Rectangular Coordinate system

Question 1: Locating Points

The Question: Choose the correct option with respect to the points $P(5, -3)$, $Q(-3, 3)$, $R(0, -100)$, and $S(-2.5, 0)$ on the rectangular coordinate system.

• Point R does not lie in any quadrant
• Points P and R lie in Quadrant III
• Points S and Q lie in Quadrant II
• Points R and S cannot be represented on the rectangular coordinate system

Core Concepts: Quadrants and Axes

To solve this, we need to know where points are located based on the signs of their x and y coordinates.

• Quadrant I: x is positive (+), y is positive (+)
• Quadrant II: x is negative (-), y is positive (+)
• Quadrant III: x is negative (-), y is negative (-)
• Quadrant IV: x is positive (+), y is negative (-)
• On an Axis: If either the x or y coordinate is 0, the point is not in a quadrant but lies on one of the axes.
  • If x = 0, the point is on the y-axis.
  • If y = 0, the point is on the x-axis.

Detailed Solution:

Let’s analyze the location of each point:

1. P(5, -3): The x-coordinate (5) is positive, and the y-coordinate (-3) is negative. A (+, -) point lies in Quadrant IV.
2. Q(-3, 3): The x-coordinate (-3) is negative, and the y-coordinate (3) is positive. A (-, +) point lies in Quadrant II.
3. R(0, -100): The x-coordinate is 0. This means the point lies directly on the y-axis. Points on an axis are not in any quadrant.
4. S(-2.5, 0): The y-coordinate is 0. This means the point lies directly on the x-axis. Points on an axis are not in any quadrant.

Now let’s evaluate the given options:

• Point R does not lie in any quadrant: This is TRUE. As we determined, R(0, -100) lies on the y-axis.
• Points P and R lie in Quadrant III: This is FALSE. P is in Quadrant IV, and R is on the y-axis.
• Points S and Q lie in Quadrant II: This is FALSE. Q is in Quadrant II, but S is on the x-axis.
• Points R and S cannot be represented on the rectangular coordinate system: This is FALSE. Any ordered pair of real numbers, including those with zero, can be precisely located on the plane.

Final Answer: The only correct option is “Point R does not lie in any quadrant”.

Question 2: Fundamentals of the Coordinate System

The Question: Which of the following is/are correct with respect to the rectangular coordinate system? (Multiple Select Question)

• The horizontal line is called Y-axis
• The point of intersection of the X and Y axes is called the origin
• The vertical line is called X-axis
• Any point on the coordinate plane can be represented as an ordered pair (x, y)

Core Concepts: Definitions

• x-axis: The horizontal number line that passes through the origin.
• y-axis: The vertical number line that passes through the origin.
• Origin: The specific point $(0, 0)$ where the x-axis and y-axis intersect.
• Ordered Pair: The standard notation $(x, y)$ that gives the unique “address” of any point by specifying its horizontal distance (x) and vertical distance (y) from the origin.

Detailed Solution:

Let’s check each statement against the definitions:

1. The horizontal line is called Y-axis: This is FALSE. The horizontal line is the x-axis.
2. The point of intersection of the X and Y axes is called the origin: This is TRUE. This is the definition of the origin.
3. The vertical line is called X-axis: This is FALSE. The vertical line is the y-axis.
4. Any point on the coordinate plane can be represented as an ordered pair (x, y): This is TRUE. This is the fundamental purpose of the coordinate system.

Final Answer: The two correct statements are “The point of intersection of the X and Y axes is called the origin” and “Any point on the coordinate plane can be represented as an ordered pair (x, y)”.

Question 3: Identifying Incorrect Representations

The Question: Identify the incorrect options for the representation of a point on the coordinate plane. (Multiple Select Question)

Core Concepts: Coordinate Conventions

This question tests the same concepts as Question 1 but asks you to find the mistakes. Let’s list the correct conventions first:

• Quadrant I: (+, +)
• Quadrant II: (-, +)
• Quadrant III: (-, -)
• Quadrant IV: (+, -)
• On X-axis: (x, 0) or (±, 0)
• On Y-axis: (0, y) or (0, ±)
• Origin: (0, 0)

Detailed Solution:

Now we will evaluate each option to see if it’s correct or incorrect. The goal is to identify the incorrect ones.

1. Quadrant I : (+, +): This statement is CORRECT.
2. Quadrant IV : (-, -): This statement is INCORRECT. Quadrant IV points have a positive x and a negative y, so the form is (+, -).
3. Quadrant II : (-, -): This statement is INCORRECT. Quadrant II points have a negative x and a positive y, so the form is (-, +).
4. Quadrant III : (-, +): This statement is INCORRECT. Quadrant III points have a negative x and a negative y, so the form is (-, -).
5. On X-axis: (0, ±): This statement is INCORRECT. For any point on the x-axis, the y-coordinate is always 0. The correct form is (±, 0).
6. On Y-axis: (±, 0): This statement is INCORRECT. For any point on the y-axis, the x-coordinate is always 0. The correct form is (0, ±).
7. Origin (0,0): This statement is CORRECT.

Final Answer: The incorrect options are:

• Quadrant IV : (-, -)
• Quadrant II : (-, -)
• Quadrant III : (-, +)
• On X-axis: (0, ±)
• On Y-axis: (±, 0)