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
4301
Which term of the GP 5, 10, 20, ... is 320?
Answer:
7th
We use the nth term formula a_n = a * r^(n-1). Here, a_n = 320, a = 5, and r = 2. So, 320 = 5 * 2^(n-1). Dividing by 5 gives 64 = 2^(n-1). Since 64 is 2^6, we have 6 = n - 1, which means n = 7. Thus, 320 is the 7th term.
4302
Find the common ratio of the GP where the first term is 3 and the 4th term is 81.
Answer:
3
Using the formula a_n = a * r^(n-1), we have a_4 = a * r^3. Given a = 3 and a_4 = 81. So, 81 = 3 * r^3. Dividing by 3 gives r^3 = 27. Taking the cube root, we find the common ratio r = 3.
4303
What is the 8th term of the Geometric Progression (GP) 2, 4, 8, 16, ...?
Answer:
256
The formula for the nth term of a GP is a_n = a * r^(n-1). Here, the first term a = 2, and the common ratio r = 4/2 = 2. We need to find the 8th term, so n = 8. a_8 = 2 * 2^(8-1) = 2 * 2^7 = 2 * 128 = 256.
4304
Find the sum of the first 20 positive odd integers.
Answer:
400
The first 20 positive odd integers form an AP: 1, 3, 5, ..., 39. The sum of the first n odd integers is simply n^2. Here, n = 20. Therefore, the sum is 20^2 = 400. Alternatively, using the sum formula: S_20 = (20/2) * [2(1) + (20 - 1)2] = 10 * [2 + 38] = 10 * 40 = 400.
4305
The 5th term of an AP is 11 and the 9th term is 23. What is the first term?
Answer:
-1
We have two equations: a + 4d = 11 and a + 8d = 23. Subtracting the first from the second gives 4d = 12, so d = 3. Substituting d back into the first equation: a + 4(3) = 11, which gives a + 12 = 11, so a = -1.
4306
If 4, x, 16 are in Arithmetic Progression, find the value of x.
Answer:
10
If a, b, c are in AP, then the middle term is the arithmetic mean of the other two: b = (a + c) / 2. Here, x = (4 + 16) / 2 = 20 / 2 = 10.
4307
For the AP where S_n = 3n^2 + 5n, find the second term.
Answer:
14
The nth term can be found using a_n = S_n - S_{n-1}. To find the second term, a_2 = S_2 - S_1. We know S_1 = 8. S_2 = 3(2)^2 + 5(2) = 12 + 10 = 22. So, a_2 = 22 - 8 = 14.
4308
If the sum of the first n terms of an AP is given by S_n = 3n^2 + 5n, what is the first term?
Answer:
8
The first term of an AP is simply the sum of the first 1 term. So, a_1 = S_1. Substitute n = 1 into the given formula: S_1 = 3(1)^2 + 5(1) = 3 + 5 = 8. The first term is 8.
4309
The sum of the first 15 terms of an AP whose first term is 5 and the common difference is 2 is:
Answer:
285
The formula for the sum of the first n terms of an AP is S_n = (n/2) * [2a + (n - 1)d]. Given n = 15, a = 5, and d = 2. S_15 = (15/2) * [2(5) + (15 - 1) * 2] = (15/2) * [10 + 14 * 2] = (15/2) * [10 + 28] = (15/2) * 38 = 15 * 19 = 285.
4310
What is the common difference of an AP in which a_18 - a_14 = 32?
Answer:
8
We know that a_n = a + (n - 1)d. So, a_18 = a + 17d and a_14 = a + 13d. The given equation is (a + 17d) - (a + 13d) = 32. This simplifies to 4d = 32, which means the common difference d = 8.