If \( p^{x} = q^{y} = r^{z} \), where \( x, y \) and \( z \) are in GP, then consider the following statements:
I. \( p, q \) and \( r \) are in AP.
II. \( \ln p, \ln q \) and \( \ln r \) are in GP.
Which of the statements given above is/are correct?
If \( A \) and \( B \) are non-empty subsets of a set, and \( A^{c} \) and \( B^{c} \) represent their complements, then which of the following is/are correct?
I. \( A - B = B^{c} - A^{c} \)
II. \( A - B^{c} = A^{c} - B \)
Select the answer using the code given below.
Let \( n \) be a natural number. The number of consecutive zeros at the end of the expansion of \( n! \) is exactly 2. How many values of \( n \) are possible?
If \( (10 + \log_{10} x) \), \( (10 + \log_{10} y) \) and \( (10 + \log_{10} z) \) are in AP, then consider the following statements:
I. The GM of \( x \) and \( z \) is \( y^{2} \).
II. The AM of \( \log_{10} x \) and \( \log_{10} z \) is \( \log_{10} y \).
Which of the statements given above is/are correct?
If
\( \omega = \frac{1 + \sqrt{-3}}{2} \)
then what is the value of \( (1 + \omega^{10} - \omega^{35})^{100} - (1 - \omega^{25} + \omega^{38})^{50} \)?
