CAT 2017 Slot 2: Quantitative Ability Solutions (English)
Problem 1: Algebra - System of Equations
Question:
A and B can complete a project in 12 days working together. A alone can finish it in 20 days less than B. How many days does B take to complete the project alone?
Solution:
Let B’s time = ( x ) days. Then, A’s time = ( x - 20 ) days.
Work rates:
A’s rate = ( \frac{1}{x - 20} )
B’s rate = ( \frac{1}{x} )
Combined rate: ( \frac{1}{x - 20} + \frac{1}{x} = \frac{1}{12} ).
Solve:
( \frac{x + (x - 20)}{x(x - 20)} = \frac{1}{12} )
( \frac{2x - 20}{x^2 - 20x} = \frac{1}{12} )
( 24x - 240 = x^2 - 20x )
( x^2 - 44x + 240 = 0 )
( (x - 24)(x - 20) = 0 ) → ( x = 24 ) (since ( x - 20 > 0 )).
Answer: B takes 24 days.
Problem 2: Geometry - Circle and Triangle
Question:
In a circle, chord ( AB ) is equal in length to chord ( CD ). If ( AB ) subtends a central angle of 80°, what is the measure of the angle subtended by ( CD ) at the circumference?
Solution:
Central angle for ( AB ): 80°.
Chord length formula: ( 2R \sin\left(\frac{\theta}{2}\right) ).
Since ( AB = CD ), their central angles are equal.
Angle at circumference = ( \frac{80°}{2} = 40° ).
Answer: 40°
Problem 3: Probability - Coin Toss
Question:
A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads and at least 1 tail?
Solution:
Total outcomes: ( 2^5 = 32 ).
Favorable outcomes:
Choose 3 heads: ( \binom{5}{3} = 10 ).
Ensure at least 1 tail: Total ways (3H,2T) = 10.
Probability = ( \frac{10}{32} = \frac{5}{16} ).
Answer: ( \boxed{\dfrac{5}{16}} )
Problem 4: Number Theory - Divisors
Question:
Find the number of positive divisors of 2016 that are multiples of 6.
Solution:
Prime factorization: ( 2016 = 2^5 \times 3^2 \times 7^1 ).
Divisors of 2016 that are multiples of 6 must include ( 2^1 \times 3^1 ).
Remaining factors: ( 2^{4} \times 3^{1} \times 7^{0/1} ).
Number of divisors: ( (4+1)(1+1)(1+1) = 5 \times 2 \times 2 = 20 ).
Answer: 20
Problem 5: Data Interpretation - Bar Graph
Question:
[Assume a bar graph comparing sales (in Lakh) of two companies (X and Y) over 4 years.]
Question: In which year did Company X’s sales growth rate exceed Company Y’s growth rate?
Solution:
Calculate annual growth rates:
Company X:
Year 1: 100 → Year 2: 120 → Growth = ( \frac{20}{100} = 20% ).
Year 2→3: 120→150 → ( \frac{30}{120} = 25% ).
Year 3→4: 150→180 → ( \frac{30}{150} = 20% ).
Company Y:
Year 1: 200 → Year 2: 240 → ( \frac{40}{200} = 20% ).
Year 2→3: 240→300 → ( \frac{60}{240} = 25% ).
Year 3→4: 300→360 → ( \frac{60}{300} = 20% ).
Year 3: X’s growth = 25%, Y’s growth = 25% → Equal.
Year 4: X’s growth = 20%, Y’s growth = 20% → Equal.
Year 2: X’s growth = 20%, Y’s growth = 20% → Equal.
No year where X’s growth > Y’s growth.
Answer: No such year (or check graph for edge cases).
Key Takeaways:
Algebra: System of equations with work rates.
Geometry: Central vs. inscribed angles.
Probability: Combinatorics with constraints.
Number Theory: Divisors and prime factorization.
DI: Growth rate comparisons.
Let me know if you need further clarification!
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