SAT Math Diagnostic Quiz
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Question 1
Which number is the solution to \(4x+7=31\)?
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Subtract 7 from both sides: \(4x=24\). Then divide by 4 to get \(x=6\).
Answer: B
Question 2
The function \(f\) is defined by \(f(x)=x^2-3x+4\). What is the value of \(f(5)\)?
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Substitute \(5\) for \(x\): \(f(5)=25-15+4=14\).
Answer: D
Question 3
The line shown passes through \((0,-1)\) and \((2,3)\). What is the slope of the line?
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Use the slope formula: \(m=\frac{3-(-1)}{2-0}=\frac{4}{2}=2\).
Answer: B
Question 4
Which statement best describes the relationship shown in the scatterplot?
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The points generally rise from left to right, so the variables have a positive association.
Answer: A
Question 5
A right triangle has legs of length 9 and 12, as shown. What is the length of the hypotenuse?
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By the Pythagorean theorem, \(c^2=9^2+12^2=81+144=225\), so \(c=15\).
Answer: C
Question 6
Which expression is equivalent to \(x^2+x-12\)?
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The two numbers must multiply to \(-12\) and add to \(1\). Those numbers are \(4\) and \(-3\), so \(x^2+x-12=(x+4)(x-3)\).
Answer: D
Question 7
What is the \(x\)-coordinate of the solution to the system \(x+y=11\) and \(x-y=3\)?
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Add the equations to eliminate \(y\): \(2x=14\). Therefore, \(x=7\).
Answer: B
Question 8
The graph shown has horizontal asymptote \(y=1\). Which equation could represent the graph?
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The graph increases exponentially and approaches \(y=1\) as \(x\) decreases. The function \(y=2^x+1\) has exactly these features.
Answer: A
Question 9
The table shows the preferred study location for 80 students. What fraction of the students who prefer the library are in grade 11?
| Library | Home | Total | |
|---|---|---|---|
| Grade 10 | 12 | 28 | 40 |
| Grade 11 | 18 | 22 | 40 |
| Total | 30 | 50 | 80 |
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There are \(12+18=30\) students who prefer the library, and 18 of them are in grade 11. Thus the fraction is \(\frac{18}{30}=\frac{3}{5}\).
Answer: D
Question 10
The formula \(P=2L+2W\) gives the perimeter of a rectangle. If \(P=50\) and \(W=8\), what is the length \(L\) of the rectangle?
Enter your answer:
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Substitute the known values: \(50=2L+16\). Then \(34=2L\), so \(L=17\).
Answer: 17
Question 11
A circle has center \(O=(2,-1)\). Point \(P=(6,2)\) lies on the circle. What is the radius of the circle?
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The radius is the distance from \(O\) to \(P\): \(\sqrt{(6-2)^2+(2-(-1))^2}=\sqrt{16+9}=5\).
Answer: C
Question 12
The expression \(2x^2-20x+61\) can be written as \(2(x-h)^2+k\), where \(h\) and \(k\) are constants. What is the value of \(k\)?
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Factor 2 from the quadratic terms and complete the square: \(2(x^2-10x)+61=2[(x-5)^2-25]+61=2(x-5)^2+11\). Thus \(k=11\).
Answer: C
Question 13
A machine produces 465 parts in 3 hours at a constant rate. At the same rate, how many parts will it produce in 7 hours?
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The rate is \(465\div3=155\) parts per hour. In 7 hours, the machine produces \(155(7)=1085\) parts.
Answer: 1085
Question 14
For what value of \(k\) does the system \(6x+4y=18\) and \(3x+2y=k\) have infinitely many solutions?
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Dividing the first equation by 2 gives \(3x+2y=9\). For infinitely many solutions, the second equation must represent the same line, so \(k=9\).
Answer: B
Question 15
When \(p(x)=x^3-4x^2+6x+5\) is divided by \(x-3\), what is the remainder?
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By the Remainder Theorem, the remainder is \(p(3)\). Compute \(p(3)=27-36+18+5=14\).
Answer: 14