Let a, b, c be non-zero real numbers such that \(b^{2} < 4ac\), and \(f(x) = ax^{2} + bx + c\). If the set S consists of all integers m such that \(f(m) < 0\), then the set S must necessarily be Options: A) the set of all positive integers B) the set of all integers C) either the empty set or the set of all integers D) the empty set

C. either the empty set or the set of all integers. Step 1. The discriminant b²−4ac is negative. Step 2. A real quadratic with negative discriminant never crosses or touches zero, and its sign is the sign of a.

\(f(x) = \dfrac{x^2 + 2x - 15}{x^2 - 7x - 18}\) is negative if and only if Options: A) \(-5 9\) D) \(x < -5\) or \(3 < x < 9\)

A. \(-5 < x < -2\) or \(3 < x < 9\). Step 1. Factor f=(x+5)(x−3)/[(x−9)(x+2)]. Step 2. Critical points−5, −2, 3, 9; poles−2, 9 are excluded.

The set of all real values of x for which (x² − |x + 9| + x) > 0, is Options: A) (−∞, −3) ∪ (3, ∞) B) (−9, −3) ∪ (3, ∞) C) (−∞, −9) ∪ (3, ∞) D) (−∞, −9) ∪ (9, ∞)

A. (−∞, −3) ∪ (3, ∞). Use X maro G Strategy put x = -5 and x = -10 Option Interval Test x = −5 Test x = −10 Status A (−∞, −3) ∪ (3, ∞) Included, LHS = 16 > 0 Included, LHS = 89 > 0 ✅ Possible B (−9, −3) ∪ (3, ∞) Included, LHS = 16 > 0 Not included ❌ Eliminated C (−∞, −9) ∪ (3, ∞) Not included Included, LHS = 89 > 0 ❌ Eliminated D (−∞, −9) ∪ (9, ∞) Not included Included, LHS = 89 > 0 ❌ Eliminated Answer & Explanation cc

CAT Inequalities Bible Top 10 Must Solve Inequalities Questions covering 80% of Syllabus

Every real CAT question on this chapter from the past papers we have, newest year first. Passages and data sets stay together on one page.

Tap Show answer, or open the detailed solution (free login). Page 1 of 11.

Start here: the repeating CAT PYQs

Study the inequality pattern in each CAT PYQ, then practise variations. The 15 selected families contain 49 of the 55 questions tagged Inequalities in the reviewed set.

CAT Inequalities Bible Top 10 PYQ Questions

Questions	CAT Area Covered
First 5 CAT Questions	52.73%
First 10 CAT Questions	80.00%
All 15 CAT Questions	89.09%

Importance of Inequalities each year of CAT exam

CAT year	Total Inequalities questions	First 6 examples	First 10 examples	All 15 examples	Outside these 15 families	Share of all Quant
2021	8	6	7	7	1	12.12%
2022	18	12	14	18	0	27.27%
2023	11	7	8	9	2	16.67%
2024	7	3	5	5	2	10.61%
2025	11	5	10	10	1	16.67%
Total	55	33	44	49	6	16.67%
Share of all 55	100%	60.00%	80.00%	89.09%	10.91%

Quick review for beginners and last day revision

Rule	Formula or what to do	Small example
Add or subtract	Adding or subtracting the same number preserves the sign.	x + 3 < 8 → x < 5
Multiply or divide	Reverse the inequality when multiplying or dividing by a negative.	−2x < 6 → x > −3
Combine bounds	Use the intersection when all conditions must hold.	x > 2 and x ≤ 5 → 2 < x ≤ 5
Endpoints	Strict < or > excludes equality; ≤ or ≥ includes it.	x² < 9 → −3 < x < 3
Product sign chart	Factor, mark zeros and poles, then test the sign between them.	(x − 1)(x − 4) < 0 → 1 < x < 4
Rational inequalities	Never cross-multiply by an expression of unknown sign without cases.	1/x > 0 requires x > 0; x = 0 is excluded.
Absolute value	For a ≥ 0, \|x\| ≤ a means −a ≤ x ≤ a.	\|x − 3\| < 2 → 1 < x < 5
Discriminant	For a real quadratic root, b² − 4ac ≥ 0.	Use a discriminant inequality to bound a parameter.
Monotonic powers and logs	A base above 1 preserves order; a base between 0 and 1 reverses it.	(1/2)ˣ < (1/2)³ → x > 3
Integer answers	Solve the real bounds first, then count allowed integers.	2 < n < 7 → n = 3, 4, 5, 6

10 Core CAT PYQ Questions

Study these first.

CAT PYQ 1: Exponential bounds and smallest integer periods

Understand: Find the allowed whole numbers from the two bounds, then count only those that keep the expression valid.

CAT 2021 Slot 2 · Quant

For all possible integers n satisfying \(2.25 \leq 2 + 2^{n+2} \leq 202\), then the number of integer values of \(3 + 3^{n+1}\) is:

CAT PYQ 2: Discriminant bounds for real roots or ranges

Understand: If the discriminant is negative, the quadratic never touches zero, so its sign is always the sign of a.

CAT 2022 Slot 1 · Quant

Let a, b, c be non-zero real numbers such that \(b^{2} < 4ac\), and \(f(x) = ax^{2} + bx + c\). If the set S consists of all integers m such that \(f(m) < 0\), then the set S must necessarily be

CAT PYQ 3: Linear allocation bounds with nonnegative counts

Understand: Write the counts as multiples of one number k, apply the total limit, and take the largest whole k.

CAT 2022 Slot 1 · Quant

Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3 : 5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is

CAT PYQ 4: Absolute value intervals and sign cases

Understand: Compare the distances from each point. The two conditions give one interval, then count the integers in it.

CAT 2021 Slot 1 · Quant

The number of integers n that satisfy the inequalities \(\mid n - 60 \mid<\mid n - 100 \mid<\mid n - 20 \mid\) is

CAT PYQ 5: Ordered values and average bounds

Understand: Use the change in the total to link the first value to N, then use the ordering to bound N.

CAT 2022 Slot 2 · Quant

The average of a non-decreasing sequence of N numbers a1, a2, ..., aN is 300. If a1 is replaced by 6a1, the new average becomes 400. Then, the number of possible values of a1 is

CAT PYQ 6: Logarithmic bounds and monotonicity

Understand: Change the logs to one base, combine them, and use the limits on the variables to cap the answer.

CAT 2024 Slot 2 · Quant

If \(a, b\) and \(c\) are positive real numbers such that \(a>10 \geq b \geq c\) and \(\frac{\log _8(a+b)}{\log _2 c}+\frac{\log _{27}(a-b)}{\log _3 c}=\frac{2}{3}\), then the greatest possible integer value of \(a\) is

CAT PYQ 7: Rational sign charts and excluded poles

Understand: Factor, mark the roots and the excluded points, and test the sign in each part.

CAT 2021 Slot 1 · Quant

\(f(x) = \dfrac{x^2 + 2x - 15}{x^2 - 7x - 18}\) is negative if and only if

CAT PYQ 8: Set overlap bounds and feasible endpoints

Understand: Add the set sizes. The overlap is at least the excess over the total, and at most the smallest set.

CAT 2022 Slot 1 · Quant

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

CAT PYQ 10: Floor intervals and endpoint inclusion

Understand: Find where the floor value changes, then see how far the sum can go before it passes the target.

CAT 2022 Slot 1 · Quant

For any real number x, let [x] be the largest integer less than or equal to x. If \(\displaystyle\sum_{n=1}^{N} \left[ \dfrac{1}{5} + \dfrac{n}{25} \right] = 25\), then N is

Variations

Same idea, extra twist.

CAT PYQ 11: Factorial size bounds

Understand: Check which factorial sizes still fit under the bound, and stop at the first one that does not.

CAT 2022 Slot 2 · Quant

For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is

CAT PYQ 12: Bounds on eligible AP terms

Understand: Find the first and last terms inside the allowed range, then use the two ends for the average.

CAT 2022 Slot 3 · Quant

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

CAT PYQ 13: Remainders with an upper population bound

Understand: Use the LCM to write the form of the number, apply the upper bound, then use the extra condition.

CAT 2022 Slot 3 · Quant

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

CAT PYQ 14: Polygon side inequality

Understand: Each side must be shorter than the sum of the other three. Solve for the fourth side.

CAT 2022 Slot 3 · Quant

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

CAT 2025 Slot 1 · QuantEasy

Let 3 ≤ x ≤ 6 and [x²] = ([x])², where [x] is the greatest integer not exceeding x.
If S represents all feasible values of x, then a possible subset of S is

CAT 2025 Slot 1 · QuantEasy

A value of c for which the minimum value of f(x) = x² – 4cx + 8c is greater than the maximum value of
g(x) = –x² + 3cx – 2c, is

CAT 2025 Slot 1 · QuantHard

For any natural number k, let aₖ = 3ᵏ. The smallest natural number m for which (a₁)¹ × (a₂)² × ... × (a₂₀)²⁰ < (a₂₁ × a₂₂ × ... × a₂₀₊ₘ)⁵ is:

CAT 2025 Slot 1 · QuantEasy

The number of distinct integers n for which log_(1/4) of (n² − 7n + 11) > 0 is _____.

CAT 2025 Slot 2 · QuantEasy

The set of all real values of x for which (x² − |x + 9| + x) > 0, is

Frequently asked questions

Where can I find CAT inequalities questions with solutions?

Every Inequalities question on this page is a real CAT question from the CAT Actual Quant Papers. Tap Show answer for the answer; the detailed step-by-step solution opens after a free login. Start with the CAT PYQs at the top, because they repeat the most.

Are there CAT inequalities questions with answers?

Yes. Every question has its answer one tap away, free for everyone: pick an option and press Submit, or tap Show answer.

Is there a PDF of CAT inequalities questions?

Yes. Tap Download PDF at the top of this page (free, after a quick OTP login) for the CAT Inequalities Bible Top 10 PYQ Questions, a free PDF of the 10 most repeated CAT inequalities questions. All the CAT Inequalities questions are also on this page to read online, with answers and, after a free login, detailed solutions.

What are the CAT inequalities previous year questions (PYQs)?

They are the real questions of the CAT Actual Quant Papers that are about inequalities: 55 questions, grouped into 15 repeating CAT PYQs, with the other questions listed after them in paper order.

Which inequalities questions should I practise first for CAT?

Start with the first 6 CAT PYQs, which cover about 60% of the Inequalities questions. Then do the first 10 (about 80%) and finally all 15 (about 90%).

How many Inequalities questions came in CAT papers?

38 Inequalities questions are on this page: 10 in CAT 2025, 7 in CAT 2024, 10 in CAT 2023, 9 in CAT 2022, 2 in CAT 2021. The same real question is shown once.

How difficult are CAT Inequalities questions?

Of the 10 Inequalities questions that carry a difficulty rating, 5 are Easy, 3 Medium and 2 Hard; the other 28 are not rated yet.

Are the solutions free?

The question and its answer are free for everyone. The detailed Cetking solution opens after a free login with your mobile number (OTP).

How is a set shown on this page?

Each passage or data set gets one page with all its questions together (2 sets here), so you never solve half of one. Questions that stand alone are shown five to a page.

Where can I practise more Inequalities?

Open all CAT papers and topics for every paper, or the Cetking Arena to practise.

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.

CAT Inequalities Bible Top 10 Must Solve Inequalities Questions covering 80% of Syllabus

Start here: the repeating CAT PYQs

10 Core CAT PYQ Questions

Solution — first steps

Solution — first steps

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Solution — first steps

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Variations

Solution — first steps

Solution — first steps

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Frequently asked questions