CSAT Solved Papers/ 2022/Q37

2022 CSAT — Q37

Quant Data sufficiency 2.5 marks Medium

Consider the Question and two Statements given below in respect of three cities $P, Q$ and $R$ in a State:

Question: How far is city $P$ from city $Q$ ?

Statement-1: City $Q$ is $18$ km from city $R$ .

Statement-2: City $P$ is $43$ km from city $R$ .

Which one of the following is correct in respect of the Question and the Statements?

A Statement-1 alone is sufficient to answer the Question
B Statement-2 alone is sufficient to answer the Question
C Both Statement-1 and Statement-2 are sufficient to answer the Question
D Both Statement-1 and Statement-2 are not sufficient to answer the Question Answer

Worked rationale

We want $PQ$ .

Statement-1 alone ( $QR = 18$ ): says nothing about $P$ . Insufficient.

Statement-2 alone ( $PR = 43$ ): says nothing about $Q$ . Insufficient.

Both together ( $QR = 18$ , $PR = 43$ ): the three cities need not be collinear. By the triangle inequality, $PQ$ can be anything in the range

|43 - 18| \le PQ \le 43 + 18,\qquad\text{i.e.}\quad 25 \le PQ \le 61,

so $PQ$ is not uniquely determined — e.g. collinear with $P,R,Q$ in a line gives $PQ = 61$ or $25$ , while a right angle at $R$ gives $PQ = \sqrt{43^2 + 18^2} \approx 46.6$ . The answer is not fixed even jointly.

Answer: (d) Both Statement-1 and Statement-2 are not sufficient.

Visual solution

The same solve, worked by hand — read it, then trace it.

Hand-drawn worked solution for UPSC 2022 CSAT Q37 — Data sufficiency — Tap the drawing to open it full size for the fine detail.

Why the other options miss

A

thought it was enough when it wasn’t: thinks one distance from $R$ locates $P$ relative to $Q$ .
B

thought it was enough when it wasn’t: the same overreach using $PR$ alone.
C

thought it was enough when it wasn’t: assumes the cities are collinear (so $PQ = 43 - 18 = 25$ or $43 + 18 = 61$ ), forcing a value the problem never guarantees.

Specialist insight

The engineered trap is assuming collinearity. Two side-lengths of a triangle fix the third only when the included angle is known; here $R$ is a shared vertex but the angle $\angle PRQ$ is free, so $PQ$ ranges over $[25, 61]$ . On distance DS, always ask “are these points forced onto a line?” — if not, two radii from a common point leave the third distance open.

The trap, in one line

$QR=18, PR=43$ leave $PQ \in [25, 61]$ (angle at $R$ free) — not unique $\Rightarrow$ (d).

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