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With the CAT 2024 exam date approaching, you may be confused about how to study effectively in just 2 months, what strategy to use, and the best sources for theory and practice. Questions about whether you can crack the CAT in 2 months, how much CAT preparation time is needed, and the best study plans are common. You might also be considering a 2-month crash course.
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For Day 14 of our CAT preparation, we’re focusing on Quadratic Equations questions. This article will help you practice and understand different types of Quadratic Equations that frequently appear on the exam.
Following the 60-day CAT exam preparation schedule we’ve provided will keep you on track with a structured plan. We’ve already covered strategies for Algebra (Linear Equations ), Direction ion (Logical Reasoning), and Future Tenses on Day 13. Today, we’ll dive into quadratic equations to further enhance your preparation.
Quadratic Equations are the 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:
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Questions that involve calculations of some unknown variable.
Questions that involve calculations of the nature of roots.
Formation of the quadratic equation.
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
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
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]
8
6
5
7
Ans: (b)
CAT 2024 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).
For this topic, students must have knowledge of
Degree of Polynomials and zeros of a polynomial.
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.
How to Prepare Quantitative Aptitude by Arun Sharma (6th edition)
2iim (You-Tube)
NCERT Class 9-10
https://iim-cat-questions-answers.2iim.com
Questions based on cubes play an important role in CAT. Here we will discuss cube CAT questions, Cube Logical Reasoning questions, cube logical reasoning tricks etc.
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
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)
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.
To Prepare the questions based on Cube, students must have the knowledge of
This topic requires basic calculations
Imagination
Analytical Approach
Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)
Cat preparation-iQuanta (YouTube Channel)
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
In previous articles, we have made strategies to prepare 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.
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.
Basic Knowledge of Nouns, Pronouns and Verbs.
Basic knowledge of Tenses.
Knowledge of the use of helping verbs.
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.
Advanced English grammar by Wren and Martin
GMATPrepNow (Youtube Channel)
How to Prepare for Verbal Ability & Reading Comprehension for CAT by Meenakshi Upadhyay and Arun Sharma (9th Edition)
Practice CAT Sample Question Papers.
The CUSAT CAT previous year question papers are available on the Careers360 website. You can find question papers from 2012 to 2023, which can be downloaded in PDF format HERE
Additionally, the official website of Cochin University of Science and Technology (CUSAT) also provides previous year question papers. To access them, you can follow these steps:
- Visit the CUSAT official website - admissions.cusat.ac.in.
- Click on the resources tab from the homepage.
- Select the previous year's question papers from the drop-down menu.
- Select the Common Admission Test (CAT) question papers link.
- Now click on the B.Tech course.
By practicing with these previous year question papers, you can get familiar with the exam pattern, identify important topics, and improve your time management skills.
Hey Shathvika,
Let's dive deep into MBA Entrance Exams (Other than CAT)
1. XAT (Xavier Aptitude Test)
Conducted by: XLRI, Jamshedpur
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Decision Making, Verbal & Logical Ability, Quantitative Ability & Data Interpretation, and General Knowledge.
- Duration: 3 hours.
- Marks: No negative marking for General Knowledge.
Key Highlights:
- The Decision-Making section tests analytical and situational judgment skills.
- General Knowledge is evaluated during interviews, not for percentile calculation.
Official Website: https://xatonline.in
---
2. SNAP (Symbiosis National Aptitude Test)
Conducted by: Symbiosis International (Deemed) University
Eligibility: Bachelor’s degree with 50% marks (45% for reserved categories).
Exam Pattern:
- Sections: General English, Quantitative, Data Interpretation & Data Sufficiency, and Analytical & Logical Reasoning.
- Duration: 1 hour.
- Marks: Total 60 marks (MCQs).
Key Highlights:
- Specifically for admission to Symbiosis institutes like SIBM Pune and SCMHRD.
- Easy-to-moderate difficulty level.
Official Website: https://snaptest.org
---
3. CMAT (Common Management Admission Test)
Conducted by: National Testing Agency (NTA)
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Quantitative Techniques & Data Interpretation, Logical Reasoning, Language Comprehension, General Awareness, and Innovation & Entrepreneurship.
- Duration: 3 hours.
- Marks: 400 marks.
Key Highlights:
- Accepted by over 1,000 AICTE-approved institutes.
- Includes an optional Innovation & Entrepreneurship section.
Official Website: https://cmat.nta.nic.in
---
4. MAT (Management Aptitude Test)
Conducted by: All India Management Association (AIMA)
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Language Comprehension, Mathematical Skills, Data Analysis & Sufficiency, Intelligence & Critical Reasoning, and Indian & Global Environment.
- Duration: 2.5 hours.
- Marks: 200 marks.
Key Highlights:
- Held multiple times a year in PBT, CBT, and IBT modes.
- Accepted by more than 600 MBA colleges in India.
Official Website: https://mat.aima.in
---
5. NMAT (NMIMS Management Aptitude Test)
Conducted by: Graduate Management Admission Council (GMAC)
Eligibility: Bachelor’s degree with 50% marks.
Exam Pattern:
- Sections: Language Skills, Quantitative Skills, and Logical Reasoning.
- Duration: 2 hours.
- Marks: 360 marks.
Key Highlights:
- Allows three attempts, with the best score considered.
- Accepted by NMIMS and partner universities worldwide.
Official Website: https://www.mba.com/exams/nmat
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6. IIFT (Indian Institute of Foreign Trade)
Conducted by: National Testing Agency (NTA)
Eligibility: Bachelor’s degree (final-year students can also apply).
Exam Pattern:
- Sections: Quantitative Analysis, Reading Comprehension & Verbal Ability, Logical Reasoning & Data Interpretation, and General Knowledge.
- Duration: 2 hours.
- Marks: 300 marks.
Key Highlights:
- Specifically for IIFT’s flagship MBA in International Business.
- General Knowledge plays a significant role in shortlisting.
Official Website: https://iift.nta.nic.in
---
7. TISSNET (Tata Institute of Social Sciences National Entrance Test)
Conducted by: Tata Institute of Social Sciences
Eligibility: Bachelor’s degree.
Exam Pattern:
- Sections: English Proficiency, Mathematics & Logical Reasoning, and General Awareness.
- Duration: 100 minutes.
- Marks: 100 marks.
Key Highlights:
- Focused on social sciences, HRM, and public policy programs.
- General Awareness is crucial (with emphasis on current affairs).
Official Website: https://admissions.tiss.edu
---
8. MICAT (MICA Aptitude Test)
Conducted by: MICA, Ahmedabad
Eligibility: Bachelor’s degree and a valid CAT/XAT/GMAT score.
Exam Pattern:
- Sections: Psychometric Test, Verbal Ability, Quantitative Ability, General Awareness, Divergent-Convergent Thinking, and Descriptive Test.
- Duration: 2.5 hours.
Key Highlights:
- Psychometric Test is a qualifying section.
- Specifically for MICA’s flagship PGDM-C program in Strategic Marketing and Communications.
Official Website: https://www.mica.ac.in
---
9. GMAT (Graduate Management Admission Test)
Conducted by: Graduate Management Admission Council (GMAC)
Eligibility: No specific eligibility; bachelor’s degree recommended.
Exam Pattern:
- Sections: Analytical Writing, Integrated Reasoning, Quantitative Reasoning, and Verbal Reasoning.
- Duration: 3 hours 7 minutes.
- Marks: 800 (scaled score).
Key Highlights:
- Globally recognized for MBA programs.
- Popular for Indian B-schools like ISB, SPJIMR, and Great Lakes.
Official Website: https://www.mba.com
---
10. MAH-CET (Maharashtra Common Entrance Test)
Conducted by: Maharashtra State CET Cell
Eligibility: Bachelor’s degree with a minimum of 50% marks (45% for reserved categories).
Exam Pattern:
- Sections: Logical Reasoning, Abstract Reasoning, Quantitative Aptitude, and Verbal Ability & Reading Comprehension.
- Duration: 2.5 hours.
- Marks: 200 marks (MCQs).
Key Highlights:
- Accepted by top Maharashtra B-schools like JBIMS, SIMSREE, and PUMBA.
- Known for its speed-intensive nature with 200 questions in 150 minutes.
Official Website: https://cetcell.mahacet.org
Let me know if you have any doubts regarding a specific exam or any other query.
Have a nice day !!
Hello Ayush,
K.R. Mangalam University utilizes a blend of MCQs, descriptive formats, and practical assessments to evaluate student performance. The continuous evaluation system ensures that students are assessed regularly throughout the semester, contributing to a more comprehensive understanding of their academic progress.
Hello,
Here are some simple tips to prepare for the CUSAT CAT exam:
1. Focus on Physics, Chemistry, and Maths (PCM).
2. Give equal time to all subjects.
3. Use NCERT Books as they are best for basics.
You can refer to following study materials for better practice :-
Hope it helps !
Hello there,
Yes, you can apply for CAT examination after completion of Doctor of Pharmacy (Pharm D) .
The Common Admission Test (CAT) has no restrictions regarding your educational background thus allowing you to submit your application following your completion of a Pharm. D degree.
The Common Admission Test permits candidates to take the exam after completing any undergraduate degree without any restrictions to the area of study. The professional doctoral degree of Pharm. D (Doctor of Pharmacy) functions as an equivalent to bachelor's level for CAT eligibility recognition.
Therefore, Candidates who received their Pharm. D degree can pursue the CAT exam application process based on their qualification standards.
Hope this will help!
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A BDE often collaborates with cross-functional teams, including marketing and product development, to align business objectives. Strong communication, negotiation, and relationship-building skills are essential for success in this role. BDE strives to expand the company's customer base, drive sales, and contribute to overall organisational growth in a dynamic and competitive business environment.
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The Sales Manager work also includes hiring and laying off sales personnel after evaluating his or her work performance. In bigger entities, sales quotas and plans are usually set at the executive level. He or she is responsible for overseeing the set target or quotas met by salespeople or upholding any policy. He or she guides his or her fellow salespeople and allows them to sell.
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