Tüm alıştırma soruları

Soru 821Soru →

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 $110^\circ\text{C}$ ?

Cevabı ve açıklamayı göster

Cevap: 15.3

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Real and Apparent Cubical Expansivity of Liquids

Tahmini Süre:1m 30s

Soru 822Soru →

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.

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Determinant Product Property and 2x2 Matrix Determinant

Tahmini Süre:1m 30s

Soru 823Soru →

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 $a - b$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 824Soru →

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

Cevabı ve açıklamayı göster

Cevap: 7.2

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Linear Expansivity and Thermal Expansion of Solids

Tahmini Süre:1m 30s

Soru 825Soru →

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 $x \cdot y$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Dimensional Homogeneity and Exponent Analysis

Soru 826Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

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 $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$.

Adım Adım Çözüm

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: $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$.

Anahtar Kavram

Combinations with restricted or fixed choices

Soru 827Soru →

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.

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 828Soru →

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 $k$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Matrix Polynomial Equations and Cayley-Hamilton Theorem

Soru 829Soru →

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

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Permutations of

n

distinct items taken

r

at a time

Tahmini Süre:1m 0s

Soru 830Soru →

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 ( $\text{nm}$ ).

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

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}

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

De Broglie Standing Wave Quantization in Bohr Atomic Model

Soru 831Soru →

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 $\det(A)$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Tahmini Süre:1m 30s

Soru 832Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 54

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Uniform Linear Acceleration

Soru 833Soru →

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 ( $\mu\text{F}$ )?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

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}

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

Series and parallel combinations of capacitors combined with electrostatic energy storage

Tahmini Süre:2m 0s

Soru 834Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 835Soru →

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

Cevabı ve açıklamayı göster

Cevap: 5.4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Relationship between Real and Apparent Cubic Expansivity of Liquids

Soru 836Soru →

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 $e = 1.6 \times 10^{-19}\text{ C}$ )

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Quantization of Electric Charge

Soru 837Soru →

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 ( $\mu\text{m}$ )?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

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}

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Angular position condition for constructive interference in double-slit diffraction

Soru 838Soru →

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

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Differentiation of exponential and trigonometric functions using the quotient rule

Soru 839Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Determining wave velocity from progressive wave equation parameters

Soru 840Soru →

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 ( $\text{eV}$ ).

Cevabı ve açıklamayı göster

Cevap: -6.8

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram