[Placement test] Digital SAT MATH

3x - 4y = 5a

12x + by = 20

In the system of equations above, a and b are constants. If the system has infinitely many solutions, What is the value of a/b ?

A) -16

B) -1/16

C) 16

D) 1/16

2) The expression 8x² + bx - 52, where b is a constant, can be rewritten as

(hx + k)(x + j)

, where h, k, and j are integer constants.

Which of the following must be an integer?

A) b/h

B) b/k

C) 52/h

D) 52/k

69x²+ (69b + a)x + ab = 0

In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab, where k is a constant. What is the value of k?

A) 1/69

B) 1/23

C) 1

D) 69

4) In the xy-plane, a line with equation 3y = k, where k is a constant, intersects a parabola at exactly one point. If the equation of the parabola is y = 4x² + 12x,

what is the value of k?

A) -27

B) -24

C) -21

D) -18

6) A print publisher paid the copywriter of a certain book $0.75 per page for the first 200 pages proofread. If the number of pages in the book exceeds 200 pages, the payment increases to $1.25 per page proofread. Which of the following functions gives the copywriter’s total payment c(p), in dollars, in terms of the number of pages proofread, p, where p > 200?

A) c(p) = 0.75 (200) + 1.25p

B) c(p) = 0.75p + 1.25p

C) c(p) = 0.75p + 1.25 (p - 200)

D) c(p) = 0.75 (200) + 1.25 (p - 200)

7) David has two part time jobs. He works as a barista, which pays $13 per hour, and he works as a waiter, which pays $8.3 per hour. Due to the employment right, he can work no more than 25 hours per week, but he wants to earn at least $190 per week. Which of the following systems of inequalities represents this situation in terms of b and w , where b is the number of hours he works as a barista and w is the number of hours he works as a waiter?

A) 13b + 8.3w ≤ 190

b + w ≥ 25

B) 13b + 8.3w ≤ 190

b + w ≤ 25

C) 13b + 8.3w ≥ 190

b + w ≥ 25

D) 13b + 8.3w ≥ 190

b + w ≤ 25

8) The population of Purdu District decreased by approximately 21% between 2000 and 2005 and then grew by approximately 24% between 2005 and 2010. If the population was 15,500 in 2005, what was the approximate change in the population of Purdu between 2000 and 2010?

A) Increase of 465 people

B) Increase of 400 people

C) Decrease of 465 people

D) Decrease of 400 people

9) In the xy-plane above, the center of the circle is (0, 0) and its radius is 2. What is the y-coordinate of point A?

A) 15/4

B) √(3/2)

C) √15 / 2

D)√17 / 2

10) The graph of a line in the xy-plane has slope 3 and contains the point (2, 5). The graph of a second line passes through the points (-1, 2) and (3, 6). If the two lines intersect at the point (a, b), what is the value of b - a?

A) 2

B) 3

C) 5

D) 7

11)

f(x) = (x - p)(x - q)

In the equation above, p and q are constants, where p ≠ q. When f is graphed in the xy-plane, what is the meaning of (p+q)/2?

A) The x-coordinate of one of the zeros of the parabola

B) The y-coordinate of one of the zeros of the parabola

C) The x-coordinate of the vertex of the parabola

D) The y-coordinate of the vertex of the parabola

12)

x	-12	-11	-10	-9
g(x)	28	24	20	16

Some values of x and their corresponding values g(x) for the linear functions g are shown in the table above. What is the x-intercept of the graph of y = g(x) in the xy-plane?

A) (-4, 0)

B) (-5, 0)

C) (-6, 0)

D) (-12, 0)

13) The amount of PM 2.5 in the downtown area at the beginning of a month is M₀ , micrograms per cubic meter (μg/m³). For every 3-day period, the amount of PM 2.5 increases by 4% from 3 days before. Which represents the amount of PM 2.5, M , in μg/m³, after t days?

14)

g(x) = -3(x - 2)² + 27

The quadratic function g is defined as shown above. In the xy-plane, the graph of
y = g(x) intersects the x-axis at points (a, 0) and (b, 0), where a and b are constants.

What is the value of a + b?

A) 3

B) 4

C) 5

D) 6

15) The complete graph of the function f is shown in the xy-plane above. What is the
y-intercept of the graph of y = f(x + 3)?

A) -2

B) -1

C) 0

D) 2

16) The mean midterm score of 10 students in a class was 25.5 points. If the highest individual score is removed, the mean midterm score of the remaining 9 students becomes 23 points. What was the highest score of the 10 students?

A) 20

B) 22.5

C) 39

D) 48

17) A scientist monitored 10 blue whales living in Monterey Bay region. The scientist measured the weight (W), in tons, and the length (L), in meters of the 10 blue whales as shown in the scatterplot above. A line of best fit for the data is also shown.

Which of the following could be the equation of the line of best fit?

A) W = 0.3L + 120

B) W = 3.33L + 120

C) W = 3.33L + 40

D) W = -3.33L + 40

18)

Number of puzzles solved correctly

	None	1 to 3	4 or more	Total
Group A	60	37	53	150
Group B	48	45	57	150
Total	108	82	110	300

A psychological researcher studies the relationship between the number of languages people can speak and their cognitive skills. In the study, 300 people were given a set of 7 puzzles to complete. The table above shows the number of puzzles they solved correctly. Group A consisted of 150 people who can speak only one language whereas Group B consisted of 150 people who can speak more than one language. If a person is chosen at random from those who solved at least 1 problem correctly, what is the probability that the person belonged to Group B?

A) 57/150

B) 102/150

C) 102/192

D) 192/300

19) In the figure shown above, segments JN and KM intersect at point L. Which additional piece of information is sufficient to conclude that triangle JKL is congruent to triangle NML?

A) The measures of angles J and K are equal.

B) The lengths of segments JL and LK are equal.

C) The measures of angles J and N are equal.

D) The lengths of segments JL and LN are equal.

20)

QR = 360

PR = 325

PQ = 485

The side lengths of the right triangle PQR are given above. Triangle PQR is similar to triangle XYZ, where X corresponds to P and Z corresponds to R.

What is the value of tanY ?

A) 65/97

B) 72/97

C) 65/72

D) 72/65

21) The circle shown above has point O as its center and the radius of 6. If the length of arc AB is 5π, what is the measure of angle ABO ?

A) 15˚

B) 20˚

C) 25˚

D) 30˚

22) A right circular cone and a sphere have the same radii of 2 inches. If the volume of the cone and the sphere are equal, what is the height, in inches, of the cone?

A) 8

B) 12

C) 16

D) 20