BE Entrance Exam Model Questions and Solutions
Mathematics - Model Questions and Solutions
Question 1: Relation in Integers
The relation \( R \) defined by \( x + 5y = 10 \) on the set of integers \( \mathbb{Z} \) is:
- A) Symmetric
- B) Non-symmetric
- C) Anti-symmetric
- D) Equivalence
Answer: B) Non-symmetric
Solution: A relation is symmetric if \( (a, b) \) implies \( (b, a) \) for all elements in the set. The given relation \( x + 5y = 10 \) does not satisfy symmetry, as switching \( x \) and \( y \) will not necessarily satisfy the equation.
Question 2: Set Theory
Which of the following statements is true?
- A) \( \Phi = \{0\} \)
- B) \( \Phi \) is a subset of \( \{0\} \)
- C) \( \Phi = \{ \Phi, \{0\} \} \)
- D) None of the above
Answer: B) \( \Phi \) is a subset of \( \{0\} \)
Solution: The empty set \( \Phi \) does not contain any elements, not even 0. So, it cannot be equal to \( \{0\} \). The correct statement is that \( \Phi \) is a subset of every set, including \( \{0\} \).
Question 3: Domain of a Function
The domain of the function \( f(x) = \frac{1}{\sqrt{x^2 - 3x + 2}} \) is:
- A) \( (0, 1) \)
- B) \( (-\infty, 1) \)
- C) \( (0, 1) \cup (2, \infty) \)
- D) \( (2, \infty) \)
Answer: C) \( (0, 1) \cup (2, \infty) \)
Solution: The expression under the square root must be positive for the function to be defined. The quadratic \( x^2 - 3x + 2 \) factors as \( (x - 1)(x - 2) \), so the expression is non-negative when \( x \in (-\infty, 1) \cup (2, \infty) \).
Question 4: Set Union
Set \( A \) has 3 elements and set \( B \) has 6 elements. Then the minimum number of elements in \( A \cup B \) is:
- A) 3
- B) 6
- C) 9
- D) 0
Answer: B) 6
Solution: The minimum number of elements in \( A \cup B \) occurs when all elements of \( A \) are in \( B \), so the minimum is the size of the larger set.
Question 5: Handshake Problem
There are 15 persons in a party and each person shakes hands with another. The total number of handshakes is:
- A) 205
- B) 105
- C) 120
- D) 15
Answer: B) 105
Solution: The number of handshakes is given by \( \frac{n(n-1)}{2} \), where \( n = 15 \). Hence, the number of handshakes is \( \frac{15(14)}{2} = 105 \).