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
3661
Find the slope of a line that is perpendicular to the line 3x - y = 8.
Answer:
-1/3
Step 1: Rewrite the given equation in slope-intercept form: y = 3x - 8. The slope is 3. Step 2: A perpendicular line will have a slope that is the negative reciprocal of 3. Step 3: The negative reciprocal of 3 is -1/3.
3662
What is the equation of the line perpendicular to x - y = 0 passing through the origin?
Answer:
x + y = 0
Step 1: Find the slope of x - y = 0. Rewrite as y = x. The slope is 1. Step 2: The perpendicular slope is the negative reciprocal: -1/1 = -1. Step 3: Using y = mx + c with m = -1 and passing through (0,0), we get y = -1x + 0, which is x + y = 0.
3663
What is the equation of the line parallel to 2x + 3y = 6 passing through the origin (0,0)?
Answer:
2x + 3y = 0
Step 1: Lines parallel to ax + by = c have the form ax + by = k. Step 2: The parallel line must be 2x + 3y = k. Step 3: Since it passes through (0,0), substitute x=0, y=0 to find k. 2(0) + 3(0) = k, so k = 0. The equation is 2x + 3y = 0.
3664
What is the mathematical condition for two non-vertical lines with slopes m1 and m2 to be perpendicular?
Answer:
m1 * m2 = -1
Step 1: Perpendicular lines intersect at a 90-degree angle. Step 2: In coordinate geometry, this relationship dictates that their slopes are negative reciprocals of one another. Step 3: Expressed algebraically, m1 = -1/m2, or m1 * m2 = -1.
3665
What is the mathematical condition for two non-vertical lines with slopes m1 and m2 to be parallel?
Answer:
m1 = m2
Step 1: Parallel lines run alongside each other and never intersect, meaning they have the exact same steepness and direction. Step 2: Steepness and direction are defined by the slope. Step 3: Thus, their slopes must be strictly equal, m1 = m2.
3666
What is the slope of a line perpendicular to the line y = 2x + 1?
Answer:
-1/2
Step 1: Identify the slope of the given line, which is m = 2. Step 2: The slopes of perpendicular lines are negative reciprocals of each other (m1 * m2 = -1). Step 3: Calculate the negative reciprocal of 2, which gives -1/2.
3667
What is the slope of a line parallel to the line y = 4x - 5?
Answer:
4
Step 1: Determine the slope of the given line. For y = 4x - 5, the slope is 4. Step 2: Parallel lines have identical slopes. Step 3: Therefore, any line parallel to this line will also have a slope of 4.
3668
The form of a linear equation written as y = mx + c is commonly known as:
Answer:
Slope-intercept form
Step 1: Identify the components of the equation y = mx + c. Step 2: 'm' represents the slope of the line, and 'c' represents the y-intercept. Step 3: Because it explicitly shows both the slope and the y-intercept, it is called the slope-intercept form.
3669
What is the equation of the line passing through (-1, 2) and (3, -2)?
Answer:
x + y = 1
Step 1: Find the slope m = (-2 - 2) / (3 - (-1)) = -4 / 4 = -1. Step 2: Use point-slope form with point (-1, 2): y - 2 = -1(x - (-1)). Step 3: Simplify: y - 2 = -x - 1, which rearranges to x + y = 1.
3670
What is the equation of the line with an x-intercept of 2 and a y-intercept of 3?
Answer:
3x + 2y = 6
Step 1: Use the intercept form of a line: x/a + y/b = 1. Step 2: Substitute a = 2 (x-intercept) and b = 3 (y-intercept): x/2 + y/3 = 1. Step 3: Multiply the entire equation by the common denominator 6 to get 3x + 2y = 6.