CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 8
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 8 of 14.
Let \(k\) be the largest integer such that the equation \((x-1)^2+2 k x+11=0\) has no real roots. If \(y\) is a positive real number, then the least possible value of \(\frac{k}{4 y}+9 y\) is
Answer: 6
Solution — first steps
Step 1. Expand (x−1)²+2kx+11=0 to x²+(2k−2)x+12=0.
Step 2. No real roots requires discriminant 0.
Step 2. No real roots requires discriminant 0.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
If a certain amount of money is divided equally among \(n\) persons, each one receives Rs 352 . However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330 . Then, the maximum possible value of \(n\) is
Answer: 16
Solution — first steps
Step 1. Total money is 352n.
Step 2. After two people receive 506 each, the remaining n−2 people each get (352n−1012)/(n−2).
Step 2. After two people receive 506 each, the remaining n−2 people each get (352n−1012)/(n−2).
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let both the series \(a_1, a_2, a_3, \ldots\) and \(b_1, b_2, b_3 \ldots\) be in arithmetic progression such that the common differences of both the series are prime numbers. If \(a_5=b_9, a_{19}=b_{19}\) and \(b_2=0\), then \(a_{11}\) equals
Answer: A. 79
Solution — first steps
Step 1. Let AP differences be prime numbers p and q.
Step 2. Using b2=0 gives b_n=(n−2)q.
Step 2. Using b2=0 gives b_n=(n−2)q.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let \(a_n\) and \(b_n\) be two sequences such that \(a_n=13+6(n-1)\) and \(b_n=15+7(n-1)\) for all natural numbers \(n\). Then, the largest three digit integer that is common to both these sequences, is
Answer: 967
Solution — first steps
Step 1. The two sequences are6n+7 and7m+8 for n,m≥1.
Step 2. Common values N satisfy N≡1 modulo6 and modulo7, hence N≡1 modulo42.
Step 2. Common values N satisfy N≡1 modulo6 and modulo7, hence N≡1 modulo42.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let \( n \) be any natural number such that \( 5^{n-1} < 3^{n+1} \).
The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?
The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?
Answer: C. 5
Solution — first steps
The least value of \( m \) is 5. For \( n = 5 \), we need \( 3^6 < 2^{5 + m} \).
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
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.