CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 4
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 4 of 14.
Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been
Answer: 139
Solution — first steps
Step 1. Let the rates of Amal,Vimal,Sunil be a,v,s. Then a+v=1/150 and v+s=1/100.
Step 2. The staged schedule gives 75a+135v+45s=1. Substituting a=1/150−v and s=1/100−v gives 19/20+15v=1.
Step 2. The staged schedule gives 75a+135v+45s=1. Substituting a=1/150−v and s=1/100−v gives 19/20+15v=1.
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P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is
Answer: 7
Solution — first steps
Step 1. Let the number of direct Q–R paths be x and direct R–S paths be y.
Step 2. P→S via Q, via R, and via Q then R give 3·4+4y+3xy=62, so 4y+3xy=50.
Step 2. P→S via Q, via R, and via Q then R give 3·4+4y+3xy=62, so 4y+3xy=50.
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A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
Answer: 340
Solution — first steps
Step 1. Let total fruit count be T; mangoes=2T/5. Write apples A and bananas B, so A+B=3T/5.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
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\( a \) and \( b \) are natural numbers, then the value of \( a + b \) is:
\[
(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}
\]
\[
(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}
\]
Answer: 8
Solution — first steps
Surd Algebra Question \( a \) and \( b \) are natural numbers, then the value of \( a + b \) is: \[ (a + b\sqrt{3})^2 = 52 + 30\sqrt{3} \] ✅ Solution: Expanding the left-hand side: \[ (a + b\sqrt{3})^2 = a^2 + 2ab\sqrt{3} + 3b^2 \] Match with the right-hand side: \[ a^2 + 3b^2 = 52 \quad \text{(1)} \\ 2ab = 30 \Rightarrow ab = 15 \quad \text{(2)} \] If \( ab = 15 \) and \( a^2 + 3b^2 = 52 \), where \( a \) and \( b \) are natural numbers, find \( a + b \). Solution using Hit and Trial: We try natural number pairs such that \( ab = 15 \).
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The number of distinct integer solutions (x, y) of the equation |x + y| + |x – y| = 2, is
Answer: 8
Solution — first steps
Since two absolute values are added, for the equation to be true we can have three combinations of \(|x + y|, |x - y|\): \[ \{2, 0\}, \{0, 2\}, \{1, 1\} \] Cases and Corresponding Solutions: Case Conditions Solution(s) Case 1 \( x + y = 2 \) and \( x - y = 0 \) \( \{x, y\} = \{1, 1\} \) Case 2 \( x + y = -2 \) and \( x - y = 0 \) \( \{x, y\} = \{-1, -1\} \) Case 3 \( x + y = 0 \) and \( x - y = 2 \) \( \{x, y\} = \{1, -1\} \) Case 4 \( x + y = 0 \) and \( x - y = -2 \) \( \{x, y\} = \{-1, 1\} \) Case 5 \( x + y = 1 \) and \( x - y = 1 \) \( \{x, y\} = \{1, 0\} \) Case 6 \( x + y = 1 \) and \( x - y = -1 \) \( \{x, y\} = \{0, 1\} \) Case 7 \( x + y = -1 \) and \( x - y = 1 \) \( \{x, y\} = \{0, -1\} \) Case 8 \( x + y = -1 \) and \( x - y = -1 \) \( \{x, y\} = \{-1, 0\} \) ✅ Hence, there are 8 distinct integer solutions for \(\{x, y\}\).
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Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 18 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Integer exponent thresholds | 8 | CAT 2021 S2 Q15 | 8 | 10.67% |
| 2 | Integer count equations with totals and bounds | 8 | CAT 2022 S2 Q15 | 16 | 21.33% |
| 3 | Prime exponent and divisor restrictions | 7 | CAT 2022 S1 Q4 | 23 | 30.67% |
| 4 | Ordered scores and counts under average bounds | 6 | CAT 2021 S3 Q7 | 29 | 38.67% |
| 5 | Integer roots and quadratic parameter bounds | 6 | CAT 2022 S1 Q15 | 35 | 46.67% |
| 6 | Factor pair enumeration with extra bounds | 6 | CAT 2024 S1 Q2 | 41 | 54.67% |
| 7 | AP terms intersections and index restrictions | 5 | CAT 2021 S2 Q10 | 46 | 61.33% |
| 8 | Integer points under modulus or linear bounds | 4 | CAT 2021 S1 Q2 | 50 | 66.67% |
| 9 | Divisibility of averages ratios and remainders | 4 | CAT 2022 S1 Q7 | 54 | 72.00% |
| 10 | Floor and logarithm intervals containing integers | 4 | CAT 2022 S2 Q14 | 58 | 77.33% |
| 11 | Digit construction and digit bounds | 3 | CAT 2021 S2 Q12 | 61 | 81.33% |
| 12 | Integer feasibility of set overlaps | 3 | CAT 2022 S1 Q17 | 64 | 85.33% |
| 13 | Whole time periods and work schedules | 3 | CAT 2022 S2 Q18 | 67 | 89.33% |
| 14 | Distribution with minimum or even allocations | 2 | CAT 2021 S2 Q1 | 69 | 92.00% |
| 15 | Balancing distinct or near equal integers | 2 | CAT 2023 S2 Q13 | 71 | 94.67% |
| 16 | Integer bounds from ratios or polygon sides | 2 | CAT 2022 S1 Q8 | 73 | 97.33% |
| 17 | Power equals one and exponent parity | 1 | CAT 2022 S2 Q17 | 74 | 98.67% |
| 18 | Integer percentage constrained by net growth | 1 | CAT 2023 S3 Q10 | 75 | 100% |
Based on CAT Actual Quant Papers.