SAT Math Full-Length Practice Test 3

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Module 1

Question 1

If \(6x + 3 = 27\), what is the value of \(x\)?




Show solution

Subtract 3 from both sides: \(6x = 24\). Divide by 6: \(x = 4\).

    1. 5: Sign error, treating the equation as \(6x - 3 = 27\) instead of \(6x + 3 = 27\), giving \(6x = 30\).
    1. 8: Correctly found \(6x = 24\) but divided by 3 instead of 6.
    1. 24: Correctly found \(6x = 24\) but forgot to divide by 6, reporting the value of \(6x\) instead of \(x\).

Answer: A

Question 2

A rectangle has a perimeter of 36 inches. If the length is twice the width, what is the width, in inches, of the rectangle?




Show solution

Let \(w\) be the width; length is \(2w\). Perimeter: \(2w + 2(2w) = 36\), so \(6w = 36\), \(w = 6\).

    1. 4: Divided the perimeter by 9 (a miscount of the total number of width-units in the perimeter expression) instead of 6.
    1. 9: Divided the perimeter by 4 directly, ignoring the length-width relationship entirely.
    1. 12: Reported the length instead of the width.

Answer: B

Question 3

Which of the following is equivalent to \(5(2x - 1) - 3x\)?




Show solution

Distribute: \(5(2x-1) = 10x - 5\). Combine like terms: \(10x - 5 - 3x = 7x - 5\).

    1. \(7x - 1\): Correctly combined the \(x\)-coefficients as \(7x\) but forgot to distribute the 5 into the constant term.
    1. \(13x - 5\): Correctly distributed to get \(10x - 5\), but added \(3x\) instead of subtracting it, giving \(13x - 5\).
    1. \(13x - 1\): Made the same \(x\)-coefficient error as C and also forgot to distribute the 5 into the constant term.

Answer: A

Question 4

Which of the following represents all values of \(x\) satisfying \(4x - 9 < 15\)?




Show solution

Add 9 to both sides: \(4x < 24\). Divide by 4: \(x < 6\).

    1. \(x < 1.5\): Correctly isolated the inequality direction but divided 24 by 16 (adding 9 and 15 instead of just 15, then dividing by 16 instead of 4).
    1. \(x > 1.5\): Solved correctly for the boundary value but flipped the inequality direction without justification.
    1. \(x > 6\): Combined both errors: used the wrong boundary value from A and flipped the inequality direction.

Answer: B

Question 5

If \(\dfrac{2x}{5} - 3 = 7\), what is the value of \(x\)?

Enter your answer:

Show solution

Add 3 to both sides: \(\dfrac{2x}{5} = 10\). Multiply both sides by 5: \(2x = 50\). Divide by 2: \(x = 25\).

Answer: 25

Question 6

If \(f(x) = 4x - 7\), what is the value of \(f(3)\)?




Show solution

Substitute \(x=3\): \(f(3) = 4(3) - 7 = 12 - 7 = 5\).

    1. 12: Forgot the negative sign on the constant, computing \(4(3) + 7\) instead of \(4(3) - 7\).
    1. 19: Reported the value of \(4(3)\) before subtracting 7.
    1. 28: Multiplied both factors by the input, computing \(4(7)(3)\) instead of \(4(3) - 7\).

Answer: A

Question 7

What is the value of \((2^4)(2^3)\)?

Enter your answer:

Show solution

Using the product rule for exponents: \((2^4)(2^3) = 2^{4+3} = 2^7 = 128\).

Answer: 128

Question 8

A jacket originally priced at \(\$65\) is marked up by \(20\%\). What is the new price of the jacket?




Show solution

New price \(= 65 + 0.20(65) = 65(1.20) = \$78\).

    1. \(\$13\): Computed only the markup amount (\(20\%\) of 65) instead of the new price.
    1. \(\$71.50\): Used a \(10\%\) markup instead of \(20\%\), computing \(65 \times 1.10\).
    1. \(\$74.75\): Used a \(15\%\) markup instead of \(20\%\), computing \(65 \times 1.15\).

Answer: D

Question 9

The weights, in pounds, of six packages are 8, 11, 6, 14, 9, and 12. What is the median weight, in pounds, of the six packages?

Enter your answer:

Show solution

Order the values: 6, 8, 9, 11, 12, 14. With six values, the median is the average of the two middle values: \(\dfrac{9+11}{2} = 10\).

Answer: 10

Question 10

The function \(h\) is defined by \(h(x) = \sqrt{2x + 6}\). What is the domain of \(h\)?




Show solution

The expression under a square root must be non-negative: \(2x + 6 \geq 0\), so \(2x \geq -6\), and \(x \geq -3\).

    1. \(x \leq -3\): Solved the inequality but flipped the direction.
    1. \(x \leq 3\): Made a sign error, solving \(2x - 6 \geq 0\) instead of \(2x + 6 \geq 0\).
    1. \(x \geq 3\): Combined both errors: sign error on setup and a flipped inequality direction.

Answer: C

Question 11

Two angles are complementary. One angle measures \(27°\). What is the measure of the other angle?

27° ?




Show solution

Complementary angles sum to \(90°\): \(90 - 27 = 63°\).

    1. 45°: Mistakenly computed half of \(90°\) instead of \(90° - 27°\).
    1. 53°: Misread the given angle as \(37°\) before subtracting from \(90°\).
    1. 60°: Arithmetic slip, subtracting \(30°\) instead of \(27°\) from \(90°\).

Answer: D

Question 12

A triangle has a base of 14 and a height of 5. What is its area?

14 5




Show solution

Area \(= \dfrac{1}{2}(\text{base})(\text{height}) = \dfrac{1}{2}(14)(5) = 35\).

    1. 19: Added the base and height instead of multiplying.
    1. 70: Forgot the \(\dfrac{1}{2}\) factor, computing base \(\times\) height only.
    1. 98: Used base\(^2\) instead of base \(\times\) height in the area formula.

Answer: B

Question 13

If \(y = 4x - 1\) and \(2x + y = 17\), what is the value of \(x\)?




Show solution

Substitute \(y = 4x-1\) into \(2x+y=17\): \(2x + (4x-1) = 17\), so \(6x - 1 = 17\), \(6x = 18\), \(x = 3\).

    1. 6: Correctly reduced the equation to \(6x = 18\) but divided by 3 instead of 6.
    1. 11: Correctly solved for \(x\) but reported the value of \(y\) instead: \(y = 4(3) - 1 = 11\).
    1. 18: Correctly reduced the equation to \(6x = 18\) but forgot to divide by 6, reporting the value of \(6x\) instead of \(x\).

Answer: A

Question 14

A landscaping company charges a \(\$45\) service fee plus \(\$30\) per hour of labor. A second company charges no service fee but \(\$45\) per hour. For how many hours of labor do the two companies charge the same total amount?




Show solution

Set the total costs equal: \(45 + 30h = 45h\). Subtract \(30h\): \(45 = 15h\). Divide by 15: \(h = 3\).

    1. 0.6: Divided the service fee by the sum of the two rates (\(30+45=75\)) instead of their difference.
    1. 1: Divided the service fee by the second company’s rate alone (45) instead of the rate difference.
    1. 1.5: Divided the service fee by the first company’s rate alone (30) instead of the rate difference.

Answer: D

Question 15

The formula for simple interest is \(I = Prt\), where \(P\) is principal, \(r\) is the annual interest rate, and \(t\) is time in years. Which of the following expresses \(r\) in terms of \(I\), \(P\), and \(t\)?




Show solution

Divide both sides of \(I = Prt\) by \(Pt\): \(r = \dfrac{I}{Pt}\).

    1. \(r = I - Pt\): Treated the equation as additive rather than multiplicative, subtracting instead of dividing.
    1. \(r = \dfrac{Pt}{I}\): Inverted the correct expression, dividing \(Pt\) by \(I\) instead of \(I\) by \(Pt\).
    1. \(r = IPt\): Multiplied all three quantities together instead of isolating \(r\) by division.

Answer: A

Question 16

What are the solutions to \(x^2 - 2x - 24 = 0\)?




Show solution

Factor: \(x^2 - 2x - 24 = (x-6)(x+4) = 0\), so \(x = 6\) or \(x = -4\).

    1. \(x = -6\) and \(x = 4\): Flipped the signs of both roots.
    1. \(x = 6\) and \(x = 4\): Correctly found 6 but dropped the negative sign on the second root.
    1. \(x = -8\) and \(x = 3\): Chose a factor pair of \(-24\) (\(-8\) and \(3\)) without checking that the pair sums to \(-2\).

Answer: B

Question 17

The function \(f\) is defined by \(f(x) = 3x^2 + 12x - 5\). Which of the following is an equivalent form of \(f(x)\) that reveals the minimum value of \(f\)?




Show solution

Factor out 3: \(f(x) = 3(x^2+4x) - 5\). Complete the square: half of 4 is 2, and \(2^2=4\). \(f(x) = 3(x^2+4x+4-4) - 5 = 3((x+2)^2 - 4) - 5 = 3(x+2)^2 - 12 - 5 = 3(x+2)^2 - 17\).

    1. \(f(x) = 3(x+2)^2 - 5\): Correctly found \(h = -2\) but forgot to update the constant term after completing the square, leaving the original \(-5\).
    1. \(f(x) = 3(x+4)^2 - 53\): Used \(h = -4\) (the coefficient of \(x\) before dividing by 2) instead of correctly halving it to get \(h=-2\).
    1. \(f(x) = 3(x+2)^2 + 7\): Correctly found \(h=-2\) but made a sign error on the final constant, adding instead of subtracting 12.

Answer: A

Question 18

A survey asked 180 students whether they own a bicycle and whether they own a skateboard. The results are shown in the table below.

Owns bicycle No bicycle Total
Owns skateboard 45 25 70
No skateboard 95 15 110
Total 140 40 180




Show solution

From the table, 70 students own a skateboard out of 180 total. \(\dfrac{70}{180} = \dfrac{7}{18}\).

    1. \(\dfrac{7}{9}\): Used the total number of bicycle owners (140) as the denominator instead of the total number of students (180).
    1. \(\dfrac{9}{20}\): Left the fraction \(\dfrac{81}{180}\) partially reduced instead of fully simplifying to lowest terms.
    1. \(\dfrac{9}{14}\): Used the total number of skateboard owners (70) as the denominator instead of the total number of students (180).

Answer: A

Question 19

A cafe’s daily coffee sales and daily high temperature over 7 days are shown below. Which of the following best describes the association between temperature and coffee sales?

50 60 70 80 90 100 0 50 100 150 200 Temperature (degrees F) Coffee Sales




Show solution

As the daily high temperature increases across the 7 days, coffee sales tend to decrease, following this downward trend closely but not perfectly. This describes a strong negative association.

    1. A strong positive association: Misread the general downward trend as upward.
    1. No association: Failed to recognize the consistent downward pattern across the data points.
    1. A perfect negative association: Overstated the strength of the relationship — the points follow a clear downward trend but are not perfectly aligned on a single line.

Answer: B

Question 20

A circle has a radius of 12. What is the area of a sector with a central angle of \(150°\)?

O 150° r = 12




Show solution

Sector area \(= \dfrac{\theta}{360°} \times \pi r^2 = \dfrac{150}{360} \times \pi (12)^2 = \dfrac{5}{12} \times 144\pi = 60\pi\).

    1. \(5\pi\): Used \(\dfrac{150}{360} \times \pi r\) instead of \(\pi r^2\), forgetting to square the radius.
    1. \(30\pi\): Used \(\dfrac{150}{720}\) (double the correct denominator) instead of \(\dfrac{150}{360}\).
    1. \(40\pi\): Misread the central angle as \(100°\) instead of \(150°\).

Answer: D

Question 21

The equation \(3x^2 - 6x + k = 0\) has exactly one real solution. What is the value of \(k\)?

Enter your answer:

Show solution

A quadratic has exactly one real solution when its discriminant equals zero: \((-6)^2 - 4(3)(k) = 0\), so \(36 - 12k = 0\), and \(k = 3\).

Answer: 3

Question 22

Triangle \(JKL\) is similar to triangle \(MNP\), where angle \(J\) corresponds to angle \(M\). If \(JK = 8\), \(KL = 10\), \(JL = 14\), and \(MN = 12\), what is the length of \(NP\)?

J L K 8 10 14 M P N 12




Show solution

The scale factor is \(\dfrac{MN}{JK} = \dfrac{12}{8} = 1.5\). Since \(KL\) corresponds to \(NP\): \(NP = KL \times 1.5 = 10 \times 1.5 = 15\).

    1. 6.67: Inverted the scale factor, using \(\dfrac{JK}{MN} = \dfrac{2}{3}\) applied backwards to \(KL\).
    1. 12.5: Simplified the scale factor \(\dfrac{12}{8}\) incorrectly as \(\dfrac{5}{4}\) instead of \(\dfrac{3}{2}\), then applied it to \(KL\): \(10 \times 1.25 = 12.5\).
    1. 21: Correctly computed the scale factor \(\dfrac{MN}{JK} = 1.5\), but applied it to \(JL\) (14) instead of \(KL\) (10).

Answer: C

Module 2

Question 1

If \(7x - 5 = 3x + 19\), what is the value of \(x\)?




Show solution

Subtract \(3x\) from both sides: \(4x - 5 = 19\). Add 5: \(4x = 24\). Divide by 4: \(x = 6\).

    1. 3: Correctly reduced the equation to \(4x=24\) but divided by 8 instead of 4.
    1. 4: Correctly reduced the equation to \(4x=24\) but divided by the wrong number (6 instead of 4).
    1. 24: Correctly reduced the equation to \(4x=24\) but forgot to divide by 4, reporting the value of \(4x\) instead of \(x\).

Answer: C

Question 2

What is the value of \(\dfrac{5^7}{5^4 \cdot 5^1}\)?




Show solution

Combine the denominator: \(5^4 \cdot 5^1 = 5^5\). Then \(\dfrac{5^7}{5^5} = 5^{7-5} = 5^2 = 25\).

    1. 5: Combined the denominator correctly to \(5^5\) but then subtracted exponents as \(7-5-1\) instead of \(7-5\).
    1. 125: Added all three exponents together (\(7+4+1=12\)) then mistakenly reported \(5^3\) instead of resolving the full computation.
    1. 625: Ignored the division and multiplied all three powers together, computing \(5^{7+4+1}\) reduced incorrectly to \(5^4\).

Answer: B

Question 3

A car travels 320 miles using 8 gallons of gas. At this rate, how many gallons of gas are needed to travel 520 miles?




Show solution

Rate: \(320\) miles per \(8\) gallons \(= 40\) miles per gallon. Gallons needed: \(520 / 40 = 13\).

    1. 16.25: Divided 520 by the car’s miles-per-gallon rate (40) but used 32 instead of 40 as the rate.
    1. 40: Computed the car’s miles-per-gallon rate (40) and reported it directly instead of using it to answer the question.
    1. 65: Divided 520 by 8 directly without accounting for the 320-mile baseline.

Answer: A

Question 4

A support cable runs from the top of a pole to a point on the ground 9 feet from the base of the pole. If the cable is 15 feet long, how tall, in feet, is the pole?

P Q R ? 9 15




Show solution

By the Pythagorean theorem: \(9^2 + h^2 = 15^2\), so \(h^2 = 225 - 81 = 144\), \(h = 12\).

    1. 6: Subtracted the two given values directly (\(15-9=6\)) instead of applying the Pythagorean theorem.
    1. 10.2: Arithmetic slip computing \(\sqrt{225-81}\).
    1. 24: Added the two given values (\(15+9=24\)) instead of applying the Pythagorean theorem.

Answer: C

Question 5

If \(5x + 3y = 26\) and \(x - 3y = 4\), what is the value of \(x\)?




Show solution

Add the two equations: \((5x+3y)+(x-3y)=26+4\), giving \(6x=30\), so \(x=5\).

    1. -5: Correctly found \(6x=30\) but divided by \(-6\) instead of \(6\) (sign error in the division step).
    1. 3.75: Correctly found \(6x=30\) but divided by 8 instead of 6.
    1. 24: Correctly found \(6x=30\) but forgot to divide by 6, reporting the value of \(6x\) instead of \(x\).

Answer: C

Question 6

If \(p(x) = x^3 + 3x^2 - 4x + 1\), what is the remainder when \(p(x)\) is divided by \((x + 2)\)?

Enter your answer:

Show solution

By the Remainder Theorem, the remainder equals \(p(-2)\): \(p(-2) = (-2)^3 + 3(-2)^2 - 4(-2) + 1 = -8 + 12 + 8 + 1 = 13\).

Answer: 13

Question 7

A survey of 18 employees found their average commute time was 24 minutes. When 6 more employees were surveyed, their average commute time was 45 minutes. What is the average commute time, in minutes, for all 24 employees?




Show solution

Total commute time \(= 18(24) + 6(45) = 432 + 270 = 702\). Average for all 24: \(702/24 = 29.25\).

    1. 28.5: Made an arithmetic slip in the weighted sum, undercounting one group’s contribution.
    1. 34.5: Computed the simple (unweighted) average of the two group averages, \(\dfrac{24+45}{2} = 34.5\).
    1. 36: Mistakenly reported the midpoint between 24 and 45 using an incorrect method.

Answer: B

Question 8

In a circle, a central angle measures \(130°\). What is the measure of the inscribed angle that intercepts the same arc?

O A B P 130°




Show solution

The Inscribed Angle Theorem states the inscribed angle is half the central angle intercepting the same arc: \(130°/2 = 65°\).

    1. 130°: Assumed the inscribed angle equals the central angle.
    1. 230°: Computed the reflex-angle relationship (\(360° - 130°\)) instead of applying the inscribed-angle theorem.
    1. 260°: Doubled the central angle instead of halving it.

Answer: A

Question 9

A bakery sells muffins for \(\$3\) each and cookies for \(\$2\) each. On a certain day, the bakery sold 90 items for a total of \(\$220\). How many muffins were sold?




Show solution

Let \(m\) = muffins, \(c\) = cookies. \(m+c=90\) and \(3m+2c=220\). Substituting \(c=90-m\): \(3m+2(90-m)=220\), so \(m+180=220\), \(m=40\).

    1. 8: Sign error distributing the 2, computing \(3m+180+2m=220\) instead of \(3m+180-2m=220\).
    1. 45: Assumed the items split evenly (90/2), ignoring the price difference.
    1. 50: Correctly solved the system but reported the number of cookies (\(90-40=50\)) instead of muffins.

Answer: B

Question 10

If \(\dfrac{5}{6}(x - 4) = 20\), what is the value of \(x\)?

Enter your answer:

Show solution

Multiply both sides by \(\dfrac{6}{5}\): \(x - 4 = 24\). Add 4: \(x = 28\).

Answer: 28

Question 11

Which of the following is equivalent to \(\dfrac{x^2 - 25}{x^2 + 2x - 15}\) for \(x \neq -5, 3\)?




Show solution

Factor: \(\dfrac{(x-5)(x+5)}{(x+5)(x-3)}\). Cancel the common factor \((x+5)\): \(\dfrac{x-5}{x-3}\).

    1. \(\dfrac{x+5}{x-3}\): Cancelled the \((x+5)\) factor from the denominator but left an \((x+5)\) in the numerator instead of \((x-5)\).
    1. \(\dfrac{x-5}{x+3}\): Made a sign error factoring the denominator, using \((x+3)\) instead of \((x-3)\).
    1. \(\dfrac{(x-5)(x+5)}{x-3}\): Factored both numerator and denominator correctly but forgot to cancel the common \((x+5)\) factor.

Answer: A

Question 12

A bag contains 6 red marbles, 4 blue marbles, and 5 green marbles. If one marble is drawn at random and not replaced, then a second marble is drawn, what is the probability that both marbles are green? (Enter your answer as a decimal, rounded to the nearest hundredth.)

Enter your answer:

Show solution

Total marbles \(= 15\). \(P(\text{both green}) = \dfrac{5}{15} \times \dfrac{4}{14} = \dfrac{20}{210} = \dfrac{2}{21} \approx 0.10\).

Answer: 0.1

Question 13

A population of bacteria is modeled by \(N(t) = 500 \cdot (1.12)^t\), where \(t\) is the number of hours and \(N(t)\) is the population size. What does the value \(1.12\) represent in this context?




Show solution

In a model \(N(t) = a \cdot b^t\), the base \(b\) is the growth factor. Here \(b = 1.12 = 1 + 0.12\), meaning the population grows by \(12\%\) each hour.

    1. The population is increasing by \(1.12\%\) each hour.: Confused the growth rate with the growth factor, misreading 1.12 as \(1.12\%\) instead of a \(12\%\) increase.
    1. The population increases by 500 bacteria every 1.12 hours.: Misassigned the roles of the two constants, treating 1.12 as a rate of change for the initial population value.
    1. The initial population was 1.12.: Confused the growth factor 1.12 with the initial population, which is actually 500.

Answer: A

Question 14

In right triangle \(ABC\), the right angle is at \(B\). If \(AB = 9\) and \(BC = 12\), what is \(\tan(C)\)?

A B C 9 12




Show solution

Angle \(C\) has opposite side \(AB=9\) and adjacent side \(BC=12\). \(\tan(C) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{9}{12} = \dfrac{3}{4}\).

    1. \(\dfrac{4}{3}\): Inverted the tangent ratio, computing adjacent over opposite instead of opposite over adjacent.
    1. \(\dfrac{9}{15}\): Computed a sine-style ratio (opposite over hypotenuse) instead of tangent, and also needed the hypotenuse.
    1. \(\dfrac{12}{15}\): Computed a cosine-style ratio (adjacent over hypotenuse) instead of tangent, and also needed the hypotenuse.

Answer: C

Question 15

For what value of \(k\) does the system of equations below have infinitely many solutions?

\[\begin{aligned} 8x - 12y &= 20 \\ 6x - 9y &= k \end{aligned}\]




Show solution

The coefficients of the second equation are \(\dfrac{3}{4}\) of the first equation’s coefficients (\(6 = \frac{3}{4}(8)\) and \(9 = \frac{3}{4}(12)\)). For the system to represent the same line, the constant terms must follow the same ratio: \(k = \dfrac{3}{4}(20) = 15\).

    1. 10: Divided 15 by 1.5 in the wrong direction instead of multiplying, effectively inverting the scale factor.
    1. 20: Assumed the constant term must match the first equation directly without accounting for the scale factor between the equations.
    1. 30: Multiplied 20 by 1.5 instead of dividing, applying the scale factor backwards.

Answer: B

Question 16

If \(c = 25\), how many distinct real solutions does the equation \(x^2 - 10x + c = 0\) have?




Show solution

The discriminant is \((-10)^2 - 4(1)(25) = 100 - 100 = 0\). A discriminant of zero means the equation has exactly one distinct real (repeated) solution.

    1. Zero: Assumed a discriminant of 0 means no real solutions, confusing it with a negative discriminant.
    1. Two: Assumed all quadratics have two solutions without checking the discriminant.
    1. Infinitely many: Confused a repeated root with an identity that holds for all \(x\).

Answer: A

Question 17

If \(f(x) = 2x - 3\) and \(g(x) = x^2 + 4\), what is the value of \(f(g(-1))\)?




Show solution

First find \(g(-1) = (-1)^2 + 4 = 1 + 4 = 5\). Then \(f(g(-1)) = f(5) = 2(5) - 3 = 7\).

    1. 2: Correctly computed \(g(-1) = 5\), but forgot the coefficient on \(f\), computing \(5 - 3\) instead of \(2(5) - 3\).
    1. 10: Added \(f(-1) + g(-1) = -5 + 5\) instead of composing the functions.
    1. -5: Evaluated only the outer function on the original input, computing \(f(-1) = 2(-1)-3 = -5\) without composing with \(g\) at all.

Answer: B

Question 18

The solution to the system of equations below is \((x, y)\). What is the value of \(3x + y\)?

\[\begin{aligned} 2x + y &= 11 \\ x - y &= 1 \end{aligned}\]

Enter your answer:

Show solution

Add the two equations: \((2x+y)+(x-y)=11+1\), giving \(3x=12\), so \(x=4\). Substitute into \(x-y=1\): \(4-y=1\), so \(y=3\). Then \(3x+y = 3(4)+3 = 15\).

Answer: 15

Question 19

The polynomial function \(m\) is given by \(m(x) = -x^4 + 3x^2 - 2\). As \(x\) approaches negative infinity, what happens to \(m(x)\)?




Show solution

End behavior is determined by the leading term, \(-x^4\). Since the degree is even and the leading coefficient is negative, both ends of the graph fall: as \(x \to -\infty\), \(m(x) \to -\infty\).

    1. \(m(x)\) approaches positive infinity.: Assumed the negative leading coefficient means the graph rises on the left without checking the degree’s effect.
    1. \(m(x)\) approaches zero.: Mistakenly assumed the polynomial’s value approaches zero as \(x\) decreases, confusing end behavior with a horizontal asymptote.
    1. \(m(x)\) approaches \(-2\).: Assumed the constant term determines long-run behavior, ignoring the dominant leading term.

Answer: C

Question 20

A study examined whether a new fertilizer increases crop yield. Researchers randomly assigned 40 plots of farmland to either receive the new fertilizer or a standard fertilizer, then measured the yield of each plot. The plots with the new fertilizer had an average yield 15% higher than the plots with standard fertilizer. Which of the following conclusions is best supported?




Show solution

The key feature is random assignment to treatment groups, which controls for confounding variables and allows a causal conclusion. This is an experiment, not an observational study.

    1. The new fertilizer caused higher yield for every single plot that received it.: Overstates the conclusion — a higher average yield doesn’t guarantee every single plot improved.
    1. The result cannot be trusted because only 40 plots were studied.: 40 plots is a reasonable sample size for a controlled experiment; the study design supports inference regardless.
    1. There is a correlation between fertilizer type and yield, but no conclusion about causation can be drawn.: This would be the appropriate conclusion for an observational study — but here random assignment was used, which does support causation.

Answer: B

Question 21

What is the value of \(x\) that satisfies \(\dfrac{5}{x-3} + \dfrac{1}{2} = 3\), given that \(x \neq 3\)? Enter your answer as a decimal.

Enter your answer:

Show solution

Subtract \(\dfrac{1}{2}\) from both sides: \(\dfrac{5}{x-3} = 2.5\). Multiply both sides by \((x-3)\): \(5 = 2.5(x-3)\). Divide by 2.5: \(x-3 = 2\), so \(x = 5\).

Answer: 5

Question 22

A circle is defined by the equation \(x^2 + y^2 - 10x + 2y + 22 = 0\). What is the radius of the circle?

Enter your answer:

Show solution

Complete the square: \((x^2-10x+25) + (y^2+2y+1) = -22+25+1\), giving \((x-5)^2+(y+1)^2=4\). The radius is \(\sqrt{4}=2\).

Answer: 2