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CAT Exam- CAT Algebra question from Polynomials, LR & DI |Day 15
Edited By Team Careers360 | Updated on Sep 19, 2024 03:18 PM IST | #CAT
Based on the CAT 2024 syllabus, we previously covered strategies for Algebra (Quadratic Equations), Cube (Logical Reasoning), and Sentence Corrections with CAT algebra questions on Day 14 of our preparation series.
Now, for Day 15, we’ll focus on strategies for tackling polynomial questions in the CAT exam.
To excel in CAT 2024, follow our 60-day preparation schedule to ensure you stay organized and on track with your study plan.
Quantitative Aptitude for CAT 2024
Polynomial (Algebra), Polynomial Questions For CAT Exam & CAT algebra questions
What is polynomials?
Polynomials are mathematical expressions that consist of variables or determinants, coefficients, exponents, and constant terms combined with mathematical expressions like addition, subtraction, or multiplication.
It is generally denoted by f(x).
The general form of a polynomial equation is:
an xn + an-1 xn-1 + an-2 xn-2 + ........ + a1 x + a0
Example:
4x2 + 2x + 3 is a polynomial where x is the variable.
An example of a polynomial equation with two variables is:
3x2 + 2xy + y2 + 4x + 5y + 7, where x and y are two variables.
Degree of polynomials
The degree of a polynomial is the highest exponent of a variable in the polynomial expression with a non-zero coefficient.
Example:
7x4 + 3x3 + 9x + 1; Degree of the polynomial = 4
x3 + 3x2y4 + 3y4 + 5; Degree of the polynomial = 6
It is also to be remembered:
A polynomial with degree 0 is called a Constant or zero polynomial.
This Story also Contains
- Quantitative Aptitude for CAT 2024
- CAT Logical Reasoning and Data Interpretation
- CAT Exam VARC
- CAT 2024 study material pdf
In other words, the degree of zero polynomial is zero.
Example: f(x) = 2
A Polynomial with degree 1 is a Linear polynomial.
Example: f(x) = 2x + 1A polynomial with degree 2 is a Quadratic polynomial.
Example: f(x) = 2x2 + x + 1A polynomial with degree 3 is a Cubic polynomial.
Example: f(x) = 2x3 + 3x2 + x + 1A polynomial with degree 4 is a Biquadratic polynomial.
Example: f(x) = 2x4 + 2y + 4z + 1
Zeros of a polynomial
A zero of a Polynomials is a value of the variable (often denoted as "x") that makes the entire polynomial equal to zero. In other words, if you substitute that value for "x" in the polynomial expression, the result will be zero. Another term for the zero of a polynomial is "root" or "solution."
In general, the number of zeroes of a polynomial is equal to its degree, and some of the zeroes may be repeated. For example, a polynomial of degree 3 will have three zeroes (counting multiplicities), and so on. Understanding the zeroes of a Polynomials is crucial in solving various mathematical problems.
There are a few ways to find the zeros of a polynomial.
The factorization method is the most commonly used.
Quadratic formula
Synthetic division
We also have to study “polynomial regression”, “Types of Polynomials ”, and “polynomial function” to learn more about this chapter.
CAT Previous Year Algebra Questions/ Polynomials CAT questions/Polynomial Questions For CAT exam
Here are some CAT previous year algebra questions shuffled up from different PYQs of the CAT exam before:
Q1. If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a – b)2 + (a – c)2 + (a – d)2 is:
(TITA, CAT exam 2017)
5
6
2
8
Solution:
Given that a + b + c + d = 30.
For the minimum possible value of (a – b)2 + (a – c )2+ (a – d)2,
(a – b) = (a – c) = (a – d) = 0
But this is not possible because 30 is not a multiple of 4.
We should find a as close to b, c, and d as possible.
We should take values of a, b, c and d close to 30/4 = 7.5
So, a, b, c, and d should take the values of 7 or 8.
For the minimum possible value of (a – b)2 + (a – c)2 + (a – d)2, the difference of a taken with b, c or d should not be greater than 1.
So, Possible sets of a, b, c and d will be
a | b | c | d | (a – b)2 + (a – c)2 + (a – d)2 |
7 | 7 | 8 | 8 | 2 |
8 | 8 | 7 | 7 | 2 |
7 | 8 | 7 | 8 | 2 |
7 | 8 | 8 | 7 | 2 |
8 | 7 | 8 | 7 | 2 |
8 | 7 | 7 | 8 | 2 |
So, the minimum possible value of (a – b)2 + (a – c)2 + (a – d)2 is 2.
Hence, the correct answer is 2.
Q2. For real x, the maximum possible value of x/√(1+x4) is:
[CAT exam 2020]
A. 1
B. ½
C. 1/√2
D. 1/√3
Solution:
Dividing both numerator and denominator by x, we get,
1/√[(1/x2) + x2]
The value of (1/x2) + x2 should be the minimum to get the original fraction maximum.
For real values of x, (1/x2) + x2 has a minimum value equal to 2.
So, the maximum value of x/√(1+x4) is 1/√2.
Hence, the correct answer is 1/√2.
Tips and Tricks to Prepare Polynomials Questions for CAT
The standard form of a Linear polynomial is ax + b.
The standard form of a Quadratic polynomial is ax2 + bx + c.
The standard form of a Cubic polynomial is ax3 + bx2 + cx + d.
The standard form of a biquadratic polynomial is ax4 + bx3 + cx2 + dx + e.
The degree of a constant polynomial is 0.
For the quadratic equation ax2 + bx + c = 0 if the roots are α and β, then
Sum of the roots, α + β = –b/a
Product of the roots, α β = c/aFor the cubic equation ax3 + bx2 + cx + d = 0, if the roots are α, β, and γ, then
Sum of the roots, α + β + γ = –b/a
Product of the roots, α β γ = –d/a
Sum of the product of roots taken two at a time, α β + β γ + α γ = c/aYou have to solve different kinds of problems based on polynomials. Practice more and more problems on polynomials for CAT.
Also, learn
(x + y)2 = x2 + y2 + 2xy
(x – y)2 = x2 + y2 – 2xy
(x + y)3 = x3 + y3 + 3xy2 + 3x2y
(x – y)3 = x3 – y3 – 3x2y + 3xy2
(x – y – z)3 = x3 – y3 – z3 – 3x2y + 3xy2 – 3y2z – 3yz2 + 3xz2 – 3x2z + 6xyz
References to Study (Theory)
NCERT Class 9-10
How to Prepare Quantitative Aptitude by Arun Sharma (7th edition)
2IIM CAT Preparation (You-Tube)
References to Study (Practice) CAT Algebra Questions
Pearson (How to prepare quantitative aptitude for CAT)
https://iim-cat-questions-answers.2iim.com
Previous Year's Question Papers to practice polynomial questions for the CAT exam
CAT Logical Reasoning and Data Interpretation
Cube Part-2 (LR) And CAT Questions On Cubes
The different kinds of questions asked in the CAT Exam on Cubes are:
Type 1 Cubes questions in CAT:
Question. A cube painted on all faces is cut into 1331 cubes of the same size.
1. How many cubes are not painted on any face?
1000
729
81
270
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 NowAnswer: (B)
Trick: Only the inner cubes
i.e., (k - 2)3, where k = dimension of larger cube/dimension of smaller cube
Here, k = 11
So, the number of cubes not painted on any face is (11 - 2)3 = 93 = 729
2. How many cubes are painted on three faces?
8
16
36
54
Answer: (A)
Trick: Corner cubes only.
A cube has 8 corners.
Hence, the correct answer is always 8.
Type 2 Cubes questions for CAT:
Question. A cube of 4 cm is painted yellow on the pair of one opposite surface, red on the pair of another opposite surface and blue on the remaining pair of opposite surfaces. The cube is now divided into smaller cubes of equal size of 1 cm each.
3. Find the number of smaller cubes with three surfaces painted.
6
8
16
27
Answer: (B)
Trick: Corner cubes only.
A cube has 8 corners.
Hence, the correct answer is always 8.
4. Find the number of smaller cubes with two surfaces painted.
24
8
16
30
Answer: (A)
Trick: Surface cubes except for the corner cubes.
k = dimension of larger cube/ dimension of smaller cube
Cubes with two face painted = 6 × (k – 2)2 = 6 × 22 = 24
5. Find the number of cubes with two surfaces painted with yellow and red colour.
10
8
9
16
Answer: (B)
Trick: Cubes at the common edges of yellow and red surfaces, except corner cubes.
Common edges of yellow and red will be 4 and 2 cubes (except at corners) at each edge.
So, the number of cubes with two surfaces painted with yellow and red colour is 4 × 2 = 8
Type 3 Cubes questions for CAT:
Question. Here is the information given below. Based on this information, answer the questions.
A cube has six sides, each of which has a different colour: black, blue, yellow, red, pink, and white.
The red side is opposite the black.
The yellow side is between the red and the black.
The blue side is adjacent to the white.
The black side is adjacent to the white.
The red side is the bottom face.
1. What are the four colours that are adjacent to the yellow?
Black, blue, pink, red
Black, blue, red, white
Black, blue, pink, white
Black, red, pink, white
Answer: A
Trick: Except for the colour opposite to the yellow face.
So, except for white, others are adjacent to yellow.
2. Which of the information can be deduced from the information given in statements I, II, and VI?
White is on the top.
Blue is on the top.
Black is on the top.
White is opposite to black.
Answer: C
Trick: Draw an open cube diagram.
Black must be at the top according to statements I, II, and VI.
References for 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
CAT Exam VARC
Subject-Verb Agreement
A sentence is a group of words connected to make meaning. A sentence must use appropriate verbs, helping verbs, punctuation etc.
The Subject and Verb must agree in number and Person (Noun/ Pronoun, Singular/ Plural). Mastering subject-verb agreement rules is crucial for effective communication and writing in the English language.
Subject-verb agreement rules with examples
Person Rule:
Person | Singular | Verb | Plural | Verb |
First Person | I | Am/was/shall | We | Are/ were/ shall |
Second Person | You | Are/ were/ will | You | Are/ were/ will |
Third Person | He/She/it | It/was/will | They | Are/ were/ will |
Number Rule: The number rule is not as simple as the person's rule. There are a few exceptions too.
There are many rules such as:
Subject and Verb should be either singular or plural.
Countable nouns can take a plural, can combine with counting quantifiers (e.g., one, hundred, every, many), and can take an indefinite article such as a or an. Examples of counting nouns are table, glass, and occasion.
Uncountable nouns: they cannot take plurals or combine with number words.
Proximity Rule: Where two or more subjects are connected by: or, either…or, neither…nor. The subject that is closest to the verb decides the number of the verb.
Students can also practice on “subject-verb agreement worksheets with answers pdf”, “40 rules of subject-verb agreement pdf”, and “subject-verb agreement exercises with answers doc” to learn more about this chapter.
VARC Questions For CAT On Subject-Verb Agreement: [Based on PYQs]
The following subject-verb agreement exercises are VARC questions for CAT, based on topic subject-verb agreement based on the previous year's question papers of the CAT exam.
Correct the sentences:
Q1. The Seven Strangers, Manav included, was supposed to meet for dinner later that night after a conference.
was supposed to meet
was supposed to have met
were suppose to be meeting
were supposed to meet
they were supposed to be meeting
Answer: (D)
Here the subject is “The seven strangers”, which is plural.
So, the plural verb “were” will be used here.
The correct sentence will be:
“The Seven Strangers, Manav included, were supposed to meet for dinner later that night after a conference.”
Q2. Without proper guidance and better support, there is no chances that our candidate will win the election.
there is no chances that
there can be no chance for
there is no chance that
there are no chances for
there will be no chances for
Answer: (C)
Here the singular verb ‘is’ does not agree with the plural subject ‘chances’.
So, replace chances with a chance.
The correct sentence will be:
“Without proper guidance and better support, there is no chance that our candidate will win the election.”
Q3. After he attended the seminar, many more resources were at Philip’s disposal, including new contacts, excellent books and pamphlets to help him improve his resume.
many more resources were at Philip’s disposal
at Philip’s disposal were many more resources
there were many more resources at Philip’s disposal
Philip, at his disposal, had many more resources
Philip had many more resources at his disposal
Answer: (E)
On Comparison, we found that ‘E’ is most suitable.
The correct sentence will be:
“After he attended the seminar, Philip had many more resources at his disposal, including new contacts, excellent books, and pamphlets to help him improve his resume.”
Tips to Prepare Subject-Verb Agreement
To identify the subject, one must spot the verb and ask a question that the verb along with the predicate is supposed to answer.
A collective noun is considered as a whole and takes a singular verb.
When the plural noun denotes some specific quantity considered as a unit, the verb used is generally singular.
Words such as glasses, pliers, and scissors are regarded as plural and require plural verbs unless they’re preceded by the phrase ‘pair of ‘and this word becomes the subject.
Write rules and revise them regularly.
Attempt Mock-Test papers.
Prerequisite:
Complete Knowledge of Nouns
Complete Knowledge of Pronouns
Knowledge of Articles
Knowledge of Verbs
Basic knowledge of Tenses.
References to Study (Theory):
Advanced English grammar by Wren and Martin
CAT Edurav (Youtube Channel)
References to Study (Practice):
How to Prepare for Verbal Ability & Reading Comprehension for CAT by Meenakshi Upadhyay and Arun Sharma (9th Edition)
https://www.800score.com/gmat-prep/prep-guide/sample-questions/
Practice Sample Question Papers.
CAT 2024 study material pdf
CAT 2024 Quantitative Aptitude Study Material PDF - Geometry and Mensuration | CAT 2024 preparation tips with 30-day study material by Experts |
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Questions related to CAT
Have a question related to CAT ?
With a CAT rank of 5419, your percentile is around 95 to 97. This means you may not get into the top IIMs like Ahmedabad, Bangalore, or Calcutta, because they usually take students above 98 or 99 percentile. But you still have good chances at newer IIMs and other top MBA colleges.
You can get admission into IIMs like Amritsar, Raipur, Jammu, Udaipur, and Bodh Gaya. These IIMs accept students with lower percentiles, usually between 90 and 94.
Apart from IIMs, you can also get colleges like MDI Gurugram, IMI Delhi, Great Lakes Chennai, and maybe even FMS Delhi or JBIMS Mumbai depending on your category and performance in interview rounds.
You should now start preparing for the interview, group discussions, and written tests that most colleges will conduct after shortlisting you based on your CAT rank.
Here are websites you can check for cut-off and admission details:
https://www.imsindia.com/blog/cat-cutoff/
Ayush Madanpotra 18 July,2025
Hello Sahitya
For FMS Delhi, your percentile should be 99%+ and it should be 99.5%+ to be on the safer side.
Now coming to the academics:
1. 10th marks: 75%+ are given 10 out of 10
2. 12th marks: 75%+ are given 10 out of 10
3. Graduation marks: FMS doesn't considers graduation marks for screening.
So if you have 75%+ in both 10th and 12th, you will get full marks in academic score.
To know more about FMS: FMS
Hope this answer helps! Thank You!!!
Yash Teotia 13 July,2025
Hello Ayush
This is definitely a myth that with 0 work-ex you can't get IIM ABC because freshers still get into them although the competition is brutal but if you have good academics (9/9/9) and a good CAT (99.5%+), you have a chance.
These are some other good colleges except IIM ABC which you can aim for:
TIER 1:
- IIM Lucknow
- IIM Kozhikode
- IIM Indore
- IIM Shillong
- FMS Delhi
- SPJIMR Mumbai
- MDI Gurgaon
- IIFT Delhi & Kolkata
Tier 2:
- NMIMS Mumbai
- SIBM Pune
- SCMHRD Pune
- IMT Ghaziabad
- XIM Bhubaneshwar
- GIM Goa
- TISS Mumbai
To know more about CAT: CAT by Careers360
Hope this answer helps! Thank You!!!
Yash Teotia 13 July,2025
Hello Aspirant,
With an AIR (All India Rank) of 1,60,000 in NEET and in the OC (Open Category) there is a very slim chance of getting an MBBS seat in a government medical college, especially in General Category seats through All India Quota (AIQ), but here are some options:
- MBBS in Private Colleges: You could get a seat in a private medical college through the state quota, especially in Tier 2/3 states where the cutoffs are relatively lower. The fee would be high compared to government colleges.
- BDS Seats: You have fair chances for BDS in both government & private dental colleges under the state quota.
- Deemed Universities: Some deemed universities may offer MBBS seats at this rank but the fee is exorbitantly high (15 to 25 lakhs/year).
- Management/NRI Quota: If you can afford, you can explore MBBS through management/NRI quota seats in private colleges.
- Alternatives: You can think of allied health science courses such as BAMS, BHMS, B.Sc Nursing etc., which also have good career scope.
You can also participate in state counselling rounds, as sometimes, seats remain vacant/unfilled, including mop-up rounds.
gopi 13 July,2025
Hello aspirant,
The question papers for CAT MGU University can be found on their official website, or you can also visit careers360 website for the same.
The link of which, I am attaching here,
https://university.careers360.com/articles/mgu-cat-2025
Regards
Mansi Satwani 07 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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