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
4241
The sum of the first n natural numbers is 210. Find the value of n.
Answer:
20
The sum formula is n(n+1)/2 = 210. This gives n(n+1) = 420. The two consecutive numbers that multiply to 420 are 20 and 21. Since n must be a positive integer, n = 20.
4242
What is the common ratio of the GP 5, -10, 20, -40, ...?
Answer:
-2
The common ratio is found by dividing any term by its preceding term. For example, -10 / 5 = -2. The ratio alternates the sign of the sequence, hence it is -2.
4243
Find the next term of the sequence 1, 4, 9, 16, 25, ...
Answer:
36
The given sequence represents the squares of the natural numbers (1^2, 2^2, 3^2, 4^2, 5^2). The next term in the sequence is the square of 6, which is 6^2 = 36.
4244
If the 3rd term of an AP is 5 and the 7th term is 13, find the common difference.
Answer:
2
a + 2d = 5 and a + 6d = 13. Subtracting the first equation from the second gives 4d = 8, so d = 2.
4245
How many terms of the AP 24, 21, 18, ... must be taken so that their sum is 78?
Answer:
4 or 13
We use S_n = (n/2)[2a + (n-1)d]. Here S_n = 78, a = 24, d = -3. So, 78 = (n/2)[48 + (n-1)(-3)] = (n/2)[51 - 3n]. Multiply by 2: 156 = 51n - 3n^2. Divide by 3: 52 = 17n - n^2, which rearranges to n^2 - 17n + 52 = 0. Factoring gives (n-4)(n-13) = 0. Both n=4 and n=13 yield the sum 78 because the terms from the 5th to the 13th add up to zero.
4246
Find the 7th term of the sequence whose nth term is given by a_n = (-1)^(n-1) * n^2.
Answer:
49
To find the 7th term, substitute n = 7 into the formula: a_7 = (-1)^(7-1) * 7^2 = (-1)^6 * 49. Since an even power of -1 is positive 1, the result is 1 * 49 = 49.
4247
Find the sum of the first 20 terms of the series: 1, 4, 7, 10, ...
Answer:
590
This is an AP with a=1 and d=3. The sum is S_20 = (20/2) * [2(1) + (20-1)3] = 10 * [2 + 19(3)] = 10 * [2 + 57] = 10 * 59 = 590.
4248
Which of the following is not true for any two distinct positive real numbers a and b?
Answer:
AM < HM
For any two distinct positive real numbers, the inequality AM > GM > HM holds true. Therefore, the statement AM < HM is definitively false. Also, the relationship AM * HM = GM^2 is a fundamental identity.
4249
If the AM and GM of roots of a quadratic equation are 8 and 5 respectively, then find the quadratic equation.
Answer:
x^2 - 16x + 25 = 0
Let the roots be α and β. AM = (α+β)/2 = 8, so sum of roots (α+β) = 16. GM = sqrt(αβ) = 5, so product of roots (αβ) = 25. The quadratic equation is x^2 - (sum)x + (product) = 0, which gives x^2 - 16x + 25 = 0.
4250
What is the harmonic mean of 4 and 6?
Answer:
4.8
The formula for the harmonic mean of a and b is HM = 2ab / (a+b). Here, a=4 and b=6. HM = (2 * 4 * 6) / (4 + 6) = 48 / 10 = 4.8.