SAT Math - Full-Length Practice Test 2
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Module 1
Question 1
If \(3x+7=25\), which of the following is equal to \(2x+1\)?
Show solution
From \(3x+7=25\), subtract 7 to get \(3x=18\), so \(x=6\).
Therefore, \(2x+1=2(6)+1=13\).
Answer: C
Question 2
The bar chart shows the number of books checked out from a library on four days. How many more books were checked out on Thursday than on Monday?
Show solution
The chart shows 21 books on Thursday and 12 on Monday.
The difference is \(21-12=9\).
Answer: B
Question 3
Which of the following is equivalent to \(x^2+7x+12\)?
Show solution
The two constants must multiply to 12 and add to 7.
The numbers 3 and 4 satisfy both conditions: \(x^2+7x+12=(x+3)(x+4)\).
Answer: D
Question 4
A triangular garden has a base of 8 meters and a perpendicular height of 5 meters, as shown. What is the area of the garden, in square meters?
Show solution
Use the triangle area formula \(A=\frac12 bh\).
Substitute \(b=8\) and \(h=5\): \(A=\frac12(8)(5)=20\) square meters.
Answer: A
Question 5
The equation \(5y-8=27\) is true. Enter the value of \(y\).
Enter your answer:
Show solution
Add 8 to both sides: \(5y=35\).
Dividing by 5 gives \(y=7\).
Answer: 7
Question 6
The graph of a quadratic function is shown. Which value is the \(x\)-coordinate of the vertex?
Show solution
The vertex is labeled \((2,-1)\).
Its \(x\)-coordinate is therefore 2.
Answer: C
Question 7
A school club can spend at most $95 on notebooks. The club has already spent $35, and each additional notebook costs $12. If \(n\) is the number of additional notebooks, which inequality represents the situation?
Show solution
The total cost is the $35 already spent plus $12 for each of \(n\) notebooks.
Because the club may spend no more than $95, the total must satisfy \(35+12n\le95\).
Answer: A
Question 8
A theater sold 240 tickets for a performance. If 35% of the tickets were sold online, how many tickets were sold online?
Enter your answer:
Show solution
Compute 35% of 240: \(0.35(240)=84\).
Answer: 84
Question 9
Two angles of a triangle measure \(58^\circ\) and \(67^\circ\). What is the measure of the third angle?
Show solution
The angles of a triangle sum to \(180^\circ\).
The third angle is \(180-58-67=55^\circ\).
Answer: D
Question 10
The function \(f\) is defined by \(f(x)=2x^2-3\). Enter the value of \(f(4)\).
Enter your answer:
Show solution
Substitute 4 for \(x\): \(f(4)=2(4^2)-3=32-3=29\).
Answer: 29
Question 11
The graph of a line contains the points \((0,2)\) and \((3,8)\), as shown. What is the slope of the line?
Show solution
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the two points: \(m=\frac{8-2}{3-0}=\frac{6}{3}=2\).
Answer: B
Question 12
A streaming service charges a one-time setup fee of $18 and then $6 per month. If \(m\) is the number of months and \(c\) is the total cost, which equation gives \(c\) in terms of \(m\)?
Show solution
The fixed cost is 18 and the monthly rate is 6.
The total cost after \(m\) months is \(c=18+6m\).
Answer: D
Question 13
The function \(f\) is defined by \(f(x)=x^2\). The function \(g\) is defined by \(g(x)=(x+3)^2-5\). Which statement correctly describes the graph of \(g\) relative to the graph of \(f\)?
Show solution
Replacing \(x\) with \(x+3\) shifts the graph 3 units left, and subtracting 5 shifts it 5 units down.
Answer: B
Question 14
The system \(2x+y=11\) and \(x-y=1\) has solution \((x,y)\). What is the value of \(x+y\)?
Show solution
Adding the equations gives \(3x=12\), so \(x=4\).
Then \(4-y=1\), giving \(y=3\).
Therefore, \(x+y=7\).
Answer: C
Question 15
A scatterplot and its line of best fit are shown. According to the line of best fit \(y=2x+5\), what is the predicted value of \(y\) when \(x=7\)?
Show solution
Substitute \(x=7\) into the line of best fit: \(y=2(7)+5=19\).
Answer: D
Question 16
At a school play, 50 tickets were sold. Adult tickets cost $12 each and student tickets cost $8 each. The total ticket revenue was $520. How many adult tickets were sold?
Show solution
Let \(a\) be adult tickets and \(s\) be student tickets: then \(a+s=50\) and \(12a+8s=520\).
Substituting \(s=50-a\) gives \(12a+8(50-a)=520\): \(4a=120\) and \(a=30\).
Answer: D
Question 17
A right triangular support has one leg of length 5 feet and a hypotenuse of length 13 feet, as shown. What is the area of the triangle, in square feet?
Show solution
Use the Pythagorean theorem to find the missing leg: \(5^2+b^2=13^2\).
Then \(25+b^2=169\), so \(b^2=169-25=144\) and \(b=12\).
Use the triangle area formula \(A=\frac12 bh\): \(A=\frac12(5)(12)=30\).
Answer: B
Question 18
The equation \(2(x-4)^2=50\) has two solutions. What is the sum of the solutions?
Show solution
Divide by 2: \((x-4)^2=\frac{50}{2}=25\).
Take square roots: \(x-4=5\) or \(x-4=-5\).
The solutions are 9 and \(-1\), whose sum is 8.
Answer: A
Question 19
The table summarizes whether students in two grade levels participate in a school club. If one 11th-grade student is selected at random, what is the probability that the student participates in the club?
| Grade | Participates | Does not participate | Total |
|---|---|---|---|
| 10th | 22 | 18 | 40 |
| 11th | 18 | 12 | 30 |
| Total | 40 | 30 | 70 |
Show solution
Use conditional probability: \(P(\text{club}\mid\text{eleventh grade})=\frac{\text{eleventh grade and club}}{\text{eleventh grade}}\).
Substitute the counts: \(P=\frac{18}{30}=\frac35\).
Answer: C
Question 20
A line has slope \(-3\) and passes through the point \((2,5)\). Which equation represents the line?
Show solution
Use \(y=mx+b\) with \(m=-3\): substituting \((2,5)\) gives \(5=-3(2)+b\), so \(b=11\).
Thus, \(y=-3x+11\).
Answer: B
Question 21
The line \(y=2x+k\) is tangent to the parabola \(y=x^2-6x+20\). Enter the value of \(k\).
Enter your answer:
Show solution
At a tangent point, the line and parabola have exactly one common solution.
Set the equations equal: \(x^2-6x+20=2x+k\), or \(x^2-8x+(20-k)=0\).
Exactly one real solution requires discriminant 0. \((-8)^2-4(1)(20-k)=0\). Thus \(64-80+4k=0\). \(4k=16\) and \(k=4\).
Answer: 4
Question 22
In the figure, \(O\) is the center of the circle, \(OA=6\), \(OB=10\), and line \(AB\) is tangent to the circle at \(A\). What is the length of \(AB\)?
Show solution
A radius drawn to a point of tangency is perpendicular to the tangent line.
triangle \(OAB\) is right with hypotenuse 10 and leg 6.
Therefore, \(AB=\sqrt{10^2-6^2}=\sqrt{64}=8\).
Answer: D
Module 2
Question 1
If \(7x-5=30\), which value is equal to \(x+4\)?
Show solution
Add 5 to get \(7x=35\), so \(x=5\): then \(x+4=9\).
Answer: B
Question 2
The line graph shows the number of visitors, in thousands, to a museum from 2020 through 2023. Between which two consecutive years was the increase greatest?
Show solution
The increases are 2 thousand from 2020 to 2021, 5 thousand from 2021 to 2022, and 1 thousand from 2022 to 2023.
The greatest increase occurred from 2021 to 2022.
Answer: C
Question 3
For \(x\ne4\), which expression is equivalent to \(\dfrac{x^2-16}{x-4}\)?
Show solution
Factor the numerator: \(x^2-16=(x-4)(x+4)\).
Since \(x\ne4\), cancel \(x-4\) to obtain \(x+4\).
Answer: A
Question 4
A circular walking path has radius 9 meters. What is the circumference of the path, in meters?
Show solution
The circumference of a circle is \(2\pi r\).
With \(r=9\), \(C=2\pi(9)=18\pi\).
Answer: C
Question 5
A culture contains 300 bacteria initially and is modeled by \(P(t)=300(1.2)^t\), where \(t\) is time in hours. Which statement best interprets the value 1.2?
Show solution
An exponential growth factor of 1.2 equals \(1+0.20\).
Therefore, the population increases by 20% each hour.
Answer: C
Question 6
A repair company charges $90 for the first 2 hours of a job and $35 for each additional hour. A customer was charged $195 for a job lasting \(h\) hours, where \(h>2\). Which equation represents this situation?
Show solution
The first 2 hours cost $90 total.
The number of additional hours is \(h-2\), and those hours cost $35 each.
Therefore, \(90+35(h-2)=195\).
Answer: A
Question 7
Triangles \(ABC\) and \(DEF\) are similar, and triangle \(DEF\) is a scale copy of triangle \(ABC\) with scale factor 1.5. Side \(AC\) has length 8. What is the length of the corresponding side \(DF\)?
Show solution
Corresponding lengths are multiplied by the scale factor.
Thus, \(DF=1.5(8)=12\).
Answer: D
Question 8
The scatterplot shows the relationship between two variables. Which description best matches the relationship shown?
Show solution
As \(x\) increases, the plotted \(y\)-values tend to increase.
This indicates a positive association.
Answer: B
Question 9
The line \(3x+2y=18\) crosses the \(x\)-axis at \((a,0)\). Enter the value of \(a\).
Enter your answer:
Show solution
At the \(x\)-intercept, \(y=0\).
Substituting gives \(3a=18\), so \(a=6\).
Answer: 6
Question 10
The equation \(x^2-10x+21=0\) has two real solutions. Enter the larger solution.
Enter your answer:
Show solution
Factor: \(x^2-10x+21=(x-3)(x-7)\): the solutions are 3 and 7.
The larger solution is 7.
Answer: 7
Question 11
In a right triangle, an acute angle \(\theta\) has adjacent side length 12 and hypotenuse length 20. Which value is equal to \(\cos\theta\)?
Show solution
Cosine is adjacent over hypotenuse: \(\cos\theta=\frac{12}{20}=\frac35\).
Answer: C
Question 12
The table shows values of a linear function \(f\). What is \(f(8)\)?
| \(x\) | \(f(x)\) |
|---|---|
| \(2\) | \(7\) |
| \(5\) | \(16\) |
Show solution
Use the slope formula \(m=\frac{\Delta y}{\Delta x}\).
From \(x=2\) to \(x=5\), \(f(x)\) increases by 9 while \(x\) increases by 3, so \(m=\frac93=3\).
Increasing \(x\) from 5 to 8 adds another \(3(3)=9\) to the output: \(16+9=25\).
Answer: D
Question 13
A student has an average score of 82 on four quizzes. What score on a fifth quiz would increase the average for all five quizzes to 85?
Enter your answer:
Show solution
The first four quizzes total \(4(82)=328\) points.
A five-quiz average of 85 requires \(5(85)=425\) total points.
The fifth score must be \(425-328=97\).
Answer: 97
Question 14
The graph of a quadratic function has \(x\)-intercepts at \(-1\) and 5, as shown. Which equation gives the axis of symmetry?
Show solution
The axis of symmetry lies halfway between the two \(x\)-intercepts.
The midpoint of \(-1\) and 5 is \(\frac{-1+5}{2}=2\).
The axis is \(x=2\).
Answer: B
Question 15
The expression \((1.4641)^{x/4}\) can be written in the form \((1+p/100)^x\), where \(p\) is a positive number. What is the value of \(p\)?
Show solution
Because \(1.4641=(1.1)^4\), \((1.4641)^{x/4}=((1.1)^4)^{x/4}=(1.1)^x\).
Thus \(1+p/100=1.1\), giving \(p=10\).
Answer: D
Question 16
A linear function is defined by \(f(x)=mx+b\), where \(m\) and \(b\) are constants. If \(f(3)=0\) and \(f(5)>0\), which statement must be true?
Show solution
Since \(f(3)=0\), the line crosses the \(x\)-axis at 3: \(f(x)=m(x-3)\).
Then \(f(5)=2m>0\), which means \(m>0\).
Expanding gives \(f(x)=mx-3m\). \(b=-3m<0\).
Answer: B
Question 17
A researcher randomly selects 200 students from all students enrolled in a school district and asks whether they prefer a later school start time. To which population can the results most appropriately be generalized?
Show solution
Because the sample was randomly selected from the district population, the results can appropriately be generalized to students in that district, but not to broader populations that were not sampled.
Answer: A
Question 18
The equation \(8^{x+1}=4^{2x-1}\) is true. Enter the value of \(x\).
Enter your answer:
Show solution
Write both sides with base 2: \(8^{x+1}=2^{3x+3}\) and \(4^{2x-1}=2^{4x-2}\).
Therefore, \(3x+3=4x-2\), which gives \(x=5\).
Answer: 5
Question 19
A circle has center \((2,-1)\) and radius 5. Point \(P=(x,3)\) lies on the circle and is to the right of the center, as shown. Enter the value of \(x\).
Enter your answer:
Show solution
The circle equation is \((x-2)^2+(y+1)^2=25\).
For \(P=(x,3)\), \((x-2)^2+4^2=25\): \((x-2)^2=9\).
Thus \(x=5\) or \(x=-1\). Since \(P\) is to the right of the center, \(x>2\), so \(x=5\).
Answer: 5
Question 20
The quantities \(x-2\) and \(y+1\) satisfy the system \(3(x-2)+2(y+1)=23\) and \((x-2)-(y+1)=1\). Enter the value of \(x\).
Enter your answer:
Show solution
Let \(u=x-2\) and \(v=y+1\): then \(3u+2v=23\) and \(u-v=1\), so \(u=v+1\).
Substitution gives \(3(v+1)+2v=23\), hence \(5v=20\) and \(v=4\).
Thus \(u=5\), so \(x-2=5\) and \(x=7\).
Answer: 7
Question 21
The graph of \(y=x^2+bx+c\) is shown. The graph has \(x\)-intercepts at \(-2\) and 4 and a \(y\)-intercept at \(-8\). What is the value of \(bc\)?
Show solution
With roots \(-2\) and 4 and leading coefficient 1, the quadratic is \((x+2)(x-4)=x^2-2x-8\).
Therefore, \(b=-2\) and \(c=-8\), so \(bc=16\).
Answer: C
Question 22
The equation \(x^2-(k+3)x+3k=0\) has two real solutions whose difference is 4. What is the sum of all possible values of \(k\)?
Show solution
For a monic quadratic, the difference between the roots has magnitude \(\sqrt{D}\), where \(D\) is the discriminant.
Here, \(D=(k+3)^2-12k=k^2-6k+9=(k-3)^2\).
A root difference of 4 requires \(\sqrt{D}=4\): \(|k-3|=4\). Thus \(k=7\) or \(k=-1\), and their sum is 6.
Answer: D