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
3611
What is the reflection of the point (3, -2) across the origin?
Answer:
(-3, 2)
Step 1: A reflection across the origin inverts both the x and y coordinates simultaneously. Step 2: The transformation rule is (x, y) becomes (-x, -y). Step 3: Applying this to (3, -2) results in (-3, 2).
3612
What is the reflection of the point (-4, 5) across the y-axis?
Answer:
(4, 5)
Step 1: Reflecting a point across the y-axis flips its horizontal position but leaves its vertical position unchanged. Step 2: The rule for reflection across the y-axis changes (x, y) to (-x, y). Step 3: Applying this to (-4, 5) yields (-(-4), 5), which is (4, 5).
3613
The circumcenter of a right-angled triangle is always located at:
Answer:
The midpoint of the hypotenuse
Step 1: A right-angled triangle can be inscribed in a semicircle. Step 2: According to Thales's theorem, the hypotenuse of the triangle acts as the diameter of that circumscribed circle. Step 3: The center of the circle (circumcenter) is precisely the midpoint of the diameter (the hypotenuse).
3614
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.
3615
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).
3616
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.
3617
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.
3618
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).
3619
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.
3620
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.