Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
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
51
If the centroid of a triangle is at the origin (0,0) and two of its vertices are (2, 3) and (-1, -1), what is the third vertex?
Answer:
(-1, -2)
Step 1: Set up the centroid equations. Let the third vertex be (x,y). (2 - 1 + x)/3 = 0 and (3 - 1 + y)/3 = 0. Step 2: Solve for x: (1 + x)/3 = 0 => 1 + x = 0 => x = -1. Step 3: Solve for y: (2 + y)/3 = 0 => 2 + y = 0 => y = -2. The vertex is (-1, -2).
52
Find the centroid of a triangle with vertices at (a, 0), (0, b), and the origin.
Answer:
(a/3, b/3)
Step 1: Identify the three vertices: (a, 0), (0, b), and (0, 0). Step 2: Apply the centroid formula: x = (a + 0 + 0) / 3 = a/3. Step 3: Apply for y: y = (0 + b + 0) / 3 = b/3. The centroid is (a/3, b/3).
53
The centroid of a triangle divides each median in the ratio of:
Answer:
2:1
Step 1: The centroid is the point where the three medians of a triangle intersect. Step 2: A core geometric theorem states that the centroid acts as a balance point on the medians. Step 3: It divides each median into two segments in a 2:1 ratio, with the larger segment connecting to the vertex.
54
What is the centroid of the triangle with vertices (1, 2), (3, -4), and (5, 8)?
Answer:
(3, 2)
Step 1: Use the centroid formula C = ((x1+x2+x3)/3, (y1+y2+y3)/3). Step 2: Sum the x-coordinates: (1 + 3 + 5)/3 = 9/3 = 3. Step 3: Sum the y-coordinates: (2 + (-4) + 8)/3 = 6/3 = 2. The centroid is (3, 2).
55
Find the centroid of the triangle with vertices (0, 0), (3, 0), and (0, 3).
Answer:
(1, 1)
Step 1: The centroid formula is ((x1+x2+x3)/3, (y1+y2+y3)/3). Step 2: Calculate x: (0 + 3 + 0) / 3 = 3 / 3 = 1. Step 3: Calculate y: (0 + 0 + 3) / 3 = 3 / 3 = 1. The centroid is (1, 1).
56
What is the reflection of the point (2, 3) across the x-axis?
Answer:
(2, -3)
Step 1: Reflecting a point across the x-axis means the horizontal position stays the same, but it flips vertically. Step 2: Therefore, the x-coordinate remains unchanged, and the y-coordinate reverses its sign. Step 3: The point (2, 3) becomes (2, -3).
57
Find the point dividing the segment between (-1, 2) and (3, -2) internally in the ratio 1:3.
Answer:
(0, 1)
Step 1: Apply the section formula with m=1, n=3. Step 2: x = (1*3 + 3*(-1)) / (1+3) = (3 - 3) / 4 = 0. Step 3: y = (1*(-2) + 3*2) / (1+3) = (-2 + 6) / 4 = 4 / 4 = 1. The point is (0, 1).
58
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.
59
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.
60
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.