Consider the following statements in respect of the expansion of \( (x + y)^{10} \):
1. Among all the coefficients of the terms, the coefficient of the 6th term has the highest value
2. The coefficient of the 3rd term is equal to coefficient of the 9th term
Which of the above statements is/are correct?
Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2} + px + q = 0 \). If \( \alpha^{3} \) and \( \beta^{3} \) are the roots of the equation \( x^{2} + mx + n = 0 \), then what is the value of \( m + n \)?
Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2} - ax - bx + ab - c = 0 \). What is the quadratic equation whose roots are \( a \) and \( b \)?
A.
\( x^{2} - \alpha x - \beta x + \alpha\beta + c = 0 \)
B.
\( x^{2} - \alpha x - \beta x + \alpha\beta - c = 0 \)
C.
\( x^{2} + \alpha x + \beta x + \alpha\beta + c = 0 \)
D.
\( x^{2} + \alpha x + \beta x + \alpha\beta - c = 0 \)
Let \( \alpha \) and \( \beta \) \( (\alpha > \beta) \) be the roots of the equation \( x^{2} - 8x + q = 0 \). If \( \alpha^{2} - \beta^{2} = 16 \), then what is the value of \( q \)?
