◆ DILR · Logical Reasoning

LR · Selection, Conditional & Ordering , rules, truth-values & schedules

Selection with conditions, binary/true-false logic, ranking, and scheduling/timetables. Translate each rule into if-then form, assume-and-propagate, and lock cells with "exactly N" constraints.

5approach cards

7CAT sets

33questions

↩ All LR Approach Sheet CAT Sets

Approach Sheet, Selection, Conditional & Ordering

How to handle conditional selection, binary logic, ranking and scheduling sets.

6Selection with conditions

"Choose a team / menu / mixture such that…", translate each rule into if-then form.
"If A then B" and its contrapositive "if not B then not A" are both usable.
Eliminate options by testing them against rules, often faster than building from scratch.
Watch quantifiers: all / some / none / at least / at most change everything.

7Ordering & ranking

Rank by height/score/age/finish. Draw a vertical or horizontal order line.
Convert comparisons into a chain: A > B, B > C ⇒ A > B > C
"X is taller than exactly two people" pins X's absolute position.
Track which gaps are fixed vs flexible; many ranking sets have multiple valid orders.

10Scheduling & timetables

Match activities/people to days, slots, rooms, or years (e.g. vendor contracts 2010-2019).
Grid: rows = entities, columns = time periods; mark contracts/sessions as spans.
Use "exactly N in period X" and "no overlap / multi-year" constraints to lock cells.
Single-year vs multi-year spans behave differently, track durations explicitly.

13Binary & conditional logic

Truth-tellers vs liars, true/false statements, two-valued (Hi/Lo, Yes/No) bids.
Assume one value, propagate consequences, check for contradiction; flip if it breaks.
Tabulate every round/statement against each player and verify the running totals.
Coded-arithmetic (cryptarithm): use carries, leading digit can't be 0, X+X gives parity clues.

15Data-sufficiency style LR

"What can be concluded?", answer only what the clues force, never assume the rest.
"Must be true / must be false / cannot be determined", test each option against ALL valid cases.
If two arrangements both satisfy the clues, anything differing between them is "cannot be determined".
Beware option-set trap: if "A, B and C" are all possible but options only pair them, pick "None of these".

Real CAT LR Sets, Selection, Conditional & Ordering

Actual CAT previous-year selection, conditional-logic, scheduling and binary-logic sets from the book. Difficulty: Easy Moderate Hard. Click any question to reveal the solution.

CAT 2005

Directions (Q. 12 to 15): Answer the questions on the basis of following information. The table below presents the revenue (in million rupees) of four firms in three states. These firms, Honest Ltd., Aggressive Ltd., Truthful Ltd. and Profitable Ltd. are disguised in the table as A, B, C and D, in no particular order.

States	Firm A	Firm B	Firm C	Firm D
UP	49	82	80	55
Bihar	69	72	70	65
MP	72	63	72	65

Further, it is known that:
• In the state of MP, Truthful Ltd. has the highest market share.
• Aggressive Ltd.'s aggregate revenue differs from Honest Ltd.'s by ₹5 million.
(Aggregate revenues: A = 49+69+72 = 190, B = 82+72+63 = 217, C = 80+70+72 = 222, D = 55+65+65 = 185. So {Aggressive, Honest} is either {A, D} or {B, C}.)

HardCAT 2005

12. What can be said regarding the following two statements? Statement 1: Profitable Ltd. has the lowest share in MP market. Statement 2: Honest Ltd.'s total revenue is more than Profitable Ltd.

(1) If Statement 1 is true then Statement 2 is necessarily true.
(2) If Statement 1 is true then Statement 2 is necessarily false.
(3) Both Statement 1 and Statement 2 are true.
(4) Neither Statement 1 nor Statement 2 is true.

Show solution

(2) If Statement 1 is true then Statement 2 is necessarily false. If Statement 1 is true, the firm with the lowest MP share is B (63), so Firm B = Profitable Ltd. Then Honest Ltd. is one of A or D (total 190 or 185), each less than B's 217. So Honest's revenue is less than Profitable's, making Statement 2 necessarily false.

HardCAT 2005

13. What can be said regarding the following two statements? Statement 1: Aggressive Ltd.'s lowest revenues are from MP. Statement 2: Honest Ltd.'s lowest revenues are from Bihar.

(1) If Statement 2 is true then Statement 1 is necessarily false.
(2) If Statement 1 is false then Statement 2 is necessarily true.
(3) If Statement 1 is true then Statement 2 is necessarily true.
(4) None of the above.

Show solution

(3) If Statement 1 is true then Statement 2 is necessarily true. If Statement 1 is true, Firm B = Aggressive Ltd. (its lowest is 63 in MP), so the {B, C} pairing holds and Firm C = Honest Ltd. Firm C's lowest revenue (70) is from Bihar. Hence Statement 2 follows.

HardCAT 2005

14. What can be said regarding the following two statements? Statement 1: Honest Ltd. has the highest share in the UP market. Statement 2: Aggressive Ltd. has the highest share in the Bihar market.

(1) Both statements could be true.
(2) At least one of the statements must be true.
(3) At most one of the statements is true.
(4) None of the above.

Show solution

(3) At most one of the statements is true. Firm B has the highest share in both UP (82) and Bihar (72). Honest and Aggressive are different firms, so they cannot both be Firm B. Hence at most one of the two statements can be true.

ModerateCAT 2005

15. If Profitable Ltd.'s lowest revenue is from UP, then which of the following is true?

(1) Truthful Ltd.'s lowest revenues are from MP.
(2) Truthful Ltd.'s lowest revenues are from Bihar.
(3) Truthful Ltd.'s lowest revenues are from UP.
(4) No definite conclusion is possible.

Show solution

(3) Truthful Ltd.'s lowest revenues are from UP. Firm D has its lowest revenue (55) from UP, so Firm D = Profitable Ltd. This forces the {A, D} pairing, leaving Firm A = Truthful Ltd. Firm A's lowest revenue (49) is from UP.

CAT 2007

Directions (Q. 21 to 24): Answer the questions on the basis of following information. A health-drink company's R&D department is trying to make various diet formulations, which can be used for certain specific purposes. It is considering a choice of 5 alternative ingredients (O, P, Q, R, and S), which can be used in different proportions in the formulations. The table below gives the composition of these ingredients. The cost per unit of each of these ingredients is O: 150, P: 50, Q: 200, R: 500, S: 100.

Ingredient	Carbohydrate %	Protein %	Fat %	Minerals %
O	50	30	10	10
P	80	20	0	0
Q	10	30	50	10
R	5	50	40	5
S	45	50	0	5

ModerateCAT 2007

21. The company is planning to launch a balanced diet required for growth needs of adolescent children. This diet must contain at least 30% each of carbohydrate and protein, no more than 25% fat and at least 5% minerals. Which one of the following combinations of equally mixed ingredients is feasible?

(1) O and P
(2) R and S
(3) P and S
(4) O and S

Show solution

(4) O and S. Equal mix of O and S gives carbohydrate 47.5%, protein 40%, fat 5%, minerals 7.5%. All four conditions (carb ≥30, protein ≥30, fat ≤25, minerals ≥5) are satisfied.

ModerateCAT 2007

22. For a recuperating patient, the doctor recommended a diet containing 10% minerals and at least 30% protein. In how many different ways can we prepare this diet by mixing at least two ingredients?

(1) One
(2) Two
(3) Three
(4) Four

Show solution

(1) One. Exactly 10% minerals with protein ≥30% is possible only when O and Q are mixed in equal proportion. So there is just one way.

HardCAT 2007

23. Which among the following is the formulation having the lowest cost per unit for a diet having 10% fat and at least 30% protein? The diet has to be formed by mixing two ingredients.

(1) P and Q
(2) P and S
(3) P and R
(4) Q and S

Show solution

(4) Q and S. Only Q&S and R&S meet fat = 10% with protein ≥30%. Q:S = 1:4 gives fat 10%, protein 46%, at cost 6/5 of base; R:S = 1:3 gives fat 10%, protein 50%, at cost 2. Q and S has the lower cost per unit.

HardCAT 2007

24. In what proportion P, Q and S should be mixed to make a diet having at least 60% carbohydrate at the lowest per unit cost?

(1) 2 : 1 : 3
(2) 4 : 1 : 2
(3) 2 : 1 : 4
(4) 4 : 1 : 1

Show solution

(4) 4 : 1 : 1. P:Q:S = 4:1:1 gives carbohydrate (320+10+45)/6 = 62.5% (≥60) at cost (200+200+100)/6 = ₹83.3/unit, versus 4:1:2 which gives carb 60% at (200+200+200)/7 = ₹85.7/unit. So 4:1:1 has the lowest cost per unit.

CAT 2017

Directions (Q. 40 to 43): Answer the questions on the basis of following information. A high security research lab requires the researchers to set a pass key sequence based on the scan of the five fingers of their left hands. When an employee first joins the lab, her fingers are scanned in an order of her choice, and then when she wants to re-enter the facility, she has to scan the five fingers in the same sequence. The lab authorities are considering some relaxations of the scan order requirements, since it is observed that some employees often get locked-out because they forget the sequence.

HardCAT 2017 · TITA

40. The lab has decided to allow a variation in the sequence of scans of the five fingers so that at most two scans (out of five) are out of place. For example, if the original sequence is Thumb (T), index finger (I), middle finger (M), ring finger (R) and little finger (L) then TLMRI is also allowed, but TMRLI is not. How many different sequences of scans are allowed for any given person's original scan? TITA

Show solution

11. At most two scans out of place means we either keep all five in place (the original sequence) or swap exactly two of the five positions. The number of ways to choose which 2 of the 5 positions are interchanged is ⁵C₂ = 10. Adding the 1 original sequence gives 10 + 1 = 11 allowed sequences.

HardCAT 2017

41. The lab has decided to allow variations of the original sequence so that input of the scanned sequence of five fingers is allowed to vary from the original sequence by one place for any of the fingers. Thus, for example, if TIMRL is the original sequence, then ITRML is also allowed, but LIMRT is not. How many different sequences are allowed for any given person's original scan?

(1) 7
(2) 5
(3) 8
(4) 13

Show solution

(3) 8. Each finger may move at most one place, so only adjacent positions can be interchanged. Counting: 1 (original) + 4 (any one of the four adjacent pairs interchanged) + 2 (the first pair (1-2) interchanged together with one of the two non-overlapping later pairs, i.e. (3-4) or (4-5)) + 1 (pairs (2-3) and (4-5) both interchanged) = 1 + 4 + 2 + 1 = 8.

HardCAT 2017 · TITA

42. The lab has now decided to require six scans in the pass key sequence, where exactly one finger is scanned twice, and the other fingers are scanned exactly once, which can be done in any order. For example, a possible sequence is TIMTRL. Suppose the lab allows a variation of the original sequence (of six inputs) where at most two scans (out of six) are out of place, as long as the finger originally scanned twice is scanned twice and other fingers are scanned once. How many different sequences of scans are allowed for any given person's original scan? TITA

Show solution

15. Take the original TIMTRL. Counting the distinct two-position swaps that keep the duplicated finger scanned twice and each other finger once: swapping the first T with another position gives 4 valid new sequences, swapping I gives 4, swapping M gives 3, swapping the second T gives 2, swapping R gives 1, a total of 14 variations. Adding the 1 original sequence gives 14 + 1 = 15.

HardCAT 2017

43. The lab has now decided to require six scans in the pass key sequence, where exactly one finger is scanned twice, and the other fingers are scanned exactly once, which can be done in any order. For example, a possible sequence is TIMTRL. Suppose the lab allows a variation of the original sequence (of six inputs) so that input in the form of scanned sequence of six fingers is allowed to vary from the original sequence by one place for any of the fingers, as long as the finger originally scanned twice is scanned twice and other fingers are scanned once. How many different sequences of scans are allowed if the original scan sequence is LRLTIM?

(1) 8
(2) 11
(3) 13
(4) 14

Show solution

(3) 13. Each finger may move at most one place, so only adjacent positions can be interchanged. For LRLTIM: 1 (original) + 5 (interchanging any single adjacent pair: LR, RL, LT, TI or IM) + 7 valid double swaps of two non-overlapping adjacent pairs (LR with LT, LR with TI, LR with IM, RL with TI, RL with IM, LT with IM, and LR&LT&IM) = 13.

CAT 2018

Directions (Q. 52 to 55): Answer the questions on the basis of following information. The following table represents addition of two six-digit numbers given in the first and the second rows, while the sum is given in the third row. In the representation, each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 has been coded with one letter among A, B, C, D, E, F, G, H, J, K, with distinct letters representing distinct digits.

	B	H	A	A	G	F
+ A	A	H	J	F	K	F
A	A	F	G	C	A	F

HardCAT 2018 · TITA

52. Which digit does the letter A represent? TITA

Show solution

1. In the units column F + F = F, which is possible only if F = 0. The result has one more digit than each addend (leading A), and the largest carry from adding two six-digit numbers is 1, so the leading digit A = 1.

HardCAT 2018 · TITA

53. Which digit does the letter B represent? TITA

Show solution

9. With F = 0 and A = 1: the column H + H ends in 0, so H = 5 (giving a carry of 1). Then the leftmost full column gives B + A + carry = B + 1 + 1, which must equal 11 to produce the leading 1 with sum-digit A = 1. Hence B = 9.

HardCAT 2018

54. Which among the digits 3, 4, 6 and 7 cannot be represented by the letter D?

(1) 3
(2) 4
(3) 6
(4) 7

Show solution

(4) 7. With the consistent assignment F = 0, A = 1, H = 5, B = 9, and G + K = 11 with J = G − 1 and C = 2, the feasible (G, K, J) triples use the value 7 (e.g. G = 4, K = 7, J = 3, or G = 7, K = 4, J = 6). So 7 is taken by other letters and D cannot be 7.

HardCAT 2018

55. Which among the digits 4, 6, 7 and 8 cannot be represented by the letter G?

(1) 4
(2) 6
(3) 7
(4) 8

Show solution

(2) 6. From G + K = 11 (with the constraint J = G − 1 and C = 2), the only feasible (G, K) pairs giving valid distinct digits are (8, 3), (4, 7) and (7, 4). G = 6 would need K = 5 = H, which is already used, so G cannot be 6.

CAT 2020

Directions (Q. 56 to 61): Read the following passage carefully and answer the questions that follows. Four institutes, A, B, C, and D, had contracts with four vendors W, X, Y, and Z during the ten calendar years from 2010 to 2019. The contracts were either multi-year contracts running for several consecutive years or single-year contracts. No institute had more than one contract with the same vendor. However, in a calendar year, an institute may have had contracts with multiple vendors, and a vendor may have had contracts with multiple institutes. It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes.

The following facts are also known about these contracts.
I. Vendor Z had at least one contract in every year.
II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that.
III. Vendor Y had contracts in 2010 and 2019. Vendor W had contracts only in 2013.
IV. There were five contracts in 2012.
V. There were exactly four multi-year contracts. Institute B had a 7-year contract, D had a 4-year contract, and A and C had one 3-year contract each. The other four contracts were single-year contracts.
VI. Institute C had one or more contracts in 2012 but did not have any contract in 2011.
VII. Institutes B and D each had exactly one contract in 2012. Institute D did not have any contract in 2010.

Reconstructed schedule (each cell = institute-vendor contract active that year):

Year	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Contract 1	B-Z	B-Z	B-Z	B-Z	B-Z	B-Z	B-Z	-	-	-
Contract 2	A-X	A-X	A-X	-	-	-	-	-	-	-
Contract 3	B-Y	-	D-X	D-X	D-X	D-X	-	-	-	-
Contract 4	-	-	C-Z*	C-W	-	-	-	-	-	D-Y

*C had a 3-year contract with Z spanning 2012-2014; A's other (3-year) contract and the single-year contracts fill the five-contract count in 2012. The grid above shows the spans used in the solutions below.

HardCAT 2020

56. In which of the following years were there two or more contracts?

(1) 2015
(2) 2018
(3) 2017
(4) 2016

Show solution

(1) 2015. In 2015 both B-Z and D-X are active (X's last possible year), giving two or more contracts; 2017, 2018 and 2016 do not have two contracts.

HardCAT 2020

57. Which of the following is true?

(1) B had a contract with Y in 2019
(2) D had a contract with X in 2011
(3) B had a contract with X in 2017
(4) D had a contract with Y in 2019

Show solution

(4) D had a contract with Y in 2019. Y has contracts only in 2010 (with B) and 2019. Since each vendor has exactly two contracts with two different institutes, the 2019 Y-contract must be with D, so D-Y in 2019.

HardCAT 2020

58. In how many years during this period was there only one contract?

(1) 3
(2) 5
(3) 2
(4) 4

Show solution

(1) 3. From the completed schedule, exactly three of the ten years have a single contract active.

HardCAT 2020

59. What BEST can be concluded about the number of contracts in 2010?

(1) At least 3
(2) At least 4
(3) Exactly 4
(4) Exactly 3

Show solution

(4) Exactly 3. In 2010 the active contracts are B-Z, A-X and B-Y (Y's 2010 contract). Institute D has no contract in 2010, so exactly 3 contracts ran in 2010.

HardCAT 2020

60. Which institutes had multiple contracts during the same year?

(1) B and C only
(2) B only
(3) A only
(4) A and B only

Show solution

(4) A and B only. Only institutes A and B ran two of their contracts overlapping within a single calendar year.

HardCAT 2020

61. Which institutes and vendors had more than one contracts in any year?

(1) A, D, W, and Z
(2) B, W, X, and Z
(3) A, B, Z, and X
(4) B, D, W, and X

Show solution

(3) A, B, Z, and X. These are precisely the institutes (A, B) and vendors (Z, X) that had more than one contract running in some single calendar year.

Directions (Q. 66 to 71): Read the following passage carefully and answer the questions that follows. The Hi-Lo game is a four-player game played in six rounds. In every round, each player chooses to bid Hi or Lo. The bids are made simultaneously. If all four bid Hi, then all four lose 1 point each. If three players bid Hi and one bids Lo, then the players bidding Hi gain 1 point each and the player bidding Lo loses 3 points. If two players bid Hi and two bid Lo, then the players bidding Hi gain 2 points each and the players bidding Lo lose 2 points each. If one player bids Hi and three bid Lo, then the player bidding Hi gains 3 points and the players bidding Lo lose 1 point each. If all four bid Lo, then all four gain 1 point each.

Four players Arun, Bankim, Charu, and Dipak played the Hi-Lo game. The following facts are known about their game:
(1) At the end of three rounds, Arun had scored 6 points, Dipak had scored 2 points, Bankim and Charu had scored -2 points each.
(2) At the end of six rounds, Arun had scored 7 points, Bankim and Dipak had scored -1 point each, and Charu had scored -5 points.
(3) Dipak's score in the third round was less than his score in the first round but was more than his score in the second round.
(4) In exactly two out of the six rounds, Arun was the only player who bid Hi.

HardCAT 2020

66. What were the bids by Arun, Bankim, Charu and Dipak, respectively in the first round?

(1) Hi, Lo, Lo, Hi
(2) Hi, Hi, Lo, Lo
(3) Lo, Lo, Lo, Hi
(4) Hi, Lo, Lo, Lo

Show solution

(1) Hi, Lo, Lo, Hi. The first three rounds reconstruct to score-vectors (Arun, Bankim, Charu, Dipak) = (2, −2, −2, 2), (3, −1, −1, −1) and (1, 1, 1, 1). Round 1 = (2, −2, −2, 2) is a 2-Hi/2-Lo round in which Arun and Dipak bid Hi (+2) and Bankim and Charu bid Lo (−2). So the bids are Hi, Lo, Lo, Hi.

HardCAT 2020 · TITA

67. In how many rounds did Arun bid Hi? TITA

Show solution

4. Across the six reconstructed rounds, Arun bid Hi in 4 of them.

HardCAT 2020 · TITA

68. In how many rounds did Bankim bid Lo? TITA

Show solution

4. Bankim bid Lo in 4 of the six rounds.

HardCAT 2020 · TITA

69. In how many rounds did all four players make identical bids? TITA

Show solution

2. Two rounds had all four players bidding the same way (one all-Lo round scoring (1,1,1,1) and one all-Hi round scoring (−1,−1,−1,−1)).

HardCAT 2020 · TITA

70. In how many rounds did Dipak gain exactly 1 point? TITA

Show solution

1. Dipak gained exactly +1 point in exactly one round (the all-Lo round, where every player gains 1).

HardCAT 2020

71. In which of the following rounds, was Arun DEFINITELY the only player to bid Hi?

(1) Third
(2) Second
(3) First
(4) Fourth

Show solution

(2) Second. The second round scored (3, −1, −1, −1): Arun gained +3 and the other three each lost 1, which happens only when exactly one player (Arun) bids Hi and three bid Lo. So the second round is definitely an "Arun-only-Hi" round.

CAT 2023

Directions (Q. 6 to 10): Answer the questions on the basis of the information given below. Faculty members in a management school can belong to one of four departments - Finance and Accounting (F&A), Marketing and Strategy (M&S), Operations and Quants (O&Q) and Behaviour and Human Resources (B&H). The numbers of faculty members in F&A, M&S, O&Q and B&H departments are 9, 7, 5 and 3 respectively.

Prof. Pakrasi, Prof. Qureshi, Prof. Ramaswamy and Prof. Samuel are four members of the school's faculty who were candidates for the post of the Dean of the school. Only one of the candidates was from O&Q. Every faculty member, including the four candidates, voted for the post. In each department, all the faculty members who were not candidates voted for the same candidate. The rules for the election are listed below.
1. There cannot be more than two candidates from a single department.
2. A candidate cannot vote for himself/herself.
3. Faculty members cannot vote for a candidate from their own department.

After the election, it was observed that Prof. Pakrasi received 3 votes, Prof. Qureshi received 14 votes, Prof. Ramaswamy received 6 votes and Prof. Samuel received 1 vote. Prof. Pakrasi voted for Prof. Ramaswamy, Prof. Qureshi for Prof. Samuel, Prof. Ramaswamy for Prof. Qureshi and Prof. Samuel for Prof. Pakrasi.

HardCAT 2023 · Slot 1

6. Which two candidates can belong to the same department?

(1) Prof. Pakrasi and Prof. Samuel
(2) Prof. Pakrasi and Prof. Qureshi
(3) Prof. Qureshi and Prof. Ramaswamy
(4) Prof. Ramaswamy and Prof. Samuel

Show solution

(2) Prof. Pakrasi and Prof. Qureshi. Working out the vote counts, the department with two candidates is M&S, and the only consistent pairing (Qureshi and Pakrasi did not vote for each other) puts Pakrasi and Qureshi together in M&S.

HardCAT 2023 · Slot 1

7. Which of the following can be the number of votes that Prof. Qureshi received from a single department?

(1) 7
(2) 8
(3) 6
(4) 9

Show solution

(4) 9. Of Qureshi's 14 votes, 1 came from Ramaswamy and 13 from non-candidates, split as 9 from F&A and 4 from O&Q. So 9 is a possible single-department count.

HardCAT 2023 · Slot 1

8. If Prof. Samuel belongs to B&H, which of the following statements is/are true? Statement A: Prof. Pakrasi belongs to M&S. Statement B: Prof. Ramaswamy belongs to O&Q.

(1) Neither statement A nor statement B
(2) Both statements A and B
(3) Only statement B
(4) Only statement A

Show solution

(2) Both statements A and B. If Samuel is in B&H, then Pakrasi must be the other M&S candidate and Ramaswamy must be the lone O&Q candidate. Both statements are true.

HardCAT 2023 · Slot 1

9. What best can be concluded about the candidate from O&Q?

(1) It was either Prof. Pakrasi or Prof. Qureshi.
(2) It was Prof. Ramaswamy.
(3) It was either Prof. Ramaswamy or Prof. Samuel.
(4) It was Prof. Samuel.

Show solution

(3) It was either Prof. Ramaswamy or Prof. Samuel. Pakrasi and Qureshi are both fixed to M&S, so the single O&Q candidate is undetermined between Ramaswamy and Samuel.

HardCAT 2023 · Slot 1

10. Which of the following statements is/are true? Statement A: Non-candidates from M&S voted for Prof. Qureshi. Statement B: Non-candidates from F&A voted for Prof. Qureshi.

(1) Only statement B
(2) Only statement A
(3) Both statements A and B
(4) Neither statement A nor statement B

Show solution

(1) Only statement B. Non-candidates from M&S voted for Ramaswamy, not Qureshi (Statement A false). Non-candidates from F&A voted for Qureshi (Statement B true). So only B is true.