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CAT Aspirant's Triad: Mastering Quadratic Equations, Introduction to Cubes, and Tenses All in One - Day 14
Edited By Team Careers360 | Updated on Jul 19, 2025 01:10 PM IST | #CAT
Ongoing Event
CAT Application Date : 01 Aug' 2025 - 13 Sep' 2025
With the CAT 2025 exam getting closer, many students start asking how to prepare smartly, what study plan to follow, and which books or mock tests to trust for both theory and practice. Doubts about how much time to give, how to structure daily study, and whether crash courses actually help are very common. In our 60-day CAT preparation plan, Day 14 is all about mastering quadratic equations. This topic builds directly on what you learned on Day 13, where you tackled linear equations, logical reasoning questions on Directions, and future tense practice in VARC. Today’s goal is to make you more confident in solving equation-based problems step by step.
This Story also Contains
- Part-1: Quantitative Aptitude for CAT 2025
- Part 2: CAT Logical Reasoning and Data Interpretation:
- Part 3: CAT Exam VARC
- Best Resources to Practice Questions for CAT 2025
- Free PDFs for CAT 2025 preparation
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]
2
3
4
1
CAT 2025: VARC, DILR, and Quant MCQs & Weightages
Comprehensive CAT prep guide with focused practice on Verbal Ability, Data Interpretation & Logical Reasoning, and Quantitative Aptitude.
Download NowSolution: (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]
8
6
5
7
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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
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
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:
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
You can assume your room is a cube to analyze the situation.
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.
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
2
3
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?
2
3
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?
2
4
6
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.
Broke
Was broken
No correction is required
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.
Meet
Meeting
No correction is required
Met
Ans: (a)
Future indefinite tense should be used.
Q.3. Preeti attended dance class every Thursday.
Attends
No correction required
Is attending
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.
Played
No correction required
Plays
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.
Leave
Was leaving
Left
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.
Frequently Asked Questions (FAQs)
1. Can a quadratic equation have more than two roots?
No, a quadratic equation cannot have more than two roots. We know a quadratic equation is a polynomial of degree 2, which means it can have a maximum of two solutions or two roots, either real or complex.
2. What is the reasoning formula for a cube?
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.
3. Why do errors appear in sentences?
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
4. How can I prepare for error correction or sentence improvement for CAT 2024?
There are several ways one can improve their error correction or sentence improvement skills, these are as follows:
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.
5. Why are quadratic equations important for CAT preparation?
Quadratic equations form a core part of Algebra in the Quant section and frequently appear in CAT with moderate to high difficulty.
6. What should I know before starting cube-related questions in CAT?
You should understand cube numbers, cube roots, and basic algebraic identities. These concepts help solve number-based and logic-based cube problems efficiently.
7. How do tenses help in the VARC section of CAT?
A good grasp of tenses improves sentence correction and comprehension accuracy, which are common in VARC grammar and usage questions.
8. Can I study quadratic equations, cubes, and tenses together in one day?
Yes, with a well-planned schedule. Spend focused time on each section and practice small question sets to build clarity and confidence.
9. Are these topics part of the official CAT syllabus?
While the CAT doesn’t have a fixed syllabus, these topics are consistently seen in past papers and are essential for strong performance.
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Have a question related to CAT ?
Hello aspirant,
The CUSAT CAT 2025 inaugural real-time spot admission will take place at Cochin University of Science and Technology (CUSAT) from August 4–6. For admission to a variety of undergraduate (B.Tech) and graduate (MSc) programs, the Common Admission Test, or CUSAT CAT, is an entrance exam. Every year, it features sections in mathematics/biology, chemistry, and physics.
For complete information, you can visit our site through following link:
https://www.careers360.com/exams/cusat-cat
Thank you
Sajal Trivedi 03 August,2025
Hello Aspirant,
Yes, if your caste is listed under Central OBC and you come under the category of Non-Creamy Layer, then you shall be eligible for the OBC-NCL reservation in the CAT exam and IIM admissions, irrespective of its being categorized as OEC in Kerala. Remember to get a valid OBC-NCL certificate obtained as per the format prescribed by the Government of India.
gopi 30 July,2025
Hey Vijay:)
Thanks for reaching out!
OEC,(Other Eligible Communities) is a state-specific reservation category used in Kerala,and it is not recognised as a separate category in central-level exams like the CAT.
For Central govt exams candidates must choose from the recognised categories: General,SC,ST,OBC-NCL,EWS or PwD.If your community is listed under OEC-SC or OEC-ST in Kerala,you should check whether it falls under SC or ST in the central list and apply accordingly.
Similarly,If your OEC community is in the central OBC list and you meet the non-creamy layer(NCL)rules,you can apply as OBC-NCL, otherwise you need to apply under the general category.
You can verify caste's status in the central list on the National Commission for Backward Classes (NCBC) website.
Hope this clears your doubt!
Taabia Fathima 30 July,2025
You can easily get CAT previous year question papers online in PDF format, with solutions too. Websites like CollegeDunia, Careers360, SelfStudys, Cracku, and Aglasem have papers from CAT 2024, 2023, 2022… even back to the 1990s.
Ankur Mehta 01 August,2025
Hi dear candidate,
The score of your CAT exam is valid for one year from the date of exam.
For example: If your CAT exam was held on 24th November 2024 then your score is valid till 31st December 2025 meaning that you can take admission in the academic year 2025 - 27.
BEST REGARDS
Aryan 27 July,2025
Popular CAT Questions
Directions for question :
M/s Deloitte Touche Tohmatsu Limited, one of the top four audit and accounting firms in the world with headquarters at London, UK, and with an operational presence in 153 countries, hires Management Trainees (MT) from all the premier management institutes of India thrice every year, in the months of January, May and September.
Each new group of Management Trainees (MT) have to go through a four month rigorous training schedule, after which they have to pass through a test consisting of a written assessment and a case-analysis. The top hundred ranked Management Trainees (MT) based on the performance in the test are confirmed as Management Executives (ME). The rest are given the opportunity of undergoing the training for four months one more time along with the next batch of Management Trainees (MT) and then passing through the subsequent test consisting of the written assessment and case-analysis. The Management Trainee (MT) who fails to get confirmed as a Management Executive (ME) the second time is fired.
The scatter-graph below depicts the number of Management Trainees (MT) at Deloitte taking the tests from January 2020 till May 2022, and the vis-à-vis hired Management Trainees (MT) at Deloitte who were fired :
It is also known that for the month of September 2019 at Deloitte, 96 hired Management Trainees (MT) failed to be confirmed as a Management Executive (ME) the first time, and that 36 hired Management Trainees (MT) were fired.
Question :
In which test did the minimum number of Management Trainees (MT) get confirmed as a Management Executive (ME) in the second attempt ?
Option: 1
September 2020
Option: 2
May 2021
Option: 3
January 2021
Option: 4
January 2022
Directions for question:
The bar-graph given below shows the foreign exchange reserves of Nepal (in million Rupees) from 2014 to 2021. Answer the following questions based on the graph :
Question:
What was the percentage increase (rounded to the nearest integer, if deemed necessary) in the foreign exchange reserves in 2020 over 2016 ?
Option: 1 None
Option: 2 None
Option: 3 None
Option: 4 None
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