Tüm alıştırma soruları

Soru 741Soru →

The critical velocity $v_c$ of a fluid flowing through a cylindrical pipe of diameter $D$ depends on the dynamic viscosity $\eta$ of the fluid, its density $\rho$ , and the pipe diameter $D$ according to the empirical relationship $v_c = R_e \eta^x \rho^y D^z$ , where $R_e$ is the dimensionless Reynolds number. Using dimensional analysis, calculate the numerical value of the sum of the exponents $x + y + z$ .

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Cevap: -1

Cevap

The sum of the exponents

x + y + z

is

-1

.

Using the principle of dimensional homogeneity, the dimensions of both sides of the formula

v_c = R_e \eta^x \rho^y D^z

must be equal. Equating the powers of Mass, Length, and Time yields

x = 1

,

y = -1

, and

z = -1

. Summing these three values gives

1 + (-1) + (-1) = -1

.

Adım Adım Çözüm

1

Determine the dimensions of all physical quantities involved

Critical velocity

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

, dynamic viscosity

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

, fluid density

[\rho] = \text{M L}^{-3}

, and diameter

[D] = \text{L}

. The Reynolds number

R_e

is dimensionless.

Dimensional analysis requires replacing physical quantities with their base SI dimensions.

2

Formulate the dimensional homogeneity equation

\text{M}^0 \text{L}^1 \text{T}^{-1} = (\text{M L}^{-1} \text{T}^{-1})^x (\text{M L}^{-3})^y (\text{L})^z = \text{M}^{x+y} \text{L}^{-x-3y+z} \text{T}^{-x}.

By the principle of dimensional homogeneity, the total exponent of each fundamental dimension must match on both sides of the equation.

3

Solve the system of simultaneous linear equations for

x

,

y

, and

z

From

\text{T}

:

-x = -1 \implies x = 1

.
From

\text{M}

:

x + y = 0 \implies y = -1

.
From

\text{L}

:

-x - 3y + z = 1 \implies -1 + 3 + z = 1 \implies z = -1

.

Equating powers of fundamental quantities yields explicit values for each dimensional power.

4

Calculate the target sum

x + y + z

x + y + z = 1 + (-1) + (-1) = -1.

Combining the calculated exponents gives the required numerical value.

Anahtar Kavram

Principle of Dimensional Homogeneity and Derivation of Physical Formulas

Soru 742Soru →

Given the simultaneous equations $2x - y = 4$ and $x^2 + y^2 = 13$ , where both $x$ and $y$ are positive real numbers, calculate the value of $x + y$ .

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Cevap: 5

Cevap

The correct value of

x + y

is 5.

Substituting

y = 2x - 4

into

x^2 + y^2 = 13

gives

5x^2 - 16x + 3 = 0

, which yields

x = 3

or

x = 0.2

. The corresponding

y

-values are

y = 2

and

y = -3.6

. Since both

x

and

y

must be positive, the valid pair is

(3, 2)

, giving

x + y = 3 + 2 = 5

.

Adım Adım Çözüm

1

Express

y

in terms of

x

using the linear equation

2x - y = 4

.

y = 2x - 4

Substitution is the standard method for solving simultaneous linear and quadratic equations.

2

Substitute

y = 2x - 4

into the quadratic equation

x^2 + y^2 = 13

.

x^2 + (2x - 4)^2 = 13

This reduces the system to a single quadratic equation in one variable.

3

Expand and collect like terms.

5x^2 - 16x + 3 = 0

Expanding

(2x - 4)^2 = 4x^2 - 16x + 16

and subtracting 13 puts the quadratic in standard form

ax^2 + bx + c = 0

.

4

Solve the quadratic equation

5x^2 - 16x + 3 = 0

for

x

.

x = 3

or

x = 0.2

Factoring

(5x - 1)(x - 3) = 0

yields two real solutions for

x

.

5

Find corresponding values of

y

and apply the positivity constraint

x > 0

and

y > 0

.

x = 3, y = 2

When

x = 0.2

,

y = -3.6

, which is not positive. Hence,

(3, 2)

is the only valid solution pair.

6

Calculate

x + y

.

5

3 + 2 = 5

.

Anahtar Kavram

Solving simultaneous linear and quadratic equations using substitution

Tahmini Süre:1m 30s

Soru 743Soru →

A curve has a gradient function defined by $\frac{dy}{dx} = 12x^3 - 6\sin(3x) + 4$ . If the curve passes through the point $(0, 15)$ , what is the value of the constant of integration $C$ ?

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Cevap: 13

Cevap

The constant of integration

C

is 13.

Integrating

12x^3 - 6\sin(3x) + 4

with respect to

x

yields

y = 3x^4 + 2\cos(3x) + 4x + C

. Substituting

x = 0

and

y = 15

gives

15 = 3(0)^4 + 2\cos(0) + 4(0) + C = 2 + C

, which simplifies to

C = 13

.

Adım Adım Çözüm

1

Integrate the gradient function to obtain the general equation of the curve.

y = 3x^4 + 2\cos(3x) + 4x + C

The integral of

12x^3

is

3x^4

, the integral of

-6\sin(3x)

is

+2\cos(3x)

, and the integral of

4

is

4x

.

2

Apply the initial condition

(0, 15)

to solve for

C

.

15 = 3(0)^4 + 2\cos(0) + 4(0) + C

Substitute

x = 0

and

y = 15

into the integrated equation.

3

Calculate the numerical value of

C

.

C = 13

Since

\cos(0) = 1

, the equation becomes

15 = 2 + C

, yielding

C = 13

.

Anahtar Kavram

Indefinite Integration of Polynomial and Trigonometric Functions with Initial Conditions

Soru 744Soru →

Given the matrices $A = \begin{pmatrix} x & 2 & 1 \\ 3 & 0 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1 \\ 5 & 3 \end{pmatrix}$ , find the value of $x$ if $\det(A) + \det(B) = 5$ .

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Cevap: 2

Cevap

The value of x is 2.

Evaluating \det(B) gives 11, and cofactor expansion of \det(A) along the top row yields 10 - 8x. Summing these determinants gives (10 - 8x) + 11 = 5, which simplifies to 8x = 16, resulting in x = 2.

Adım Adım Çözüm

1

Calculate the determinant of the 2x2 matrix B

\det(B) = 11

For a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad - bc. Thus, \det(B) = (2)(3) - (-1)(5) = 6 + 5 = 11.

2

Calculate the determinant of the 3x3 matrix A in terms of x

\det(A) = 10 - 8x

Expanding along the first row: \det(A) = x\begin{vmatrix} 0 & 2 \\ 4 & 1 \end{vmatrix} - 2\begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix} + 1\begin{vmatrix} 3 & 0 \\ 1 & 4 \end{vmatrix} = x(0 - 8) - 2(3 - 2) + 1(12 - 0) = -8x - 2 + 12 = 10 - 8x.

3

Set up and solve the linear equation \det(A) + \det(B) = 5

x = 2

Substituting the computed determinants gives (10 - 8x) + 11 = 5, which simplifies to 21 - 8x = 5. Rearranging yields 8x = 16, so x = 2.

Anahtar Kavram

Determinants of 2x2 and 3x3 matrices and linear equation solving

Soru 745Soru →

A small charged sphere of mass $2.0 \times 10^{-4}\text{ kg}$ carrying a positive charge of $+4.0 \times 10^{-8}\text{ C}$ is suspended by a light insulating string between two vertical parallel plates. When a uniform horizontal electric field of magnitude $E$ is applied between the plates, the string deflects and comes to equilibrium at an angle of $45^\circ$ to the vertical. Taking the acceleration due to gravity $g = 10\text{ m}\cdot\text{s}^{-2}$ , calculate the magnitude of the electric field intensity $E$ in $\text{N}\cdot\text{C}^{-1}$ .

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Cevap: 50000

Cevap

The magnitude of the electric field intensity is

50000\text{ N}\cdot\text{C}^{-1}

(or

5.0 \times 10^4\text{ N}\cdot\text{C}^{-1}

).

In electrostatic equilibrium, the sphere experiences three forces: weight (

mg

) vertically downward, electrostatic force (

qE

) horizontally, and tension (

T

) along the thread at

45^\circ

to the vertical. Balancing components gives

T \sin 45^\circ = qE

and

T \cos 45^\circ = mg

. Dividing these yields

\tan 45^\circ = \frac{qE}{mg} = 1

, which gives

qE = mg

. Substituting the given values gives

E = \frac{2.0 \times 10^{-3}}{4.0 \times 10^{-8}} = 50000\text{ N}\cdot\text{C}^{-1}

.

Adım Adım Çözüm

1

Calculate the weight of the charged sphere.

W = mg = (2.0 \times 10^{-4}\text{ kg})(10\text{ m}\cdot\text{s}^{-2}) = 2.0 \times 10^{-3}\text{ N}

.

The weight provides the downward vertical force in equilibrium.

2

Relate the electrostatic force to the weight using the angle of deflection.

\tan(45^\circ) = \frac{F_e}{W} \implies 1 = \frac{F_e}{2.0 \times 10^{-3}\text{ N}} \implies F_e = 2.0 \times 10^{-3}\text{ N}

.

In electrostatic equilibrium, the ratio of the horizontal force to the vertical force equals the tangent of the angle with the vertical.

3

Calculate the electric field strength

E

using

F_e = qE

.

E = \frac{F_e}{q} = \frac{2.0 \times 10^{-3}\text{ N}}{4.0 \times 10^{-8}\text{ C}} = 50000\text{ N}\cdot\text{C}^{-1}

.

The electric field intensity is the electric force per unit charge.

Anahtar Kavram

Equilibrium of a charged body in a uniform electric field

Tahmini Süre:2m 0s

Soru 746Soru →

A trader bought a set of solar panels for $\text{₦}160,000$ . He marked up the cost price by $25\%$ to fix the marked price. During a trade fair, he offered a $10\%$ discount on the marked price. What is his net profit in Naira?

Cevabı ve açıklamayı göster

Cevap: 20000

Cevap

The net profit made by the trader is ��20,000.

First, find the marked price by adding a 25% markup to the cost price: ₦160,000 × 1.25 = ₦200,000. Next, calculate the selling price after a 10% discount on the marked price: ₦200,000 × 0.90 = ₦180,000. Finally, subtract the cost price from the selling price to find the net profit: ₦180,000 - ₦160,000 = ₦20,000.

Adım Adım Çözüm

1

Calculate the marked price

Marked Price = ₦200,000

The marked price is 125% of the original cost price of ₦160,000.

2

Calculate the selling price after discount

Selling Price = ₦180,000

A 10% discount reduces the marked price by ₦20,000.

3

Determine the net profit

Net Profit = ₦20,000

Net profit is the difference between the selling price (₦180,000) and the cost price (₦160,000).

Anahtar Kavram

Markup, Discount, and Profit Calculations

Soru 747Soru →

A pie chart illustrates the annual allocation of funds for an agricultural research station across four sectors: Crop Research, Livestock, Irrigation, and Equipment Maintenance. The sector angles for Crop Research, Livestock, and Irrigation are $135^\circ$ , $90^\circ$ , and $60^\circ$ respectively. If the total annual budget is $\text{N}7,200,000$ , what is the amount allocated to Equipment Maintenance in Naira?

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Cevap: 1500000

Cevap

The amount allocated to Equipment Maintenance is 1,500,000 Naira.

A complete pie chart has a total angle of

360^\circ

, representing the full amount of

\text{N}7,200,000

. The angle corresponding to Equipment Maintenance is

360^\circ - (135^\circ + 90^\circ + 60^\circ) = 75^\circ

. The dollar/naira allocation is given by

\frac{75^\circ}{360^\circ} \times 7,200,000 = 1,500,000

.

Adım Adım Çözüm

1

Calculate the sum of the given sector angles

Sum of known angles = 285 degrees

Determining the total angular measure already accounted for by the three known sectors.

2

Subtract the sum of known angles from 360 degrees

Sector angle for Equipment Maintenance = 75 degrees

The sum of all sector angles in a pie chart is always 360 degrees.

3

Multiply the fraction of the pie chart by the total funds

1,500,000 Naira

Converting the sector angle representation to its corresponding quantitative value.

Anahtar Kavram

Pie chart sector angle computation and value conversion

Soru 748Soru →

In a hydrogen atom modeled according to Bohr's theory, an electron undergoes a transition from an excited state with an energy of $-1.51\text{ eV}$ to a lower energy state of $-3.40\text{ eV}$ . Calculate the energy of the emitted photon in electron-volts ( $\text{eV}$ ).

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Cevap: 1.89

Cevap

The energy of the emitted photon is

1.89\text{ eV}

.

According to Bohr's atomic model, when an electron drops from an initial higher energy level

E_i

to a final lower energy level

E_f

, a photon is emitted carrying energy

E = E_i - E_f

. Substituting the given values gives

E = -1.51\text{ eV} - (-3.40\text{ eV}) = 1.89\text{ eV}

.

Adım Adım Çözüm

1

Identify the initial and final energy states of the electron.

E_i = -1.51\text{ eV}

and

E_f = -3.40\text{ eV}

.

The electron moves from a higher (less negative) energy state to a lower (more negative) energy state.

2

Apply Bohr's energy quantization formula for photon emission

E_{\text{photon}} = E_i - E_f

.

E_{\text{photon}} = -1.51 - (-3.40) = 1.89\text{ eV}

.

By energy conservation, the energy lost by the transitioning electron equals the energy of the emitted photon.

Anahtar Kavram

Bohr's Energy Transition Postulate

Tahmini Süre:1m 0s

Soru 749Soru →

A binary operation $\star$ on the set of real numbers $\mathbb{R} \setminus \{1\}$ is defined by $a \star b = a + b - ab$ . If $y^{-1}$ denotes the inverse of an element $y$ under $\star$ , find the value of $x$ such that $(x \star 3)^{-1} = 2$ .

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The value of

x

is

0.5

.

To solve for

x

, first calculate the identity element

e = 0

using

a \star e = a

. Next, use

y \star y^{-1} = 0

to get the inverse formula

y^{-1} = \frac{y}{y - 1}

. Substituting

y = x \star 3 = 3 - 2x

into the inverse formula yields

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

, which simplifies to

2x = 1

, giving

x = 0.5

.

Adım Adım Çözüm

1

Find the identity element

e

of the operation

\star

.

e = 0

By definition of the identity element,

a \star e = a \implies a + e - ae = a \implies e(1 - a) = 0

, so

e = 0

for all

a \neq 1

.

2

Derive the general expression for the inverse element

y^{-1}

of

y

.

y^{-1} = \frac{y}{y - 1}

By definition of inverse element,

y \star y^{-1} = e \implies y + y^{-1} - y y^{-1} = 0 \implies y^{-1}(1 - y) = -y \implies y^{-1} = \frac{y}{y - 1}

.

3

Evaluate the inner operation

x \star 3

.

x \star 3 = 3 - 2x

Using the operational rule

a \star b = a + b - ab

, we obtain

x \star 3 = x + 3 - 3x = 3 - 2x

.

4

Set up and solve the equation

(x \star 3)^{-1} = 2

.

x = 0.5

Letting

y = 3 - 2x

, its inverse is

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

. Equating this to

2

gives

\frac{3 - 2x}{2 - 2x} = 2 \implies 3 - 2x = 4 - 4x \implies 2x = 1 \implies x = 0.5

.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 750Soru →

A binary operation $\oplus$ is defined on the set of real numbers $\mathbb{R}$ by $a \oplus b = 2a + 3b - 5$ . What is the value of $(4 \oplus 1) \oplus 2$ ?

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Cevap: 13

Cevap

The value of

(4 \oplus 1) \oplus 2

is 13.

First, evaluate the inner expression

4 \oplus 1

using

a = 4

and

b = 1

, which gives

2(4) + 3(1) - 5 = 8 + 3 - 5 = 6

. Then, substitute this result into the outer expression to evaluate

6 \oplus 2

using

a = 6

and

b = 2

, yielding

2(6) + 3(2) - 5 = 12 + 6 - 5 = 13

.

Adım Adım Çözüm

1

Evaluate the inner operation

4 \oplus 1

6

Substitute

a = 4

and

b = 1

into the operational rule

a \oplus b = 2a + 3b - 5

.

2

Evaluate the outer operation using the result from Step 1:

6 \oplus 2

13

Substitute

a = 6

and

b = 2

into the operational rule

a \oplus b = 2a + 3b - 5

.

Anahtar Kavram

Evaluation of Binary Operations

Soru 751Soru →

If $5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Converting all terms to base

5

gives

5^{2x - 1} \times 5^{2(x + 1)} = 5^{3(x + 2)}

. Simplifying the exponents yields

5^{2x - 1 + 2x + 2} = 5^{3x + 6}

, which reduces to

5^{4x + 1} = 5^{3x + 6}

. Setting the exponents equal to each other gives

4x + 1 = 3x + 6

, resulting in

x = 5

.

Adım Adım Çözüm

1

Express all bases in terms of prime base 5

5^{2x - 1} \times (5^2)^{x + 1} = (5^3)^{x + 2}

All terms must share the same base to combine exponents using index laws.

2

Apply power of a power rule

(a^m)^n = a^{mn}

5^{2x - 1} \times 5^{2x + 2} = 5^{3x + 6}

Multiplication of inner and outer powers simplifies composite exponent expressions.

3

Apply the product rule

a^m \times a^n = a^{m+n}

5^{4x + 1} = 5^{3x + 6}

Adding the exponents on the left-hand side produces a single exponential term.

4

Equate exponents and solve for

x

x = 5

Equal bases imply equal exponents:

4x + 1 = 3x + 6

.

Anahtar Kavram

Solving exponential equations using common base conversion and laws of indices

Tahmini Süre:1m 30s

Soru 752Soru →

A plane contains $12$ distinct points such that exactly $5$ of them lie on the same straight line, while no other subset of three points is collinear. How many distinct straight lines can be formed by joining pairs of these points?

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Cevap: 57

Cevap

The total number of distinct straight lines that can be formed is 57.

The total number of distinct straight lines is obtained by taking all possible pairs of points from 12, which is

^{12}C_2 = 66

, subtracting the pairs formed among the 5 collinear points (

^5C_2 = 10

), and adding 1 back to account for the single straight line on which those 5 points lie. This gives

66 - 10 + 1 = 57

.

Adım Adım Çözüm

1

Calculate the total possible lines by selecting any 2 points out of 12.

^{12}C_2 = \frac{12 \times 11}{2} = 66

Any pair of non-collinear points uniquely determines a straight line.

2

Calculate the combinations of 2 points selected from the 5 collinear points.

^5C_2 = \frac{5 \times 4}{2} = 10

These 10 point pairs all lie on the exact same single straight line instead of forming 10 distinct lines.

3

Adjust the count to find the net number of distinct straight lines.

66 - 10 + 1 = 57

Subtracting the 10 overcounted combinations and adding back 1 for the single line passing through all 5 collinear points gives the correct total.

Anahtar Kavram

Combinations with Collinear Constraints

Soru 753Soru →

A search-and-rescue helicopter leaves a central station $P$ and flies $16\text{ km}$ on a bearing of $050^\circ$ to reach a waypoint $Q$ . It then changes course and flies $12\text{ km}$ on a bearing of $140^\circ$ to reach a mountain rescue site $R$ . From the central station $P$ , the angle of elevation to the helicopter hovering vertically above point $R$ is $45^\circ$ . What is the vertical height of the helicopter above the horizontal plane of station $P$ , in kilometers?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The vertical height of the helicopter above the horizontal plane of station

P

is

20\text{ km}

.

The horizontal journey forms a right-angled triangle

PQR

with side lengths

16\text{ km}

and

12\text{ km}

, yielding a hypotenuse (horizontal distance

PR

) of

20\text{ km}

. Since the angle of elevation from

P

to the hovering helicopter is

45^\circ

, the vertical height is equal to

20 \times \tan(45^\circ) = 20\text{ km}

.

Adım Adım Çözüm

1

Find the back bearing of station

P

from waypoint

Q

Back bearing =

050^\circ + 180^\circ = 230^\circ

To determine the enclosed interior angle at point

Q

, the reverse direction from

Q

to

P

must be calculated.

2

Calculate the interior angle

\angle PQR

\angle PQR = 230^\circ - 140^\circ = 90^\circ

Subtracting the forward bearing of

R

from the back bearing of

P

yields the right angle between the two paths.

3

Calculate the horizontal displacement distance

PR

PR = \sqrt{16^2 + 12^2} = 20\text{ km}

Since

\triangle PQR

is right-angled at

Q

, the distance

PR

is obtained using the Pythagorean theorem.

4

Determine the vertical altitude using trigonometry

\text{Height} = PR \times \tan(45^\circ) = 20 \times 1 = 20\text{ km}

In the vertical right-angled triangle formed by

P

, the ground projection of

R

, and the helicopter,

\tan(45^\circ) = \frac{\text{Height}}{\text{Horizontal Distance}}

.

Anahtar Kavram

Combining 3-point bearings in 2D with right-triangle trigonometry for 3D angles of elevation.

Soru 754Soru →

What is the value of $y$ at the local minimum stationary point of the curve $y = x^3 - 3x^2 - 9x + 15$ for $x > 0$ ?

Cevabı ve açıklamayı göster

Cevap: -12

Cevap

The value of

y

at the local minimum stationary point is

-12

.

To find the local minimum point of

y = x^3 - 3x^2 - 9x + 15

, set the derivative

\frac{dy}{dx} = 3x^2 - 6x - 9

equal to

0

. Solving

3(x - 3)(x + 1) = 0

yields

x = 3

for

x > 0

. The second derivative

\frac{d^2y}{dx^2} = 6x - 6

equals

12 > 0

at

x = 3

, confirming a local minimum. Substituting

x = 3

back into the original curve equation yields

y = (3)^3 - 3(3)^2 - 9(3) + 15 = -12

.

Adım Adım Çözüm

1

Differentiate

y = x^3 - 3x^2 - 9x + 15

with respect to

x

\frac{dy}{dx} = 3x^2 - 6x - 9

Stationary points occur where the slope (first derivative) is zero.

2

Set

\frac{dy}{dx} = 0

and solve for

x

x = 3 \text{ or } x = -1

Factoring

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

gives

(x - 3)(x + 1) = 0

.

3

Apply the second derivative test at

x = 3

(since

x > 0

)

\frac{d^2y}{dx^2} = 6(3) - 6 = 12 > 0

A positive second derivative confirms that

x = 3

is a local minimum.

4

Evaluate

y

at

x = 3

in the original curve equation

y = (3)^3 - 3(3)^2 - 9(3) + 15 = -12

Substituting

x = 3

into

y(x)

gives the

y

-coordinate of the minimum point.

Anahtar Kavram

Stationary Points and Local Minima of Polynomial Curves

Soru 755Soru →

If $y = \int (4x^3 - 2\sin(x)) \, dx$ and $y = 5$ when $x = 0$ , what is the value of the constant of integration $C$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The constant of integration

C

is 3.

Integrating

4x^3 - 2\sin(x)

gives

y = x^4 + 2\cos(x) + C

. Substituting

x = 0

and

y = 5

yields

5 = 0 + 2(1) + C

, so

C = 3

.

Adım Adım Çözüm

1

Integrate the polynomial and trigonometric terms

y = x^4 + 2\cos(x) + C

The integral of

4x^3

with respect to

x

is

x^4

, and the integral of

-2\sin(x)

is

2\cos(x)

.

2

Apply the given initial condition

y(0) = 5

5 = 0^4 + 2\cos(0) + C \implies 5 = 2 + C

Evaluating the expression at

x = 0

uses the trigonometric property

\cos(0) = 1

.

3

Solve the linear equation for

C

C = 3

Subtracting 2 from both sides isolates

C

.

Anahtar Kavram

Determining the Constant of Integration from Initial Boundary Conditions

Soru 756Soru →

A biased four-sided die numbered $1$ , $2$ , $3$ , and $4$ is rolled $200$ times in a probability experiment. The frequency of each outcome is recorded in the table below:

Outcome	Frequency
$1$	$45$
$2$	$60$
$3$	$55$
$4$	$40$

What is the experimental probability of rolling an odd number?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

0.5

The correct experimental probability is calculated by dividing the total frequency of observed odd outcomes (

45 + 55 = 100

) by the total number of trials (

200

), which gives

\frac{100}{200} = 0.5

.

Adım Adım Çözüm

1

Identify the favorable outcomes for rolling an odd number

The odd outcomes on the die are

1

and

3

.

Odd numbers are integers that are not divisible by

2

.

2

Calculate the total frequency of the favorable outcomes

Frequency of

1

is

45

and frequency of

3

is

55

. Total frequency

= 45 + 55 = 100

.

The total number of times an odd number was rolled is the sum of the frequencies of outcome

1

and outcome

3

.

3

Compute the experimental probability

\text{Experimental Probability} = \frac{\text{Frequency of odd outcomes}}{\text{Total number of trials}} = \frac{100}{200} = 0.5

.

Experimental probability measures the relative frequency of an event occurring in an actual experiment.

Anahtar Kavram

Experimental Probability

Soru 757Soru →

Find the sum of all values of $x$ (in degrees) in the interval $0^\circ \le x \le 180^\circ$ that satisfy the trigonometric equation $\cos(3x - 45^\circ) = -\frac{\sqrt{2}}{2}$ .

Cevabı ve açıklamayı göster

Cevap: 330

Cevap

The sum of all values of

x

satisfying the equation in the domain

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

is 330.

Transforming the domain

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

gives

-45^\circ \le 3x - 45^\circ \le 495^\circ

. The angles within this range where the cosine value equals

-\frac{\sqrt{2}}{2}

are

135^\circ

,

225^\circ

, and

495^\circ

. Solving

3x - 45^\circ

for each of these angles gives

x = 60^\circ

,

90^\circ

, and

180^\circ

. Summing these three roots yields

330^\circ

.

Adım Adım Çözüm

1

Determine the interval of the transformed angle

\theta = 3x - 45^\circ

.

-45^\circ \le 3x - 45^\circ \le 495^\circ

Applying the linear transformation

3x - 45^\circ

to the given domain

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

establishes the exact boundaries for the argument of the cosine function.

2

Find all values of

\theta

within

[-45^\circ, 495^\circ]

satisfying

\cos \theta = -\frac{\sqrt{2}}{2}

.

\theta \in \{135^\circ, 225^\circ, 495^\circ\}

Cosine is negative in Quadrants II and III. The reference angle is

45^\circ

, giving base solutions

135^\circ

and

225^\circ

. Adding

360^\circ

to

135^\circ

gives

495^\circ

, which lies exactly on the upper boundary.

3

Solve for

x

by setting

3x - 45^\circ

equal to each valid

\theta

.

x \in \{60^\circ, 90^\circ, 180^\circ\}

Isolating

x

yields

x = \frac{\theta + 45^\circ}{3}

. All three resulting values lie within

[0^\circ, 180^\circ]

.

4

Sum the valid solution values.

60^\circ + 90^\circ + 180^\circ = 330^\circ

The problem asks specifically for the sum of all solution angles in degrees.

Anahtar Kavram

Solving multi-angle trigonometric equations with phase shifts across a specified domain

Soru 758Soru →

If $x$ and $y$ satisfy the simultaneous equations $x - 2y = 1$ and $x^2 - xy - y^2 = 5$ , what is the positive value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive value of

x

is 3.

Substituting

x = 2y + 1

into the non-linear equation yields

(2y + 1)^2 - (2y + 1)y - y^2 = 5

, which simplifies to

y^2 + 3y - 4 = 0

. Solving this gives

y = 1

or

y = -4

. Substituting

y = 1

into

x = 2y + 1

gives

x = 3

, which is the positive value of

x

.

Adım Adım Çözüm

1

Make

x

the subject of the linear equation

x = 2y + 1

Substitution is the standard method for solving linear-quadratic simultaneous systems.

2

Substitute

x = 2y + 1

into

x^2 - xy - y^2 = 5

(2y + 1)^2 - (2y + 1)y - y^2 = 5

Eliminates

x

to create a single quadratic equation in terms of

y

.

3

Expand and simplify to standard quadratic form

y^2 + 3y - 4 = 0

Simplifying algebraic expressions allows factorization.

4

Solve for

y

by factoring

y = 1

or

y = -4

Factors of

-4

that sum to

3

are

+4

and

-1

.

5

Calculate corresponding values for

x

x = 3

when

y = 1

, and

x = -7

when

y = -4

Substitute

y

back into the linear expression for

x

.

6

Select the positive value of

x

3

The question explicitly asks for the positive value of

x

.

Anahtar Kavram

Simultaneous Linear and Quadratic Equations

Soru 759Soru →

A solid rectangular wooden block measures $10\text{ cm}$ by $14\text{ cm}$ by $12\text{ cm}$ . A cylindrical hole of radius $3.5\text{ cm}$ is drilled straight through the block along its height of $12\text{ cm}$ . Taking $\pi = \frac{22}{7}$ , what is the volume of the remaining solid in $\text{cm}^3$ ?

Cevabı ve açıklamayı göster

Cevap: 1218

Cevap

The volume of the remaining solid is

1218\text{ cm}^3

.

The initial total volume of the rectangular block is

10 \times 14 \times 12 = 1680\text{ cm}^3

. The volume of the cylindrical hole drilled through it is

\frac{22}{7} \times (3.5)^2 \times 12 = 462\text{ cm}^3

. Subtracting the removed cylindrical volume from the block gives

1680 - 462 = 1218\text{ cm}^3

.

Adım Adım Çözüm

1

Calculate the total volume of the rectangular block.

Volume of block =

1680\text{ cm}^3

The initial volume of the cuboid before drilling is calculated by multiplying its length, width, and height:

10 \times 14 \times 12 = 1680\text{ cm}^3

.

2

Calculate the volume of the cylindrical hole removed from the block.

Volume of cylinder =

462\text{ cm}^3

The cylindrical hole has radius

r = 3.5\text{ cm} = \frac{7}{2}\text{ cm}

and height

h = 12\text{ cm}

. Using

V = \pi r^2 h

, we get

\frac{22}{7} \times \frac{49}{4} \times 12 = 462\text{ cm}^3

.

3

Subtract the volume of the cylindrical hole from the total volume of the block.

Remaining volume =

1218\text{ cm}^3

Because material is removed by drilling, the remaining volume is

1680 - 462 = 1218\text{ cm}^3

.

Anahtar Kavram

Volume of a composite solid (cuboid with a cylindrical cavity)

Soru 760Soru →

What is the period, in degrees, of the trigonometric function $y = 7\sin(5x) - 2$ ?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The period of the trigonometric function is 72 degrees.

For any function of the form

y = A\sin(Bx) + D

, the period

T

in degrees is calculated using

T = \frac{360^\circ}{|B|}

. For the given equation

y = 7\sin(5x) - 2

, the value of

B

is 5. Substituting this into the formula gives

T = \frac{360^\circ}{5} = 72^\circ

.

Adım Adım Çözüm

1

Identify the coefficient

B

of the variable

x

in the given function

y = 7\sin(5x) - 2

.

Here,

A = 7

,

B = 5

, and

D = -2

.

The period of a sine function depends on the angular frequency parameter

B

multiplying the input variable

x

.

2

Apply the standard formula for finding the period

T

of a sine function in degrees:

T = \frac{360^\circ}{|B|}

.

T = \frac{360^\circ}{5} = 72^\circ

.

Dividing the standard full revolution of

360^\circ

by the multiplier

5

determines the angle needed for one full cycle.

Anahtar Kavram

Period of a Trigonometric Graph

Tahmini Süre:45s

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