Let \( p, q \) and 3 be respectively the first, third and fifth terms of an AP. Let \( d \) be the common difference. If the product \( (pq) \) is minimum, then what is the value of \( d \)?
Consider the following statements in respect of the roots of the equation \( x^{3} - 8 = 0 \):
1. The roots are non-collinear.
2. The roots lie on a circle of unit radius.
Which of the above statements is/are correct?
Consider the following statements in respect of sets:
1. The union over intersection of sets is distributive.
2. The complement of union of two sets is equal to intersection of their complements.
3. If the difference of two sets is equal to empty set, then the two sets must be equal.
Which of the above statements are correct?
Consider three sets \( X, Y \) and \( Z \) having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let \( S = (X - Y) \cup Z \). How many proper subsets does \( S \) have?
Consider the following statements in respect of relations and functions:
1. All relations are functions but all functions are not relations.
2. A relation from \( A \) to \( B \) is a subset of Cartesian product \( A \times B \).
3. A relation in \( A \) is a subset of Cartesian product \( A \times A \).
Which of the above statements are correct?
