All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
3671
What is the equation of the line passing through (2, 3) and parallel to the y-axis?
Answer:
x = 2
Step 1: A line parallel to the y-axis is a vertical line. Step 2: All points on a vertical line share the same x-coordinate. Step 3: Since it passes through (2, 3), the x-coordinate must always be 2. The equation is x = 2.
3672
What is the equation of the line parallel to the x-axis and at a distance of 3 units above it?
Answer:
y = 3
Step 1: A line parallel to the x-axis has a constant y-value. Step 2: Since it is 3 units above the x-axis, all points on this line will have a y-coordinate of 3. Step 3: Thus, the equation of the line is y = 3.
3673
What is the equation of the y-axis?
Answer:
x = 0
Step 1: The y-axis consists of all points where the horizontal distance from the origin is zero. Step 2: This means the x-coordinate for every point on the y-axis is 0. Step 3: Therefore, the equation representing the y-axis is x = 0.
3674
What is the equation of the x-axis?
Answer:
y = 0
Step 1: The x-axis consists of all points where the vertical distance from the origin is zero. Step 2: This means the y-coordinate for every point on the x-axis is 0. Step 3: Therefore, the equation representing the x-axis is y = 0.
3675
Find the equation of a line passing through the point (1, 1) with a slope of -1.
Answer:
x + y = 2
Step 1: Use the point-slope form: y - y1 = m(x - x1). Step 2: Substitute m = -1 and (x1, y1) = (1, 1): y - 1 = -1(x - 1). Step 3: Expand and rearrange: y - 1 = -x + 1, which simplifies to x + y = 2.
3676
What is the equation of the line passing through the origin (0, 0) and the point (2, 4)?
Answer:
y = 2x
Step 1: Find the slope m = (4 - 0) / (2 - 0) = 4 / 2 = 2. Step 2: Since it passes through the origin, the y-intercept c = 0. Step 3: Using y = mx + c, the equation is y = 2x.
3677
What is the equation of the line with a slope of 2 and a y-intercept of 3?
Answer:
y = 2x + 3
Step 1: Use the slope-intercept form of a linear equation: y = mx + c. Step 2: Here, the slope (m) is 2 and the y-intercept (c) is 3. Step 3: Substitute the values to get the equation: y = 2x + 3.
3678
What is the slope of a line making an angle of 45 degrees with the positive direction of the x-axis?
Answer:
1
Step 1: The slope m of a line is equal to the tangent of the angle it makes with the positive x-axis. Step 2: Formula: m = tan(θ). Step 3: Substitute 45 degrees: m = tan(45°) = 1.
3679
If the points (1, 2), (2, 3), and (3, y) are collinear, what is the value of y?
Answer:
4
Step 1: Collinear points share the same slope between any pair of points. Step 2: Slope between (1,2) and (2,3) is (3-2)/(2-1) = 1. Step 3: Slope between (2,3) and (3,y) must also be 1. So, (y-3)/(3-2) = 1, meaning y - 3 = 1, giving y = 4.
3680
Find the slope of the line represented by the equation 3x - 4y = 12.
Answer:
3/4
Step 1: Express the equation in slope-intercept form (y = mx + c). Step 2: Subtract 3x from both sides: -4y = -3x + 12. Step 3: Divide by -4: y = (3/4)x - 3. The slope (m) is 3/4.