Bluebook SAT Mathematics 2

Question 1

The equation (x^2 + y^2 - 6)^2 = 4(x^2 + y^2) + 21 represents a set of points in the xy-plane. Which of the following best describes this set of points?

Accepted Answer

Two concentric circles

Answer

This question tests the ability to recognize algebraic structure. Let u = x^2 + y^2. The equation becomes (u - 6)^2 = 4u + 21. Expanding the left side gives u^2 - 12u + 36 = 4u + 21. Rearranging into a standard quadratic form gives u^2 - 16u + 15 = 0. Factoring this quadratic yields (u - 15)(u - 1) = 0. Therefore, u = 15 or u = 1. Substituting back x^2 + y^2 for u gives two separate equations: x^2 + y^2 = 15 and x^2 + y^2 = 1. These are the equations of two circles centered at the origin, one with a radius of sqrt(15) and one with a radius of 1. Thus, the set of points is two concentric circles.

Question 2

In the xy-plane, the graph of the function f(x) = 2x^2 - 18 is intersected by the line y = p. For exactly one value of x, the line intersects the function. What is the value of p?

Accepted Answer

-18

Answer

The function f(x) = 2x^2 - 18 describes a parabola that opens upwards. A horizontal line y = p will intersect a parabola at two points, one point, or zero points. For the line to intersect the parabola at exactly one point, the line must pass through the vertex of the parabola. The vertex of a parabola in the form y = ax^2 + c is at (0, c). For f(x) = 2x^2 - 18, the vertex is at (0, -18). Therefore, the horizontal line that intersects the parabola at exactly one point is y = -18. So, p = -18.

Question 3

For a polynomial p(x), the value of p(5) is -2. Which of the following must be true about p(x)?

Accepted Answer

The remainder when p(x) is divided by (x - 5) is -2.

Answer

This question applies the Remainder Theorem, which states that if a polynomial p(x) is divided by a linear factor (x - a), the remainder is p(a). In this case, we are given that p(5) = -2. According to the Remainder Theorem, the remainder when p(x) is divided by (x - 5) is p(5). Therefore, the remainder is -2. The other options are incorrect interpretations: (x-5) would be a factor if p(5)=0, not -2.

Question 4

The equation (kx - 7)^2 = 49 has solutions x = 0 and x = 2. What is the value of k?

Accepted Answer

7

Answer

We are given two solutions. Let's use the non-zero solution, x = 2, to solve for k. Substitute x = 2 into the equation: (k(2) - 7)^2 = 49. Taking the square root of both sides gives two possibilities: 2k - 7 = 7 or 2k - 7 = -7. From 2k - 7 = 7, we get 2k = 14, so k = 7. From 2k - 7 = -7, we get 2k = 0, so k = 0. We must check which value of k also satisfies the x=0 solution. If k=0, the equation becomes (-7)^2 = 49, which is true for all x, not just x=0 and x=2. If k=7, the equation is (7x-7)^2 = 49. Let's check the solutions. If x=0, (7(0)-7)^2 = (-7)^2 = 49. This works. If x=2, (7(2)-7)^2 = (14-7)^2 = 7^2 = 49. This also works. Therefore, the correct value is k=7.

Question 5

A function f has the property that f(x) = f(x/2) + 3 for all x > 0. If f(16) = 14, what is the value of f(1)?

Accepted Answer

2

Answer

We are given f(16) = 14 and the recursive relationship f(x) = f(x/2) + 3. We can work backwards from f(16) to find f(1). 
From the rule, f(x/2) = f(x) - 3.
 f(8) = f(16/2) = f(16) - 3 = 14 - 3 = 11.
 f(4) = f(8/2) = f(8) - 3 = 11 - 3 = 8.
 f(2) = f(4/2) = f(4) - 3 = 8 - 3 = 5.
 f(1) = f(2/2) = f(2) - 3 = 5 - 3 = 2.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Mathematics 2