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
4271
A ball is dropped from a height of 10 meters. It rebounds to half its previous height after every bounce. What is the total distance traveled by the ball until it comes to rest?
Answer:
30 m
The ball falls 10m. Then it bounces up 5m and down 5m. Then up 2.5m and down 2.5m, and so on. The total distance is 10 + 2(5 + 2.5 + 1.25 + ...). The term in parentheses is an infinite GP with a=5 and r=1/2. Its sum is S = 5 / (1 - 1/2) = 10. Total distance = 10 + 2(10) = 30 meters.
4272
Find the sum of the first n terms of the series 1 + 2 + 4 + 8 + ...
Answer:
2^n - 1
This is a Geometric Progression with first term a = 1 and common ratio r = 2. The formula for the sum of n terms is S_n = a(r^n - 1) / (r - 1). So, S_n = 1 * (2^n - 1) / (2 - 1) = 2^n - 1.
4273
What is the relationship between Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) of two distinct positive numbers?
Answer:
A > G > H
For any two distinct positive numbers, their Arithmetic Mean is always the largest, the Harmonic Mean is the smallest, and the Geometric Mean lies between them. Thus, A > G > H. Furthermore, G^2 = A * H.
4274
The harmonic mean (HM) of two numbers a and b is given by:
Answer:
2ab/(a+b)
The harmonic mean of two numbers a and b is the reciprocal of the arithmetic mean of their reciprocals. So, HM = 1 / [ (1/a + 1/b) / 2 ] = 2 / [ (a+b)/ab ] = 2ab / (a+b).
4275
If AM and GM of two numbers are 5 and 4 respectively, what are the numbers?
Answer:
2 and 8
Let the numbers be a and b. AM = (a+b)/2 = 5, so a+b = 10. GM = sqrt(ab) = 4, so ab = 16. We need two numbers that add to 10 and multiply to 16. These numbers are 2 and 8.
4276
Insert three geometric means between 2 and 162.
Answer:
6, 18, 54
The sequence is a GP with 5 terms: 2, G1, G2, G3, 162. Here a_1 = 2 and a_5 = 162. Using a_n = a * r^(n-1), we have 162 = 2 * r^4. This gives r^4 = 81, so r = 3 (taking the real positive root). The means are 2*3=6, 6*3=18, 18*3=54.
4277
Insert two arithmetic means between 3 and 18.
Answer:
8, 13
We need an AP with 4 terms: 3, A1, A2, 18. Here, a_1 = 3 and a_4 = 18. Using the formula a_n = a + (n-1)d, we get 18 = 3 + 3d, so 15 = 3d, making d = 5. The means are A1 = 3+5 = 8 and A2 = 8+5 = 13.
4278
What is the arithmetic mean (AM) of 12 and 30?
Answer:
21
The arithmetic mean of two numbers a and b is simply their average, calculated as (a + b) / 2. Here, AM = (12 + 30) / 2 = 42 / 2 = 21.
4279
If x, 2x+2, 3x+3 are in GP, then what is the 4th term?
Answer:
-13.5
For terms to be in GP, the square of the middle term equals the product of the first and third. (2x+2)^2 = x(3x+3). Expanding gives 4x^2 + 8x + 4 = 3x^2 + 3x. Rearranging yields x^2 + 5x + 4 = 0. Factoring gives (x+4)(x+1) = 0. If x=-1, terms are -1, 0, 0, which is a trivial GP with r=0. If x=-4, terms are -4, -6, -9. Here r = -6/-4 = 1.5. The 4th term is -9 * 1.5 = -13.5.
4280
The sum of an infinite geometric series is 15, and its first term is 5. What is the common ratio?
Answer:
2/3
The formula for the sum of an infinite GP is S = a / (1 - r). We are given S = 15 and a = 5. So, 15 = 5 / (1 - r). This means 1 - r = 5 / 15 = 1/3. Therefore, r = 1 - 1/3 = 2/3.