Let \( p, q \) \( (p > q) \) be the roots of the quadratic equation \( x^{2} + bx + c = 0 \) where \( c > 0 \). If \( p^{2} + q^{2} - 11pq = 0 \), then what is \( p - q \) equal to?
Let \( A = \{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\} \) and let \( f: A \rightarrow \mathbb{N} \) be defined by \( f(x) = \) the highest prime factor of \( x \). How many elements are there in the range of \( f \)?
Let \( R \) be a relation from \( \mathbb{N} \) to \( \mathbb{N} \) defined by \( R = \{(x, y): x, y \in \mathbb{N} \) and \( x = y^{2}\} \). Which of the following are not correct?
1. \( (x, x) \in R \) for all \( x \in \mathbb{N} \)
2. \( (x, y) \in R \Rightarrow (y, x) \in R \)
3. \( (x, y) \in R \) and \( (y, z) \in R \Rightarrow (x, z) \in R \)
Select the correct answer using the code given below:
Consider the following:
1. \( A \cap B = A \cap C \Rightarrow B = C \)
2. \( A \cup B = A \cup C \Rightarrow B = C \)
Which of the above is/are correct?
Directions (Qn. 16-18): Consider the following for the next three (03) items that follow:
Let \( z = \frac{1 + i \sin \theta}{1 - i \sin \theta} \), where \( i = \sqrt{-1} \).
Directions (Qn. 19-21): Consider the following for the next three (03) items that follow:
Let \( P \) be the sum of first \( n \) positive terms of an increasing arithmetic progression A. Let \( Q \) be the sum of first \( n \) positive terms of another increasing arithmetic progression B. Let \( P : Q = (5n + 4) : (9n + 6) \).
What is the ratio of the first term of A to that of B?
