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
4181
A rubber ball is dropped from a height of 20 meters. It bounces back to 3/4 of the height of every fall. Find the total distance it travels before coming to rest.
Answer:
140 m
The distance traveled is D = 20 + 2(20 * 3/4) + 2(20 * (3/4)^2) + ... This forms an infinite GP for the bounces. D = 20 + 2 * [ 15 / (1 - 3/4) ] = 20 + 2 * [ 15 / (1/4) ] = 20 + 2 * 60 = 20 + 120 = 140 meters. (Shortcut formula: D = H * (1+r)/(1-r) = 20 * (1+3/4)/(1-3/4) = 20 * (7/4)/(1/4) = 20 * 7 = 140).
4182
How many terms of the GP 3, 6, 12, ... must be taken to make a sum of 381?
Answer:
7
This is a GP with a=3 and r=2. We need S_n = 381. S_n = a(r^n - 1) / (r - 1) => 381 = 3(2^n - 1) / (2 - 1). Divide by 3: 127 = 2^n - 1. So 2^n = 128. Since 2^7 = 128, n = 7.
4183
If the third term of a GP is 3, what is the product of its first 5 terms?
Answer:
243
Let the terms be a/r^2, a/r, a, ar, ar^2. The 3rd term is 'a', so a = 3. The product of the 5 terms is a^5. Thus, the product is 3^5 = 243.
4184
If the sum of an AP is 3n^2 + n, what is the common difference?
Answer:
6
The sum of n terms of an AP is of the form S_n = (d/2)n^2 + (a - d/2)n. Comparing this with the given formula S_n = 3n^2 + n, we can see that the coefficient of n^2 is d/2. Therefore, d/2 = 3, which implies the common difference d = 6. Alternatively, calculate a_1 = S_1 = 4 and a_2 = S_2 - S_1 = 14 - 4 = 10. The difference is 10 - 4 = 6.
4185
Find the sum of the series: 0.2 + 0.02 + 0.002 + ... to infinity.
Answer:
2/9
This is an infinite GP with a = 0.2 and r = 0.1. The sum is S = a / (1 - r) = 0.2 / (1 - 0.1) = 0.2 / 0.9 = 2/9.
4186
If 1+2+3+...+n = 120, what is the value of n?
Answer:
15
The sum of the first n natural numbers is n(n+1)/2. So, n(n+1)/2 = 120, which gives n(n+1) = 240. The two consecutive integers that multiply to 240 are 15 and 16. Therefore, n = 15.
4187
In an AP, the sum of the first 3 terms is equal to the sum of the first 6 terms. Which term of the AP is necessarily zero?
Answer:
5th term
If S_3 = S_6, it implies that the sum of the extra terms (4th, 5th, and 6th) must be zero. So, a_4 + a_5 + a_6 = 0. Because it's an AP, a_4 + a_6 = 2 * a_5. Substituting this in gives 2*a_5 + a_5 = 0, which means 3*a_5 = 0, so the 5th term a_5 must be zero.
4188
The 10th term of a GP is 9 and its 4th term is 4. Find its 7th term.
Answer:
6 or -6
We are given a_10 = ar^9 = 9 and a_4 = ar^3 = 4. Dividing them gives r^6 = 9/4. The 7th term is a_7 = ar^6 = (ar^3) * r^3. Since r^6 = 9/4, r^3 = ±sqrt(9/4) = ±3/2. Therefore, a_7 = 4 * (±3/2) = ±6. So it can be 6 or -6.
4189
What is the common difference of an AP whose nth term is given by 5 - 3n?
Answer:
-3
The common difference d is the difference between consecutive terms: a_n - a_{n-1}. So, d = (5 - 3n) - (5 - 3(n-1)) = 5 - 3n - 5 + 3n - 3 = -3. Alternatively, in a linear nth term formula An + B, the coefficient of n is the common difference.
4190
Find the sum of the first 20 terms of the sequence 1, -2, 3, -4, 5, -6, ...
Answer:
-10
Group the terms in pairs: (1 - 2) + (3 - 4) + (5 - 6) + ... There are 20 terms, so there are 10 pairs. Each pair sums to -1. Total sum = 10 * (-1) = -10.