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
191
The sum of an AP is given by S_n = 4n^2 - 3n. What is the common difference?
Answer:
8
First, find the first two terms. S_1 = a_1 = 4(1)^2 - 3(1) = 1. Next, S_2 = a_1 + a_2 = 4(2)^2 - 3(2) = 16 - 6 = 10. So, a_2 = S_2 - S_1 = 10 - 1 = 9. The common difference d = a_2 - a_1 = 9 - 1 = 8. (Shortcut: for S_n = An^2 + Bn, the common difference is always 2A. Here 2*4 = 8).
192
Find the sum of the series 2 + 5 + 8 + ... to 20 terms.
Answer:
610
This is an AP with a = 2, d = 3, and n = 20. The sum S_20 = (n/2)[2a + (n-1)d] = (20/2)[2(2) + (19)3] = 10[4 + 57] = 10[61] = 610.
193
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.
194
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.
195
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.
196
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).
197
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.
198
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.
199
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.
200
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.