A mapping \( f: A \rightarrow B \) defined as \( f(x) = \frac{2x + 3}{3x + 5} \), \( x \in A \). If \( f \) is to be onto, then what are \( A \) and \( B \) equal to?
A.
\( A = \mathbb{R} \setminus \left\{-\frac{5}{3}\right\} \) and \( B = \mathbb{R} \setminus \left\{-\frac{2}{3}\right\} \)
B.
\( A = \mathbb{R} \) and \( B = \mathbb{R} \setminus \left\{-\frac{5}{3}\right\} \)
C.
\( A = \mathbb{R} \setminus \left\{-\frac{5}{3}\right\} \) and \( B = \mathbb{R} \setminus \{0\} \)
D.
\( A = \mathbb{R} \setminus \left\{-\frac{5}{3}\right\} \) and \( B = \mathbb{R} \setminus \left\{\frac{2}{3}\right\} \)
D. \( A = \mathbb{R} \setminus \left\{-\frac{5}{3}\right\} \) and \( B = \mathbb{R} \setminus \left\{\frac{2}{3}\right\} \)
23.
\( \alpha \) and \( \beta \) are distinct real roots of the quadratic equation \( x^{2} + ax + b = 0 \). Which of the following statements is/are sufficient to find \( \alpha \)?
1. \( \alpha + \beta = 0, \alpha^{2} + \beta^{2} = 2 \)
2. \( \alpha\beta^{2} = -1, a = 0 \)
Select the correct answer using the code given below:
Consider the following statements for a fixed natural number \( n \):
1. \( C(n, r) \) is greatest if \( n = 2r \)
2. \( C(n, r) \) is greatest if \( n = 2r - 1 \) and \( n = 2r + 1 \)
Which of the statements given above is/are correct?
Let \( x \) be the number of permutations of the word ‘PERMUTATIONS’ and \( y \) be the number of permutations of the word ‘COMBINATIONS’. Which one of the following is correct?
