CAT 2022 Slot 3: Strategic Solutions to Complex Game Problems
The CAT (Common Admission Test) 2022 Slot 3 featured a series of intricate logical and analytical game problems designed to test candidates' critical thinking and quantitative reasoning. Below is a detailed breakdown of key question patterns, solving strategies, and solutions specific to this slot.
1. Seating Arrangement with Constraints
Problem Statement:
8 people (A, B, C, D, E, F, G, H) are seated around a circular table.
A and E sit directly opposite each other.
B and F are neighbors of A.
C and G sit next to each other but not adjacent to D or H.
D sits between E and H.
Question: How many distinct seating arrangements are possible?
Solution:
Fix A's position to avoid counting rotations (circular permutations).
E must be directly opposite A (1 possible position).
B and F are adjacent to A (2 possible orders: B-F or F-B).
D must sit between E and H. Since E is fixed opposite A, D can be between E and H in 2 ways: E-D-H or H-D-E.
C and G must sit together but not next to D or H. The remaining seats are 2 adjacent spots not adjacent to D/H.
Possible pairs: G-C or C-G (2 permutations).
Remaining person (H or D) fills the last seat.
Total Arrangements:
2 (B-F/F-B) × 2 (D-H/H-D) × 2 (C-G/G-C) = 8 arrangements.
2. Probability of Drawing Cards
Problem Statement:
A deck of 52 cards has 13 cards each of Spades, Hearts, Clubs, and Diamonds.
4 cards are drawn at random.
Question: What is the probability that all 4 cards are of different suits?
Solution:
Total ways to draw 4 cards: ( \binom{52}{4} = 270,725 ).
Favorable outcomes:
Choose 4 suits out of 4: ( \binom{4}{4} = 1 ).
Pick 1 card from each chosen suit: ( 13^4 = 28,561 ).
Probability: ( \frac{28,561}{270,725} \approx 0.1055 ) (10.55%).
Key Insight: Use combinatorial counting to avoid overcounting suits.
3. Work-Rate Problem
Problem Statement:
Pipe X can fill a tank in 6 hours, Pipe Y in 4 hours, and Pipe Z can empty it in 3 hours.
Question: If all pipes are opened simultaneously, how long will it take to fill the tank?
Solution:
Rate of Pipe X: ( \frac{1}{6} ) tank/hour.
Rate of Pipe Y: ( \frac{1}{4} ) tank/hour.
Rate of Pipe Z: ( -\frac{1}{3} ) tank/hour (emptying).
Combined rate: ( \frac{1}{6} + \frac{1}{4} - \frac{1}{3} = \frac{2 + 3 - 4}{12} = \frac{1}{12} ) tank/hour.
Time to fill: ( \frac{1}{\frac{1}{12}} = 12 ) hours.
Common Mistake: Forgetting to subtract Pipe Z’s emptying rate.
4. Algebraic Inequalities
Problem Statement:
If ( x + y = 10 ) and ( x^2 + y^2 = 50 ), find the value of ( xy ).
Solution:
Use the identity ( (x + y)^2 = x^2 + 2xy + y^2 ).
Substitute values: ( 10^2 = 50 + 2xy ).
Solve: ( 100 = 50 + 2xy ) → ( 2xy = 50 ) → ( xy = 25 ).
Application: Useful for solving quadratic equations and optimization problems.
5. Data Interpretation (DI) with Tables
Problem Statement:
A table shows sales (in $10,000s) for 3 products (A, B, C) across 4 quarters (Q1-Q4).
Q1: A = 20, B = 30, C = 40.
Q2: A increased by 15%, B decreased by 10%, C doubled.
Q3: A = Q2’s A + 10, B = Q2’s B - 5, C = Q1’s C + 25.
Q4: A = Q3’s A × 0.8, B = Q3’s B + 15, C = Q2’s C.
Question: What is the total sales in Q4?
Solution:
Q2 Calculations:
A: ( 20 \times 1.15 = 23 ).
B: ( 30 \times 0.9 = 27 ).
C: ( 40 \times 2 = 80 ).
Q3 Calculations:
A: ( 23 + 10 = 33 ).
B: ( 27 - 5 = 22 ).
C: ( 40 + 25 = 65 ).
Q4 Calculations:
A: ( 33 \times 0.8 = 26.4 ).
B: ( 22 + 15 = 37 ).
C: ( 80 ) (same as Q2).
Total Q4 Sales: ( 26.4 + 37 + 80 = 143.4 ) (in 10,000s) → 1,434,000.
Tip: Break down each quarter step-by-step to avoid errors.
Conclusion
CAT 2022 Slot 3 emphasized speed, logical deduction, and 精细 mathematical manipulation. Focus on mastering combinatorial reasoning, algebraic identities, and systematic data interpretation to excel in similar exams. For future slots, practice time-bound problem-solving and review common traps (e.g., circular permutations, work-rate negatives).
Good luck to all aspirants! 🚀
|
