Tüm alıştırma soruları

If $\log_3(2x - 1) = 2$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Converting the logarithmic equation

\log_3(2x - 1) = 2

to its equivalent exponential form yields

3^2 = 2x - 1

. Simplifying the exponent gives

9 = 2x - 1

. Adding

1

to both sides results in

10 = 2x

, and dividing by

2

gives

x = 5

.

Adım Adım Çözüm

1

Rewrite the logarithmic equation in exponential form.

2x - 1 = 3^2

By definition,

\log_b(y) = z

is equivalent to

b^z = y

.

2

Evaluate the exponent

3^2

.

2x - 1 = 9

Calculating

3

squared yields

9

.

3

Solve the linear equation for

x

.

2x = 10

, which simplifies to

x = 5

Adding

1

to both sides and then dividing by

2

isolates

x

.

Anahtar Kavram

Converting logarithmic equations to exponential equations

Tahmini Süre:45s

Soru 182Soru →

A smart home heating system has two operating modes: Eco mode and Comfort mode. In a certain week, the system was active in one of these two modes for a total of $168$ hours. The number of hours the system was in Eco mode was $24$ hours less than three times the number of hours it was in Comfort mode. For how many hours was the heating system in Comfort mode during that week?

Cevabı ve açıklamayı göster

Cevap: 48

Cevap

The heating system was in Comfort mode for 48 hours.

The correct answer is obtained by representing the hours in Comfort mode as

c

and Eco mode as

3c - 24

. Since their sum is

168

, we write the equation

c + 3c - 24 = 168

. Solving this linear equation gives

4c = 192

, which yields

c = 48

.

Adım Adım Çözüm

1

Define variables for the Comfort and Eco modes.

Let

c

represent the hours in Comfort mode and

e

represent the hours in Eco mode.

To represent the unknown quantities algebraically.

2

Translate the given information into a system of equations.

c + e = 168

and

e = 3c - 24

The sum of the hours in both modes is

168

, and the relationship between Eco and Comfort hours is described.

3

Substitute the expression for

e

into the total hours equation.

c + (3c - 24) = 168

To create a single linear equation with one variable.

4

Solve for the variable

c

.

4c - 24 = 168 \Rightarrow 4c = 192 \Rightarrow c = 48

Simplifying the equation gives the value of

c

, which is the hours spent in Comfort mode.

Anahtar Kavram

Translating and Solving Algebraic Word Problems

Soru 183Soru →

In the standard $(x,y)$ coordinate plane, the graph of the linear equation $y - 7 = -2(x + 3)$ crosses the $y$ -axis. What is the $y$ -coordinate of this intersection point?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The

y

-coordinate of the intersection point is

1

.

To find where the graph of the linear equation crosses the

y

-axis, substitute

x = 0

into the equation

y - 7 = -2(x + 3)

. This gives

y - 7 = -2(0 + 3)

, which simplifies to

y - 7 = -6

. Adding

7

to both sides results in

y = 1

. Thus, the

y

-coordinate of the intersection point is

1

.

Adım Adım Çözüm

1

Substitute

x = 0

into the linear equation.

y - 7 = -2(0 + 3)

The graph of an equation crosses the

y

-axis (which defines the

y

-intercept) when the

x

-coordinate is equal to

0

.

2

Simplify the expression on the right side of the equation.

y - 7 = -6

Evaluating

-2(0 + 3)

yields

-2(3) = -6

.

3

Solve the simplified linear equation for

y

.

y = 1

Adding

7

to both sides isolates

y

, giving

y = -6 + 7 = 1

.

Anahtar Kavram

Finding the

y

-intercept of a line from its point-slope form equation

Soru 184Soru →

A manufacturing company produces two types of metal alloys. The production cost, in dollars per ton, of Alloy A is modeled by the expression $800 - 15(x - 5)$ , where $x$ is the amount of stabilizer added in kilograms. The production cost of Alloy B, in dollars per ton, is modeled by the expression $520 + 5(x + 3)$ . If the company requires the production cost of Alloy A to be strictly less than the production cost of Alloy B, what is the minimum integer amount of stabilizer $x$ , in kilograms, that must be added?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The minimum integer amount of stabilizer that must be added is

18

kg.

The inequality representing the condition is

800 - 15(x - 5) < 520 + 5(x + 3)

. Expanding both sides gives

875 - 15x < 535 + 5x

. Isolating

x

yields

-20x < -340

. Dividing by

-20

and flipping the inequality sign results in

x > 17

. The minimum integer value that is strictly greater than

17

is

18

.

Adım Adım Çözüm

1

Set up the linear inequality using the cost expressions for Alloy A and Alloy B.

800 - 15(x - 5) < 520 + 5(x + 3)

The cost of Alloy A must be strictly less than the cost of Alloy B.

2

Distribute and combine like terms to simplify both sides of the inequality.

875 - 15x < 535 + 5x

Simplifying the expressions makes it easier to isolate the variable.

3

Isolate the variable term on one side of the inequality.

-20x < -340

Subtracting

5x

and

875

from both sides moves all variable terms to the left and constant terms to the right.

4

Divide both sides by the negative coefficient

-20

and reverse the inequality sign.

x > 17

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

5

Identify the smallest integer that satisfies the inequality.

18

The solution requires a strict inequality

x > 17

, so the smallest integer value that is strictly greater than

17

is

18

.

Anahtar Kavram

Solving multi-step linear inequalities with variables on both sides, including reversing the inequality sign when dividing by a negative number.

Soru 185Soru →

If $x$ and $y$ are real numbers such that $2^x \cdot 3^y = 24$ and $3^x \cdot 2^y = 54$ , what is the value of $x^2 + y^2$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

Multiplying the two equations gives

(2^x \cdot 3^y)(3^x \cdot 2^y) = 24 \cdot 54 \implies 6^{x+y} = 1296 = 6^4

, which yields

x + y = 4

. Dividing the first equation by the second equation gives \frac{2^x \cdot 3^y}{3^x \cdot 2^y} = \frac{24}{54} \implies (\frac{2}{3})^{x-y} = \frac{4}{9} = (\frac{2}{3})^2

, which yields

x - y = 2

. Solving the system of equations

x + y = 4

and

x - y = 2

gives

x = 3

and

y = 1

. The value of

x^2 + y^2

is therefore

3^2 + 1^2 = 10$.

Adım Adım Çözüm

1

Multiply the two equations together.

6^{x+y} = 1296

, which simplifies to

x + y = 4

.

Multiplying the equations groups bases of 2 and 3 together to form base 6, allowing us to find the sum of the variables.

2

Divide the first equation by the second equation.

(\frac{2}{3})^{x-y} = \frac{4}{9}

, which simplifies to

x - y = 2

.

Dividing the equations groups bases of 2 and 3 to form base

2/3

, allowing us to find the difference of the variables.

3

Solve the system of equations for

x

and

y

.

x = 3

and

y = 1

.

Solving the linear system of equations

x + y = 4

and

x - y = 2

gives the individual values of

x

and

y

.

4

Calculate

x^2 + y^2

.

10

Substitute the values of

x

and

y

into the target expression.

Anahtar Kavram

Solving systems of exponential equations using properties of exponents and bases

Alternatif Yöntem

Take the logarithm of both sides of each equation to convert them into a system of linear equations in terms of

x

and

y

:

x \log 2 + y \log 3 = \log 24

and

x \log 3 + y \log 2 = \log 54

. Solving this system using elimination or substitution yields

x = 3

and

y = 1

, so

x^2 + y^2 = 10

.

Tahmini Süre:2m 30s

Soru 186Soru →

Let matrix $A = \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix}$ and matrix $B = \begin{bmatrix} -2 & 3 \\ 2 & -3 \end{bmatrix}$ . If matrix $C$ is defined by the equation $C = 3A - 2B$ , what is the value of the element in the second row and first column of $C$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The element in the second row and first column of matrix

C

is 11.

To find the element in the second row and first column of matrix

C

, we apply the operations defined by

C = 3A - 2B

directly to the corresponding elements of

A

and

B

. The element in row 2, column 1 of matrix

A

is

5

, and of matrix

B

is

2

. Computing

3(5) - 2(2)

yields

15 - 4 = 11

.

Adım Adım Çözüm

1

Identify the elements in the second row and first column of matrices

A

and

B

.

A_{21} = 5

and

B_{21} = 2

.

To find a specific element of the resulting matrix

C = 3A - 2B

, we only need to perform the operations on the elements in the corresponding position.

2

Set up the equation for the element in the second row and first column of

C

.

C_{21} = 3A_{21} - 2B_{21}

Matrix addition, subtraction, and scalar multiplication are performed element-wise.

3

Substitute the identified values into the equation and compute the result.

C_{21} = 3(5) - 2(2) = 15 - 4 = 11

Evaluating the expression gives the value of the target element.

Anahtar Kavram

Matrix scalar multiplication and element-wise subtraction

Soru 187Soru →

Let $P(x) = (3x^2 - 2x + 4)(2x - 5) - (4x^3 - 7x^2 + x - 3)$ . When $P(x)$ is simplified and written in standard form, what is the coefficient of the $x^2$ term?

Cevabı ve açıklamayı göster

Cevap: -12

Cevap

The coefficient of the

x^2

term is -12.

Expanding

(3x^2 - 2x + 4)(2x - 5)

yields

6x^3 - 19x^2 + 18x - 20

. Distributing the negative sign across the second polynomial yields

-4x^3 + 7x^2 - x + 3

. Combining the

x^2

terms gives

-19x^2 + 7x^2 = -12x^2

, so the coefficient of the

x^2

term is -12.

Adım Adım Çözüm

1

Expand the product of the trinomial and the binomial:

(3x^2 - 2x + 4)(2x - 5)

6x^3 - 19x^2 + 18x - 20

To find the expanded form of the first polynomial component before subtraction

2

Distribute the negative sign across the second polynomial:

-(4x^3 - 7x^2 + x - 3)

-4x^3 + 7x^2 - x + 3

To prepare the second polynomial for combination of like terms

3

Combine the like terms from the two expanded components

2x^3 - 12x^2 + 17x - 17

To write the entire polynomial in standard form and identify the coefficient of

x^2

Anahtar Kavram

Operations on Polynomials

Soru 188Soru →

Let the functions $f$ and $g$ be defined by $f(x) = (x - 3)^2$ and $g(x) = 2x + 1$ . What is the value of the composite function $f(g(-2))$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

To evaluate the composite function

f(g(-2))

, evaluate from the inside out. First, evaluate the inner function

g(-2) = 2(-2) + 1 = -3

. Next, substitute this output value into the outer function

f(x)

to get

f(-3) = (-3 - 3)^2 = (-6)^2 = 36

.

Adım Adım Çözüm

1

Evaluate the inner function

g(x)

at

x = -2

g(-2) = -3

Before evaluating the outer function

f

, we must determine the output of the inner function

g

at the given input value.

2

Evaluate the outer function

f(x)

at the result of the inner function

f(-3) = 36

Substitute the inner output

-3

as the input for

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

, giving

(-3 - 3)^2 = (-6)^2 = 36

.

Anahtar Kavram

Function composition and evaluation

Tahmini Süre:1m 0s

Soru 189Soru →

If the quadratic equation $x^2 + kx + 25 = 0$ has exactly one real solution, and $k > 0$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

k

is

10

.

For any quadratic equation in standard form,

ax^2 + bx + c = 0

, the equation has exactly one real solution when the discriminant

b^2 - 4ac

is equal to

0

. Here, the equation is

x^2 + kx + 25 = 0

, so

a = 1

,

b = k

, and

c = 25

. Setting the discriminant to

0

gives

k^2 - 4(1)(25) = 0

, which simplifies to

k^2 - 100 = 0

. Solving for

k

yields

k = \pm 10

. Since we are given that

k > 0

, the only valid solution is

10

.

Adım Adım Çözüm

1

Set the discriminant of the quadratic equation to zero.

b^2 - 4ac = 0

A quadratic equation has exactly one real solution if and only if its discriminant is zero.

2

Identify coefficients

a

,

b

, and

c

from the equation

x^2 + kx + 25 = 0

and substitute them into the discriminant equation.

k^2 - 4(1)(25) = 0

For

x^2 + kx + 25 = 0

, the coefficients are

a=1

,

b=k

, and

c=25

.

3

Simplify the equation and solve for

k

.

k^2 = 100

, so

k = 10

or

k = -10

Simplifying

k^2 - 100 = 0

gives

k^2 = 100

.

4

Apply the constraint

k > 0

to find the final value.

k = 10

The problem specifies that

k

must be greater than zero.

Anahtar Kavram

Using the discriminant (

b^2 - 4ac

) of a quadratic equation to determine when there is exactly one real solution.

Tahmini Süre:45s

Soru 190Soru →

A line with a positive slope passes through the point $(0, -5)$ and is tangent to the parabola $y = x^2 - 6x + 11$ . What is the slope of this line?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The slope of the line is 2.

The correct slope is 2. Representing the line as

y = mx - 5

and setting it equal to the parabola

y = x^2 - 6x + 11

results in the quadratic equation

x^2 - (6 + m)x + 16 = 0

. For the line to be tangent, this equation must have exactly one real solution, meaning its discriminant must equal zero:

(6 + m)^2 - 4(1)(16) = 0

. Solving this gives

6 + m = 8

or

6 + m = -8

, which results in

m = 2

or

m = -14

. Since the problem specifies that the slope is positive, the value of

m

must be 2.

Adım Adım Çözüm

1

Write the equation of the line in slope-intercept form.

y = mx - 5

, where

m > 0

.

The line passes through

(0, -5)

, which represents the y-intercept of the line.

2

Equate the line and the parabola to set up the system of equations.

x^2 - 6x + 11 = mx - 5

Setting the two expressions equal allows us to find the x-coordinates of any intersection points.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - (6 + m)x + 16 = 0

Writing the equation in standard form identifies the coefficients

a = 1

,

b = -(6 + m)

, and

c = 16

needed for the discriminant.

4

Apply the condition for tangency by setting the discriminant to zero.

b^2 - 4ac = (-(6 + m))^2 - 4(1)(16) = 0 \implies (6 + m)^2 - 64 = 0

A line is tangent to a parabola if they touch at exactly one point, meaning the quadratic equation has exactly one real root (discriminant equals zero).

5

Solve the quadratic equation for the slope

m

and filter for the positive value.

(6+m)^2 = 64 \implies 6+m = \pm 8

, yielding

m = 2

or

m = -14

. Since the slope is positive,

m = 2

.

Solving the equation gives two possible slope values for tangent lines, and we select the positive slope as specified in the problem statement.

Anahtar Kavram

Determining tangency between a linear and quadratic equation by setting the discriminant of the intersection equation to zero.

Soru 191Soru →

A line graphed on the $(x, y)$ coordinate plane has a $y$ -intercept of $-1$ and passes through the point $(3, 8)$ . If the point $(a, 14)$ also lies on this line, what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The line has a y-intercept of

-1

, which corresponds to the point

(0, -1)

. The slope is calculated as

m = \frac{8 - (-1)}{3 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting

(a, 14)

into this equation gives

14 = 3a - 1

, which simplifies to

15 = 3a

, and solving for

a

yields

5

.

Adım Adım Çözüm

1

Identify the coordinate representation of the y-intercept.

The point is

(0, -1)

.

A y-intercept of

-1

means the line crosses the y-axis at the point where

x = 0

, which is

(0, -1)

.

2

Calculate the slope of the line.

m = 3

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the points

(0, -1)

and

(3, 8)

, we get

m = \frac{8 - (-1)}{3 - 0} = 3

.

3

Write the equation of the line and solve for

a

.

a = 5

The equation in slope-intercept form is

y = 3x - 1

. Substituting the point

(a, 14)

yields

14 = 3a - 1

. Adding 1 to both sides gives

15 = 3a

, so

a = 5

.

Anahtar Kavram

Finding the equation of a line from two points and solving for a missing coordinate.

Tahmini Süre:1m 30s

Soru 192Soru →

Matrices $C$ and $D$ are defined as follows:

C = \begin{bmatrix} 4 & 1 \\ -2 & k \end{bmatrix}, \quad D = \begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}

If the element in the second row and first column of the product matrix $CD$ is $-11$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The value of

k

is

-1

.

To find the element in the second row and first column of the product matrix

CD

, we take the dot product of the second row of

C

, which is

\begin{bmatrix} -2 & k \end{bmatrix}

, and the first column of

D

, which is

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

. This gives

(-2)(3) + (k)(5) = -6 + 5k

. Setting this equal to the given value of

-11

yields the equation

-6 + 5k = -11

. Adding

6

to both sides gives

5k = -5

, and dividing by

5

results in

k = -1

.

Adım Adım Çözüm

1

Identify the second row of matrix

C

and the first column of matrix

D

.

Row 2 of

C

is

\begin{bmatrix} -2 & k \end{bmatrix}

and Column 1 of

D

is

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

.

To find the element in the second row and first column of the product matrix

CD

, we must compute the dot product of the second row of the first matrix (

C

) and the first column of the second matrix (

D

).

2

Multiply the corresponding elements of the row and column and add the products.

The element at row 2, column 1 of

CD

is

(-2)(3) + (k)(5) = -6 + 5k

.

This defines the matrix multiplication rule for that specific position in the resulting matrix.

3

Set the expression

-6 + 5k

equal to the given value of

-11

and solve the linear equation for

k

.

-6 + 5k = -11 \implies 5k = -5 \implies k = -1

.

Solving the equation yields the value of the unknown variable

k

.

Anahtar Kavram

2x2 Matrix Multiplication

Soru 193Soru →

If $\log_3(x + 4) = 4$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 77

Cevap

The value of

x

is 77.

The correct answer is 77. To solve the equation

\log_3(x + 4) = 4

, convert the equation from its logarithmic form to its exponential form. Since

\log_b(a) = c

means

b^c = a

, the equation becomes

3^4 = x + 4

. Calculating

3^4

yields 81. Thus,

81 = x + 4

. Subtracting 4 from both sides gives

x = 77

.

Adım Adım Çözüm

1

Rewrite the logarithmic equation in exponential form.

x + 4 = 3^4

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

2

Calculate the value of the exponential expression.

3^4 = 81

3^4 = 3 \times 3 \times 3 \times 3 = 81

.

3

Solve for

x

by isolating the variable.

x = 77

Subtract 4 from both sides of the equation:

81 - 4 = 77

.

Anahtar Kavram

Converting logarithmic equations to exponential form

Soru 194Soru →

For a real constant $k$ , the quadratic equation $x^2 - 2(k - 2)x + (k^2 - 3k + 2) = 0$ has two real roots, $r_1$ and $r_2$ . What is the value of $k$ that minimizes the sum of the squares of these roots, $r_1^2 + r_2^2$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

that minimizes the sum of the squares of the roots is 2.

Applying Vieta's formulas and algebraic identities, the sum of the squares of the roots is expressed as

2k^2 - 10k + 12

. The discriminant condition for the roots to be real requires

k \leq 2

. Since the vertex of the upward-opening parabola

2k^2 - 10k + 12

is at

k = 2.5

, the function is strictly decreasing for all

k \leq 2

. Thus, the minimum value on the interval

k \leq 2

occurs at the boundary

k = 2

.

Adım Adım Çözüm

1

Use Vieta's formulas to find the sum and product of the roots.

r_1 + r_2 = 2(k - 2)

and

r_1 r_2 = k^2 - 3k + 2

Vieta's formulas relate the roots of a quadratic equation

Ax^2 + Bx + C = 0

to its coefficients by

r_1 + r_2 = -B/A

and

r_1 r_2 = C/A

.

2

Express the sum of the squares of the roots,

r_1^2 + r_2^2

, in terms of

k

.

r_1^2 + r_2^2 = 2k^2 - 10k + 12

Since

r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2

, substituting the Vieta relations gives

[2(k-2)]^2 - 2(k^2 - 3k + 2) = 4(k^2 - 4k + 4) - 2k^2 + 6k - 4 = 2k^2 - 10k + 12

.

3

Determine the condition for the quadratic equation to have real roots using the discriminant.

\Delta = -4k + 8 \geq 0 \implies k \leq 2

For the roots

r_1

and

r_2

to be real, the discriminant

\Delta = B^2 - 4AC

must be greater than or equal to 0.

4

Minimize the quadratic function

f(k) = 2k^2 - 10k + 12

subject to the constraint

k \leq 2

.

k = 2

The parabola

f(k)

opens upwards with its vertex at

k = 2.5

. For

k \leq 2

, the function is strictly decreasing, meaning its minimum value on this interval occurs at the upper boundary,

k = 2

.

Anahtar Kavram

Quadratic Equations and the Quadratic Formula

Soru 195Soru →

A linear function contains the points shown in the table below:

$x$	$y$
$k - 2$	$5$
$k + 1$	$14$
$2k + 3$	$35$

What is the $y$ -intercept of the line representing this function?

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The

y

-intercept of the line representing this function is

-4

.

By using the first two coordinates, the constant slope of the linear function is determined to be

3

. Equating the slope between the second and third coordinates to

3

yields the parameter

k = 5

. Substituting this back into the coordinate expressions gives the points

(3, 5)

and

(6, 14)

. Using the point-slope form, the equation of the line is

y = 3x - 4

, meaning the

y

-intercept is

-4

.

Adım Adım Çözüm

1

Calculate the slope of the line using the first two points:

(k-2, 5)

and

(k+1, 14)

.

m = \frac{14 - 5}{(k+1) - (k-2)} = \frac{9}{3} = 3

Since the function is linear, the rate of change (slope) remains constant between any two points.

2

Express the slope using the second and third points,

(k+1, 14)

and

(2k+3, 35)

, set it equal to

3

, and solve for

k

.

\frac{35 - 14}{(2k+3) - (k+1)} = 3 \implies \frac{21}{k+2} = 3 \implies 3(k+2) = 21 \implies k+2 = 7 \implies k = 5

The slope between the second and third points must also equal the constant slope of

3

.

3

Substitute

k = 5

back into the coordinates to determine the actual points on the line.

The points are

(3, 5)

,

(6, 14)

, and

(13, 35)

.

This provides concrete coordinates that can be used to write the equation of the line.

4

Use the point-slope form with the point

(3, 5)

and slope

m = 3

to write the equation of the line, then convert to slope-intercept form.

y - 5 = 3(x - 3) \implies y - 5 = 3x - 9 \implies y = 3x - 4

Converting to slope-intercept form (

y = mx + b

) directly gives the

y

-intercept as the constant term

b

.

Anahtar Kavram

Determining the equation and intercepts of a line using the constant slope property of linear functions.

Alternatif Yöntem

Once the slope

m = 3

and a point such as

(3, 5)

are established, substitute these values directly into the slope-intercept equation

y = mx + b

to solve for

b

. This gives

5 = 3(3) + b \implies 5 = 9 + b \implies b = -4

.

Tahmini Süre:2m 30s

Soru 196Soru →

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $y = \frac{1}{12}x^2 - 2x + 13$ . What is the $y$ -coordinate of the focus of this parabola?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The y-coordinate of the focus is 4.

The standard form of the parabola is

y - 1 = \frac{1}{12}(x - 12)^2

. Comparing this to

y - k = \frac{1}{4p}(x - h)^2

gives the vertex

(h, k) = (12, 1)

and

4p = 12 \implies p = 3

. Since the parabola opens upward, the focus is at

(12, 1 + 3) = (12, 4)

, making the

y

-coordinate 4.

Adım Adım Çözüm

1

Complete the square to rewrite the equation in standard vertex form.

y = \frac{1}{12}(x - 12)^2 + 1

Rewriting the general quadratic equation into standard form allows us to directly identify the vertex and focal parameters.

2

Equate the coefficients to find the focal distance

p

and vertex

(h, k)

.

Vertex

(h, k) = (12, 1)

and

p = 3

The standard vertex form of a vertical parabola is

y - k = \frac{1}{4p}(x - h)^2

. Setting

\frac{1}{4p} = \frac{1}{12}

gives

p = 3

.

3

Calculate the focus coordinates

(h, k + p)

.

Focus

= (12, 4)

, so the

y

-coordinate is

4

For an upward-opening parabola, the focus is located

p

units directly above the vertex.

Anahtar Kavram

Rewriting a quadratic equation into standard vertex form to find the properties of a parabola, including its vertex and focus.

Soru 197Soru →

If the expression

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

is equivalent to $x^2 y$ for all non-zero real numbers $x$ and $y$ , what is the value of the exponent $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of the exponent

k

is 3.

Applying exponent rules simplifies the numerator of the expression to

x^5 y^4

. Dividing this by the denominator

x^k y^3

yields

x^{5-k} y

. Equating the exponent of

x

to 2 in the target expression

x^2 y

gives

5 - k = 2

, which solves to

k = 3

.

Adım Adım Çözüm

1

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

to simplify each factor in the numerator.

(x^3 y^2)^3 = x^9 y^6

and

(x^{-2} y^{-1})^2 = x^{-4} y^{-2}

To raise a product to a power, raise each factor to that power by multiplying the exponents.

2

Multiply the two simplified factors in the numerator together by adding the exponents of like bases.

x^9 y^6 \cdot x^{-4} y^{-2} = x^5 y^4

When multiplying exponential expressions with the same base, add their exponents:

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

.

3

Divide the numerator by the denominator by subtracting the exponents of like bases.

\frac{x^5 y^4}{x^k y^3} = x^{5-k} y

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{a^m}{a^n} = a^{m-n}

.

4

Set the exponent of

x

in the simplified expression equal to the exponent of

x

in the target expression

x^2 y

, and solve for

k

.

5 - k = 2 \implies k = 3

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of like bases must be equal.

Anahtar Kavram

Properties of exponents (power of a power, product of powers, and quotient of powers rules)

Tahmini Süre:1m 30s

Soru 198Soru →

If $x$ and $y$ are positive real numbers such that $\log_x(y) = 2$ and $\log_4(x) + \log_2(y) = 5$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

Applying the definition of logarithms to

\log_x(y) = 2

yields

y = x^2

. Substituting this relationship into the second equation gives

\log_4(x) + \log_2(x^2) = 5

. Using the change of base formula, we rewrite

\log_4(x)

as

\frac{\log_2(x)}{\log_2(4)} = \frac{1}{2}\log_2(x)

, and using the power property, we rewrite

\log_2(x^2)

as

2\log_2(x)

. Combining the terms results in

\frac{5}{2}\log_2(x) = 5

, which simplifies to

\log_2(x) = 2

. Converting this back to exponential form gives

x = 2^2 = 4

. Substituting this value back into the relation

y = x^2

yields

y = 4^2 = 16

.

Adım Adım Çözüm

1

Apply the definition of a logarithm to the first equation.

y = x^2

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

2

Substitute

y = x^2

into the second equation.

\log_4(x) + \log_2(x^2) = 5

Substitution reduces the system to a single equation in terms of

x

.

3

Express both logarithmic terms using base 2 properties.

\frac{1}{2}\log_2(x) + 2\log_2(x) = 5

The change of base formula gives

\log_4(x) = \frac{\log_2(x)}{\log_2(4)} = \frac{1}{2}\log_2(x)

, and the power property gives

\log_2(x^2) = 2\log_2(x)

.

4

Combine the coefficients and solve for

x

.

\log_2(x) = 2 \Rightarrow x = 2^2 = 4

Adding the coefficients gives

\frac{5}{2}\log_2(x) = 5

. Multiplying by

\frac{2}{5}

isolates

\log_2(x)

to solve for

x

.

5

Calculate the value of

y

using the relation from Step 1.

y = 4^2 = 16

Since

y = x^2

and

x = 4

, evaluating the square gives the final answer.

Anahtar Kavram

Solving systems of logarithmic equations using base conversion and definition of logarithms

Tahmini Süre:2m 0s

Soru 199Soru →

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $y^2 = 12x$ . What is the $x$ -coordinate of the focus of this parabola?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The

x

-coordinate of the focus is 3.

The equation

y^2 = 12x

represents a parabola with vertex at the origin opening to the right. The standard form for such a parabola is

y^2 = 4px

, where the focus is located at

(p, 0)

. By setting

4p = 12

, we find

p = 3

. Thus, the focus is

(3, 0)

, and its

x

-coordinate is 3.

Adım Adım Çözüm

1

Identify the standard form of the parabola's equation.

The equation

y^2 = 12x

fits the standard form of a horizontal parabola with its vertex at the origin,

y^2 = 4px

.

This allows us to relate the given equation to the coordinate of the focus, which is located at

(p, 0)

.

2

Solve for the parameter

p

.

4p = 12

, which gives

p = 3

.

By equating the coefficients of

x

from the given equation and the standard form, we can find the value of

p

.

3

Determine the focus coordinate.

The focus is at

(3, 0)

, so the

x

-coordinate is 3.

The focus of a parabola of the form

y^2 = 4px

has coordinates

(p, 0)

.

Anahtar Kavram

Focus of a Parabola

Soru 200Soru →

In the standard $(x,y)$ coordinate plane, two opposite vertices of a square are $(1, 2)$ and $(4, 6)$ . If all four vertices of the square lie in the first quadrant, what is the $x$ -coordinate of the vertex that is closest to the $y$ -axis?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The correct answer is

0.5

. The vertex closest to the

y

-axis is

(0.5, 5.5)

, which has an

x

-coordinate of

0.5

.

The diagonals of a square are perpendicular, equal in length, and bisect each other. Using the given opposite vertices

(1, 2)

and

(4, 6)

, we find the midpoint to be

(2.5, 4)

. The vector between them is

(3, 4)

with length

5

. A perpendicular vector of length

5

is

(-4, 3)

. Adding and subtracting half of this vector,

(2, -1.5)

, from the midpoint yields the other two vertices:

(0.5, 5.5)

and

(4.5, 2.5)

. Since all four vertices are in the first quadrant, we compare their

x

-coordinates:

1

,

4

,

0.5

, and

4.5

. The smallest

x

-coordinate is

0.5

, which represents the vertex closest to the

y

-axis.

Adım Adım Çözüm

1

Find the midpoint of the given diagonal.

The midpoint is

M(2.5, 4)

.

The diagonals of a square bisect each other at their common midpoint.

2

Determine the vector representing the given diagonal

AB

and its length.

\vec{AB} = (3, 4)

and its length is

5

.

The vector is found by subtracting coordinates:

(4 - 1, 6 - 2) = (3, 4)

, and its length is

\sqrt{3^2 + 4^2} = 5

.

3

Find a perpendicular vector of the same length to represent the other diagonal.

A perpendicular vector is

(-4, 3)

.

The dot product of

(3, 4)

and

(-4, 3)

is

3(-4) + 4(3) = 0

, and its length is

\sqrt{(-4)^2 + 3^2} = 5

.

4

Calculate the coordinates of the other two vertices of the square.

The vertices are

C(0.5, 5.5)

and

D(4.5, 2.5)

.

The vertices are located at

M \pm \frac{1}{2}\vec{CD}

, which gives

(2.5, 4) \pm ( -2, 1.5 )

.

5

Determine which of the four vertices is closest to the

y

-axis and identify its

x

-coordinate.

The vertex closest to the

y

-axis is

C(0.5, 5.5)

, and its

x

-coordinate is

0.5

.

The distance to the

y

-axis is the

x

-coordinate of the point. Comparing the

x

-coordinates

1

,

4

,

0.5

, and

4.5

, the smallest value is

0.5

.

Anahtar Kavram

Properties of diagonals of a square on a coordinate plane, including midpoint and perpendicularity.

Alternatif Yöntem

Instead of using vectors, one can set up a system of equations. Let

(x, y)

be one of the unknown vertices. Since it forms a right isosceles triangle with the midpoint

(2.5, 4)

and has distance

2.5

from it along a line with slope

-3/4

, we can write the equation of the line as

y - 4 = -0.75(x - 2.5)

and use the distance formula

(x - 2.5)^2 + (y - 4)^2 = 2.5^2

to solve for

x

and

y

.

Tahmini Süre:3m 0s

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Alternatif Yöntem

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Alternatif Yöntem

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