Explanations
Explanations

1.      D     

Since the line in the image is vertical, the slope is undefined, and there is no coefficient of the x in the slope-intercept form, y = mx + b, where m is the slope and b the y-intercept of the line. So we know that this line is in the format of x = c, where c is a constant. Since the line passes through the point (4, 0), the equation of the line is x = 4.

2.      C     

Before trying to answer the question, you need to put the equation into the slope-intercept form, y = mx + b, where m is the slope and b the y-intercept. Divide both sides of the equation by 2 to get y = 2x + 9/ 2. Therefore, the slope is 2. A line perpendicular to this line has a slope that is the opposite of the reciprocal of 2, which is –1 /2. The only answer choice with this slope is y = – 1/2 x + 6.

3.      C     

By definition, we know that all points on the circle are of equal distance from its center, in this case, (0, 0). Note that both (0, 0) and (3, 3) lie on the line y = x. Because a straight line is always the shortest path between two points, one point where the circle and line intersect must be the closest point to (3, 3). Substituting y for x in the equation (or vice versa) you see that the line and the circle intersect at (–1, –1) and (1, –1). The point (1, 1) will obviously be closer to (3, 3), so you need to find out how far away those two points are using the distance formula.

4.      A     

Before trying to calculate the distance, first find the equation for the line through (1, 14) and (3, 25). The slope of the line is defined by the equation:

The slope of this line and the slope of line y = 3x + 4, which is 3, are not equal, which means that they are not parallel and must cross. Therefore, the least difference between the two lines is 0.

5.      B     

In order to find the perimeter of this triangle, you need to know the length of each side. The side lengths are the distance between each two vertices. The distance formula is all you need. For (1, 0) and (4, 0):

For (1, 0) and (2, ), we get

For (4, 0) and (2, ), we get

The perimeter is the sum of the lengths of all three sides: 3 + 2 + 2.65 = 7.65.

6.      C     

Use the three-dimensional distance formula to find the distance between these two points. Recall that the distance, d, between the point (x1, y1, z1) and the point (x2, y2, z2) is given by d = .

Plugging the points from the question into the equation gives you:

7.      E     

To multiply a vector by a scalar, multiply each component of the vector by the scalar. So, 2 = 2(4, 2) = (8, 4). Then, to subtract these vectors, subtract their respective components: = 2 = (8, 4) – (–5, 3) = (8 –(–5), 4 – 3) = (13, 1). The magnitude of a vector is equal to the square root of the sum of the squares of its components. In this case, the magnitude of is ≈ 13.04.

8.      A     

First, we need to convert the points from polar coordinates to rectangular. To find the rectangular coordinates of a point (r, ), use the conversion formulas, x = r cos and y = r sin . In this case, for point a, x = 3 cos π/4 ≈ 2.12 and y = 3 sin π /4 ≈ 2.12. The rectangular coordinates for point a are (2.12, 2.12). For point b, x = 3 cos /3 = –1.5 and y = 3 sin /3 ≈ 2.6, so the rectangular coordinates for point b are (–1.5, 2.6).

To find the length of chord AB, use the distance formula to find the distance from point a to point b: AB = ≈ 3.65.

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