CAT Exam- CAT Algebra question from Polynomials, LR & DI |Day 15

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:

a_n xⁿ + a_n-1 x^n-1 + a_n-2 x^n-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.

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 + 1

• A polynomial with degree 2 is a Quadratic polynomial.
  Example: f(x) = 2x² + x + 1

• A polynomial with degree 3 is a Cubic polynomial.
  Example: f(x) = 2x³ + 3x² + x + 1

• A 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)

1. 5

2. 6

3. 2

4. 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

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

1. The standard form of a Linear polynomial is ax + b.

2. The standard form of a Quadratic polynomial is ax² + bx + c.

3. The standard form of a Cubic polynomial is ax³ + bx² + cx + d.

4. The standard form of a biquadratic polynomial is ax⁴ + bx³ + cx² + dx + e.

5. The degree of a constant polynomial is 0.

6. 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/a

7. For 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/a

8. You have to solve different kinds of problems based on polynomials. Practice more and more problems on polynomials for CAT.

9. 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?

1. 1000

2. 729

3. 81

4. 270

Answer: (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?

1. 8

2. 16

3. 36

4. 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.

1. 6

2. 8

3. 16

4. 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.

1. 24

2. 8

3. 16

4. 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.

1. 10

2. 8

3. 9

4. 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?

1. Black, blue, pink, red

2. Black, blue, red, white

3. Black, blue, pink, white

4. 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?

1. White is on the top.

2. Blue is on the top.

3. Black is on the top.

4. 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:

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:

1. Subject and Verb should be either singular or plural.

2. 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.

3. Uncountable nouns: they cannot take plurals or combine with number words.

4. 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.

1. was supposed to meet

2. was supposed to have met

3. were suppose to be meeting

4. were supposed to meet

5. 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.

1. there is no chances that

2. there can be no chance for

3. there is no chance that

4. there are no chances for

5. 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.

1. many more resources were at Philip’s disposal

2. at Philip’s disposal were many more resources

3. there were many more resources at Philip’s disposal

4. Philip, at his disposal, had many more resources

5. 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

1. To identify the subject, one must spot the verb and ask a question that the verb along with the predicate is supposed to answer.

2. A collective noun is considered as a whole and takes a singular verb.

3. When the plural noun denotes some specific quantity considered as a unit, the verb used is generally singular.

4. 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.

5. Write rules and revise them regularly.

6. 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