CAT 2004 — QA Question 31
Answer the following question based on the information given below.
In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.
Let u = (log2 x)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has
Answer & solution
- A
no solution for x
exactly one solution for x
- C
exactly two distinct solutions for x
- D
exactly three distinct solutions for x
u = (log2x)2 – 6log2x + 12 ...(1)
xu = 256
Taking log on both sides,
ulog2x = log2256 = 8
Let log2x = m
∴ u × m = 8
Substituting in (1),
= m2 - 6m + 12
⇒ m3 – 6m2 + 12m – 8 = 0
⇒ (m – 2)3 = 0
⇒ m = 2
⇒ log2x = 2
⇒ x = 4
∴ The given equation has exactly one solution for x.
Hence, option (b).