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
4231
Determine the sum of the first 5 terms of the geometric sequence 1, -2, 4, -8, ...
Answer:
11
This is a GP with a=1 and r=-2. The 5 terms are 1, -2, 4, -8, 16. Their sum is 1 - 2 + 4 - 8 + 16 = 11. Alternatively using the formula: S_5 = 1( (-2)^5 - 1) / (-2 - 1) = (-32 - 1) / -3 = -33 / -3 = 11.
4232
In a GP, if the 3rd term is 24 and the 6th term is 192, find the common ratio.
Answer:
2
We have ar^2 = 24 and ar^5 = 192. Dividing the second equation by the first gives (ar^5)/(ar^2) = 192/24. This simplifies to r^3 = 8. Taking the cube root gives r = 2.
4233
If the sum of the first p terms of an AP is ap^2 + bp, find its common difference.
Answer:
2a
S_1 = a+b (this is the first term). S_2 = 4a + 2b (sum of first two terms). The second term is S_2 - S_1 = (4a+2b) - (a+b) = 3a+b. The common difference d is the second term minus the first term: (3a+b) - (a+b) = 2a.
4234
The 7th term of an AP is 20 and its 13th term is 32. Find the 20th term.
Answer:
46
We have a+6d = 20 and a+12d = 32. Subtracting the first from the second gives 6d = 12, so d = 2. Then a+12=20, so a=8. The 20th term a_20 = a + 19d = 8 + 19(2) = 8 + 38 = 46.
4235
What is the 100th term of the AP 1, 1, 1, 1, ...?
Answer:
1
This is an AP with a constant difference of 0. Since every term is 1, the 100th term is also 1.
4236
Which of the following forms an AP?
Answer:
5, 10, 15, 20
An AP must have a constant difference between terms. In 5, 10, 15, 20, the difference is always 5. The others are GP, sequence of squares, and HP respectively.
4237
The sum of the infinite series 1/3 + 1/9 + 1/27 + ... is:
Answer:
1/2
This is an infinite GP with a = 1/3 and r = 1/3. The sum S_inf = a / (1 - r) = (1/3) / (1 - 1/3) = (1/3) / (2/3) = 1/2.
4238
A sum of Rs. 1000 is to be used to give 5 cash prizes to students. If each prize is Rs. 20 less than its preceding prize, find the value of the highest prize.
Answer:
240
The prizes form an AP with n=5 and d=-20. The sum is S_5 = 1000. Formula: S_n = (n/2)[2a + (n-1)d]. 1000 = (5/2)[2a + 4(-20)] = 2.5[2a - 80]. Divide by 2.5: 400 = 2a - 80. So 2a = 480, yielding a = 240. The highest prize is Rs. 240.
4239
Find the sum of the first 10 terms of the GP 2, 4, 8, 16, ...
Answer:
2046
Here a = 2, r = 2, and n = 10. The sum S_10 = a(r^10 - 1) / (r - 1) = 2(2^10 - 1) / (2 - 1) = 2(1024 - 1) = 2(1023) = 2046.
4240
If the nth term of a sequence is 3 * 2^(n-1), what kind of sequence is it?
Answer:
GP
The formula represents an exponential function of n, which is the defining characteristic of a Geometric Progression (GP) where a=3 and r=2.