CAT 2023 Slot 2 — DILR Question 1

Mixed PracticeEasy

Passage / Data

Answer the following questions based on the information given below:

Odsville has five firms – Alfloo, Bzygoo, Czechy, Drjbna and Elavalaki. Each of these firms was founded in some year and also closed down a few years later.

Each firm raised Rs. 1 crore in its first and last year of existence. The amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down. No firm raised the same amount of money in two consecutive years. Each annual increase and decrease was either by Rs. 1 crore or by Rs. 2 crores.

The table below provides partial information about the five firms.

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For which firm(s) can the amounts raised by them be concluded with certainty in each year?

Answer & solution

A
Only Czechy
B
Only Bzygoo and Czechy and Drjbna
C
Only Drjbna
Only Czechy and Drjbna

Solution

Easy

This is a "build every possible year-by-year sequence" set. The five rules pin each firm to just one or two number patterns, and the same master grid answers all five questions. We build it once here.

The given table (partial information):

Firm	First year	Last year	Total raised (Rs. cr)
Alfloo	2009	2016	21
Bzygoo	2012	2015	—
Czechy	2013	—	9
Drjbna	2011	2015	10
Elavalaki	2010	—	13

The rules in one line: each firm raises $1$ in its first and last year; the yearly amount strictly rises to a single peak, then strictly falls; every step is $+1,+2$ (up) or $-1,-2$ (down); no two consecutive years are equal. So each firm's sequence is a "mountain" $1 \to \dots \to \text{peak} \to \dots \to 1$ whose terms sum to the firm's total.

Alfloo — 2009 to 2016 is $8$ years, total $21$ . We need $8$ numbers, first and last $=1$ , summing to $21$ , rising then falling in $\pm1/\pm2$ steps. Only two mountains work:

\begin{aligned} &\text{Case 1: } 1,2,3,4,5,3,2,1\\ &\text{Case 2: } 1,2,3,5,4,3,2,1 \end{aligned}

(Both sum to $21$ ; the difference is whether the peak $5$ sits in 2013 or 2012.)

Bzygoo — 2012 to 2015 is $4$ years (total not given). Endpoints are $1$ , middle two distinct and one is the peak:

\begin{aligned} &\text{Case 1: } 1,2,3,1\\ &\text{Case 2: } 1,3,2,1 \end{aligned}

Czechy — starts 2013, total $9$ , last year unknown. The only mountain that sums to $9$ with unit/double steps is

1,2,3,2,1 \quad\Rightarrow\quad \text{2013–2017},\ \text{uniquely determined.}

Drjbna — 2011 to 2015 is $5$ years, total $10$ . The only mountain is

1,2,4,2,1 \quad\Rightarrow\quad \text{2011–2015, uniquely determined.}

Elavalaki — starts 2010, total $13$ , last year unknown. Three mountains sum to $13$ :

\begin{aligned} &\text{Case 1: } 1,2,4,3,2,1 \quad (\text{2010–2015})\\ &\text{Case 2: } 1,2,3,4,2,1 \quad (\text{2010–2015})\\ &\text{Case 3: } 1,3,5,3,1 \quad\ \ (\text{2010–2014}) \end{aligned}

Master grid of amount raised each year (a "/" separates the alternative cases):

Year	Alfloo	Bzygoo	Czechy	Drjbna	Elavalaki
2009	1	—	—	—	—
2010	2	—	—	—	1
2011	3	—	—	1	2 / 3
2012	4 / 5	1	—	2	4 / 3 / 5
2013	5 / 4	2 / 3	1	4	3 / 4 / 3
2014	3	3 / 2	2	2	2 / 2 / 1
2015	2	1	3	1	1 / 1 / —
2016	1	—	2	—	—
2017	—	—	1	—	—

This question: "certain in every year" means the firm has a single possible sequence. From the work above, only Czechy and Drjbna are uniquely fixed; Alfloo, Bzygoo and Elavalaki each have two or three live cases.

Only Czechy and Drjbna can be concluded with certainty in every year — option (d).