Bluebook SAT Mathematics 2 — Questions and Answers

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?

A single point
A single circle
Two concentric circles (Correct answer)
A parabola

Correct answer: Two concentric circles

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?

9
-18 (Correct answer)
0
2

Correct answer: -18

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)?

(x + 2) is a factor of p(x).
x = 5 is a root of the equation p(x) = 0.
The remainder when p(x) is divided by (x - 5) is -2. (Correct answer)
The remainder when p(x) is divided by (x + 2) is 5.

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

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?

14
0
7 (Correct answer)
2

Correct answer: 7

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)?

-1
2 (Correct answer)
5
8

Correct answer: 2

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.

Question 6: In a right triangle, one angle measures x° and another angle measures y°, where sin(x°) = cos(y°). If x = 4k - 22 and y = 6k - 18, what is the value of k?

13 (Correct answer)
15
11.2
21

Correct answer: 13

In any right triangle, the two acute angles are complementary, meaning their sum is 90°. The trigonometric co-function identity states that sin(x°) = cos(90° - x°). Since we are given sin(x°) = cos(y°), it must be that y = 90 - x, or x + y = 90. We can substitute the given expressions for x and y into this equation: (4k - 22) + (6k - 18) = 90. Combining like terms gives 10k - 40 = 90. Adding 40 to both sides gives 10k = 130. Dividing by 10 gives k = 13.

Bluebook SAT Mathematics 2

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?