Below are some sample questions along with detailed explanations to help you understand the concepts better. To get complete MTH101, PHY101, CSC Past-Question
Questions and Explanations
1. Universal Set
A universal set \( U = \{1, 2, 3, \dots, 10\} \). \( A \) and \( B \) are subsets of \( U \) such that:
\( A = \{1, 3, 4, 10\} \), and \( B = \{2, 5, 6, 7\} \). Find \( A’ \cap B \).
A. \( \{1, 2, 3, \dots, 10\} \)
B. \( \{1, 2, 3, 4, 10\} \)
C. \( \{2, 5, 6, 7\} \)
D. \( \{2, 3, 4, 10\} \)
Answer: C
Explanation: The complement of \( A \) (\( A’ \)) is the set of all elements in \( U \) that are not in \( A \). Thus, \( A’ = \{2, 5, 6, 7, 8, 9\} \). The intersection \( A’ \cap B \) gives \( \{2, 5, 6, 7\} \).
2. Power Set
Which of the following is a member of the power set of \( \{1, 2, 3, 4\} \)?
A. \( \{1, 2, 3, 4, 10\} \)
B. \( \{16\} \)
C. \( \{1, 4\} \)
D. \( \{1, 3, 5\} \)
Answer: C
Explanation: The power set of a set \( S \) is the set of all subsets of \( S \). Since \( \{1, 4\} \) is a valid subset of \( \{1, 2, 3, 4\} \), it is a member of the power set. The other options are either not subsets or include elements outside \( S \).
3. Real Numbers
Which of the following is NOT a real number?
A. \( 23 \)
B. \( 2.12 \)
C. \( 2^3 \)
D. \( \sqrt{-4} \)
Answer: D
Explanation: Real numbers include all rational and irrational numbers. However, \( \sqrt{-4} \) is not a real number because it results in an imaginary number \( 2i \).
4. Sum of Odd Integers
Find the sum of the first 50 odd integers.
A. \( 200 \)
B. \( 2500 \)
C. \( 130 \)
D. \( 50 \)
Answer: B
Explanation: The sum of the first \( n \) odd integers is given by \( n^2 \). For \( n = 50 \), the sum is \( 50^2 = 2500 \).
5. Complex Numbers
Given \( i = \sqrt{-1} \), evaluate \( i^{11} \).
A. \( -i \)
B. \( -1 \)
C. \( 1 \)
D. \( i^2 \)
Answer: A
Explanation: Powers of \( i \) repeat cyclically: \( i, -1, -i, 1 \). For \( i^{11} \), divide 11 by 4 (remainder = 3). Thus, \( i^{11} = i^3 = -i \).
