Bluebook SAT Mathematics 3

Question 1

In the xy-plane, the graph of the equation y = -x^2 + 8x - 15 intersects a circle at exactly one point. If the circle's center is at the vertex of the parabola, what is the radius of the circle?

Accepted Answer

0

Answer

First, find the vertex of the parabola y = -x^2 + 8x - 15. The x-coordinate of the vertex is given by the formula x = -b/(2a). Here, a = -1 and b = 8, so x = -8/(2*(-1)) = 4. To find the y-coordinate, substitute x=4 back into the equation: y = -(4)^2 + 8(4) - 15 = -16 + 32 - 15 = 1. So, the vertex is at (4, 1). Since the parabola opens downward (because a < 0), and the circle intersects the parabola at exactly one point, that point must be the vertex. This means the circle is tangent to the parabola at its vertex. However, for a circle centered at the vertex to be tangent to the parabola at that same point, the circle must have a radius of 0, which represents a single point.

Question 2

Two data sets, A and B, are created. Data set A consists of 10 consecutive even integers. Data set B is created by multiplying each element of data set A by -2 and then adding 5. Which of the following statements correctly compares the standard deviations of the two data sets?

Accepted Answer

The standard deviation of set B is twice the standard deviation of set A.

Answer

Standard deviation measures the spread or dispersion of a data set. Adding a constant to every value in a data set does not change the spread, so the '+5' operation has no effect on the standard deviation. However, multiplying each value by a constant 'c' multiplies the standard deviation by the absolute value of that constant, |c|. In this case, each element of set A is multiplied by -2. Therefore, the standard deviation of set B will be |-2| times the standard deviation of set A. This means the standard deviation of B is 2 times the standard deviation of A.

Question 3

In the xy-plane, a point P lies on the unit circle. The angle formed by the positive x-axis and the line segment from the origin to P is θ. If sin(θ) = -cos(θ) and the angle θ is in Quadrant IV, what is the value of θ in radians?

Accepted Answer

7π/4

Answer

The equation sin(θ) = -cos(θ) is equivalent to tan(θ) = sin(θ)/cos(θ) = -1. The tangent function is negative in Quadrants II and IV. The reference angle for which tan(θ) = 1 is π/4. In Quadrant II, the angle would be π - π/4 = 3π/4. In Quadrant IV, the angle would be 2π - π/4 = 7π/4. Since the problem specifies that θ is in Quadrant IV, the correct value is 7π/4.

Question 4

A circle in the xy-plane is defined by the equation x^2 + y^2 - 10x + 6y + c = 0. For what value of c does the radius of the circle equal 1?

Accepted Answer

33

Answer

To find the radius, we must first rewrite the equation of the circle in standard form: (x - h)^2 + (y - k)^2 = r^2. We can do this by completing the square. Rearrange the terms: (x^2 - 10x) + (y^2 + 6y) = -c. To complete the square for x, take half of -10, square it, and add it to both sides: (-10/2)^2 = 25. To complete the square for y, take half of 6, square it, and add it to both sides: (6/2)^2 = 9. The equation becomes (x^2 - 10x + 25) + (y^2 + 6y + 9) = -c + 25 + 9. This simplifies to (x - 5)^2 + (y + 3)^2 = 34 - c. In this form, the right side is equal to r^2. We are given that the radius r = 1, so r^2 = 1. Therefore, 34 - c = 1. Solving for c gives c = 33.

Question 5

The function g is defined by g(x) = a(x+b)^2, where a and b are constants. If g(x) has a maximum value of 5 and one of its x-intercepts is at x = -3 + sqrt(5), what is the value of the other x-intercept?

Accepted Answer

-3 - sqrt(5)

Answer

The function g(x) = a(x+b)^2 is a parabola in vertex form. The vertex is at (-b, 0), but since it has a maximum value of 5, the vertex form should be g(x) = a(x-h)^2 + k, where the vertex is (h, k). So the vertex is at (h, 5). The x-coordinate of the vertex is the axis of symmetry. The x-intercepts of a parabola are symmetric with respect to its axis of symmetry. Let the two x-intercepts be x1 and x2. The axis of symmetry is x = h = (x1 + x2)/2. We are given one intercept, x1 = -3 + sqrt(5). From the vertex form g(x) = a(x+b)^2, we can see the axis of symmetry is at x = -b. Therefore, h = -b. The given x-intercept is -3 + sqrt(5). The axis of symmetry must be x = -3. So, -3 = ((-3 + sqrt(5)) + x2) / 2. Solving for x2: -6 = -3 + sqrt(5) + x2. This gives x2 = -3 - sqrt(5).

Bluebook SAT Practice Test Practice Test

Bluebook SAT Practice Test Practice Test

Bluebook SAT Mathematics 3