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
41
What is the distance of the point (3, 4) from the origin?
Answer:
5
Step 1: Use the distance formula for a point from the origin: d = √(x² + y²). Step 2: Substitute the coordinates (3, 4): d = √(3² + 4²). Step 3: Simplify: d = √(9 + 16) = √25 = 5.
42
Find the center of the circle given by the general equation x² + y² - 4x - 6y + 9 = 0.
Answer:
(2, 3)
Step 1: The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the center is (-g, -f). Step 2: Here, 2g = -4 so g = -2, and 2f = -6 so f = -3. Step 3: The center is (-(-2), -(-3)) which simplifies to (2, 3).
43
Which of the following represents a circle with a center at (1, 1) and a radius of 1?
Answer:
(x - 1)² + (y - 1)² = 1
Step 1: The standard equation for a circle is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius. Step 2: Substitute h=1, k=1, and r=1. Step 3: The equation becomes (x - 1)² + (y - 1)² = 1² or simply 1.
44
What is the radius of the circle defined by the equation x² + y² = 25?
Answer:
5
Step 1: Match the equation to the standard form x² + y² = r². Step 2: We can see that r² = 25. Step 3: Take the positive square root to find the radius: r = √25 = 5.
45
What is the center of the circle described by the equation (x - 2)² + (y + 3)² = 16?
Answer:
(2, -3)
Step 1: Compare the given equation to the standard circle equation: (x - h)² + (y - k)² = r². Step 2: Here, -h = -2, meaning h = 2. Step 3: Similarly, -k = 3, meaning k = -3. The center (h, k) is (2, -3).
46
The equation of a circle with its center at the origin and radius r is:
Answer:
x² + y² = r²
Step 1: A circle is defined as the set of points at a distance 'r' from the center. Step 2: Using the distance formula from the origin (0,0) to a point (x,y), we have √(x² + y²) = r. Step 3: Squaring both sides yields the standard equation x² + y² = r².
47
What is the distance between the two parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 5 = 0?
Answer:
2
Step 1: The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is d = |c1 - c2| / √(a² + b²). Step 2: Substitute the values: d = |-5 - 5| / √(3² + 4²). Step 3: Simplify: d = |-10| / √25 = 10 / 5 = 2.
48
What is the perpendicular distance from the origin (0,0) to the line 3x + 4y - 10 = 0?
Answer:
2
Step 1: The distance formula from a point (x1, y1) to a line ax + by + c = 0 is d = |ax1 + by1 + c| / √(a² + b²). Step 2: Substitute (0,0) into the line equation: d = |3(0) + 4(0) - 10| / √(3² + 4²). Step 3: Simplify: d = |-10| / √25 = 10 / 5 = 2.
49
The centroid of a triangle represents the intersection point of its:
Answer:
Medians
Step 1: A median is a line segment drawn from a vertex to the midpoint of the opposite side. Step 2: Every triangle has three medians. Step 3: The single point where all three medians intersect is defined geometrically as the centroid.
50
The incenter of a triangle represents the intersection point of its:
Answer:
Angle bisectors
Step 1: Different centers of a triangle are formed by different intersecting lines. Step 2: The centroid is from medians, orthocenter from altitudes, and circumcenter from perpendicular bisectors. Step 3: The incenter, the center of the inscribed circle, is the intersection of the angle bisectors.