Tüm alıştırma soruları

Evaluate the numerical expression $\frac{0.00054 \times 0.002}{0.00036}$ and state the final result as a decimal.

Cevabı ve açıklamayı göster

Cevap: 0.003

Cevap

The correct value as a decimal is 0.003.

Converting all terms to scientific notation yields

\frac{(5.4 \times 10^{-4}) \times (2 \times 10^{-3})}{3.6 \times 10^{-4}} = \frac{10.8 \times 10^{-7}}{3.6 \times 10^{-4}} = 3 \times 10^{-3} = 0.003

.

Adım Adım Çözüm

1

Convert each decimal in the expression to scientific notation (standard form)

0.00054 = 5.4 \times 10^{-4}

,

0.002 = 2 \times 10^{-3}

,

0.00036 = 3.6 \times 10^{-4}

Converting decimals with leading zeros to powers of 10 prevents errors in decimal point alignment during multiplication and division.

2

Simplify the numerator by multiplying coefficients and adding exponents

(5.4 \times 10^{-4}) \times (2 \times 10^{-3}) = 10.8 \times 10^{-7}

According to the laws of indices,

10^a \times 10^b = 10^{a+b}

, so

-4 + (-3) = -7

.

3

Divide the simplified numerator by the denominator

\frac{10.8 \times 10^{-7}}{3.6 \times 10^{-4}} = \left(\frac{10.8}{3.6}\right) \times 10^{-7 - (-4)} = 3.0 \times 10^{-3}

Dividing the coefficients gives

10.8 \div 3.6 = 3

, and subtracting the exponents gives

-7 - (-4) = -3

.

4

Express the result in standard decimal form

3.0 \times 10^{-3} = 0.003

Shifting the decimal point 3 positions to the left converts

10^{-3}

to standard decimal representation.

Anahtar Kavram

Simplifying Decimal Expressions using Standard Form and Laws of Indices

Soru 7742Soru →

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 7743Soru →

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 7744Soru →

Match each type of wave listed on the left with its correct classification and propagation characteristic on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Sound wave in air matches Mechanical longitudinal wave requiring a material medium; Radio wave in vacuum matches Electromagnetic transverse wave capable of traveling without a medium; Water ripple on a lake surface matches Mechanical transverse wave propagating along a liquid surface.

Each wave is correctly paired based on whether it needs a physical medium to propagate (mechanical waves require a medium, electromagnetic waves do not) and whether the displacement is parallel (longitudinal) or perpendicular (transverse) to the direction of energy propagation.

Adım Adım Çözüm

1

Identify the medium requirement and vibration direction for a sound wave in air.

Sound waves require a material medium (air) and vibrate parallel to the direction of wave movement, making them mechanical longitudinal waves.

Classification depends on whether a physical medium is needed and how particles oscillate relative to energy transport.

2

Identify the medium requirement and vibration direction for a radio wave in a vacuum.

Radio waves can travel through empty space without a material medium and consist of field oscillations perpendicular to propagation, making them electromagnetic transverse waves.

Electromagnetic waves propagate via mutually perpendicular electric and magnetic field oscillations and require no medium.

3

Identify the medium requirement and vibration direction for a water ripple.

Ripples require a material medium (water) and displace surface water up and down perpendicular to wave travel, making them mechanical transverse waves.

Surface water waves exhibit transverse displacement characteristics in a physical liquid medium.

Anahtar Kavram

Classification of waves based on medium requirement (mechanical vs. electromagnetic) and particle vibration direction relative to propagation (transverse vs. longitudinal).

Soru 7745Soru →

A wheel is divided into $10$ equal sectors numbered $1$ through $10$ . In an initial experiment, the wheel is spun $400$ times, and a prime number is recorded $180$ times. Additional spins are to be conducted, all of which result in non-prime numbers. How many additional spins must be performed so that the overall experimental relative frequency of landing on a prime number equals the theoretical probability of landing on a prime number?

50

100

200

450

Cevabı ve açıklamayı göster

Cevap: $50$

Cevap

50 additional spins

The correct value is 50. The theoretical probability of landing on a prime number from 1 to 10 is 4/10 = 2/5, since the prime numbers are 2, 3, 5, and 7. With 180 prime outcomes in 400 spins, adding N non-prime spins yields an experimental relative frequency of 180 / (400 + N). Setting 180 / (400 + N) = 2/5 yields 800 + 2N = 900, giving N = 50.

Adım Adım Çözüm

1

Determine the theoretical probability of landing on a prime number.

The prime numbers between

1

and

10

are

2, 3, 5,

and

7

. Thus, there are

4

favorable outcomes out of

10

, giving

P(\text{theoretical}) = \frac{4}{10} = \frac{2}{5}

.

Theoretical probability is the ratio of favorable outcomes to the total number of equally likely outcomes.

2

Formulate the expression for experimental relative frequency after

N

additional non-prime spins.

The number of prime occurrences remains

180

, while the total number of spins becomes

400 + N

. Thus,

P(\text{experimental}) = \frac{180}{400 + N}

.

Since all

N

additional spins produce non-prime numbers, the count of prime outcomes does not increase, but the total number of trials increases by

N

.

3

Equate the experimental relative frequency to the theoretical probability and solve for

N

.

\frac{180}{400 + N} = \frac{2}{5} \implies 2(400 + N) = 180 \times 5 \implies 800 + 2N = 900 \implies 2N = 100 \implies N = 50.

Setting the experimental relative frequency equal to the theoretical probability allows determining the exact number of additional non-prime spins needed.

Anahtar Kavram

Theoretical vs Experimental Probability

Tahmini Süre:2m 0s

Soru 7746Soru →

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 7747Soru →

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 7748Soru →

A harmonic wave traveling through an initial elastic medium is represented by the wave equation $y = 0.04 \cos\left(50\pi t - \frac{\pi}{4} x\right)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. When this wave passes into a second medium, its propagation speed doubles. What is the wavelength of the wave in the second medium?

4.0\text{ m}

8.0\text{ m}

16.0\text{ m}

32.0\text{ m}

Cevabı ve açıklamayı göster

Cevap: $16.0\text{ m}$

Cevap

The wavelength of the wave in the second medium is

16.0\text{ m}

.

The wave's angular frequency

\omega = 50\pi\text{ rad/s}

corresponds to a source frequency of

f = 25\text{ Hz}

, and its wave number

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

corresponds to an initial wavelength

\lambda_1 = 8.0\text{ m}

. The wave speed in the first medium is

v_1 = f \lambda_1 = 200\text{ m/s}

. In the second medium, the wave speed doubles to

v_2 = 400\text{ m/s}

. Because frequency is determined by the source and remains invariant during refraction across media boundaries (

f_2 = f_1 = 25\text{ Hz}

), the new wavelength becomes

\lambda_2 = \frac{v_2}{f} = \frac{400\text{ m/s}}{25\text{ Hz}} = 16.0\text{ m}

.

Adım Adım Çözüm

1

Extract angular frequency

\omega

and wave number

k

from the general wave equation

y = A \cos(\omega t - k x)

.

\omega = 50\pi\text{ rad/s}

and

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

.

Matching coefficients in the standard wave equation provides the temporal and spatial frequencies of the wave.

2

Calculate the frequency

f

and wavelength

\lambda_1

in the first medium.

f = \frac{\omega}{2\pi} = \frac{50\pi}{2\pi} = 25\text{ Hz}

and

\lambda_1 = \frac{2\pi}{k} = \frac{2\pi}{\pi/4} = 8.0\text{ m}

.

Fundamental wave relationships connect angular frequency to frequency and wave number to wavelength.

3

Determine the wave speed

v_1

in the first medium and

v_2

in the second medium.

v_1 = f \lambda_1 = 25 \times 8.0 = 200\text{ m/s}

. Therefore,

v_2 = 2 \times v_1 = 400\text{ m/s}

.

The problem states that wave propagation speed doubles upon entering the second medium.

4

Calculate the wavelength

\lambda_2

in the second medium using constant frequency

f = 25\text{ Hz}

.

\lambda_2 = \frac{v_2}{f} = \frac{400}{25} = 16.0\text{ m}

.

When a wave crosses a boundary between two media, its frequency is determined solely by the source and remains constant.

Anahtar Kavram

Frequency invariance across media boundaries and the wave speed equation

v = f \lambda

Tahmini Süre:1m 30s

Soru 7749Soru →

The first, third, and seventh terms of a non-constant arithmetic progression (AP) form the first three consecutive terms of a geometric progression (GP). If the first term of the AP is $4$ , what is the sum of the first $4$ terms of the geometric progression?

60

28

124

92

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The sum of the first

4

terms of the geometric progression is

60

.

The first three terms of the GP are

T_1 = 4

,

T_3 = 4 + 2d

, and

T_7 = 4 + 6d

. Equating

(4 + 2d)^2 = 4(4 + 6d)

yields

4d^2 - 8d = 0

, giving

d = 2

. The first four terms of the GP are

4, 8, 16, 32

, which sum to

4 + 8 + 16 + 32 = 60

.

Adım Adım Çözüm

1

Express the terms of the arithmetic progression in terms of first term

a

and common difference

d

.

First term

T_1 = 4

, third term

T_3 = 4 + 2d

, seventh term

T_7 = 4 + 6d

.

The

n^{\text{th}}

term of an AP is defined as

T_n = a + (n-1)d

.

2

Set up the geometric progression condition

(T_3)^2 = T_1 \cdot T_7

to solve for

d

.

(4 + 2d)^2 = 4(4 + 6d) \implies 16 + 16d + 4d^2 = 16 + 24d \implies 4d^2 - 8d = 0 \implies d = 2

(since the AP is non-constant,

d \neq 0

).

Three terms

x, y, z

form a GP if and only if

y^2 = xz

.

3

Determine the terms and common ratio

r

of the GP.

First term

G_1 = 4

, second term

G_2 = 4 + 2(2) = 8

. Thus,

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

.

The common ratio

r

is the quotient of consecutive terms of the GP.

4

Calculate the sum of the first

4

terms of the GP using

S_n = \frac{a(r^n - 1)}{r - 1}

.

S_4 = \frac{4(2^4 - 1)}{2 - 1} = \frac{4(16 - 1)}{1} = 60

.

Formula for the sum of the first

n

terms of a geometric progression.

Anahtar Kavram

Arithmetic and Geometric Progression Inter-relationships

Soru 7750Soru →

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 7751Soru →

Two capacitors with capacitances of $3.0\text{ }\mu\text{F}$ and $6.0\text{ }\mu\text{F}$ are connected in parallel across a direct-current source. What is the equivalent capacitance of this combination?

9.0\text{ }\mu\text{F}

2.0\text{ }\mu\text{F}

4.5\text{ }\mu\text{F}

18.0\text{ }\mu\text{F}

Cevabı ve açıklamayı göster

Cevap: $9.0\text{ }\mu\text{F}$

Cevap

The equivalent capacitance of the parallel combination is

9.0\text{ }\mu\text{F}

.

When capacitors are connected in parallel, each capacitor experiences the full potential difference of the voltage source, and the total charge stored is the sum of individual charges (

Q_{total} = Q_1 + Q_2

). Thus, the equivalent capacitance is the direct sum of the individual capacitances:

C_{eq} = 3.0\text{ }\mu\text{F} + 6.0\text{ }\mu\text{F} = 9.0\text{ }\mu\text{F}

.

Adım Adım Çözüm

1

Identify the combination rule for parallel capacitors.

The total capacitance is given by

C_{eq} = C_1 + C_2

.

Capacitors in parallel share the same potential difference, so total charge stored is the sum of individual charges.

2

Substitute the given values into the parallel capacitance formula.

C_{eq} = 3.0\text{ }\mu\text{F} + 6.0\text{ }\mu\text{F} = 9.0\text{ }\mu\text{F}

.

Direct addition yields the total equivalent capacitance.

Anahtar Kavram

Parallel Combination of Capacitors

Tahmini Süre:45s

Soru 7752Soru →

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 7753Soru →

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 7754Soru →

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 7755Soru →

Match each physical quantity on the left with its corresponding SI unit expressed in terms of fundamental (base) units on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Frequency matches

\text{s}^{-1}

, Electric charge matches

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

, Mass density matches

\text{kg}\cdot\text{m}^{-3}

, and Acceleration matches

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

.

Frequency (

f = 1/T

) is expressed in reciprocal seconds (

\text{s}^{-1}

). Electric charge (

Q = I \cdot t

) is current times time, giving

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

. Mass density (

\rho = m/V

) is mass per volume, giving

\text{kg}\cdot\text{m}^{-3}

. Acceleration (

a = \Delta v / \Delta t

) is rate of velocity change, giving

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

.

Adım Adım Çözüm

1

Identify the defining formula for each physical quantity

Frequency

f = \frac{1}{T}

, Charge

Q = I \cdot t

, Density

\rho = \frac{m}{V}

, Acceleration

a = \frac{\Delta v}{\Delta t}

.

Relating derived quantities to their defining equations allows reduction into fundamental quantities.

2

Substitute the SI base units for mass (

\text{kg}

), length (

\text{m}

), time (

\text{s}

), and current (

\text{A}

)

Frequency:

\text{s}^{-1}

; Charge:

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

; Density:

\frac{\text{kg}}{\text{m}^3} = \text{kg}\cdot\text{m}^{-3}

; Acceleration:

\frac{\text{m/s}}{\text{s}} = \text{m}\cdot\text{s}^{-2}

.

This expresses each derived unit strictly in terms of fundamental SI units.

3

Match each physical quantity to its calculated base unit representation

Frequency

\rightarrow \text{s}^{-1}

, Electric charge

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

, Mass density

\rightarrow \text{kg}\cdot\text{m}^{-3}

, Acceleration

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

.

Completes the pairing verification.

Anahtar Kavram

Expressing derived physical quantities in terms of SI fundamental (base) units

Tahmini Süre:45s

Soru 7756Soru →

Two identical air-filled parallel-plate capacitors, $C_1$ and $C_2$ , each of capacitance $C$ , are connected in series across a direct-current voltage source of potential difference $V$ . While the circuit remains connected to the voltage source, a dielectric slab of relative permittivity $\varepsilon_r = 3$ is fully inserted into $C_1$ , completely filling the space between its plates. What is the ratio of the electrostatic energy stored in $C_1$ after inserting the dielectric to the energy stored in $C_1$ before the insertion?

3 : 4

1 : 3

3 : 1

27 : 4

Cevabı ve açıklamayı göster

Cevap: $3 : 4$

Cevap

The ratio of the energy stored in the first capacitor after dielectric insertion to before insertion is

3 : 4

(or

0.75

).

Initially, the two identical capacitors divide the total source voltage

V

equally, giving

V_1 = V/2

and initial energy

U_{1,i} = \frac{1}{8} C V^2

. When the dielectric of relative permittivity

3

is inserted, the capacitance of the first capacitor becomes

3C

. In a series circuit connected to a constant voltage source, the total charge becomes

Q = C_{eq}V = \frac{3}{4}CV

, which reduces the voltage across the modified capacitor to

V_1' = \frac{Q}{3C} = \frac{V}{4}

. The new stored energy is

U_{1,f} = \frac{1}{2}(3C)(\frac{V}{4})^2 = \frac{3}{32} C V^2

. Dividing

U_{1,f}

by

U_{1,i}

yields

\frac{3/32}{1/8} = \frac{3}{4}

.

Adım Adım Çözüm

1

Calculate initial capacitance and voltage across

C_1

Initial capacitance

C_{1,i} = C

. Since

C_1

and

C_2

are identical and in series, initial potential difference across

C_1

is

V_{1,i} = \frac{V}{2}

.

Equal capacitors in series divide total voltage equally.

2

Calculate initial electrostatic energy stored in

C_1

U_{1,i} = \frac{1}{2} C_{1,i} V_{1,i}^2 = \frac{1}{2} C \left(\frac{V}{2}\right)^2 = \frac{1}{8} C V^2

.

Formula for energy stored in a capacitor is

U = \frac{1}{2} C V^2

.

3

Determine final capacitance of

C_1

and new voltage division

New capacitance

C_{1,f} = \varepsilon_r C = 3C

. Total equivalent capacitance

C_{eq} = \frac{3C \cdot C}{3C + C} = \frac{3}{4} C

. Total charge supplied

Q = C_{eq} V = \frac{3}{4} C V

. Final voltage across

C_1

is

V_{1,f} = \frac{Q}{C_{1,f}} = \frac{\frac{3}{4} C V}{3C} = \frac{V}{4}

.

Dielectric increases capacitance by factor

\varepsilon_r

, altering equivalent capacitance and potential distribution in series.

4

Calculate final electrostatic energy in

C_1

and compute the ratio

U_{1,f} = \frac{1}{2} (3C) \left(\frac{V}{4}\right)^2 = \frac{3}{32} C V^2

. Ratio

\frac{U_{1,f}}{U_{1,i}} = \frac{\frac{3}{32} C V^2}{\frac{1}{8} C V^2} = \frac{3}{4}

.

Divide final stored energy by initial stored energy.

Anahtar Kavram

Series combination of capacitors with dielectric insertion under constant battery voltage

Soru 7757Soru →

In $\Delta KLM$ , side $k = 5\text{ cm}$ , side $l = 5\sqrt{3}\text{ cm}$ , and $\angle K = 30^\circ$ . If $\angle L$ is an obtuse angle, what is the measure of $\angle L$ ?

60^\circ

120^\circ

135^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $120^\circ$

Cevap

The measure of angle

L

is

120^\circ

.

Applying the Sine Rule

\frac{k}{\sin K} = \frac{l}{\sin L}

gives

\sin L = \frac{5\sqrt{3} \cdot \sin 30^\circ}{5} = \frac{\sqrt{3}}{2}

. The inverse sine gives an acute angle of

60^\circ

. Because the problem specifies that angle

L

is obtuse, we find its supplementary angle in the second quadrant:

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

.

Adım Adım Çözüm

1

Apply the Sine Rule relating sides

k, l

and angles

K, L

.

\frac{k}{\sin K} = \frac{l}{\sin L}

The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.

2

Substitute the given values into the Sine Rule equation.

\frac{5}{\sin 30^\circ} = \frac{5\sqrt{3}}{\sin L} \implies \frac{5}{0.5} = \frac{5\sqrt{3}}{\sin L} \implies 10 = \frac{5\sqrt{3}}{\sin L}

Known values are

k = 5\text{ cm}

,

l = 5\sqrt{3}\text{ cm}

, and

\sin 30^\circ = \frac{1}{2}

.

3

Solve for

\sin L

.

\sin L = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}

Isolating

\sin L

yields the principal ratio.

4

Determine the obtuse angle solution for

L

.

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

Since

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

, the acute solution is

60^\circ

, so the supplementary obtuse solution is

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

.

Anahtar Kavram

Ambiguous Case of the Sine Rule

Tahmini Süre:1m 15s

Soru 7758Soru →

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 7759Soru →

Which of the following is the inverse of the matrix $M = \begin{pmatrix} 5 & 2 \\ 7 & 3 \end{pmatrix}$ ?

\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

\begin{pmatrix} 3 & 2 \\ 7 & 5 \end{pmatrix}

\begin{pmatrix} -5 & 2 \\ 7 & -3 \end{pmatrix}

\begin{pmatrix} 5 & 7 \\ 2 & 3 \end{pmatrix}

Cevabı ve açıklamayı göster

Cevap: $\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}$

Cevap

\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

For any non-singular 2x2 matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

, the inverse is

\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

. Here,

ad - bc = (5)(3) - (2)(7) = 1

, so the inverse matrix is

\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

.

Adım Adım Çözüm

1

Calculate the determinant of matrix M

\det(M) = (5)(3) - (2)(7) = 15 - 14 = 1

For a 2x2 matrix

M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

, the determinant is given by

\det(M) = ad - bc

.

2

Find the adjugate matrix of M

\text{adj}(M) = \begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

The adjugate of a 2x2 matrix is obtained by swapping the main diagonal elements (

a

and

d

) and negating the off-diagonal elements (

b

and

c

).

3

Apply the matrix inverse formula

M^{-1} = \frac{1}{\det(M)} \text{adj}(M) = \frac{1}{1}\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix} = \begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

Multiplying the adjugate matrix by the reciprocal of the determinant gives the exact inverse matrix.

Anahtar Kavram

Inverse of a 2x2 Matrix

Soru 7760Soru →

In a secondary school class of 45 students, 28 study Chemistry, 25 study Physics, and 6 study neither of the two subjects. How many students study both Chemistry and Physics?

Cevabı ve açıklamayı göster

Cevap: 14; 14 students

Cevap

14 students study both Chemistry and Physics.

First, find the number of students taking at least one subject by subtracting those taking neither from the total: 45 - 6 = 39. Then, apply the formula n(C ∪ P) = n(C) + n(P) - n(C ∩ P). Substituting the values gives 39 = 28 + 25 - n(C ∩ P), which simplifies to n(C ∩ P) = 53 - 39 = 14.

Adım Adım Çözüm

1

Calculate the number of students taking at least one of the two subjects.

n(Chemistry ∪ Physics) = 45 - 6 = 39

Students who study neither subject must be subtracted from the total class population to find the cardinality of the union.

2

Set up the two-set inclusion-exclusion formula.

n(Chemistry ∪ Physics) = n(Chemistry) + n(Physics) - n(Chemistry ∩ Physics)

The sum of individual set cardinalities overcounts elements present in both sets.

3

Substitute the known values into the equation and solve for the intersection.

39 = 28 + 25 - n(Chemistry ∩ Physics) ⇒ n(Chemistry ∩ Physics) = 53 - 39 = 14

Subtracting 39 from 53 gives the exact number of students taking both subjects.

Anahtar Kavram

Two-set inclusion-exclusion principle and complement of a set

Tahmini Süre:1m 30s

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Anahtar Kavram