CAT Exam: Maxima & Minima Solved questions, Algebra Quant, LR & DI - Day 19

CAT Exam Part-A: Quantitative Aptitude for CAT 2025

Quantitative Aptitude for CAT 2025 tests your mathematical skills, problem-solving abilities, and numerical accuracy. You have to solve questions on Arithmetic, Algebra, Geometry, and Number Systems to boost your score and improve your overall CAT percentile.

Algebra and Maxima and Minima Questions for the CAT 2025 Exam

Maxima and Minima CAT Questions are used to find the maximum or minimum possible value of a function.

Because the shape of the graph depends on these values.

Example:

In a polynomial function, the maxima is the highest point and the minima is the lowest point and the graph looks like a parabola.

How to find maxima and minima?

Depending on the function, there are several ways to find the maximum and minimum value of a function.

Two main methods are:

• Graphical method: In the graphical method, we have to plot the polynomial function on a coordinate plane and visually identify the highest and lowest points on the graph.

• Algebraic method: In the algebraic method, we use calculus to find the critical points and then determine whether these points are maxima, minima, or neither.

In maxima minima aptitude questions, the function given can be:

• Quadratic function

• Linear function

• Polynomial function etc.

For such functions, we need to find the maximum or minimum values.

We need to study “local Maxima and Minima ”, “maxima and minima formula”, “absolute maxima and minima”, “Maxima and Minima calculator”, and “global maxima and minima” for more details.

Tips To Solve The Maxima and Minima Questions for CAT

Here are some tips and tricks to solve maxima minima aptitude questions:

1. If the sum of positive numbers is constant then their product is maximum when all numbers are equal.
   Example:
   a + b + c = 36, find maximum value of a.b.c
   The answer is $(36/3)^3 = 1728$

2. Arithmetic Mean ≥ Geometric Mean
   $a + b + c = 11$; find Maximum value of $a^3 × b^4 × c^5$
   $a + b + c = a/3 + a/3 + a/3 + b/4 + b/4 + b/4 + b/4 + c/5 + c/5 +c/5 + c/5 + c/5 = 11$
   Now, AM ≥ GM
   So, $(a/3 + a/3 + a/3 + b/4 + b/4 + b/4 + b/4 + c/5 + c/5 +c/5 + c/5 + c/5)/12 ≥ (a/3 .a/3 .a/3 .b/4 .b/4 .b/4 .b/4 .c/5 .c/5 .c/5 .c/5 .c/5)^{1/12}$
   ⇒ $(11/12)^12 ≥ (a/3)^3 .(b/4)^4 . (c/5)^5$
   ⇒ $a^3. b^4. c^5 ≤ 3^3 × 4^4 × 5^5 × (11/12)^12$

Trick: To solve Maxima and Minima Questions, we can use the following trick.
Divide 11 in the ratio 3 : 4 : 5
i.e., a = (11/12)³; b = (11/12)⁴; c = (11/12)⁵
So, a³. b⁴. c⁵ = 3³ × 4⁴ × 5⁵ × (11/12)¹² (After putting the values of a, b and c)

3. For a quadratic equation:
   If the minimum value is to be found, convert the equation in the form of (a – b)² + k.
   (Minimum value is k)
   
   Example: find the minimum value of a² + 26 – 4a
   a² + 26 – 4a = (a – 2)² + 22;

Minimum value is 22

4. To find the maximum value of a function f(x) can be calculated with the help of a method of derivative.
   Step 1: find f’(x) i.e., the first derivative.
   
   Step 2: Put f’(x) = 0, and find values of x for which function is either maximum or minimum.
   Let the values of x be x1 and x2.
   
   Step 3: Find f’’(x) i.e., the second derivative.
   If f’’(x) > 0 at x = x1 then the function gives minimum value at x = x₁.
   If f’’(x) < 0 at x = x2 then the function gives the minimum value at x = x₂.
   These are local maxima and local minima.

5. Learn the above concepts well and apply them to solve questions.

6. Learn basic rules of differentiation.

7. Use the graphical method to visually identify the maxima and minima.

8. If the problem involves a specific interval, evaluate the polynomial at the endpoints and compare it with critical points within the interval.

CAT Exam: Maxima Minima Questions for CAT exam From CAT Previous Year Papers

Q1. If three sides of a rectangular park have a total length of 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is:

[CAT Exam 2017]

Solution:

Let one side be a and the other side be b.

So, a + 2b = 400
⇒ 2b = 400-a
⇒ b = 400-a/2

Area = ab = a (400-a)/2 = (400a – a²)/2

To find maximum value of (400a – a2); differentiate with respect to a

dA/da = 400 – 2a = 0 (For maxima)
⇒ a = 200

So, b = 100

Therefore, the longer side will be 200 ft for maximum area.

Q2. An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group?

[CAT exam 2017, TITA]

Solution:

Each person must weigh 53 Kg or more.

So, remaining weight = 630 – 53 – 57 = 520

Therefore, no of remaining persons n ≤ 520/53

So, n ≤ 10 (For maximum value n = 9)

Therefore, total persons in the elevator = 9 + 2 = 11

Q3. If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum possible values of (a + b) is:

[CAT 2022 Slot 2]

1. 3

2. 4

3. 3.5

4. 4.5

Solution:

a + 2b = 6

From the above equation, we can say that the maximum value b can take is 3 and the minimum value b can take is 0.
a + b + b = 6
a + b = 6 - b
a + b is maximum when b is minimum, i.e. b = 0

Maximum value of a + b = 6 - 0 = 6

a + b is minimum when b is maximum, i.e., b = 3

Minimum value of a + b = 6 - 3 = 3

∴ Average = (6 + 3) / 2 = 4.5

Hence, the correct answer is 4.5.

Q4. The minimum possible value of (x² – 6x + 10) / (3 – x), for x < 3 is:

1. –½

2. 2

3. ½

4. –2

Solution:

(x² – 6x + 10) / (3 – x)
= (x² – 6x + 9 +1) / (3 – x)
= (x² – 6x + 9)/ (3 – x) + 1 / (3 – x)

= (x – 3)² / (3 – x) + 1/ (3 – x)

As, (x – 3)² = (3 – x)²,

We can write,

(3 – x)² / (3 – x) + 1/ (3 – x)

= (3 – x) + 1/ (3 – x)

Since x < 3, (3 – x) is positive.

Let k = 3 – x

Since k is positive, the minimum value of k + 1/k = 2

Hence, the correct answer is 2.

CAT Exam: Prerequisite to Maxima Minima Questions Maths

Before practising Maxima and Minima Questions for CAT, students must be well-versed with

• Function and Graph

• Derivatives

• Arithmetic and Geometric Progression

• Basic arithmetic identities,

CAT Exam: References to Study (Theory)

• NCERT Class 12 (Chapter: Functions, Derivative, Maxima and Minima)

• Functions and Graph (Arihant Publication)

• 2IIM CAT Preparation

CAT Exam: References to Practise Maxima and Minima Questions for CAT 2025

• R D Sharma

• Previous Year's Question Papers

Part-B: CAT Logical Reasoning And Data Interpretation

Logical Reasoning and Data Interpretation (LRDI) is a crucial section of the CAT exam. It tests a candidate's ability to analyse data, identify patterns, and solve complex puzzle problems.

Circular arrangement questions for CAT 2025

The topic covered for the day and Logical Reasoning puzzles for the CAT exam are given below:

Problem Solving Questions in Maths/ Puzzles Solving Questions (Circular Arrangement questions or CAT Part-1)

Placing objects or persons in a circular table following a specific rule is a Circular arrangement.

According to a given set of rules or instructions, persons are to be arranged circularly.

Aspirants can also download the following circular arrangement questions for CAT pdf. for further references.

Concepts Involved and Importance of Circular Arrangement

Puzzles Solving Questions of Circular Arrangements involves the concept of Ranking , Blood Relation , Direction Sense etc.

It carries a good weightage in the CAT exam/SNAP and other MBA entrance Exams as seen in previous years.

Problem-solving questions in maths are generally based on:

1. Facing towards centre

2. Facing outward the centre

3. Facing in two different directions.

Knowledge of solving problems enables you:

1. To identify the unique position of a person in an arrangement.

2. To identify respective positions of persons.

3. Relations among the persons/ objects.

In the CAT exam, linear and circular arrangement questions for CAT are very common.

Students can also study “circular arrangement questions pdf” and “Logical Reasoning circular arrangement questions” to know further about this topic.

Circular Arrangement Questions from CAT Previous Year Papers

Based on the information given below, answer the questions: (SNAP)

Ten friends Matt, Sam, Pat, Tom, Sid, Alex, Kat, Jim, Jane and John are having dinner on a rectangular table. Eight face each other along the length of the table while two face each other along the smaller side of the table. Pat is sitting diagonally opposite to Kat, Alex is facing Jim, John is to the right of Jane, Tom is sitting between Sam and Jim, Pat is to the extreme left of Jane who is sitting on the extreme right along the length of the table. Sid is facing Tom and Matt is on the right of Sam.

Q1. Who do you think is sitting in front of each other along the small sides of the table? (CAT exam)

1. Matt and John

2. Matt and Sam

3. Matt and Kat

4. John and Sam

Answer: A

Q2. Sid is sitting between which of the following two? (CAT exam)

1. Sam and Kat

2. Pat and Alex

3. Alex and Jane

4. Kat and Jane

Answer: B

Q3. Who is sitting diagonally opposite to each other? (CAT exam)

1. Sid and Tom

2. Sam and Jane

3. Alex and Jim

4. Jane and Kat

Answer: B

Q4. Who is sitting to the immediate right of Jim (CAT exam)

1. Sam

2. Jane

3. Alex

4. Tom

Answer: D

Q5. Who is sitting in two places left to Jane? (CAT exam)

1. Pat

2. Alex

3. Sid

4. Sam

Answer: C

Solution:

Based on the information given above; the following arrangement be formed

For this read every piece of information very carefully and place the persons.

Prerequisite to Logical Reasoning Puzzles for CAT Exam

1. Knowledge of Ranking , Direction Sense and Blood Relation .

2. Knowledge of Linear Arrangement.

Tips and Tricks for Puzzle Solving Questions of Circular Arrangement

Tips and Tricks for Puzzle Solving Questions of Circular Arrangement

Circular arrangement puzzles are a popular question type in the Logical Reasoning and Data Interpretation (LRDI) section of the CAT exam. These CAT puzzle questions test your logical skills and accuracy in solving seating arrangement problems based on specific conditions.

1. There is no specific formula to solve CAT Circular arrangement questions.

2. First, find the number of persons in the arrangement. Draw a circle or rectangle (as per the given condition) and mark their positions.

3. If the persons are facing away from the centre, the left of the persons/ objects will be in the anticlockwise direction and the right of the persons/ objects will be in the clockwise direction.

4. If the persons are facing towards the centre, the right of the persons/ objects will be in the anticlockwise direction and the left of the persons/ objects will be in the clockwise direction.

5. Questions on rectangular arrangements are to be solved in the same way.

6. Remember the information by short notations.

CAT Exam: References to Study Theory (CAT circular arrangement questions)

• Verbal and Non-Verbal Reasoning by R S Agrawal

• Cetking (YouTube Channel)

• MBAP CAT EBook (Concept Theory)

CAT Exam: References to Practise Problem-Solving Questions in Maths

• LRDI for CAT and GMAT by Nishit K Sinha. (Advance)

• Previous Year's Question Papers

Part-C: CAT Exam Verbal Ability and Reading Comprehension

Verbal Ability and Reading Comprehension (VARC) of the CAT exam assesses a candidate’s proficiency in English language, comprehension, and reasoning. Here, in this section, you can check out the questions based on articles, practice questions, and tips to prepare:

Sentence Improvement Based on Articles

Correct Use of Articles is one of the most significant skills one should be aware of. “The” article is commonly used with an uncountable or countable noun, indefinite or definite noun, and plural or singular noun.

Sentence improvement questions based on articles are important for both the CAT exam and real life.

It is effective in the following ways:

• If you want to rack your skills up in the verbal ability domain, then practise for sentence improvement based on Articles that should be a ‘not to be missed’ topic in your life.

• Correct use of articles gives precise specifications about the noun.

• With the knowledge of articles, one can easily enhance the quality of a sentence.

• Adding articles to your sentence can help you to speak fluently.

Students can also practice “sentence improvement questions and answers pdf”, “sentence improvement rules pdf”, and “sentence improvement mcq” to grasp more about this topic.

CAT Exam: Practice Questions on Articles

Mastering the use of articles is crucial for scoring well in the VARC section of the CAT exam. Practising questions on articles helps improve grammar accuracy, comprehension skills, and overall verbal ability.

Direction: Write the correct form of the underlined phrase.

Q1. Union Carbide Corporation (UCC) was asked to build a plant for manufacture of Sevin, a pesticide commonly used throughout Asia.

1. of manufacturing the Sevin,

2. As to manufacture of the Sevin,

3. for the manufacture of Sevin,

4. None of these

Answer: C

‘the’ should be used before “manufacture”.

Q2. CAT exam candidates are not allowed to use calculators during their examinations, although they are permitted to do so if certain models meet the requirements.

1. are not allowed using calculator

2. do not allowed using the calculator

3. are not allowed to use a calculator

4. No correction required

Answer: C

‘a’ should be used before the “calculator”.

Q3. After Geeta fought with Seeta, she moved to a second apartment.

1. had moved to second apartment

2. moved to the second apartment

3. had been moved to the second apartment

4. No correction required

Answer: B

‘the’ should be used before ‘second’.

Q4. The prime necessity, as the Director of ICRIER said, is to neutralise the amount of imports and exports in the country in a non-discriminatory manner.

1. neutralising amount of the imports and the exports

2. neutralise the amount of the imports and exports

3. neutralise the amount of imports and exports

4. No correction required

Answer: B

‘the’ should be used before “imports and exports”.

CAT Exam: Prerequisite

• Basic Knowledge of Sentence Structure.

• Basic Knowledge of the Usage of Articles.

Tips to Prepare Sentence Improvement on Articles and Uses

1. Make sure you pay close attention to words whose first letters are vowels, but when they are pronounced, they have the sound of a consonant or vice versa. This is the case with words like university, an honest person, etc. For more details, you can head over to the Day-6 strategy.

2. Practice error-based questions daily.

3. Read newspapers, and listen to Ted Talks so that you will get more clarity about the uses of articles.

CAT Exam: References to Study (Theory)

• How to Prepare for Verbal Ability & Reading Comprehension for CAT by Meenakshi Upadhyay and Arun Sharma (9th Edition)

• Mastering A, An, The - English Articles Solved by Douglas Porter.

• A' and 'The' Explained: A learner's guide to definite and indefinite articles by Seonaid Beckwith.

CAT Exam: References to Study (Practice)

• The Article Book: Practice Toward Mastering a, an, and the by Tom Cole.

• Oxford Practice Grammar (sixth edition)

• engvid.com (Web)

Study resources for CAT 2025

Preparing for CAT 2025 requires access to reliable and comprehensive study resources that cover all three sections, VARC, DILR, and Quant. The right mix of books, online courses, mock tests, and previous papers can significantly boost your preparation. Get the link for the best resources for CAT 2025 examination below:

Top Ebooks to Ace CAT 2025 Preparation

High-quality eBooks for CAT VARC preparation are essential for mastering Reading Comprehension, Para Jumbles, Vocabulary, and Grammar. These CAT preparation eBooks by Careers360 offer topic-wise study material, expert tips, and ample practice questions designed to boost accuracy and speed. Ideal for focused learning, they help aspirants strengthen core concepts and improve performance in the Verbal Ability and Reading Comprehension section. Get the links to top ebooks to aid your CAT 2025 exam preparations: