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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 Sep 19, 2024 03:22 PM IST | #CAT
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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.
Part-1: Quantitative Aptitude for CAT 2024
Topic: Quadratic Equation (Algebra) And Quadratic Equation Questions For CAT
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:
Questions that involve calculations of some unknown variable.
Questions that involve calculations of the nature of roots.
Formation of the quadratic equation.
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Download EBookHow 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
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).
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, cube logical reasoning 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
Topic: How to solve problems of sentence correction based on Tenses.
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.
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.
Free PDFs for CAT 2024 preparation
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Questions related to CAT
Have a question related to CAT ?
The student cannot join the postgraduate MA program directly in the second year of the MA after completing the MBA.
Universities generally expect students to begin the first year of a masters' program even though they had qualifications at the level of a bachelor's degree. The university's curriculum along with the courses are planned to give an in-depth knowledge of the subjects.
But you could opt for:
Postgraduate Studies:
Research Career: In case research interests you, then you can pursue a Ph.D. in your specific interest.
Specialized Master's Programs: Find any specific master's course that supports the career goal.
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Make a professional certification in CFA, CA, or FRM to enhance the professional skills and better opportunities in the career field.
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In the end, it would depend on what you would like and what you are particularly interested in. It will be better if you take a suggestion from your academic advisor or career counselor.
Samprikta Mondal 13 November,2024
Hello Pratik,
If your family income exceeds 8 LPA, it will affect your eligibility for the NC-OBC category under the CAT reservation rules. The NC-OBC category requires that the family income should be below 8 LPA to qualify. If you anticipate that your income will exceed this threshold, you should update your status and notify the CAT authorities .
Here's what you can do:
-
Email the CAT authorities : Inform them about the potential change in your income status and request to be considered under the General category, if applicable. You should do this before the examination or any further processing, as your eligibility for the OBC category depends on the income criteria.
-
Documents and Proof : Make sure you have a valid OBC certificate and any relevant documentation regarding your family income, in case they require verification. Also, be aware that if your income exceeds 8 LPA, you might need to apply under the General category, as the OBC-NC reservation is no longer applicable.
-
General Category Consideration : If your application is accepted as General, you will be evaluated based on the General category criteria, including cutoffs and ranking.
To summarize, notifying the CAT authorities is the correct approach to avoid discrepancies later, and if your income is above the threshold, you will be treated as a General candidate for the admissions process.
For more details about CAT exam you can check out an article by Careers360 whose link is given below:
Link: https://bschool.careers360.com/exams/cat
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
kumardurgesh1802 08 November,2024
Hi Shreya ,
Having a tattoo on your wrist should not create a problem during the CAT exam as long as it does not contain any content that violates exam rules , such as communication-related symbols , unauthorised messages or any text resembling formulas or answers .
For a safe side , it is recommended :
1. Inform the Invigilators : Let them know about your tattoo during the identification check , so there are no misunderstandings .
2. Follow the Guidelines : Ensure you follow all instructions regarding permissible items and dress code on the exam day .
Hope this helps you .
ALL THE BEST .
Mitali Sharma 09 November,2024
To know about this please prefer our college predictor
Sumit Mishra 07 November,2024
Hello,
Yes, you can also join IIMs two years after graduation, depending upon the justification in interviews and your performance in entrance exams. If you are not sure about CAT, try those other management exams less hard:
Common Management Admission Test: Easier than CAT; 4-5 months of continuous preparation will suffice.
MAT : Much easier and takes place thrice a year.
SNAP (Symbiosis National Aptitude Test): A bit easier than CAT, apt for Symbiosis Institutes.
Xavier Aptitude Test or XAT : Hence, easier than CAT but the unique decision-making section.
Use exams like CMAT and MAT as a challenge to be competitively prepared. This can be given about 4-5 months, along with a few rounds of mock tests and smart studying.
Deepti Singh 06 November,2024
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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