2018-Q1) If \(x\) is a positive quantity such that \(2^x = 3^{\log_5 2}\), then \(x\) is equal to:

• (a) \(\log_5 8\)
• (b) 1 + \(\log_3\left(\frac{5}{3}\right)\)
• (c) \(\log_5 9\)
• (d) 1 + \(\log_5\left(\frac{3}{5}\right)\)

2018-Q2) Let \(f(x) = \min(2x^2, 52 - 5x)\) where \(x\) is any positive real number. Then the maximum possible value of \(f(x)\) is:

2018-Q3) If \(\log_{12} 81 = p\), then \(3\left(\frac{4 - p}{4 + p}\right)\) is equal to:

• (a) \(\log_4 16\)
• (b) \(\log_6 16\)
• (c) \(\log_2 8\)
• (d) \(\log_6 8\)

2018-Q4) If \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers \(x\), and \(f(11) = 91\), \(f(15) = 617\), then \(f(10)\) equals:

2018-Q5) Let \(x, y, z\) be three positive real numbers in a geometric progression such that \(x < y < z\). If \(5x, 16y, 12z\) are in an arithmetic progression, then the common ratio of the geometric progression is:

• (a) \(\frac{5}{6}\)
• (b) \(\frac{1}{6}\)
• (c) \(\frac{5}{2}\)
• (d) \(\frac{2}{3}\)

2018-Q6) Given that \(x^{2018}y^{2017} = \frac{1}{2}\), and \(x^{2016}y^{2019} = 8\), then value of \(x^2 + y^3\) is:

• (a) \(\frac{31}{4}\)
• (b) \(\frac{35}{4}\)
• (c) \(\frac{37}{4}\)
• (d) \(\frac{33}{4}\)

2018-Q7) If \(\log_2(5 + \log_3 a) = 3\) and \(\log_5(4a + 12 + \log_2 b) = 3\), then \(a + b\) is equal to:

• (a) 59
• (b) 40
• (c) 32
• (d) 67

2018-Q8) The number of integers \(x\) such that \(0.25 \leq 2^x \leq 200\) and \(2^x + 2\) is perfectly divisible by either 3 or 4, is:

• (a) 3
• (b) 4
• (c) 5
• (d) 6

2018-Q9) Let \( f(x) = \max(5x,\ 52 - 2x^2) \), where \( x \) is any positive real number. Then the minimum possible value of \( f(x) \) is:

2018-Q10) The smallest integer \( n \) such that \[ n^3 - 11n^2 + 32n - 28 > 0 \] is:

2018-Q11) If \( N \) and \( x \) are positive integers such that \[ N^N = 2^{160} \quad \text{and} \quad N^2 + 2^N \] is an integral multiple of \( 2^x \), then the largest possible \( x \) is:

2018-Q12) The arithmetic mean of \( x, y \) and \( z \) is 80, and that of \( x, y, z, u \) and \( v \) is 75, where \( u = \frac{x+y}{2} \) and \( v = \frac{y+z}{2} \). If \( x \geq z \), then the minimum possible value of \( x \) is:

2018-Q13) If \( a \) and \( b \) are integers such that \[ 2x^2 - ax + 2 \geq 0 \quad \text{and} \quad x^2 - bx + 8 \geq 0 \] for all real numbers \( x \), then the largest possible value of \( 2a - 6b \) is:

2018-Q14) Let \( t_1, t_2, \ldots \) be real numbers such that \[ t_1 + t_2 + \cdots + t_n = 2n^2 + 9n + 13 \] for every positive integer \( n \geq 2 \). If \( t_k = 103 \), then \( k \) equals:

2018-Q15) Let \( a_1, a_2, \ldots, a_{52} \) be positive integers such that \( a_1 < a_2 < \cdots < a_{52} \). Suppose their arithmetic mean is one less than the arithmetic mean of \( a_2, a_3, \ldots, a_{52} \). If \( a_{52} = 100 \), then the largest possible value of \( a_1 \) is:

1. 48
2. 20
3. 23
4. 45

2018-Q16) If \( A = \{6^{2n} - 35n - 1\} \), where \( n = 1,2,3,\ldots \) and \( B = \{35(n-1)\} \), where \( n = 1,2,3,\ldots \), then which of the following is true?

1. Every member of A is in B and at least one member of B is not in A
2. Neither every member of A is in B nor every member of B is in A
3. Every member of B is in A
4. At least one member of A is not in B

2018-Q17) The smallest integer \( n \) for which \( 4^n > 17^{19} \) holds is closest to:

1. 37
2. 35
3. 33
4. 39

2018-Q18) \( \frac{1}{\log_2 100} + \frac{1}{\log_4 100} + \frac{1}{\log_5 100} + \frac{1}{\log_{10} 100} + \frac{1}{\log_{20} 100} - \frac{1}{\log_{25} 100} + \frac{1}{\log_{50} 100} = ? \)

1. \( \frac{1}{2} \)
2. 10
3. 0
4. -4

2018-Q19) If \( p^3 = q^4 = r^5 = s^6 \), then the value of \( \log_s(pqr) \) is equal to:

1. \( \frac{47}{10} \)
2. \( \frac{24}{5} \)
3. \( \frac{16}{5} \)
4. 1