Question

Each of the following questions is followed by two statements, I and II. Answer as follows:
Mark option (1) if the question can be answered using statement I alone, but not using statement II alone.
Mark option (2) if the question can be answered using statement II alone, but not using statement I alone.
Mark option (3) if the question can be answered using both statements I and II together, but not by using either statement alone.
Mark option (4) if the question cannot be answered even using both statements I and II together.

A, B, C and D are points on a straight line. Is AB = BC = CD?
I. AC = 2 CD
II. AB = BC

• A. The question can be answered using statement I alone, but not using statement II alone.
• B. The question can be answered using statement II alone, but not using statement I alone.
• C. The question can be answered using both statements I and II together, but not by using either statement alone.
• D. The question cannot be answered even using both statements I and II together.

Hint:

Write AC = AB + BC and turn each statement into an equation in three unknowns; check if one equation alone forces all three lengths equal, or if you need both together.

Answer 1

The Correct Option is C

Step 1: Set up the line.
A, B, C, D lie on a straight line in that order, so the whole segment AD splits into three parts: AB, BC, and CD.
Also \( AC = AB + BC \), since AC is made up of AB and BC together.
We need to check if AB = BC = CD.

Step 2: Test statement I alone.
Statement I says \( AC = 2\,CD \), that is, \( AB + BC = 2\,CD \).
This is one equation with three unknowns (AB, BC, CD), so many different combinations of AB, BC, CD can satisfy it without AB, BC, CD being equal.
For example, AB = 1, BC = 3, CD = 2 gives AC = 4 = 2(2), which fits, but AB, BC, CD are not all equal here.
So statement I alone is not enough.

Step 3: Test statement II alone.
Statement II says \( AB = BC \), but it says nothing about CD.
CD could be any length, so we cannot conclude AB = BC = CD from this alone.
So statement II alone is not enough either.

Step 4: Combine statement I and statement II.
From statement II, \( AB = BC \), so \( AC = AB + BC = 2\,AB \).
From statement I, \( AC = 2\,CD \).
Equating the two expressions for AC: \( 2\,AB = 2\,CD \), so \( AB = CD \).
Since we already have \( AB = BC \) and now \( AB = CD \), it follows that \( AB = BC = CD \).

Final Answer:
Both statements are needed together to confirm AB = BC = CD; neither alone is enough.
\[ \boxed{\text{Option (3)}} \]