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: Assign variables.
Let $AB = p$, $BC = q$, $CD = r$, all positive lengths, with A, B, C, D lying on the line in that order, so $AC = p + q$.
The question is whether $p = q = r$ must hold.


Step 2: Check statement I on its own.
Statement I says $AC = 2CD$, so $p + q = 2r$.
Try $p = 2$, $q = 2$, $r = 2$: this fits since $4 = 2 \times 2$, and here $p = q = r$.
Try $p = 1$, $q = 3$, $r = 2$: this also fits since $4 = 2 \times 2$, but here $p \ne q$.
Two different sets of values satisfy the same equation and give opposite answers to the question, so statement I alone cannot settle it.


Step 3: Check statement II on its own.
Statement II says $AB = BC$, so $p = q$.
This says nothing about $r$. $r$ could equal $p$ or be any other length, so we still cannot decide if $p = q = r$.
Statement II alone also fails.


Step 4: Use both statements together.
From statement II, $p = q$, so $AC = p + q = 2p$.
From statement I, $AC = 2r$.
Setting these equal: $2p = 2r$, so $p = r$.
Combined with $p = q$, we now get $p = q = r$, which means $AB = BC = CD$ always holds once both statements are used.


Final Answer:
Neither statement alone fixes the answer, but together they force $AB = BC = CD$ every time.
\[ \boxed{\text{Option (3): both statements together are needed}} \]