Examkin

Intermediate Algebra

272 soru

Tüm türler272Çoktan seçmeli162Sayısal101Boşluk doldurma9
Soru 101Soru →

A community theater sells adult tickets for $15\$15 each and student tickets for $10\$10 each. The number of tickets sold for the evening performances on Friday and Saturday is represented by the matrix AA:

A=[804511060]A = \begin{bmatrix} 80 & 45 \\ 110 & 60 \end{bmatrix}

where the rows represent Friday and Saturday, respectively, and the columns represent adult and student tickets, respectively.

The ticket prices are represented by the matrix BB:

B=[1510]B = \begin{bmatrix} 15 \\ 10 \end{bmatrix}

The product matrix C=ABC = AB represents the total ticket revenue for each night. What is the element in the second row and first column of matrix CC, which represents the total ticket revenue, in dollars, for Saturday's performance?

Cevabı ve açıklamayı göster

Cevap: 2250

Cevap

The correct answer is 22502250 dollars, which represents the ticket revenue for Saturday's performance.
The correct answer is 22502250 dollars. The element in the second row and first column of the product matrix C=ABC = AB is obtained by multiplying the elements of the second row of matrix AA (Saturday's ticket sales: 110110 adult, 6060 student) by the corresponding elements of the first column of matrix BB (ticket prices: $15\$15 adult, $10\$10 student) and summing the products: 110×15+60×10=1650+600=2250110 \times 15 + 60 \times 10 = 1650 + 600 = 2250.

Adım Adım Çözüm

1
Identify the row of matrix AA representing Saturday's sales and the column of matrix BB representing the prices.
The second row of matrix AA is [11060]\begin{bmatrix} 110 & 60 \end{bmatrix} (representing 110110 adult tickets and 6060 student tickets sold on Saturday). Matrix BB is [1510]\begin{bmatrix} 15 \\ 10 \end{bmatrix} (representing the prices of $15\$15 and $10\$10 respectively).
Since the question asks for Saturday's performance (the second row of matrix AA) and the revenue is calculated by multiplying sales by prices, we use these specific components for the matrix multiplication.
2
Calculate the dot product of Saturday's sales row and the price column.
110×15+60×10=1650+600=2250110 \times 15 + 60 \times 10 = 1650 + 600 = 2250.
This calculation determines the sum of the revenue from adult tickets (110×15=1650110 \times 15 = 1650) and student tickets (60×10=60060 \times 10 = 600) to yield the total revenue.

Anahtar Kavram

Matrix Multiplication
Soru 102Soru →

For the quadratic equation 3x2−6x+2=03x^2 - 6x + 2 = 0, what is the value of the discriminant?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The discriminant of the quadratic equation is 12.
The discriminant of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is determined by the expression b2−4acb^2 - 4ac. By substituting the coefficients a=3a = 3, b=−6b = -6, and c=2c = 2 from the given equation 3x2−6x+2=03x^2 - 6x + 2 = 0, we calculate (−6)2−4(3)(2)=36−24=12(-6)^2 - 4(3)(2) = 36 - 24 = 12.

Adım Adım Çözüm

1
Identify the coefficients of the quadratic equation
a=3a = 3, b=−6b = -6, and c=2c = 2
The standard form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0.
2
Substitute the coefficients into the discriminant formula
D=(−6)2−4(3)(2)D = (-6)^2 - 4(3)(2)
The discriminant formula is D=b2−4acD = b^2 - 4ac.
3
Simplify the expression to find the final value
36−24=1236 - 24 = 12
Squaring −6-6 gives 3636 and multiplying 4×3×24 \times 3 \times 2 gives 2424. Subtracting 2424 from 3636 gives 1212.

Anahtar Kavram

Calculating the discriminant of a quadratic equation
Tahmini Süre:45s
Soru 103Soru →

A system of equations consists of a parabola with the equation y=x2y = x^2 and a line with the equation y=3x−2y = 3x - 2. The line and the parabola intersect at two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2). What is the product of the yy-coordinates, y1⋅y2y_1 \cdot y_2, of these two points?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The product of the y-coordinates of the intersection points is 4.
Solving the system of equations by setting x2=3x−2x^2 = 3x - 2 leads to the quadratic equation x2−3x+2=0x^2 - 3x + 2 = 0, which factors as (x−1)(x−2)=0(x - 1)(x - 2) = 0. The x-coordinates of the intersection points are x=1x = 1 and x=2x = 2. Substituting these into y=x2y = x^2 gives the corresponding y-coordinates as y=12=1y = 1^2 = 1 and y=22=4y = 2^2 = 4. The product of these y-coordinates is 1⋅4=41 \cdot 4 = 4.

Adım Adım Çözüm

1
Equate the two equations to find the x-coordinates of the intersection points.
x2=3x−2x^2 = 3x - 2
Since both equations are solved for y, setting them equal to each other allows us to find the x-coordinates where they intersect.
2
Move all terms to one side to set the quadratic equation to zero.
x2−3x+2=0x^2 - 3x + 2 = 0
This puts the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0, which can be solved by factoring.
3
Factor the quadratic equation.
(x−1)(x−2)=0(x - 1)(x - 2) = 0, so x1=1x_1 = 1 and x2=2x_2 = 2
Factoring shows that the values of x that satisfy the equation are the roots of the quadratic.
4
Substitute the x-coordinates back into the equation of the parabola to find the y-coordinates.
For x1=1x_1 = 1, y1=12=1y_1 = 1^2 = 1. For x2=2x_2 = 2, y2=22=4y_2 = 2^2 = 4.
Substituting x into either of the original equations gives the corresponding y-coordinates of the intersection points.
5
Calculate the product of the y-coordinates.
1⋅4=41 \cdot 4 = 4
The question asks for the product of the y-coordinates, so we multiply y1y_1 and y2y_2.

Anahtar Kavram

Solving systems of linear and quadratic equations by substitution.
Soru 104Soru →

What is the set of all real numbers yy that make the inequality 2−3∣1−2y∣≥−102 - 3|1 - 2y| \ge -10 a true statement?

Cevabı ve açıklamayı göster

Cevap: −1.5≤y≤2.5-1.5 \le y \le 2.5

Cevap

−1.5≤y≤2.5-1.5 \le y \le 2.5
To solve the inequality 2−3∣1−2y∣≥−102 - 3|1 - 2y| \ge -10, we first isolate the absolute value term by subtracting 2 from both sides to get −3∣1−2y∣≥−12-3|1 - 2y| \ge -12, and then dividing both sides by −3-3. Since we divide by a negative number, the inequality sign reverses, giving ∣1−2y∣≤4|1 - 2y| \le 4. We rewrite this as the compound inequality −4≤1−2y≤4-4 \le 1 - 2y \le 4. Subtracting 1 from all parts yields −5≤−2y≤3-5 \le -2y \le 3. Finally, dividing by −2-2 and reversing the inequality signs gives 2.5≥y≥−1.52.5 \ge y \ge -1.5, which is rewritten from least to greatest as −1.5≤y≤2.5-1.5 \le y \le 2.5.

Adım Adım Çözüm

1
Subtract 2 from both sides of the inequality to begin isolating the absolute value term.
−3∣1−2y∣≥−12-3|1 - 2y| \ge -12
Isolating the absolute value expression allows us to rewrite it as a standard inequality.
2
Divide both sides by −3-3 and reverse the inequality sign because of division by a negative number.
∣1−2y∣≤4|1 - 2y| \le 4
Dividing by a negative value requires reversing the inequality direction to preserve the truth of the statement.
3
Rewrite the absolute value inequality as a compound inequality.
−4≤1−2y≤4-4 \le 1 - 2y \le 4
An inequality of the form ∣u∣≤c|u| \le c (where c>0c > 0) is equivalent to the compound inequality −c≤u≤c-c \le u \le c.
4
Subtract 1 from all parts of the compound inequality.
−5≤−2y≤3-5 \le -2y \le 3
This is the next step to isolate the variable yy in the middle.
5
Divide all parts of the compound inequality by −2-2 and reverse the inequality signs.
2.5≥y≥−1.52.5 \ge y \ge -1.5, which is equivalent to −1.5≤y≤2.5-1.5 \le y \le 2.5
Dividing by the negative coefficient −2-2 requires reversing the direction of all inequality signs.

Anahtar Kavram

Solving absolute value inequalities involving negative coefficients by isolating the absolute value and reversing inequality signs when multiplying or dividing by negative numbers.
Tahmini Süre:1m 30s
Soru 105Soru →

For what positive integer value of xx is the rational expression 2x−53x2−10x−8\frac{2x - 5}{3x^2 - 10x - 8} undefined?

Aşağıdaki boşlukları doldurun

The expression is undefined when xx is equal to .
Cevabı ve açıklamayı göster

Cevap

The expression is undefined when the positive integer x is equal to 4.
The rational expression is undefined when its denominator is 0. Setting the denominator 3x2−10x−83x^2 - 10x - 8 equal to 0 and factoring gives (3x+2)(x−4)=0(3x + 2)(x - 4) = 0. The roots of this equation are x=−23x = -\frac{2}{3} and x=4x = 4. Since the question restricts the target value to a positive integer, the correct value is 4.

Adım Adım Çözüm

1
Set the denominator of the rational expression equal to 0.
3x2−10x−8=03x^2 - 10x - 8 = 0
A rational expression is undefined when its denominator is equal to 0 because division by zero is undefined.
2
Factor the quadratic equation 3x2−10x−8=03x^2 - 10x - 8 = 0.
(3x+2)(x−4)=0(3x + 2)(x - 4) = 0
Factoring the quadratic trinomial helps identify the values of x that satisfy the equation.
3
Solve for the roots by setting each factor to 0.
x=−23x = -\frac{2}{3} or x=4x = 4
According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.
4
Filter the solutions based on the given constraint.
x=4x = 4
The question specifies that x must be a positive integer. Since −23-\frac{2}{3} is a negative fraction, we reject it and select 4.

Anahtar Kavram

Finding values that make a rational expression undefined by setting the denominator to zero and solving the resulting quadratic equation.
Tahmini Süre:1m 30s
Soru 106Soru →

The matrices AA and BB are defined as follows:

A=[2013],B=[3102]A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}

What is the product matrix ABAB?

Cevabı ve açıklamayı göster

Cevap: [6237]\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

Cevap

[6237]\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}
The correct matrix is obtained by performing row-by-column matrix multiplication, which yields row 1 entries of 6 and 2, and row 2 entries of 3 and 7.

Adım Adım Çözüm

1
Multiply the first row of AA by the first column of BB to find the element in the first row, first column of the product matrix.
2(3)+0(0)=62(3) + 0(0) = 6
The row-by-column rule dictates that the entry at row ii, column jj of the product is the dot product of row ii of the first matrix and column jj of the second matrix.
2
Multiply the first row of AA by the second column of BB to find the element in the first row, second column.
2(1)+0(2)=22(1) + 0(2) = 2
This determines the top-right entry of the resulting matrix.
3
Multiply the second row of AA by the first column of BB to find the element in the second row, first column.
1(3)+3(0)=31(3) + 3(0) = 3
This determines the bottom-left entry of the resulting matrix.
4
Multiply the second row of AA by the second column of BB to find the element in the second row, second column.
1(1)+3(2)=1+6=71(1) + 3(2) = 1 + 6 = 7
This determines the bottom-right entry of the resulting matrix.

Anahtar Kavram

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second matrix.
Soru 107Soru →

What are the solutions to the quadratic equation x2−52x+1=0x^2 - \frac{5}{2}x + 1 = 0?

Cevabı ve açıklamayı göster

Cevap: x=12x = \frac{1}{2} and x=2x = 2

Cevap

The solutions are x=12x = \frac{1}{2} and x=2x = 2.
The correct answer provides the exact solutions to the equation. Multiplying the equation by 22 yields 2x2−5x+2=02x^2 - 5x + 2 = 0. Factoring this expression gives (2x−1)(x−2)=0(2x - 1)(x - 2) = 0, which yields the solutions x=12x = \frac{1}{2} and x=2x = 2.

Adım Adım Çözüm

1
Multiply the entire equation by 22 to clear the fraction.
2x2−5x+2=02x^2 - 5x + 2 = 0
Converting the equation to integer coefficients simplifies factoring or applying the quadratic formula.
2
Identify the coefficients aa, bb, and cc to use in the quadratic formula.
a=2a = 2, b=−5b = -5, c=2c = 2
These coefficients correspond to the standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.
3
Calculate the discriminant b2−4acb^2 - 4ac.
(−5)2−4(2)(2)=25−16=9(-5)^2 - 4(2)(2) = 25 - 16 = 9
The discriminant determines the number and type of solutions.
4
Substitute the coefficients and the discriminant into the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
x=−(−5)±92(2)=5±34x = \frac{-(-5) \pm \sqrt{9}}{2(2)} = \frac{5 \pm 3}{4}
This formula provides the exact solutions to any quadratic equation.
5
Evaluate the two cases for the plus/minus sign.
x=5+34=2x = \frac{5 + 3}{4} = 2 and x=5−34=12x = \frac{5 - 3}{4} = \frac{1}{2}
Splitting the formula into positive and negative branches yields the two distinct real roots.

Anahtar Kavram

Solving quadratic equations using the quadratic formula
Soru 108Soru →

A retail electronics store sells two models of wireless headphones: Solo, which sells for 80,andStudio,whichsellsfor80, and Studio, which sells for 150. During a holiday sale, the number of headphones sold at the Downtown and Uptown store locations is represented by the matrix NN below, where the first row represents the Downtown store, the second row represents the Uptown store, the first column represents Solo headphones, and the second column represents Studio headphones:

N=[40253035]N = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix}

The prices of the headphones are represented by the matrix PP below:

P=[80150]P = \begin{bmatrix} 80 \\ 150 \end{bmatrix}

Which of the following matrices represents the total revenue, in dollars, generated from headphone sales at each of the two stores, where the first row represents the Downtown store and the second row represents the Uptown store?

Cevabı ve açıklamayı göster

Cevap: [6,9507,650]\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

Cevap

The column matrix with first row 6,950 and second row 7,650
The total revenue for each store is found by calculating the product of the sales matrix NN and the price matrix PP. This multiplication is performed by taking the dot product of each row in NN with the column in PP. For the first row (Downtown store), the calculation is 40×80+25×150=3,200+3,750=6,95040 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950. For the second row (Uptown store), the calculation is 30×80+35×150=2,400+5,250=7,65030 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650. Combining these results yields the column matrix containing 6,950 and 7,650.

Adım Adım Çözüm

1
Set up the matrix multiplication to calculate the revenue, multiplying the sales matrix NN by the price matrix PP.
NP=[40253035][80150]NP = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix} \begin{bmatrix} 80 \\ 150 \end{bmatrix}
To find the total revenue for each store, the number of units sold (organized by store in rows and product type in columns) must be multiplied by the unit prices (organized by product type in rows).
2
Calculate the revenue for the Downtown store by taking the dot product of the first row of NN and the column of PP.
40×80+25×150=3,200+3,750=6,95040 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950
This combines the units sold of each headphone model at the Downtown store with their respective prices to find the store's total revenue.
3
Calculate the revenue for the Uptown store by taking the dot product of the second row of NN and the column of PP.
30×80+35×150=2,400+5,250=7,65030 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650
This combines the units sold of each headphone model at the Uptown store with their respective prices to find the store's total revenue.
4
Combine the results into the final product matrix.
[6,9507,650]\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}
The resulting matrix contains the total revenues for the Downtown store in the first row and the Uptown store in the second row.

Anahtar Kavram

Matrix Multiplication and Modeling

Alternatif Yöntem

Instead of setting up the entire matrix multiplication, calculate the total revenue for the first store (Downtown) individually: 40×80+25×150=6,95040 \times 80 + 25 \times 150 = 6,950. This single calculation is enough to eliminate the options that do not have 6,950 in the first row.
Tahmini Süre:2m 0s
Soru 109Soru →

Matrices AA, BB, and CC are defined such that A=[x3−24]A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatrix}, B=[2−1y5]B = \begin{bmatrix} 2 & -1 \\ y & 5 \end{bmatrix}, and C=[1411422]C = \begin{bmatrix} 14 & 11 \\ 4 & 22 \end{bmatrix}. If AB=CAB = C, what is the value of x+yx + y?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of x+yx + y is 6.
The correct answer is 6 because performing the matrix multiplication ABAB yields the matrix [2x+3y−x+15−4+4y22]\begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}. Equating this to matrix CC gives the equations −x+15=11-x + 15 = 11, which yields x=4x = 4, and −4+4y=4-4 + 4y = 4, which yields y=2y = 2. The sum of these values is 4+2=64 + 2 = 6. This is verified by checking the remaining equation 2(4)+3(2)=142(4) + 3(2) = 14.

Adım Adım Çözüm

1
Multiply matrix AA and matrix BB to express the product ABAB algebraically.
AB=[x(2)+3(y)x(−1)+3(5)−2(2)+4(y)−2(−1)+4(5)]=[2x+3y−x+15−4+4y22]AB = \begin{bmatrix} x(2) + 3(y) & x(-1) + 3(5) \\ -2(2) + 4(y) & -2(-1) + 4(5) \end{bmatrix} = \begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}
To solve for the variables, we must first determine the resulting matrix from the multiplication of AA and BB.
2
Set up equations by equating the entries of ABAB with the corresponding entries of matrix CC.
2x+3y=142x + 3y = 14, −x+15=11-x + 15 = 11, and −4+4y=4-4 + 4y = 4
Two matrices are equal if and only if all of their corresponding elements are equal.
3
Solve the single-variable equation from the first row, second column for xx.
−x+15=11  ⟹  −x=−4  ⟹  x=4-x + 15 = 11 \implies -x = -4 \implies x = 4
Isolating xx gives its numeric value.
4
Solve the single-variable equation from the second row, first column for yy.
−4+4y=4  ⟹  4y=8  ⟹  y=2-4 + 4y = 4 \implies 4y = 8 \implies y = 2
Isolating yy gives its numeric value.
5
Verify consistency with the remaining equation and calculate the final sum x+yx + y.
2(4)+3(2)=8+6=142(4) + 3(2) = 8 + 6 = 14 (consistent), and x+y=4+2=6x + y = 4 + 2 = 6.
Checking the first row, first column confirms the values are correct, and adding them yields the requested sum.

Anahtar Kavram

Matrix multiplication and matrix equality
Soru 110Soru →

In the standard (x,y)(x, y) coordinate plane, the line defined by the equation 3x−4y=k3x - 4y = k is tangent to the circle defined by the equation x2+y2−2x−4y=4x^2 + y^2 - 2x - 4y = 4. If k>0k > 0, what is the value of kk?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10
The correct answer is 10. Completing the square for the circle's equation gives (x−1)2+(y−2)2=9(x - 1)^2 + (y - 2)^2 = 9, showing the center is (1,2)(1, 2) and the radius is 33. The distance from (1,2)(1, 2) to the line 3x−4y−k=03x - 4y - k = 0 is ∣3(1)−4(2)−k∣32+(−4)2=∣k+5∣5\frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|k + 5|}{5}. For tangency, this distance must equal the radius: ∣k+5∣5=3\frac{|k + 5|}{5} = 3, which gives ∣k+5∣=15|k + 5| = 15. Solving this absolute value equation gives k=10k = 10 or k=−20k = -20. Since kk must be positive, the value is 10.

Adım Adım Çözüm

1
Complete the square for the circle's equation x2+y2−2x−4y=4x^2 + y^2 - 2x - 4y = 4.
(x−1)2+(y−2)2=9(x - 1)^2 + (y - 2)^2 = 9, which represents a circle with center (1,2)(1, 2) and radius R=3R = 3.
To find the center and radius of the circle, which are needed to use the distance formula.
2
Express the distance dd from the center (1,2)(1, 2) to the line 3x−4y−k=03x - 4y - k = 0 using the formula d=∣Ax0+By0+C∣A2+B2d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}.
d=∣3(1)−4(2)−k∣32+(−4)2=∣−5−k∣5=∣k+5∣5d = \frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|-5 - k|}{5} = \frac{|k + 5|}{5}.
A line is tangent to a circle if and only if the distance from the center of the circle to the line equals the radius.
3
Set the distance equal to the radius (33) and solve for kk.
∣k+5∣5=3⇒∣k+5∣=15\frac{|k + 5|}{5} = 3 \Rightarrow |k + 5| = 15, which yields k+5=15⇒k=10k + 5 = 15 \Rightarrow k = 10, or k+5=−15⇒k=−20k + 5 = -15 \Rightarrow k = -20.
To find the values of kk that make the line tangent to the circle.
4
Select the positive value of kk.
k=10k = 10.
The problem specifies that k>0k > 0.

Anahtar Kavram

Systems of Linear and Non-Linear Equations
Tahmini Süre:2m 30s
Soru 111Soru →

For a certain geometric sequence, the first term is 232^3 and the common ratio is 222^2. Which of the following expressions represents the 3rd term of this sequence?

Cevabı ve açıklamayı göster

Cevap: 272^7

Cevap

The 3rd term of the sequence is 272^7.
The correct answer is 272^7. The nn-th term of a geometric sequence is given by an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. For a first term a1=23a_1 = 2^3 and a common ratio r=22r = 2^2, the 3rd term is a3=23⋅(22)3−1=23⋅24a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot 2^4. Applying the exponent addition rule for multiplying bases of the same value yields 23+4=272^{3+4} = 2^7.

Adım Adım Çözüm

1
Identify the formula for the nn-th term of a geometric sequence.
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
To find any specific term in a geometric sequence, the general term formula is used.
2
Substitute the given values (a1=23a_1 = 2^3, r=22r = 2^2, and n=3n = 3) into the formula.
a3=23⋅(22)3−1=23⋅(22)2a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot (2^2)^2
This sets up the calculation for the 3rd term of the sequence.
3
Simplify the expression using exponent rules: first compute (22)2(2^2)^2, then multiply by 232^3.
a3=23⋅24=23+4=27a_3 = 2^3 \cdot 2^4 = 2^{3+4} = 2^7
Power of a power rule states (xa)b=xab(x^a)^b = x^{ab}, and product of powers rule states xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}.

Anahtar Kavram

Finding a specific term of a geometric sequence using the general term formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} and applying laws of exponents.
Tahmini Süre:45s
Soru 112Soru →

The functions ff and gg are defined for all real numbers by f(x)=2x−5f(x) = 2x - 5 and g(x)=(x+3)2g(x) = (x + 3)^2. What is the value of f(g(−1))f(g(-1))?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3
To evaluate the composite function f(g(−1))f(g(-1)), first evaluate the inner function g(−1)=(−1+3)2=4g(-1) = (-1 + 3)^2 = 4. Then, substitute this result into the outer function to get f(4)=2(4)−5=3f(4) = 2(4) - 5 = 3.

Adım Adım Çözüm

1
Evaluate the inner function g(x)g(x) at x=−1x = -1.
g(−1)=4g(-1) = 4
In the composition f(g(−1))f(g(-1)), the inner function must be evaluated first. Substituting −1-1 into g(x)=(x+3)2g(x) = (x + 3)^2 gives g(−1)=(−1+3)2=22=4g(-1) = (-1 + 3)^2 = 2^2 = 4.
2
Evaluate the outer function f(x)f(x) at the output of the inner function.
f(4)=3f(4) = 3
Substitute the output g(−1)=4g(-1) = 4 into f(x)=2x−5f(x) = 2x - 5, which yields f(4)=2(4)−5=8−5=3f(4) = 2(4) - 5 = 8 - 5 = 3.

Anahtar Kavram

Function Composition and Evaluation
Tahmini Süre:1m 0s
Soru 113Soru →
A clothing boutique sells two types of shirts: t-shirts and polo shirts. The number of shirts sold at the boutique's two locations, Downtown and Uptown, on a weekend is represented by matrix SS:
S = \begin{pmatrix} 40 & 30 \\ 50 & 20 \\end{pmatrix}
where the rows represent the locations (Downtown and Uptown, respectively) and the columns represent the shirt types (t-shirts and polo shirts, respectively).
The selling price and the production cost per shirt (in dollars) are represented by matrix CC:
C = \begin{pmatrix} 15 & 6 \\ 25 & 10 \\end{pmatrix}
where the rows represent the shirt types (t-shirts and polo shirts, respectively) and the columns represent the selling price and production cost, respectively.

Which of the following matrices represents the total selling revenue and total production cost for the two locations?

Cevabı ve açıklamayı göster

Cevap: \begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

Cevap

The matrix representing the total selling revenue and total production cost for the two locations is the matrix with top row 1350 and 540, and bottom row 1250 and 500.
The correct matrix is obtained by multiplying the sales matrix SS by the price/cost matrix CC. Standard matrix multiplication pairs the quantity of each shirt type sold at each store with its respective selling price and production cost, yielding the correct total revenue and total production cost for both locations.

Adım Adım Çözüm

1
Identify the dimensions and layout of the sales matrix SS and the price/cost matrix CC.
Matrix SS is a 2×22 \times 2 matrix representing sales. Matrix CC is a 2×22 \times 2 matrix representing price and cost. The product SCSC is defined because the number of columns in SS (2) equals the number of rows in CC (2).
Before multiplying, we must verify that the dimensions are compatible and that the product yields the desired real-world quantities.
2
Calculate the entries for the first row of the resulting matrix SCSC (Downtown location) by taking the dot product of the first row of SS with the columns of CC.
Revenue (Row 1, Column 1): 40×15+30×25=600+750=135040 \times 15 + 30 \times 25 = 600 + 750 = 1350.
Cost (Row 1, Column 2): 40×6+30×10=240+300=54040 \times 6 + 30 \times 10 = 240 + 300 = 540.
This yields the total revenue and production cost for the Downtown location by combining sales and prices/costs for both shirt types.
3
Calculate the entries for the second row of the resulting matrix SCSC (Uptown location) by taking the dot product of the second row of SS with the columns of CC.
Revenue (Row 2, Column 1): 50×15+20×25=750+500=125050 \times 15 + 20 \times 25 = 750 + 500 = 1250.
Cost (Row 2, Column 2): 50×6+20×10=300+200=50050 \times 6 + 20 \times 10 = 300 + 200 = 500.
This yields the total revenue and production cost for the Uptown location.

Anahtar Kavram

Matrix multiplication
Soru 114Soru →

An infinite geometric series of positive terms has a sum of 99. The sum of the first two terms of the series is 88. What is the first term of this series?

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Cevap: 6

Cevap

The first term of the series is 6.
To find the first term of the geometric series, we set up a system of equations using the formulas for the sum of the first two terms, S2=a(1+r)=8S_2 = a(1+r) = 8, and the sum of an infinite geometric series, S∞=a1−r=9S_\infty = \frac{a}{1-r} = 9. Expressing the first term as a=9(1−r)a = 9(1-r) and substituting this into the first equation yields 9(1−r)(1+r)=89(1-r)(1+r) = 8, which simplifies to 9(1−r2)=89(1-r^2) = 8. Solving for the ratio yields r2=19r^2 = \frac{1}{9}, which means r=13r = \frac{1}{3} since all terms must be positive. Substituting r=13r = \frac{1}{3} back into a=9(1−r)a = 9(1-r) gives the first term as 66.

Adım Adım Çözüm

1
Define variables and identify the given formulas.
Let the first term of the geometric series be aa and the common ratio be rr. Since the series has positive terms, we require a>0a > 0 and 0<r<10 < r < 1.
This sets up the system of equations using standard geometric sequence notation.
2
Translate the given problem conditions into algebraic equations.
Equation 1 (sum of the first two terms): S2=a+ar=a(1+r)=8S_2 = a + ar = a(1+r) = 8.
Equation 2 (sum to infinity): S∞=a1−r=9S_\infty = \frac{a}{1-r} = 9.
We must express the two mathematical relationships given in the problem statement.
3
Express the first term aa in terms of the common ratio rr using Equation 2.
a=9(1−r)a = 9(1-r)
This allows for substitution into Equation 1 to solve for rr.
4
Substitute the expression for aa into Equation 1 and simplify.
9(1−r)(1+r)=8  ⟹  9(1−r2)=8  ⟹  9−9r2=89(1-r)(1+r) = 8 \implies 9(1-r^2) = 8 \implies 9 - 9r^2 = 8.
Substituting simplifies the system from two variables to a single quadratic variable in terms of rr.
5
Solve the quadratic equation for rr.
9r2=1  ⟹  r2=19  ⟹  r=139r^2 = 1 \implies r^2 = \frac{1}{9} \implies r = \frac{1}{3} (since r>0r > 0).
Finding the value of the common ratio is the final step before calculating the first term.
6
Substitute the value of rr back into the expression for aa.
a=9(1−13)=9(23)=6a = 9\left(1 - \frac{1}{3}\right) = 9\left(\frac{2}{3}\right) = 6.
This yields the value of the first term aa to complete the problem.

Anahtar Kavram

Solving systems of non-linear equations using geometric sequence term and infinite sum formulas.

Alternatif Yöntem

Instead of substituting a=9(1−r)a = 9(1-r) into a(1+r)=8a(1+r) = 8, we can divide the two equations: a(1+r)a/(1−r)=89  ⟹  (1−r)(1+r)=89  ⟹  1−r2=89\frac{a(1+r)}{a/(1-r)} = \frac{8}{9} \implies (1-r)(1+r) = \frac{8}{9} \implies 1 - r^2 = \frac{8}{9}. This directly yields r2=19r^2 = \frac{1}{9} without needing to isolate aa first.
Tahmini Süre:2m 0s
Soru 115Soru →

Consider the function ff defined on the domain [23,∞)[\frac{2}{3}, \infty) by the equation f(x)=2+3x−2f(x) = 2 + \sqrt{3x - 2}. If f−1f^{-1} represents the inverse of ff, what is the only real value of xx for which f(x)=f−1(x)f(x) = f^{-1}(x)?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The only real value of xx for which f(x)=f−1(x)f(x) = f^{-1}(x) is 6.
For a strictly increasing function, the intersection of f(x)f(x) and f−1(x)f^{-1}(x) must occur on the line y=xy = x. Equating f(x)=xf(x) = x gives 2+3x−2=x2 + \sqrt{3x - 2} = x. Isolating the radical term yields 3x−2=x−2\sqrt{3x - 2} = x - 2. Squaring both sides produces 3x−2=x2−4x+43x - 2 = x^2 - 4x + 4, which simplifies to the quadratic equation x2−7x+6=0x^2 - 7x + 6 = 0. Factoring this equation gives (x−6)(x−1)=0(x - 6)(x - 1) = 0, yielding the solutions x=6x = 6 and x=1x = 1. Substituting these back into the original equation shows that x=6x = 6 is a valid solution (2+16=62 + \sqrt{16} = 6), whereas x=1x = 1 is extraneous because it results in 2+1=3≠12 + 1 = 3 \neq 1. Furthermore, the domain of f−1f^{-1} is the range of ff, which is [2,∞)[2, \infty), meaning f−1(1)f^{-1}(1) is undefined. Thus, the only real solution is 6.

Adım Adım Çözüm

1
Equate the function to xx using properties of increasing functions and their inverses.
Since f(x)f(x) is strictly increasing on its domain [23,∞)[\frac{2}{3}, \infty), the graph of f(x)f(x) and the graph of its inverse f−1(x)f^{-1}(x) can only intersect on the line of symmetry y=xy = x. Therefore, the equation f(x)=f−1(x)f(x) = f^{-1}(x) is equivalent to f(x)=xf(x) = x.
This simplifies the relation by eliminating the need to solve a high-degree polynomial equation derived from direct composition or substitution.
2
Set up the equation f(x)=xf(x) = x and isolate the radical term.
2+3x−2=x  ⟹  3x−2=x−22 + \sqrt{3x - 2} = x \implies \sqrt{3x - 2} = x - 2
Isolating the square root term is a necessary prerequisite step before squaring both sides.
3
Square both sides and rewrite the equation as a standard quadratic equation.
3x−2=(x−2)2  ⟹  3x−2=x2−4x+4  ⟹  x2−7x+6=03x - 2 = (x - 2)^2 \implies 3x - 2 = x^2 - 4x + 4 \implies x^2 - 7x + 6 = 0
Squaring eliminates the radical to yield a standard quadratic equation that can be solved analytically.
4
Solve the quadratic equation by factoring.
(x−6)(x−1)=0  ⟹  x=6 or x=1(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1
Factoring the quadratic trinomial yields the two candidate values for xx.
5
Check the candidate solutions in the original equation to eliminate extraneous roots.
For x=1x = 1: 2+3(1)−2=3≠12 + \sqrt{3(1)-2} = 3 \neq 1 (extraneous). For x=6x = 6: 2+3(6)−2=2+4=62 + \sqrt{3(6)-2} = 2 + 4 = 6 (valid). Also, the domain of f−1f^{-1} is the range of ff, which is [2,∞)[2, \infty), thus excluding x=1x = 1.
Squaring both sides can introduce extraneous roots. We must verify that the solutions satisfy the original radical equation and fall within the domains of both ff and f−1f^{-1}.

Anahtar Kavram

Applying the symmetry of inverse functions about the line y=xy=x to solve composition-based equations, while rigorously accounting for domain restrictions and extraneous roots.

Alternatif Yöntem

Find the algebraic formula for f−1(x)f^{-1}(x) by setting y=2+3x−2y = 2 + \sqrt{3x-2}. Subtracting 2 and squaring both sides gives (y−2)2=3x−2(y - 2)^2 = 3x - 2 for y≥2y \geq 2. Solving for xx yields f−1(x)=(x−2)2+23f^{-1}(x) = \frac{(x - 2)^2 + 2}{3} for x≥2x \geq 2. Equating f(x)=f−1(x)f(x) = f^{-1}(x) results in 2+3x−2=(x−2)2+232 + \sqrt{3x - 2} = \frac{(x - 2)^2 + 2}{3}. Multiplying by 3 and isolating the radical term gives 33x−2=x2−4x3\sqrt{3x - 2} = x^2 - 4x. Squaring both sides results in a fourth-degree polynomial equation: 9(3x−2)=(x2−4x)2  ⟹  x4−8x3+16x2−27x+18=09(3x - 2) = (x^2 - 4x)^2 \implies x^4 - 8x^3 + 16x^2 - 27x + 18 = 0. This factors into (x−6)(x−1)(x2−x+3)=0(x - 6)(x - 1)(x^2 - x + 3) = 0. Since the domain of f−1(x)f^{-1}(x) is restricted to x≥2x \geq 2, the root x=1x = 1 is rejected, and the quadratic factor x2−x+3=0x^2 - x + 3 = 0 has no real roots, leaving x=6x = 6 as the unique real solution.
Tahmini Süre:3m 0s
Soru 116Soru →

A right triangle has an area of 66 square units. The lengths of its legs, aa and bb, satisfy the quadratic equation a2−2.75ab+1.5b2=0a^2 - 2.75ab + 1.5b^2 = 0. If the length of the hypotenuse is an integer, what is the perimeter of the triangle?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The perimeter of the triangle is 12.
The correct answer is 12. By solving the quadratic relationship between the legs, we find two possible ratios: one where one leg is twice the other, and one where one leg is 0.75 times the other. Using the area of 6, the first case yields non-integer side lengths, while the second case yields legs of length 3 and 4. This results in an integer hypotenuse of 5, giving a perimeter of 3 + 4 + 5 = 12.

Adım Adım Çözüm

1
Express the relation a2−2.75ab+1.5b2=0a^2 - 2.75ab + 1.5b^2 = 0 as a quadratic in terms of the ratio r=abr = \frac{a}{b}.
The equation becomes r2−2.75r+1.5=0r^2 - 2.75r + 1.5 = 0.
To find the relationship between the two legs of the triangle by solving for their ratio.
2
Apply the quadratic formula to solve for rr.
r=2r = 2 or r=0.75r = 0.75, meaning either a=2ba = 2b or a=0.75ba = 0.75b.
To determine the two possible linear relationships between the legs of the right triangle.
3
Substitute each ratio case into the area formula Area=12ab=6\text{Area} = \frac{1}{2}ab = 6, which simplifies to ab=12ab = 12.
For a=2ba = 2b, we get b=6b = \sqrt{6} and a=26a = 2\sqrt{6}. For a=0.75ba = 0.75b, we get b=4b = 4 and a=3a = 3.
To calculate the actual leg lengths for both geometric cases.
4
Calculate the hypotenuse c=a2+b2c = \sqrt{a^2 + b^2} for both cases to check which yields an integer value.
The first case yields c=30c = \sqrt{30}, which is not an integer. The second case yields c=32+42=5c = \sqrt{3^2 + 4^2} = 5, which is an integer.
To satisfy the constraint that the hypotenuse must be an integer, identifying the correct leg lengths as 33 and 44.
5
Calculate the perimeter of the triangle for the valid case.
The perimeter is 3+4+5=123 + 4 + 5 = 12.
To find the final requested value.

Anahtar Kavram

Solving quadratic relationships and applying the quadratic formula in geometric constraints.
Tahmini Süre:3m 0s
Soru 117Soru →

If 3x+19−x=3\sqrt{3x + 19} - x = 3, what is the value of xx?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of xx is 2.
To solve the equation, isolate the radical to obtain 3x+19=x+3\sqrt{3x + 19} = x + 3. Squaring both sides yields 3x+19=x2+6x+93x + 19 = x^2 + 6x + 9. Rearranging terms results in the quadratic equation x2+3x−10=0x^2 + 3x - 10 = 0, which factors into (x+5)(x−2)=0(x + 5)(x - 2) = 0. The potential solutions are x=2x = 2 and x=−5x = -5. Testing these in the original equation shows that 22 is a valid solution because 3(2)+19−2=3\sqrt{3(2) + 19} - 2 = 3. Testing −5-5 results in 3(−5)+19−(−5)=7\sqrt{3(-5) + 19} - (-5) = 7, which does not equal 33. Thus, the only real solution is 22.

Adım Adım Çözüm

1
Isolate the radical expression on one side of the equation.
3x+19=x+3\sqrt{3x + 19} = x + 3
Isolating the radical allows us to eliminate it by squaring both sides.
2
Square both sides of the equation.
3x+19=x2+6x+93x + 19 = x^2 + 6x + 9
Squaring a square root removes the radical. Remember to expand (x+3)2(x+3)^2 correctly as x2+6x+9x^2 + 6x + 9.
3
Rearrange the terms into standard quadratic form, ax2+bx+c=0ax^2 + bx + c = 0.
x2+3x−10=0x^2 + 3x - 10 = 0
Moving all terms to one side sets up the quadratic equation for factoring.
4
Factor the quadratic expression.
(x+5)(x−2)=0(x + 5)(x - 2) = 0
Factoring allows us to find the potential values of xx by setting each factor to zero.
5
Solve for the potential values of xx.
x=−5x = -5 or x=2x = 2
Setting x+5=0x + 5 = 0 gives x=−5x = -5, and setting x−2=0x - 2 = 0 gives x=2x = 2.
6
Check both potential solutions in the original equation to identify any extraneous solutions.
For x=2x = 2: 3(2)+19−2=25−2=5−2=3\sqrt{3(2) + 19} - 2 = \sqrt{25} - 2 = 5 - 2 = 3 (valid). For x=−5x = -5: 3(−5)+19−(−5)=4+5=2+5=7≠3\sqrt{3(-5) + 19} - (-5) = \sqrt{4} + 5 = 2 + 5 = 7 \neq 3 (extraneous).
Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Anahtar Kavram

Solving radical equations and checking for extraneous solutions.
Soru 118Soru →

Let matrix X=[5−321]X = \begin{bmatrix} 5 & -3 \\ 2 & 1 \end{bmatrix} and matrix Y=[14−23]Y = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}. If Z=X−YZ = X - Y, what is the value of the element in the first row and second column of ZZ?

Cevabı ve açıklamayı göster

Cevap: -7

Cevap

The correct answer is −7-7.
To find the element in the first row and second column of matrix Z=X−YZ = X - Y, we subtract the element in the first row and second column of matrix YY from the corresponding element in matrix XX. The element in the first row and second column of XX is −3-3, and the element in the first row and second column of YY is 44. Subtracting these values gives −3−4=−7-3 - 4 = -7.

Adım Adım Çözüm

1
Identify the elements in the first row and second column for both matrices.
x1,2=−3x_{1,2} = -3 and y1,2=4y_{1,2} = 4
To find the element in the first row and second column of the resulting matrix ZZ, we must use the corresponding elements from matrices XX and YY.
2
Subtract the element of YY from the element of XX.
−3−4=−7-3 - 4 = -7
Since Z=X−YZ = X - Y, each element zi,jz_{i,j} of the resulting matrix is calculated as xi,j−yi,jx_{i,j} - y_{i,j}.

Anahtar Kavram

Matrix Subtraction
Soru 119Soru →

What is the sum of the two solutions to the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0?

Cevabı ve açıklamayı göster

Cevap: 3.5

Cevap

The sum of the solutions is 3.53.5.
The sum of the solutions of the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0 is 3.53.5. According to Vieta's formulas, the sum of the roots of a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0 is given by −ba-\frac{b}{a}. Here, a=2a = 2 and b=−7b = -7, so the sum is −−72=3.5-\frac{-7}{2} = 3.5. Alternatively, solving the equation using the quadratic formula yields the roots 44 and −0.5-0.5, which sum to 3.53.5.

Adım Adım Çözüm

1
Identify the coefficients from the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0.
a=2a = 2, b=−7b = -7, and c=−4c = -4.
A quadratic equation in standard form is written as ax2+bx+c=0ax^2 + bx + c = 0.
2
Apply the sum of roots formula −ba-\frac{b}{a}.
Sum =−−72=3.5= -\frac{-7}{2} = 3.5.
By Vieta's formulas, the sum of the roots of ax2+bx+c=0ax^2 + bx + c = 0 is −ba-\frac{b}{a}.

Anahtar Kavram

Sum of roots of a quadratic equation using Vieta's formulas

Alternatif Yöntem

Solve the quadratic equation by factoring or using the quadratic formula. Factoring 2x2−7x−4=02x^2 - 7x - 4 = 0 yields (2x+1)(x−4)=0(2x + 1)(x - 4) = 0, which gives solutions x=−0.5x = -0.5 and x=4x = 4. Adding these solutions together gives −0.5+4=3.5-0.5 + 4 = 3.5.
Tahmini Süre:1m 0s
Soru 120Soru →

A system of equations is given below:

y=(x−1)2−6y=x−1\begin{aligned} y &= (x - 1)^2 - 6 \\ y &= x - 1 \end{aligned}

If (x,y)(x, y) is a solution to this system in the first quadrant, what is the value of x+yx + y?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The sum of the coordinates of the first quadrant solution is 7.
To solve the system, substitute the expression for yy from the linear equation into the quadratic equation to get x−1=(x−1)2−6x - 1 = (x - 1)^2 - 6. Substituting u=x−1u = x - 1 yields u2−u−6=0u^2 - u - 6 = 0. Factoring the quadratic expression gives (u−3)(u+2)=0(u - 3)(u + 2) = 0, so u=3u = 3 or u=−2u = -2. Since u=x−1u = x - 1, this means x=4x = 4 or x=−1x = -1. Evaluating both cases gives the intersection points (4,3)(4, 3) and (−1,−2)(-1, -2). Only (4,3)(4, 3) is in the first quadrant. Adding these coordinates together yields 4+3=74 + 3 = 7.

Adım Adım Çözüm

1
Substitute y=x−1y = x - 1 into the quadratic equation to set up an equation in terms of xx.
x−1=(x−1)2−6x - 1 = (x - 1)^2 - 6
To find the xx-coordinates of the intersection points.
2
Solve the equation for xx by substituting u=x−1u = x - 1.
u=u2−6⇒u2−u−6=0⇒(u−3)(u+2)=0u = u^2 - 6 \Rightarrow u^2 - u - 6 = 0 \Rightarrow (u - 3)(u + 2) = 0. This gives u=3u = 3 or u=−2u = -2.
To find the values of the substituted variable uu.
3
Find the corresponding xx and yy values for both cases.
Case 1: x−1=3⇒x=4x - 1 = 3 \Rightarrow x = 4, which gives y=3y = 3. Point is (4,3)(4, 3). Case 2: x−1=−2⇒x=−1x - 1 = -2 \Rightarrow x = -1, which gives y=−2y = -2. Point is (−1,−2)(-1, -2).
To determine the full coordinates of all intersection points.
4
Identify the first quadrant solution and calculate the sum of its coordinates.
The first quadrant solution is (4,3)(4, 3). The sum of the coordinates is 4+3=74 + 3 = 7.
To answer the question's requirement for the sum of coordinates in the first quadrant.

Anahtar Kavram

Solving systems of linear and quadratic equations by substitution and identifying quadrant-specific solutions.
ÖncekiSayfa 6 / 14Sonraki
Intermediate Algebra Alıştırma Soruları — ACT — Sayfa 6 | Examkin