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
4291
Find the sum of the first 100 natural numbers.
Answer:
5050
The sum of the first n natural numbers is given by the formula n(n+1)/2. For n=100, the sum is 100(101)/2 = 50 * 101 = 5050.
4292
If the sum of three numbers in an AP is 15 and their product is 80, what is the largest number?
Answer:
8
Let the three numbers in AP be a-d, a, and a+d. Their sum is 3a = 15, so a = 5. Their product is (5-d)(5)(5+d) = 80, which gives 25 - d^2 = 16, so d^2 = 9 and d = 3 (or -3). The numbers are 2, 5, and 8. The largest number is 8.
4293
If a, b, c are in AP, then what can be said about 2^a, 2^b, 2^c?
Answer:
They are in GP
If a, b, c are in AP, then b - a = c - b = d (a constant difference). For the terms 2^a, 2^b, 2^c, the ratio of consecutive terms is (2^b)/(2^a) = 2^(b-a) = 2^d and (2^c)/(2^b) = 2^(c-b) = 2^d. Since the ratio is constant, the terms form a Geometric Progression (GP).
4294
The 10th term of the harmonic progression (HP) 1/2, 1/5, 1/8, 1/11, ... is:
Answer:
1/29
A sequence is a Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression. The reciprocals are 2, 5, 8, 11, ... which is an AP with a = 2 and d = 3. The 10th term of this AP is a_10 = 2 + (10 - 1) * 3 = 2 + 27 = 29. Therefore, the 10th term of the HP is the reciprocal of this, which is 1/29.
4295
What is the geometric mean of 4 and 16?
Answer:
8
The geometric mean of two numbers a and b is given by sqrt(a * b). Therefore, the geometric mean of 4 and 16 is sqrt(4 * 16) = sqrt(64) = 8.
4296
If the third term of a GP is 12 and the sixth term is 96, find the first term.
Answer:
3
We have a_3 = a * r^2 = 12 and a_6 = a * r^5 = 96. Dividing the second equation by the first: (a * r^5) / (a * r^2) = 96 / 12, which simplifies to r^3 = 8. So, r = 2. Substitute r back into the first equation: a * 2^2 = 12 => 4a = 12 => a = 3.
4297
Find the sum of the infinite geometric series: 9 - 3 + 1 - 1/3 + ...
Answer:
6.75
This is an infinite GP with first term a = 9 and common ratio r = -3/9 = -1/3. The sum to infinity is S_inf = a / (1 - r) = 9 / (1 - (-1/3)) = 9 / (1 + 1/3) = 9 / (4/3) = 27 / 4 = 6.75.
4298
The sum to infinity of the geometric series 1 + 1/2 + 1/4 + 1/8 + ... is:
Answer:
2
For an infinite GP where the absolute value of the common ratio |r| < 1, the sum to infinity is given by S_inf = a / (1 - r). Here, a = 1 and r = 1/2. Therefore, S_inf = 1 / (1 - 1/2) = 1 / (1/2) = 2.
4299
If 3, y, 27 are in Geometric Progression, what is the value of y?
Answer:
9
If a, b, c are in GP, the middle term is the geometric mean of the other two: b^2 = ac. So, y^2 = 3 * 27 = 81. Taking the square root gives y = 9 (assuming a positive sequence) or y = -9. The option given is 9.
4300
Find the sum of the first 6 terms of the GP: 1, 3, 9, 27, ...
Answer:
364
The sum of the first n terms of a GP where r > 1 is S_n = a(r^n - 1) / (r - 1). Here a = 1, r = 3, and n = 6. S_6 = 1 * (3^6 - 1) / (3 - 1) = (729 - 1) / 2 = 728 / 2 = 364.