Sets Practice Questions - NDA Mathematics Question Bank

Question 1 of 10 Authentic UPSC NDA Question

For the set of numbers

x, x, x + 2, x + 3, x + 10

where x is a natural number, which of the following is/are correct?

1. Mean > Mode
2. Median > Mean

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Question 2 of 10 Authentic UPSC NDA Question

Let A=(–3, –2, –1, 0, 1, 2, 3} and B= {0, 1, 4, 9). How many elements does the subset of A × B corresponding to the relation R = {(x, y): |x|< y} have, where x ∈ A and y ∈ B?

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Question 3 of 10 Authentic UPSC NDA Question

Set X contains 3n elements and set Y contains 2n elements, and they have n elements in common elements does (X - Y)×(Y - X) have?

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Question 4 of 10 Authentic UPSC NDA Question

In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

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Question 5 of 10 Authentic UPSC NDA Question

If A and B are non-empty subsets of a set, and A^c and B^c represent their complements, then which of the following is/are correct?

I. A - B = B^c - A^c

II. A - B^c = A^c - B

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Question 6 of 10 Authentic UPSC NDA Question

If A and B are two non-empty sets having 10 elements in common, then how many elements do A × B and B × A have in common?

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Question 7 of 10 Authentic UPSC NDA Question

Let P and Q be two non-void relations on a set A. Which of the following statements are correct?

I. P and Q are reflexive Þ P Q is reflexive.

II. P and Q are symmetric Þ P union Q is symmetric.

III. P and Q are transitive Þ P intersection Q is transitive.

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Question 8 of 10 Authentic UPSC NDA Question

How many 4-digit numbers are there having all digits as odd?

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Question 9 of 10 Authentic UPSC NDA Question

Consider the following statements:

(I) The set of all irrational numbers between √12 and √15 is an infinite set.

(II) The set of all odd integers less than 1,000 is a finite set.

Which of the statements given above is/are correct?

Step-by-Step Logic & Solution

Between any two irrational numbers, an infinite number of irrational numbers are there.

∴ The set of all irrational numbers between √12 and √15 is an infinite set.

Set of all odd integers less than 1,000 is {...–5, –3, 1, 3... 999} which is an infinite set.

Question 10 of 10 Authentic UPSC NDA Question

A set S contains (2n + 1) elements. If the number of subsets of S which contain atmost n elements is 1024, then what is thevalue of n?

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Solution

Option (d) is correct.

Explanation:

\[ |S| = 2n + 1 \]

The number of subsets of \(S\) containing at most \(n\) elements is given by:

\[ \sum_{k=0}^{n} \binom{2n+1}{k} = 1024 \]

Using the binomial coefficient property,

\[ \sum_{k=0}^{2n+1} \binom{2n+1}{k} = 2^{2n+1} \]

Thus,

\[ \sum_{k=n+1}^{2n+1} \binom{2n+1}{k} = 1024 \Rightarrow \frac{1}{2} \cdot 2^{2n+1} = 1024 \]

This represents half of the total subsets. Therefore,

\[ 2^{2n} = 1024 \]

Solving for \(2n\):

\[ 2^n = 32 \Rightarrow n = 5 \]

📌 Hints / Properties Used:

Binomial theorem.
Subset calculation via binomial coefficients.
Summation of binomial coefficients.

📊 Visual Diagram Suggestion:

A binomial coefficient table highlighting the values \( \binom{n}{k} \) for subsets up to \(n\) elements. This could illustrate the combinatorial aspects and how they sum to a specified value.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter 8 (Binomial Theorem)

Sets Questions

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