CAT Inequalities Bible Top 10 Must Solve Inequalities Questions covering 80% of Syllabus – Page 2
All 38 Inequalities questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 2 of 11.
Suppose a, b, c are three distinct natural numbers, such that 3ac = 8(a + b).
Then, the smallest possible value of 3a + 2b + c is _______
Then, the smallest possible value of 3a + 2b + c is _______
Answer: 12
Solution — first steps
Final Answer: 12 Given: 3ac = 8(a + b) ⇒ 8b = 3ac − 8a ⇒ b = (3ac/8) − a Since a, b, c are natural numbers, 3ac must be divisible by 8. Now try small values of a (since we want the minimum value of 3a + 2b + c).
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
For real values of x, the range of the function f(x) = (2x - 3)/(2x² + 4x - 6)
Answer: C. (-oo , 1/8] U [1/2 , oo)
Solution — first steps
Correct Answer: 3 We can factor the denominator: 2x² + 4x - 6 = 2(x + 3)(x - 1) So, the function becomes: f(x) = (2x - 3) / (2(x + 3)(x - 1)) , thus x ≠ -3 , 1 Range of f(x) f(0) = 1/2 f(1) = not possible f(2) = 1/10 f(3) = 1/8 Only option 3 includes f(0) and f(3) Thus option 3 is the answer
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is equal to half the number of students who chose both mathematics and physics. What is the maximum possible number of students who chose physics but not mathematics?
Answer: B. 35
Solution — first steps
Given Information: Physics students = 75, Mathematics students = 111 , Chemistry students = 40 Assume: x - number of students who chose both Physics and Mathematics , y - students who chose both Mathematics and Chemistry , z - student who chose both Physics and Chemistry , t - number of students who chose all three subjects And z = y , y = x/2 (Given) Using Set formula (the total number of students who chose at least one subject) |P ∪ M ∪ C| = |P| + |M| + |C| − |P ∩ M| − |M ∩ C| − |P ∩ C| + |P ∩ M ∩ C| Substitute the values: 150 = 75 + 111 + 40 - x - y - z + t Now, substitute z = y and y = x/2: 150 = 75 + 111 + 40 - x - (x/2) - (x/2) + t 150 = 226 - x - x + t 150 = 226 - 2x + t 2x - 76 = t (Equation 1) We want to maximize 75 - x , (students who chose only Physics but not mathematics). 30 , 75 - x = 30 , x = 45 , t = 14 Since 75 - x is maximum at option 2 thus option 2 is the answer
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose values lies between 150 and 500.If p is less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is: _____
Answer: 397
Solution — first steps
3 × 320 = 96 Pmax + P min = 301 + 96 = 397 Answer 397
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
In a school with 1500 students , each students chooses one of the streams out of science , arts and commerce by paying a fee of rs 1100 , 1000 and 800 respectively. The total fees paid by all the students is RS. 15,50000.If the number of science students is not more than the number of arts students , then the maximum possible number of science students in school is : [TITA] _______
Answer: 700
Solution — first steps
Correct Answer: 700 Given S - science students , A - arts students , C - commerce students S + A + C = 1500 —- I 1100S + 1000A + 800C = 1550000 becomes 11S + 10A + 8C = 15500 —---- II Where S ≤ A , since we have to maximise S this S = A Multiplying I by 8 : 8S + 8A + 8C = 12000 —--- III Subtracting III from I 3S + 2A = 3500 , Since S = A 5S = 3500 , S = 700 Answer : 700
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 21 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Exponential bounds and smallest integer periods | 7 | CAT 2021 S2 Q15 | 7 | 12.73% |
| 2 | Discriminant bounds for real roots or ranges | 6 | CAT 2022 S1 Q5 | 13 | 23.64% |
| 3 | Linear allocation bounds with nonnegative counts | 6 | CAT 2022 S1 Q8 | 19 | 34.55% |
| 4 | Absolute value intervals and sign cases | 5 | CAT 2021 S1 Q2 | 24 | 43.64% |
| 5 | Ordered values and average bounds | 5 | CAT 2022 S2 Q8 | 29 | 52.73% |
| 6 | Logarithmic bounds and monotonicity | 4 | CAT 2024 S2 Q9 | 33 | 60.00% |
| 7 | Rational sign charts and excluded poles | 3 | CAT 2021 S1 Q1 | 36 | 65.45% |
| 8 | Set overlap bounds and feasible endpoints | 3 | CAT 2022 S1 Q17 | 39 | 70.91% |
| 9 | Finite integer enumeration within bounds | 3 | CAT 2025 S1 Q17 | 42 | 76.36% |
| 10 | Floor intervals and endpoint inclusion | 2 | CAT 2022 S1 Q12 | 44 | 80.00% |
| 11 | Factorial size bounds | 1 | CAT 2022 S2 Q13 | 45 | 81.82% |
| 12 | Bounds on eligible AP terms | 1 | CAT 2022 S3 Q17 | 46 | 83.64% |
| 13 | Remainders with an upper population bound | 1 | CAT 2022 S3 Q19 | 47 | 85.45% |
| 14 | Polygon side inequality | 1 | CAT 2022 S3 Q22 | 48 | 87.27% |
| 15 | Speed bound with a palindrome restriction | 1 | CAT 2023 S1 Q8 | 49 | 89.09% |
| 16 | Translate time bounds to work rate bounds | 1 | CAT 2023 S3 Q8 | 50 | 90.91% |
| 17 | Quadratic percentage growth inequality | 1 | CAT 2023 S3 Q10 | 51 | 92.73% |
| 18 | Geometric regions defined by inequalities | 1 | CAT 2024 S1 Q4 | 52 | 94.55% |
| 19 | Square bound with an equality condition | 1 | CAT 2024 S1 Q8 | 53 | 96.36% |
| 20 | Compare quadratic extrema | 1 | CAT 2025 S1 Q8 | 54 | 98.18% |
| 21 | Range by completing the square and reciprocal bounds | 1 | CAT 2021 S2 Q13 | 55 | 100% |
Based on CAT Actual Quant Papers.