SAT Math - Full-Length Practice Test 2
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Module 1
Question 1
A line has slope \(-3\) and \(y\)-intercept 7. Which equation represents the line?
Show solution
In slope-intercept form, \(y=mx+b\).
Here \(m=-3\) and \(b=7\): \(y=-3x+7\).
Answer: B
Question 2
For \(f(x)=x^2-4x+1\), what is \(f(3)\)?
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\(f(3)=3^2-4(3)+1=9-12+1=-2\).
Answer: A
Question 3
A right triangle has legs of lengths 9 and 12, as shown. What is its area, in square units?
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Use the triangle area formula \(A=\frac12 bh\).
The perpendicular legs can be used as the base and height, so \(A=\frac12(9)(12)=54\).
Answer: C
Question 4
If \(7x-11=38\), enter \(x\).
Enter your answer:
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Add 11 to get \(7x=49\), then divide by 7: thus \(x=7\).
Answer: 7
Question 5
A recipe uses 3 cups of flour for every 2 cups of milk. At this rate, how many cups of flour are needed for 8 cups of milk?
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The milk amount is multiplied by 4, from 2 to 8.
The flour amount is also multiplied by 4: \(3(4)=12\).
Answer: D
Question 6
Which expression is equivalent to \(x^2-9x+20\)?
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The constants must multiply to 20 and add to \(-9\).
They are \(-4\) and \(-5\).
The factorization is \((x-4)(x-5)\).
Answer: B
Question 7
Which value of \(x\) satisfies \(4x+5<21\)?
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Subtract 5: \(4x<16\), so \(x<4\).
Since \(3<4\), the answer is 3.
Answer: A
Question 8
The circle shown has diameter 14. What is its circumference?
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Use the circumference formula \(C=\pi d\).
Substitute \(d=14\): \(C=\pi(14)=14\pi\).
Answer: C
Question 9
The line through \((1,5)\) and \((4,11)\) has what slope?
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Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the two points: \(m=\frac{11-5}{4-1}=\frac63=2\).
Answer: B
Question 10
A jacket originally costs $80 and is discounted by 25%. What is the discounted price, in dollars?
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A 25% discount is \(0.25(80)=20\).
The discounted price is \(80-20=60\).
Answer: 60
Question 11
A quantity is modeled by \(P(t)=6(2)^t\). What is the value of \(P(3)\)?
Enter your answer:
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Substitute \(t=3\): \(P(3)=6(2^3)=6(8)=48\).
Answer: 48
Question 12
A taxi ride costs a fixed fee of $4 plus $2.50 per mile. Which function gives the total cost \(C\), in dollars, for \(m\) miles?
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The per-mile rate is the coefficient of \(m\), and the fixed fee is the constant.
Thus \(C=2.5m+4\).
Answer: D
Question 13
The bar chart shows the number of volunteer hours completed by four teams. What fraction of the total hours was completed by Team C?
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The total is \(12+18+15+15=60\) hours.
Team C completed 15.
The fraction is \(15/60=1/4\).
Answer: B
Question 14
At a school event, 18 adult and student tickets were sold for a total of $156. Adult tickets cost $12 and student tickets cost $6. How many adult tickets were sold?
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Let \(a\) be adult tickets and \(s\) student tickets: then \(a+s=18\) and \(12a+6s=156\).
Dividing the second equation by 6 gives \(2a+s=26\).
Subtracting the first equation gives \(a=8\).
Answer: C
Question 15
The graph of \(y=(x-2)^2-5\) has vertex at which point?
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In vertex form \(y=(x-h)^2+k\), the vertex is \((h,k)\).
Here it is \((2,-5)\).
Answer: D
Question 16
A rectangular screen is 15 inches wide and 8 inches high. What is the length, in inches, of its diagonal?
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The diagonal is the hypotenuse: \(d^2=15^2+8^2=225+64=289\), so \(d=17\).
Answer: B
Question 17
For the system \(2x+y=11\) and \(x-y=1\), which number is equal to \(x+y\)?
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Adding the equations gives \(3x=12\), so \(x=4\).
Then \(4-y=1\), so \(y=3\).
Thus \(x+y=7\).
Answer: A
Question 18
In a group of 80 students, 50 study Spanish, 32 play a school sport, and 20 do both. If a student who studies Spanish is selected at random, what is the probability that the student also plays a school sport?
| Sport | No sport | Total | |
|---|---|---|---|
| Spanish | 20 | 30 | 50 |
| No Spanish | 12 | 18 | 30 |
| Total | 32 | 48 | 80 |
Show solution
Use conditional probability: \(P(\text{sport}\mid\text{Spanish})=\frac{\text{Spanish and sport}}{\text{Spanish}}\).
Substitute the counts: \(P=\frac{20}{50}=\frac25\).
Answer: C
Question 19
The graph of a quadratic function has \(x\)-intercepts at \(-3\) and 7. Which equation gives its axis of symmetry?
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The axis of symmetry lies halfway between the intercepts: \(x=\frac{-3+7}{2}=2\).
Answer: A
Question 20
The table shows values of a linear function \(f\). If \(f(2)=11\) and \(f(6)=23\), which equation defines \(f\)?
| \(x\) | \(f(x)\) |
|---|---|
| \(2\) | \(11\) |
| \(6\) | \(23\) |
Show solution
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
The slope is \(m=\frac{23-11}{6-2}=\frac{12}{4}=3\).
Using \(f(2)=11\), \(11=3(2)+b\), so \(b=5\).
Thus \(f(x)=3x+5\).
Answer: D
Question 21
The polynomial \(p(x)=x^3-2x^2+5x+9\) is divided by \(x-2\). What is the remainder?
Enter your answer:
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By the Remainder Theorem, the remainder is \(p(2)=8-8+10+9=19\).
Answer: 19
Question 22
Two similar triangles have corresponding side lengths 8 and 14. If the area of the smaller triangle is 96 square units, what is the area of the larger triangle?
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The linear scale factor is \(14/8=7/4\).
The area scale factor is \((7/4)^2=49/16\).
The larger area is \(96(49/16)=6(49)=294\).
Answer: D
Module 2
Question 1
A line rises 9 units for every 3 units it moves to the right. What is the slope of the line?
Show solution
Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\).
Here \(m=9/3=3\).
Answer: C
Question 2
A printer produces 84 pages in 6 minutes at a constant rate. How many pages does it produce per minute?
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The unit rate is \(84/6=14\) pages per minute.
Answer: B
Question 3
If \(g(x)=3x^2-2\), what is \(g(-2)\)?
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\(g(-2)=3(-2)^2-2=3(4)-2=10\).
Answer: D
Question 4
Two parallel lines are cut by a transversal. One acute angle measures \(58^\circ\). What is the measure of any obtuse angle formed by the lines and transversal?
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Each obtuse angle is supplementary to a \(58^\circ\) acute angle.
Its measure is \(180-58=122^\circ\).
Answer: A
Question 5
The system \(x+y=17\) and \(2x-y=7\) has solution \((x,y)\). Enter \(y\).
Enter your answer:
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Adding the equations gives \(3x=24\), so \(x=8\).
Then \(y=17-8=9\).
Answer: 9
Question 6
The formula \(P=2L+2W\) gives the perimeter \(P\) of a rectangle with length \(L\) and width \(W\). Which equation expresses \(L\) in terms of \(P\) and \(W\)?
Show solution
Subtract \(2W\) from both sides: \(P-2W=2L\).
Divide by 2: \(L=\frac{P-2W}{2}\).
Answer: B
Question 7
Points \(A(-2,1)\) and \(B(4,5)\) are endpoints of a segment. What is the \(x\)-coordinate of the midpoint of \(\overline{AB}\)?
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The midpoint \(x\)-coordinate is the average: \((-2+4)/2=1\).
Answer: C
Question 8
A line of best fit for a data set is \(y=2.4x+7\). If every \(y\)-value in the data set is multiplied by 3, which equation is the corresponding transformed line of best fit?
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Multiplying every response value by 3 multiplies both the slope and intercept by 3: \(y=3(2.4x+7)=7.2x+21\).
Answer: D
Question 9
A storage company charges $42 for the first month and $18 for each additional month. Which expression gives the total cost for \(m\) months, where \(m\ge1\)?
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The first month costs \(42\).
The number of additional months is \(m-1\), each at \(18\), giving \(42+18(m-1)\).
Answer: A
Question 10
The equation \((x-4)^2=25\) has two solutions. What is the positive solution?
Enter your answer:
Show solution
Taking square roots gives \(x-4=\pm5\): \(x=9\) or \(x=-1\).
The positive solution is 9.
Answer: 9
Question 11
In a right triangle, angle \(A\) is acute, the side opposite \(A\) has length 12, and the hypotenuse has length 20. What is \(\sin A\)?
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For angle \(A\), sine is opposite over hypotenuse.
Thus \(\sin A=12/20=3/5\).
Answer: B
Question 12
For \(x\ne3\), which expression is equivalent to \(\frac{x^2-9}{x-3}\)?
Show solution
Factor the numerator: \(x^2-9=(x-3)(x+3)\).
Canceling \(x-3\) gives \(x+3\) for \(x\ne3\).
Answer: C
Question 13
A town had 24,000 residents. Its population increased by 7.5%. How many residents were added?
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The number added is \(0.075(24{,}000)=1{,}800\).
Answer: 1800
Question 14
For what value of \(k\) does the system \(6x+9y=15\) and \(2x+3y=k\) have infinitely many solutions?
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Dividing the first equation by 3 gives \(2x+3y=5\).
For infinitely many solutions, the second equation must be identical, so \(k=5\).
Answer: C
Question 15
A quadratic function has zeros at \(-4\) and 6 and passes through \((0,-48)\). Which equation defines the function?
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With zeros \(-4\) and 6, \(f(x)=a(x+4)(x-6)\).
Using \((0,-48)\) gives \(-24a=-48\), so \(a=2\).
Answer: A
Question 16
A circle has radius 15. From a point \(P\) outside the circle, a tangent segment touches the circle at \(T\). If \(OP=17\), where \(O\) is the center, what is the length of \(PT\)?
Enter your answer:
Show solution
A radius to a point of tangency is perpendicular to the tangent: \(\triangle OPT\) is right with hypotenuse 17 and leg 15.
Thus \(PT=\sqrt{17^2-15^2}=\sqrt{64}=8\).
Answer: 8
Question 17
Which equation is equivalent to \(x^2-8x+7=0\)?
Show solution
Move 7 to the right: \(x^2-8x=-7\).
Add 16 to both sides to complete the square: \((x-4)^2=9\).
Answer: C
Question 18
A researcher wants to estimate the proportion of all students at a large high school who support a new schedule. Which method is most likely to produce a representative sample?
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A simple random sample from the full roster gives every student a chance to be selected and is least likely to systematically favor one subgroup.
Answer: B
Question 19
For \(x\ne0\) and \(x\ne6\), \(\frac{1}{x}+\frac{1}{x-6}=\frac12\). What is the sum of the solutions?
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Multiply by \(2x(x-6)\): \(2(x-6)+2x=x(x-6)\).
This gives \(x^2-10x+12=0\).
By Vieta’s formulas, the sum of the solutions is 10.
Answer: B
Question 20
Let \(u=x-2\) and \(v=y+1\). The equations \(3u+2v=19\) and \(u-v=3\) are true. Enter \(x+y\).
Enter your answer:
Show solution
From \(u-v=3\), \(u=v+3\).
Substitute into the first equation: \(3(v+3)+2v=19\), so \(5v=10\) and \(v=2\).
Then \(u=5\). Since \(x-2=5\), \(x=7\); since \(y+1=2\), \(y=1\). Therefore \(x+y=8\).
Answer: 8
Question 21
The equation \(x^2-10x+c=0\) has two real solutions that differ by 6. What is the value of \(c\)?
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The roots have sum 10.
their midpoint is 5.
If they differ by 6, the roots are 2 and 8.
Their product is \(c=16\).
Answer: D
Question 22
A population is modeled by \(P(t)=500(1.44)^{t/2}\). The model can also be written as \(P(t)=500(1+r)^t\). What is the value of \(100r\)?
Enter your answer:
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Because \((1.44)^{t/2}=(\sqrt{1.44})^t=(1.2)^t\), we have \(1+r=1.2\), so \(r=0.2\).
Therefore \(100r=20\).
Answer: 20