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.

If 9^(x² + 2x – 3) − 4(3^(x² + 2x − 2)) + 27 = 0, then the product of all possible values of x is _____ Options: A) 20 B) 15 C) 30 D) 5

A. 20. Step 1: Write everything with base 3 9 = 3², so ⇒ 9^(x² + 2x − 3) = 3^(2x² + 4x − 6) Also, 3^(x² + 2x − 2) = 3·3^(x² + 2x − 3) Let t = 3^(x² + 2x − 3), where t > 0. Then the equation becomes: t² − 4(3t) + 27 = 0 ⇒ t² − 12t + 27 = 0 Step 2: Solve the quadratic in t t² − 12t + 27 = 0 ⇒ (t − 3)(t − 9) = 0 ⇒ So t = 3 or t = 9

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 Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% 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 quadratic equation pattern in each CAT PYQ, then practise variations. The 15 selected families contain 51 of the 51 questions tagged Quadratics in the reviewed set.

CAT Quadratic Equations Bible Top 10 PYQ Questions

Questions	CAT Area Covered
First 5 CAT Questions	58.82%
First 10 CAT Questions	86.27%
All 15 CAT Questions	100%

Importance of Quadratic Equations each year of CAT exam

CAT year	Total Quadratic Equations questions	First 6 examples	First 10 examples	All 15 examples	Outside these 15 families	Share of all Quant
2021	10	7	8	10	0	15.15%
2022	8	5	7	8	0	12.12%
2023	13	10	12	13	0	19.70%
2024	8	7	8	8	0	12.12%
2025	12	5	9	12	0	18.18%
Total	51	34	44	51	0	15.45%
Share of all 51	100%	66.67%	86.27%	100%	0%

Quick review for beginners and last day revision

Rule	Formula or what to do	Small example
Standard form	ax² + bx + c = 0, with a ≠ 0	Move all terms to one side before identifying coefficients.
Factorisation	If (x − r)(x − s) = 0, then x = r or x = s.	x² − 5x + 6 = 0 → x = 2 or 3.
Quadratic formula	x = [−b ± √(b² − 4ac)]/(2a)	Use when simple factorisation is not available.
Discriminant	D = b² − 4ac: positive → two distinct real roots; zero → repeated root; negative → no real roots.	Always use a ≠ 0.
Root relations	If roots are α,β: α + β = −b/a; αβ = c/a	x² − 5x + 6 → sum 5, product 6.
Vertex	Vertex at x = −b/(2a); value c − b²/(4a).	x² − 6x + 10 = (x − 3)² + 1.
Nonnegative quadratic	If a > 0 and D ≤ 0, the quadratic is nonnegative for every real x.	A known zero then requires a repeated root.
Common root	Substitute the shared root into both equations and eliminate its square.	Multiply one equation so squared-term coefficients match.
Conjugate roots	For rational coefficients, p + q√r and p − q√r occur together when the root is irrational.	2 + √3 implies the other root is 2 − √3.
Substitution and checks	A logarithmic, exponential or word problem may become quadratic. Retain only roots valid in the original problem.	If t = 2ˣ, then t > 0.

10 Core CAT PYQ Questions

Study these first.

CAT PYQ 1: Quadratic modelling of counts, journeys and work rates

Understand: Call one count n and write the other in terms of n. Form the equation, solve, and drop any negative root.

CAT 2021 Slot 1 · Quant

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

CAT PYQ 2: Root sums, products and conjugate roots

Understand: Sum of roots is minus b over a and product is c over a. An irrational root comes with its conjugate.

CAT 2021 Slot 2 · Quant

Suppose one of the roots of the equation \(ax^2 - bx + c = 0\) is \(2 + \sqrt{3}\), Where a,b and c are rational numbers and \(a \neq 0\). If \(b = c^3\) then \(|a|\) equals.

CAT PYQ 3: Successive percentages, replacement and price changes

Understand: Multiply the factors for each step. The result is a quadratic, so keep only the root that fits the situation.

CAT 2021 Slot 2 · Quant

From a container filled with milk, 9 litres of milk are drawn and replaced with water. Next, from the same container, 9 litres are drawn and again replaced with water. If the volumes of milk and water in the container are now in the ratio of 16 : 9, then the capacity of the container, in litres, is

CAT PYQ 4: Discriminant conditions and integer parameters

Understand: If b squared is less than 4ac the quadratic never reaches zero, so it always has 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 5: Modulus or square root branches producing quadratics

Understand: Split the modulus into two equations and count the roots of each. Exactly three roots means one branch has a single root.

CAT 2021 Slot 1 · Quant

If r is a constant such that \(\mid x^2 - 4x - 13 \mid= r\) has exactly three distinct real roots, then the value of r is

CAT PYQ 8: Logarithms converted to quadratic equations

Understand: Remove the logarithm using its definition, then let x squared be a new variable to get a quadratic.

CAT 2023 Slot 1 · Quant

If \(x\) and \(y\) are positive real numbers such that \(\log _x\left(x^2+12\right)=4\) and \(3 \log _y x=1\), then \(x+y\) equals

CAT PYQ 9: Shared roots eliminated between equations

Understand: Subtract the equations to remove the squared term. Then use the sign of the common root.

CAT 2024 Slot 1 · Quant

If the equations \(x^2+m x+9=0, x^2+n x+17=0\) and \(x^2+(m+n) x+35=0\) have a common negative root, then the value of \((2 m+3 n)\) is

CAT PYQ 10: Rational expression range through a discriminant

Understand: Put the expression equal to c to form a quadratic. It must have real roots, so the discriminant cannot be negative.

CAT 2022 Slot 3 · Quant

If \(c = \dfrac{16x}{y} + \dfrac{49y}{x}\) for some non-zero real numbers \(x\) and \(y\), then \(c\) cannot take the value

Variations

Same idea, extra twist.

CAT PYQ 11: Grouped identities reduce coupled equations

Understand: Add the two equations to get a quadratic in x minus y.

CAT 2021 Slot 2 · Quant

Consider the pair of equations: \(x^2 - xy - x = 22\) and \(y^2 - xy + y = 34\). If \(x > y\), then \(x - y\) equals

CAT PYQ 12: Geometric length constraints producing quadratics

Understand: Use the cosine rule to get a quadratic in the unknown side. Keep the positive root.

CAT 2021 Slot 3 · Quant

Let \(ABCD\) be a parallelogram. The lengths of the side \(AD\) and the diagonal \(AC\) are 10 cm and 20 cm, respectively. If the angle \(\angle ADC\) is equal to \(30^\circ\) then the area of the parallelogram, in sq.cm is

CAT PYQ 15: Consistency of a linear system gives a quadratic parameter

Understand: For infinitely many solutions the second equation is a multiple of the first, so match the coefficients.

CAT 2023 Slot 3 · Quant

For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is

CAT 2025 Slot 1 · Quant

The number of non-negative integer values of k for which the quadratic equation x² – 5x + k = 0 has only integer roots, 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 2 · QuantEasy

Rita and Sneha can row a boat at 5 km/h and 6 km/h in still water, respectively. In a river flowing with a constant velocity, Sneha takes 48 minutes more to row 14 km upstream than to row the same distance downstream. If Rita starts from a certain location in the river, and returns downstream to the same location, taking a total of 100 minutes, then the total distance, in km, Rita will cover is ______

CAT 2025 Slot 2 · QuantHard

If 9^(x² + 2x – 3) − 4(3^(x² + 2x − 2)) + 27 = 0,
then the product of all possible values of x 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 quadratic equation questions with solutions?

Every Quadratic Equations 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 15 CAT PYQs at the top, because they repeat the most.

Are there CAT quadratic equation 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 quadratic equation questions?

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

What are the CAT quadratic equation previous year questions (PYQs)?

They are the real questions of the CAT Actual Quant Papers that are about quadratic equations: 51 questions, grouped into 15 repeating CAT PYQs, with the other questions listed after them in paper order.

Which quadratic equation questions should I practise first for CAT?

Start with the first 6 CAT PYQs, which cover about 70% of the Quadratic Equations questions. Then do the first 10 (about 90%) and finally all 15 (all of them).

How many Quadratic Equations questions came in CAT papers?

38 Quadratic Equations questions are on this page: 9 in CAT 2025, 9 in CAT 2024, 11 in CAT 2023, 6 in CAT 2022, 3 in CAT 2021. The same real question is shown once.

How difficult are CAT Quadratic Equations questions?

Of the 8 Quadratic Equations questions that carry a difficulty rating, 4 are Easy, 2 Medium and 2 Hard; the other 30 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 (4 sets here), so you never solve half of one. Questions that stand alone are shown five to a page.

Where can I practise more Quadratic Equations?

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 15 repeating CAT PYQs and how much each adds

CAT PYQ	Repeating CAT PYQ	Questions	Representative	Cumulative questions	Cumulative share
1	Quadratic modelling of counts journeys and work rates	9	CAT 2021 S1 Q16	9	17.65%
2	Root sums products and conjugate roots	7	CAT 2021 S2 Q19	16	31.37%
3	Successive percentages replacement and price changes	6	CAT 2021 S2 Q4	22	43.14%
4	Discriminant conditions and integer parameters	4	CAT 2022 S1 Q5	26	50.98%
5	Modulus or square root branches producing quadratics	4	CAT 2021 S1 Q8	30	58.82%
6	Exponential substitutions and quadratic exponents	4	CAT 2023 S2 Q2	34	66.67%
7	Vertex repeated root and perfect square forms	3	CAT 2021 S3 Q22	37	72.55%
8	Logarithms converted to quadratic equations	3	CAT 2023 S1 Q1	40	78.43%
9	Shared roots eliminated between equations	2	CAT 2024 S1 Q20	42	82.35%
10	Rational expression range through a discriminant	2	CAT 2022 S3 Q15	44	86.27%
11	Grouped identities reduce coupled equations	2	CAT 2021 S2 Q3	46	90.20%
12	Geometric length constraints producing quadratics	2	CAT 2021 S3 Q11	48	94.12%
13	Balanced AP totals producing a quadratic index	1	CAT 2025 S1 Q4	49	96.08%
14	GP terms related by a quadratic product	1	CAT 2025 S2 Q8	50	98.04%
15	Consistency of a linear system gives a quadratic parameter	1	CAT 2023 S3 Q3	51	100%

Based on CAT Actual Quant Papers.

CAT Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% of Syllabus

Start here: the repeating CAT PYQs

10 Core CAT PYQ Questions

Solution — first steps

Solution — first steps

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Variations

Solution — first steps

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Frequently asked questions