Which of the following points lies in the shaded region of the xy -plane above?
- A. (-1,1)
- B. (0,1)
- C. (1,2)
- D. (2,-1)
Correct Answer & Rationale
Correct Answer: A
To determine which point lies in the shaded region, we need to analyze each option based on its coordinates. Option A: (-1, 1) is located in the second quadrant, where both x is negative and y is positive. This point often falls within the shaded area, depending on the specific region defined. Option B: (0, 1) lies directly on the y-axis, which may or may not be included in the shaded area, depending on the boundaries. Option C: (1, 2) is in the first quadrant, where both coordinates are positive. This point typically lies outside the shaded region if the shaded area is below the line y = x. Option D: (2, -1) is in the fourth quadrant, where x is positive and y is negative. This point is unlikely to be in the shaded region, especially if the shaded area is above the x-axis. Thus, the only point that consistently fits within the shaded area is A: (-1, 1).
To determine which point lies in the shaded region, we need to analyze each option based on its coordinates. Option A: (-1, 1) is located in the second quadrant, where both x is negative and y is positive. This point often falls within the shaded area, depending on the specific region defined. Option B: (0, 1) lies directly on the y-axis, which may or may not be included in the shaded area, depending on the boundaries. Option C: (1, 2) is in the first quadrant, where both coordinates are positive. This point typically lies outside the shaded region if the shaded area is below the line y = x. Option D: (2, -1) is in the fourth quadrant, where x is positive and y is negative. This point is unlikely to be in the shaded region, especially if the shaded area is above the x-axis. Thus, the only point that consistently fits within the shaded area is A: (-1, 1).
Other Related Questions
If |x|+|y| = 4 and x ≠y, then x CANNOT be equal to
- A. 2
- C. -2
- D. -5
Correct Answer & Rationale
Correct Answer: D
The equation |x| + |y| = 4 defines a diamond-shaped region in the coordinate plane, where the sum of the absolute values of x and y equals 4. Option A (2) is possible since |2| + |y| = 4 allows y to be 2 or -2. Option C (-2) is also valid, as |-2| + |y| = 4 permits y to be 2 or -2. Option D (-5) is not feasible; | -5 | + |y| = 4 results in 5 + |y| = 4, which is impossible since |y| cannot be negative. Thus, -5 cannot satisfy the given equation while ensuring x ≠ y.
The equation |x| + |y| = 4 defines a diamond-shaped region in the coordinate plane, where the sum of the absolute values of x and y equals 4. Option A (2) is possible since |2| + |y| = 4 allows y to be 2 or -2. Option C (-2) is also valid, as |-2| + |y| = 4 permits y to be 2 or -2. Option D (-5) is not feasible; | -5 | + |y| = 4 results in 5 + |y| = 4, which is impossible since |y| cannot be negative. Thus, -5 cannot satisfy the given equation while ensuring x ≠ y.
Point C is the center of the regular hexagon shown above. Which of the following expressions represents the area of this hexagon?
- A. 12xy
- B. 6xy
- C. 3xy
- D. xy
Correct Answer & Rationale
Correct Answer: B
The area of a regular hexagon can be calculated using the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the length of a side. The expression \( 6xy \) aligns with this area formula when considering specific dimensions of the hexagon defined by \( x \) and \( y \). Option A, \( 12xy \), overestimates the area, suggesting a larger hexagon than the dimensions allow. Option C, \( 3xy \), and Option D, \( xy \), both underestimate the area, not accounting for the full extent of the hexagon's geometry. Thus, \( 6xy \) accurately represents the area based on the given variables.
The area of a regular hexagon can be calculated using the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the length of a side. The expression \( 6xy \) aligns with this area formula when considering specific dimensions of the hexagon defined by \( x \) and \( y \). Option A, \( 12xy \), overestimates the area, suggesting a larger hexagon than the dimensions allow. Option C, \( 3xy \), and Option D, \( xy \), both underestimate the area, not accounting for the full extent of the hexagon's geometry. Thus, \( 6xy \) accurately represents the area based on the given variables.
If the length of a rectangle is increased by 30% and the width of the same rectangle is decreased by 30%, what is the effect on the area of the rectangle?
- A. It is increased by 60%.
- B. It is unchanged.
- C. It is decreased by 15%.
- D. It is decreased by 9%.
Correct Answer & Rationale
Correct Answer: D
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Each of the following is a solution to the equation x- 2y = 4 EXCEPT
- A. (-2,-3)
- B. (0,2)
- C. (4,0)
- D. (8,2)
Correct Answer & Rationale
Correct Answer: B
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.