CAT Aspirant's Triad: Mastering Quadratic Equations, Introduction to Cubes, and Tenses All in One - Day 14

Part-1: Quantitative Aptitude for CAT 2025

In this section, we will focus on Quantitative Aptitude specifically for CAT 2025. You’ll learn key concepts, problem-solving methods, and useful shortcuts. You can also go through practice questions to strengthen your skills and build confidence in handling quantitative problems.

Quadratic Equation Questions For CAT

Quadratic Equations are equations with degree 2. Quadratic Equations are an important part of Algebra as well as for the CAT exam. You can expect 1 or 2 algebra questions for the CAT Exam (based on analysis of previous year's question papers).
Quadratic Equations are of the form of ax2 + bx + c =0.
Here ‘x’ denotes variables (unknown) and a, b, and c are constants and a≠ 0.
To prepare CAT Quadratic Equations questions, you should invest 4-5 hours. Here we will discuss the quadratic equation CAT questions, quadratic equation formula, quadratic equation solutions, roots of quadratic equations, Quadratic Equations examples etc.
Quadratic equation problems for CAT can be categorised into the following types:

• Questions that involve calculations of some unknown variable.

• Questions that involve calculations of the nature of roots.

• Formation of the quadratic equation.

How to solve quadratic Equations problems for CAT and quadratic equation questions for CAT.

There are three methods which are generally used to find the roots (zeros) of a quadratic equation.

Method of factorization
Example: 2x2 + 5x + 3 = 0
⇒ 2x2 + 2x + 3x + 3 = 0
⇒ 2x(x + 1) + 3(x + 1) = 0
⇒ (x + 1) (2x + 3) = 0
⇒ x = -1 or x = -3/2.
This method is suitable only if factors of the quadratic polynomial are possible.

Quadratic Formula (Sridharacharya Formula)

For equation ax2 + bx + c =0.
x = [-b ± √D] /2a
where D is discriminant which is equal to (b2 - 4ac).
Example: 2x2 + 5x + 3 = 0
D = 52 - 4 x 2 x 3 = 1
x = [-5 ± √1] /4 = [-5 ± 1] /4
⇒ x = -1 or -3/2.

Method of completing the square:

Convert the given equation in the form of either (A+B)2 or (A-B)2
Example:
2x2+ 5x + 3 = 0
⇒ x2 + 5x/2 + 3/2 = 0
⇒ x2 + 2 x 5x/4 + 25/16 – 25/16 + 3/2 = 0
⇒ (x + 5/4)2 – 1/16 = 0
⇒ (x + 5/4)2 = 1/16
⇒ (x + 5/4) = ± ¼
⇒ x = ± ¼ - 5/4
⇒ x = -1 or -3/2

CAT quadratic equation questions with solutions:

The two Previously Asked Questions from Quadratic Equations in the CAT exam are given below, aspirants can also download the CAT Previous Year Algebra Questions pdf for more practice and references.

Q.1. Suppose one of the roots of the equation ax2 - bx + c = 0 is 2 + √3, where a, b, and c are rational numbers and a ≠ 0. If b = c3 then |a| equals [CAT question paper - 2021]

1. 2

2. 3

3. 4

4. 1

Solution: (a)

For a quadratic equation, ax2 + bx + c = 0.
Since the roots, a, b, and c are all rational and if one of the roots is irrational, the other root of the equation will also be irrational and a conjugate of the first root.
So, the other root is 2 - √ 3.
Sum of roots = −(-b)/a = b/a = (2 + √ 3) + (2 - √ 3) = 4
And Product of roots = c/a = (2 + √ 3) (2 - √ 3) = 4 - 3 = 1
But here the quadratic equation is ax2 - bx + c = 0.
Therefore, b = 4a and c = a
It is given that b = c3
4a = a3
So, a = 2 or -2.
So, |a| = 2.

Q.2. Let m and n be positive integers, If x2 + mx + 2n = 0 and x2 + 2nx + m = 0 have real roots, then the smallest possible value of m + n is [CAT Question Paper 2020]

1. 8

2. 6

3. 5

4. 7

Ans: (b)
CAT 2025 quadratic equations questions with answers
For real roots b2 ≥ 4ac
From first equation: m2 ≥ 4 (2n)
⇒ m2 ≥ 8n
From second equation: (2n)2 ≥ 4m
⇒ 4n2 ≥ 4m
⇒ n2 ≥ m
If n = 1; m2≥ 8; m ≥ 3
If m ≥ 3 at n = 1, n2 ≥ m is not true.
If n = 2; m2 ≥ 16; m ≥ 4
If m ≥ 4 at n = 2, n2 ≥ m which is a valid result, if m = 4 and n = 2
So, the minimum sum of m and n is 4+2 = 6.

Q.3. The sum of twice a number and thrice its reciprocal is 25/2. What is the number?
(a) 7 (b) 6 (c) 5 (d) 4

Ans: (b)

Given: The sum of twice a number and thrice its reciprocal is 25/2.

Let the number be x.

According to the question,

⇒ 2x + 3/x = 25/2

⇒ 4x2 + 6 = 25x

⇒ 4x2 – 25x + 6 = 0

⇒ 4x2 – 24x – x + 6 = 0

⇒ (4x – 1)(x – 6) = 0

⇒ (4x–1) = 0 or (x–6) = 0

⇒ x = 6 or ¼

After checking the options, we get, x=6

Hence, the correct answer is option (b).

Q.4. If (x – 2)(x – p) = x2 – ax + 6, then the value of (a – p) is:

(a) 0 (b) 1 (c) 2 (d) 3

Ans: (c)
Given: (x – 2)(x – p) = x2 – ax + 6

⇒ x2– px – 2x + 2p=x2 – ax + 6

⇒ x2 – (p + 2)x + 2p=x2 –ax + 6

By comparing both sides of the equation,

p + 2 = a

⇒ a – p = 2

Hence, the correct answer is option (c).
Q.5. If x2 + y2 + 2x + 1 = 0, then the value of x31 + y35 is:

(a) –1 (b) 1 (c) 2 (d) 0

Ans: (a)

Given:

x2 + y2 + 2x + 1 = 0

⇒ x2 + 2x + 1 + y2 = 0

⇒ (x+1)2 + y2 = 0

Since the addition of the squares of the two terms is zero,

So, both the terms are individually zero.

Therefore, (x + 1)2 = 0 and y2 = 0

⇒ x = –1 and y = 0

Now, x31 + y35

= (–1)31 + (0)35

= –1

Hence, the correct answer is option (a).

Prerequisite to Solve Quadratic Equation Questions For CAT

For this topic, students must have knowledge of

• Linear Equations

• Factors

• Degree of Polynomials and zeros of a polynomial.

Tips to Prepare CAT previous year quadratic equations questions:

• Learn all the methods of solving Quadratic Equations and use those methods according to the situation.

• If the sum of zeros of a quadratic equation is ‘s’ and the product of zeros is ‘p’. Then the quadratic equation will be x2 - sx + p = 0

• The nature of zeros is calculated by just finding the value of D.
  If D = 0, zeros/ roots are real and equal.
  If D > 0, zeros/ roots are real and unequal.
  If D < 0, zeros/ roots are imaginary.

• The sum of the roots = -b/a; Product of roots = c/a.

• Always check for the feasibility of solutions.

References to Study (Theory):

• How to Prepare Quantitative Aptitude by Arun Sharma (6th edition)

• 2iim (You-Tube)

References to Study (Practice) Quadratic Equation Problems For CAT:

• NCERT Class 9-10

• https://iim-cat-questions-answers.2iim.com

• CAT Previous Year's Question Papers

Part 2: CAT Logical Reasoning and Data Interpretation:

Questions based on cubes play an important role in CAT. Here we will discuss cube CAT questions, Cube Logical Reasoning questions and tricks etc.

Topic: Cube Part-1 (LR) And CAT Questions On Cubes

A cube is a 3-D figure which consists of 6 faces, 12 edges, and 8 vertices.

CAT questions on Cube are based on the following concepts:

• Painted surface

• Opposite faces

• Consecutive faces

• Number of cuts to make smaller cubes of equal dimension

Approaches and Tips to Solve Cube Questions Logical Reasoning:

1. If all the faces of a cube (n x n x n) painted with the same colour are cut into (n x n x n) equal parts.

• Number of cubes with 3 faces painted = 8 cubes

• Number of cubes with exactly 2 faces painted = (n-2) x 12

• Number of cubes with exactly 1 face painted = (n – 2)2 x 6

• Number of cubes with 0 faces painted = (n-2)3

2. You can assume your room is a cube to analyze the situation.

3. You can use the approach of the open cube to solve the question.

Six faces can be seen like this
(1,6), (2,4), and (3,5) form opposite pairs.
Adjacent faces to face 3 are 1,2,4 and 6. Similarly, we can find the others.

4. Number of cuts to small cubes of equal size (n) from a larger cuboid of dimension (a x b x c) = (a/2 -1) + (b/2 -1) + (c/2 -1)

Questions that can help you to build the concept of Cubes (Based on CAT EXAM):

An Iron cube is coloured black on four adjoining surfaces while the opposite sides are coloured red. The cube is then cut two times vertically and four times horizontally at right angles. Based on the same answer to the next question.

Q.1. How many cubes (cuboids) have exactly one face coloured black?

1. 1

2. 2

3. 3

4. 4

Ans: D

One cube/cuboid at each black face has exactly one face black colour.

Q.2. How many cubes/cuboids have only one face painted that too in red?

1. 2

2. 3

3. 4

4. 5

Ans: A
The cube/cuboid with each red face has exactly one face red coloured.

Q.3. How many cubes/cuboids have more than 2 sides painted?

1. 2

2. 4

3. 6

4. 8

Ans: D
More than two faces painted means 3 faces painted i.e. a number of cubes formed at each corner.

Prerequisite to solve Cube And Cuboid Questions:

To Prepare the questions based on Cube, students must have the knowledge of

• This topic requires basic calculations

• Imagination

• Analytical Approach

References to Study (Theory):

• Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)

• Cat preparation-iQuanta (YouTube Channel)

References to Study (Practice):

• Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)

• LRDI for CAT and GMAT by Nishit K Sinha. (Advance)

• Previous Year's Question Papers

Part 3: CAT Exam VARC

Get the practice questions, best references and tips to prepare for the CAT Verbal Ability and Reading Comprehension.

Sentence Correction Questions Based on Tenses

In previous articles, we have made strategies to prepare for tenses. Today we are going to discuss how to solve problems of sentence correction based on Tenses. Also, we will explain error correction based on tenses, grammatical error correction, error correction examples etc.

Errors in sentences can be due to any of the following reasons:

• Confusion in present tenses

• Confusion in past tenses

• Confusion in future tenses

• Use of the wrong verb

You have to identify which type of error is there in the sentence and prepare them accordingly.

Practice Questions on Sentence correction that can boost your preparation: [Based on PYQs]

CAT Verbal Ability Questions
Here are some CAT verbal ability questions curated for aspirants to practice:
Find the correct phrase for underlining.

Q.1. The glass of the car broke while the players were playing football.

1. Broke

2. Was broken

3. No correction is required

4. Was broke

Ans: a
The action was done in the past but the time is not known. So, Past indefinite tense is used.

Q.2. I will be meeting you in the morning.

1. Meet

2. Meeting

3. No correction is required

4. Met

Ans: (a)
Future indefinite tense should be used.

Q.3. Preeti attended dance class every Thursday.

1. Attends

2. No correction required

3. Is attending

4. Will attending

Ans: (a)
Action is in the present time at a regular interval of time. Present indefinite should be used.

Q.4. Rani was playing when her father entered the room.

1. Played

2. No correction required

3. Plays

4. Is playing

Ans: (b)
Past continuous tense is to be used as the other sentence is in Past indefinite tense.

Q.5. It had stopped raining when I was leaving the office.

1. Leave

2. Was leaving

3. Left

4. Leaves

Ans: (c)
Past perfect tense should be used as the first phrase in the past perfect tense.

Prerequisite to solve problems of sentence correction based on tenses:

• Basic Knowledge of Nouns, Pronouns and Verbs.

• Basic knowledge of Tenses.

• Knowledge of the use of helping verbs.

Tips to Prepare Error Correction in Tenses:

• Practice a lot to improve your grammar.

• Write rules and revise them on a regular basis.

• During the examination, check on every possibility of error.

• Attempt Mock-Test papers and solutions after the test to check your mistakes.

References to Study (Theory):

• Advanced English grammar by Wren and Martin

• GMATPrepNow (Youtube Channel)

References to Study (Practice):

• How to Prepare for Verbal Ability & Reading Comprehension for CAT by Meenakshi Upadhyay and Arun Sharma (9th Edition)

• Practice CAT Sample Question Papers.

Best Resources to Practice Questions for CAT 2025

Practising with CAT 2025 sample papers, previous year question papers, and mock tests is crucial for effective exam preparation. These resources help you understand the exam pattern, difficulty level, and question types. Go through the section below to know about the best resources to prepare for the CAT exam 2025:

Sample Papers

CAT sample papers are designed to mirror the actual exam pattern and question types, offering candidates a clear understanding of what to expect. These papers help in familiarising oneself with the structure and distribution of topics across sections. By practising sample papers regularly, aspirants can develop problem-solving techniques, improve their confidence, and fine-tune their preparation strategies based on recurring question trends and sectional weightages.

Previous Year Question Papers

Practising CAT previous year question papers is one of the best ways to understand the actual exam format and difficulty level. These papers provide insight into the types of questions frequently asked, helping identify important topics. They also assist in recognising changes in trends over the years. Solving them under timed conditions boosts speed and accuracy, making candidates better equipped for the real CAT exam environment.

Mock Tests

Mock tests are essential tools in CAT 2025 preparation, offering a realistic simulation of the actual exam. Taking full-length mocks helps in developing effective time management skills and building exam stamina. Detailed performance analysis after each CAT mock test highlights strengths and weaknesses, enabling focused revision. Regular mock test practice also reduces exam anxiety, improves confidence, and ensures that aspirants are mentally and strategically prepared for exam day.

Free PDFs for CAT 2025 preparation

Aspirants aiming for strong performance in CAT 2025 can benefit greatly from structured resources available in PDF format. These free PDFs for CAT 2025 preparation provide focused practice and conceptual clarity across all sections. One such valuable resource is the CAT study material 2025 for Geometry and Mensuration, which is part of a well-designed, expert-recommended CAT 60-day study material. These PDFs follow the updated exam pattern and include practice questions with detailed solutions. Aspirants can use them for both revision and daily topic-wise practice. Regular use of these resources can significantly improve problem-solving speed and accuracy.