Question

If \( abcd \neq 0 \) and \( 0 < c < b < a < 1 \), is \( \frac{a^4bc}{d^2} < 1 \)? Statement (I): \( a = \sqrt{d} \)
Statement (II): \( d > 0 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

For numbers between 0 and 1, multiplication decreases the value, while division by a small number can increase it significantly.

Answer 1

The Correct Option is A

Concept: Numbers lying between 0 and 1 become smaller when multiplied together. We need to determine whether \[ \frac{a^4bc}{d^2}<1 \] must always be true.

Step 1: Analyze Statement (I). Given \[ a=\sqrt d \] Therefore, \[ d=a^2 \] Substituting into the expression, \[ \frac{a^4bc}{d^2} = \frac{a^4bc}{(a^2)^2} = \frac{a^4bc}{a^4} = bc \] Since \[ 0<c<b<a<1 \] both \(b\) and \(c\) lie between 0 and 1. Hence \[ bc<1 \] Therefore the answer to the question is always YES. Statement (I) alone is sufficient.

Step 2: Analyze Statement (II). Statement (II) merely states \[ d>0 \] No relationship between \(d\) and the other variables is given. The expression may be less than 1 for some values and greater than 1 for others. Therefore Statement (II) alone is insufficient. Final conclusion. \[ \boxed{\text{Statement (I) alone is sufficient}} \]