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
3631
The point designated as (0, y) will always lie on the:
Answer:
y-axis
Step 1: A coordinate pair is written as (x, y). Step 2: The given point has an x-coordinate of exactly 0, which means it has no horizontal deviation from the center. Step 3: All such points lie entirely on the vertical axis, which is the y-axis.
3632
In what ratio does the y-axis divide the line segment joining the points (-2, 3) and (4, 5)?
Answer:
1:2
Step 1: Any point on the y-axis has an x-coordinate of 0. Step 2: Let the ratio be k:1. Use the section formula for the x-coordinate: x = (k*4 + 1*(-2)) / (k+1) = 0. Step 3: 4k - 2 = 0, so 4k = 2, yielding k = 1/2. The ratio is 1:2.
3633
In what ratio does the x-axis divide the line segment joining the points (2, -3) and (5, 6)?
Answer:
1:2
Step 1: Any point on the x-axis has a y-coordinate of 0. Step 2: Let the ratio be k:1. Apply the section formula for the y-coordinate: y = (k*6 + 1*(-3)) / (k+1) = 0. Step 3: 6k - 3 = 0, meaning 6k = 3, so k = 1/2. The ratio is 1:2.
3634
Find the point dividing the segment connecting (1, 1) and (4, 4) internally in the ratio 2:1.
Answer:
(3, 3)
Step 1: Use the section formula x = (mx2 + nx1)/(m+n). m=2, n=1. x1=1, x2=4. Step 2: x = (2*4 + 1*1) / (2+1) = (8+1)/3 = 9/3 = 3. Step 3: Since both points have identical x and y coordinates, the y calculation is identical. The point is (3, 3).
3635
The midpoint formula is a special case of the internal section formula where the ratio m:n is:
Answer:
1:1
Step 1: The midpoint exactly bisects a line segment into two equal halves. Step 2: This means the lengths of the two parts are identical, yielding a ratio of 1 to 1. Step 3: Plugging m=1, n=1 into the section formula produces the midpoint formula.
3636
Which formula computes the coordinates of a point dividing a line segment EXTERNALLY in the ratio m:n?
Answer:
((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n))
Step 1: The internal section formula uses addition for both the numerator and denominator products. Step 2: The external section formula modifies this by using subtraction. Step 3: The correct formula is ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n)).
3637
What point divides the line segment between (2, 0) and (0, 2) in the ratio 1:1?
Answer:
(1, 1)
Step 1: A 1:1 ratio means finding the midpoint of the segment. Step 2: Using the midpoint formula: x = (2 + 0) / 2 = 1. Step 3: y = (0 + 2) / 2 = 1. The point is (1, 1).
3638
Find the coordinates of the point dividing the segment joining (0, 0) and (3, 0) internally in the ratio 1:2.
Answer:
(1, 0)
Step 1: Use the section formula x = (mx2 + nx1)/(m+n). Step 2: Here, m=1, n=2, (x1,y1)=(0,0), (x2,y2)=(3,0). Step 3: x = (1*3 + 2*0)/(1+2) = 3/3 = 1. Since both y-coordinates are 0, y remains 0. The point is (1, 0).
3639
Are the points (1, 2), (2, 4), and (3, 6) collinear?
Answer:
Yes
Step 1: Calculate the slope between the first two points: m1 = (4-2)/(2-1) = 2. Step 2: Calculate the slope between the second and third points: m2 = (6-4)/(3-2) = 2. Step 3: Because the slopes are identical and they share a common point, they are collinear.
3640
Find the area of the triangle with vertices (2, 3), (-1, 0), and (2, -3).
Answer:
9
Step 1: Notice vertices (2,3) and (2,-3) form a vertical base. The length of this base is 3 - (-3) = 6. Step 2: The x-coordinate of the base is 2. The opposite vertex is (-1, 0). The perpendicular height (horizontal distance) is 2 - (-1) = 3. Step 3: Area = (1/2) * base * height = (1/2) * 6 * 3 = 9.