Pre Semester 2 Math Test For Grade 8(Part Two)

If a straight line is parallel to the X-axis, it means that the line is horizontal and has no vertical change. In other words, the line does not rise or fall, and therefore the ratio of the vertical distance to the horizontal distance is zero. This ratio is known as the gradient or slope of the line. Therefore, the correct answer is that if a straight line is parallel to the X-axis, its gradient is zero.

The given line has a slope of 5. To find a parallel line, we need to keep the same slope. The equation 5x - y + 3 = 0 has a slope of 5 and is therefore parallel to the given line.

The equation of a line can be determined using the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient. In this case, the point (-1, -5) and the gradient of 3 are given. Plugging these values into the point-slope form, we get y - (-5) = 3(x - (-1)), which simplifies to y + 5 = 3x + 3. Rearranging the equation, we get 3x - y + 2 = 0. Therefore, the correct answer is d. 5x - y + 3 = 0.

The maximum height of the stone can be found by finding the vertex of the quadratic function y = 40x - 5x^2. The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the x-coordinate -b/2a. In this case, a = -5 and b = 40, so the x-coordinate of the vertex is -40/(2*(-5)) = 4. Plugging this value into the function, we get y = 40(4) - 5(4)^2 = 80. Therefore, the maximum height of the stone is 80 meters.

The slope of a line is calculated by finding the change in y-coordinates divided by the change in x-coordinates between two points on the line. For pair I, the slope is (2-(-2))/(3-0) = 4/3, which is not equal to 2/3. For pair II, the slope is (2-0)/(8-5) = 2/3, which is equal to 2/3. For pair III, the slope is (-1-3)/(-3-3) = -4/-6 = 2/3, which is equal to 2/3. Therefore, the pairs II and III have a slope of 2/3.

The gradient of a line is the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. In this case, the change in y is 7 - (-3) = 10 and the change in x is 3 - 1 = 2. Therefore, the gradient is 10/2 = 5.

The given line is parallel to the line 4x - 2y = 5, which means it has the same slope. To find the slope of the given line, we can rearrange the equation in slope-intercept form (y = mx + b), where m is the slope.
Rearranging 4x - 2y = 5, we get -2y = -4x + 5, and dividing by -2 gives y = 2x - 5/2.
Since the line AB is parallel to this line, it will also have a slope of 2.
Using the point-slope form of a linear equation (y - y1 = m(x - x1)), where (x1, y1) is the given point (-1, -1), we can plug in the values to find the equation of AB.
Thus, the equation of AB is y - (-1) = 2(x - (-1)), which simplifies to y + 1 = 2x + 2, or y = 2x + 1. Therefore, the correct answer is b. 2x + 1.

The equation of a line that is perpendicular to another line can be found by taking the negative reciprocal of the slope of the given line. The given line has a slope of -2/5, so the perpendicular line will have a slope of 5/2. Using the point of intersection of the two given lines, we can use the point-slope form of a line to find the equation of the perpendicular line. Plugging in the values, we get the equation 2x - 5y = -1. Therefore, the correct answer is b. 2x - 5y = -1.

The equation of a line passing through two points can be found using the formula y - y1 = m(x - x1), where m is the slope of the line.
Using the given points (-2, 4) and (3, -2), we can find the slope: m = (y2 - y1)/(x2 - x1) = (-2 - 4)/(3 - (-2)) = -6/5.
Plugging in the slope and one of the points into the formula, we get y - 4 = (-6/5)(x - (-2)), which simplifies to y - 4 = (-6/5)(x + 2).
Expanding and rearranging the equation, we get 6x + 5y - 8 = 0. Therefore, the correct answer is c. 6x + 5y – 8 = 0.

The solution set of the given system of equations can be found by solving the equations simultaneously. By using elimination or substitution method, we can find that x = 5 and y = -1 satisfy both equations. Therefore, the correct answer is c. { 5, -1 }.