Basic Mathematics and Logarithm (Questions)

Single Correct Type Questions

1. Let a, b, c and d be positive real numbers such that a + b + c + d = 11. If the maximum value of a⁵b³c²d is 3750_β, then the value of β is:
   (a) 90   (b) 110   (c) 55   (d) 108
2. Let A = {x ∈ R : [x+3] + [x+4] < 3}, B = {x ∈ R : ...}, where [t] denotes greatest integer function. Then:
   (a) A ∩ B = ∅   (b) A = B   (c) B ⊂ C, A + B   (d) A ⊂ B, A + B
3. The integer k for which inequality x² – 2(3k – 1)x + 8k² – 7 > 0 is valid for every x ∈ R:
   (a) 3   (b) 2   (c) 4   (d) 0
4. If {p} denotes fractional part of p, then 3200/8 is equal to:
   (a) 1/8   (b) 7/8   (c) 3/8   (d) 5/8
5. The number of real roots of equation 5 + |2x – 1| = 2x(2x – 2):
   (a) 3   (b) 2   (c) 4   (d) 1

Integer Type Questions

10. Let a, b, c be distinct positive reals such that (2a)log_ea = (bc)log_eb and blog_e2 = alog_ec. Then 6a + 5bc = ?
11. If sum of all roots of e^2x – 11e^x – 45e + 81 = 0 is log p, then p = ?
12. Number of solutions of log(x – 1) = log₂(x – 3).