the first excited state is

-6.80\text{ eV}

.

In Bohr's model of the hydrogen atom, an electron held in a circular orbit by electrostatic attraction has a kinetic energy

K = -E

and an electric potential energy

U = 2E

. Given a total energy

E = -3.40\text{ eV}

, multiplying by 2 yields

U = -6.80\text{ eV}

.

Step-by-Step Solution

1

Identify the relationship between total energy and electric potential energy in the Bohr atomic model.

U = 2E

, where

U

is potential energy and

E

is total energy.

For an inverse-square electrostatic force binding an electron in a circular orbit, the virial theorem dictates that kinetic energy

K = -E

and potential energy

U = 2E

.

2

Substitute the given value of total energy

E = -3.40\text{ eV}

to solve for

U

.

U = 2 \times (-3.40\text{ eV}) = -6.80\text{ eV}

.

Direct multiplication yields the exact electric potential energy of the bound electron.

Key Concept

Energy components (kinetic, potential, and total energy) of an electron in Bohr's atomic model

Answer

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