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
101
Find the sum of all natural numbers between 1 and 50 which are divisible by 4.
Answer:
312
The numbers are 4, 8, 12, ..., 48. This is an AP with a=4, d=4, l=48. Number of terms n = 48/4 = 12. Sum = (n/2)(a + l) = (12/2)(4 + 48) = 6 * 52 = 312.
102
What is the 15th term of the AP -10, -5, 0, 5, ...?
Answer:
60
Here, a = -10 and d = 5. The nth term is a_n = a + (n-1)d. For n=15, a_15 = -10 + (15-1)5 = -10 + 14 * 5 = -10 + 70 = 60.
103
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).
104
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.
105
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.
106
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.
107
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.
108
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.
109
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.
110
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.