Conquering Functions & Graphs, Puzzle Questions for Linear Arrangement, and Conditionals- Day 18

CAT Exam Part-A: Quantitative Aptitude for CAT 2025

CAT Exam Part-A: Quantitative Aptitude for CAT 2025 tests your mathematical skills, logical reasoning, and problem-solving ability. Mastering key topics like algebra, arithmetic, geometry, and number systems is essential to score well and boost your overall percentile.

Topic: Algebra (Functions and Graph) and CAT Problem-Solving Questions and Answers

Definition: In a system, the output is a function of the input or, we can say function describes a relation between input and output.
For example: f(x) =x2, describes output (f(x)) varies as square of the input (x).
Functions can be well represented by graphs and with the help of graphs, solutions can be drawn effectively. The graph shows the variation of output with the inputs.

Tips and Tricks to Prepare Functions and Graphs for CAT

1. To solve the questions of function and graph for CAT, you have to learn some specific functions CAT questions and their graphs. A few of them are here with examples.

Odd Function: If f (– x) = – f (x)
For example: f (x) = x3 + sin x; 0 ≤ x ≤ 2π
f(–x) = (–x)3 + sin (-x) = – x3 – sin x = – f(x)
So, the given function is an odd function.
(Graph of odd functions are symmetrical about the origin)

Even function: If f (– x) = f (x)
For example: f (x) = x2 + cos x; 0 ≤ x ≤ 2π
f(–x) = (–x)2 + cos (–x) = x2 + cos x = f(x)
So, the given function is an even function.
(Graphs of even functions are symmetrical about the y-axis).
Functions can neither be odd nor even but can be expressed as the sum of even and odd functions.

Composite Functions: It is a function of some other function.
(g o f) (x) = g [f(x)]
(f o g) (x) = f [g(x)]

2. Understand the concept of Domain and Range:
   Domain: Set of input values represented on the x-axis of the graph.
   Range: Set of output values represented on the y-axis of the graph.

3. Learn the graphs of some functions as shown and that will be very helpful.

Functions Questions For CAT From CAT Previous Year Papers

Q1. If f(x) = x2 − 7x and g(x) = x + 3, then the minimum value of f(g(x)) − 3x is: [CAT exam 2021, slot-3]

(a) – 20 (b) – 15 (c) – 12 (d) – 16

Ans: (d)

f (g(x)) – 3x
= (x + 3)2 – 7(x + 3) – 3x
= x2 + 9 + 6x – 7x – 21 – 3x
= x2– 4x – 12
= x2 – 4x + 4 – 4 – 12
= (x – 2)2 – 16
The minimum value of this function will be -16 for x =2.

Q2. If f(x + y) = f(x).f(y) and f(5) = 4, then f(10) – f(–10) is equal to [CAT exam 2025, slot-3]

(a) 3 (b) 0 (c) 14.0625 (d) 15.9375

Ans: (d)

It is given that f(x+y) = f(x) × f(y)
For y = 0, f(x) = f(x) × f(0) ⇒ f(0) = 1.
For, x = 5, f (5) = 4
Now, f(10) = f(5 + 5) = f(5) × f(5) = 4 × 4 = 16
⇒ f(10) = 16
We can write;
f(0) = f(10 - 10) = f(10 + (-10)) = f(10) × f(-10) = 1
f(-10) = 1/16
f(10) - f(-10) = 16 – 1/16 = 15.9375

Q3. For all real values of x, the range of the function f(x) = (x2 + 2x + 4)/(2x2 + 4x + 9) is:
(a) [4/9, 8/9] (b) [3/7, 8/9) (c) (3/7, 1/3) (d) [3/7, 1/2)

Ans: (d)
f(x) = (x2 + 2x + 4)/(2x2 + 4x + 9)
⇒ f(x) = (x2 + 2x + 4)/[2(x2 + 2x + 4) + 1]
Let x2 + 2x + 4 = (x + 1)2 + 3 = k
The minimum value of k is 3.
Now, f(x) = k/(2k + 1)
So, the minimum value of f(x) = 3/(2×3 + 1) = 3/7
For the maximum value of f(x), as the value of k increases, the value in the denominator also increases, which decreases the value of f(x). Thus, the fraction approaches 1/2 but never equals 1/2.

Hence, the correct answer is [3/7,1/2).

Q4. If f(x) = – 4x + 9 and g(x) = 2x – 7, then f(g(x)) = ?

(a) 6x – 16 (b) – 8x + 37 (c) 2x – 16 (d) – 2x + 2

Ans: (b)

Here f(x) = – 4x + 9 and g(x) = 2x – 7

So, f(g(x)) = f(2x – 7) = – 4(2x – 7) + 9 = – 8x + 28 + 9 = – 8x + 37

Hence, the correct answer is (b) – 8x + 37.

CAT EXAM: Prerequisite to Problem-Solving Questions For Entrance Exam based on functions and graphs for CAT

• Linear Equations.

• Quadratic Equations.

• Polynomials Equations of Geometrical figures.

CAT EXAM: References to Study (Theory):

• Functions and Graph (Arihant Publication)

• How to prepare Quantitative aptitude for CAT by Arun Sharma.

• 2IIM CAT Preparation

CAT EXAM: References to Study (Practice):

• The Pearson Guide to Verbal Ability and Reading Comprehension for CAT by Nishit Sinha (2020)

• https://iim-cat-questions-answers.2iim.com

• Previous Year's Question Papers

Part-B: CAT Logical Reasoning and Data Interpretation:

The topic covered for the day is given below: Students can refer to Linear Arrangement Part 1 for Day 16

Topic: Problem-Solving/ Puzzles (Linear Arrangement Part-2)

In this section, we will discuss the strategies to solve Puzzle questions for CAT exam based on the linear arrangement with the help of examples:

Type 1: Simple Linear Arrangement ( Puzzle questions for CAT exam)

Directions for Questions 1 to 4:

There are six friends (Andrew, Baithi, Charles, Duke, Elena, and Flora) seated in a row with numbered chairs from 1 to 6. They follow the following Seating Arrangement :

1. There are two places between Charles and Duke

2. There is one place between Andrew and Charles

3. There is one person between Flora and Elena

4. Elena is to be seated before Flora

5. Baithi is to be seated before Andrew, not necessarily immediately

6. The row does not start with Baithi

Q.1. The row would start with:

1. Andrew

2. Flora

3. Duke

4. Cannot be determined

Q.2. Elena will be seated on which seat number?

1. 1

2. 2

3. 3

4. Cannot be determined

Q.3. Who is seated at place number 6?

1. Andrew

2. Flora

3. Duke

4. Cannot be determined

Q.4. Baithi’s position is

1. 6

2. 5

3. 3

4. 2

The solution strategy for the above problem will be:

1. While solving, use Initials for each Name, for example,

Write A for Andrew, B for Baithi etc.

2. Mark the 6 places and fill them according to the given instructions.

Given statements are

There are two places between Charles and Duke

C _____ _____ D or D ______ ______C

There is one place between Andrew and Charles

A _____ C or C _______ A

There is one person between Flora and Elena, Elena is to be seated before Flora

E _______ F

Baithi is to be seated before Andrew, not necessarily immediately

The row does not start with Baithi

3. Eliminate the wrong possibilities

• Find the person in the first place.

• From above, In the first place either A, C, D or E will come.

• We cannot start with A since B should come before B. and B cannot be in the first place.

• If we start with C

• C ___ A D __ ___ (we cannot place E and F in this arrangement since E and F must have one person in between them)

• If we start with D

• D ___ ____ C __ A (we cannot place E and F in this arrangement since E and F must have one person in between them)
  After placing E, F and B according to the given condition
  D B E C F A

• If we start with E

• E ___ F ___ __ A (in this arrangement C must be just after F. So, we will not be able to place D. Therefore, this arrangement cannot be true)

• The correct combination should be

Answer: 1 (c), 2 (c), 3 (a), 4 (d)

Type 2: When two arrangements are given

Directions: Study the following information to answer the given questions:

Twelve people are sitting in two parallel rows containing six people each, in such a way that there is an equal distance between adjacent persons.

In row – 1, A, B, C, D, E and F are seated and all of them are facing south.

In row – 2, P, Q, R, S, T and V are seated and all of them are facing north.

Additional information is also given as

• Each member seated in a row faces another member of the other row.

• V sits third to the right of S.

• S faces F and F does not sit at any of the extreme ends of the line.

• D sits third to the right of C. R faces C.

• The one facing E sits third to the right of P.

• B and P do not sit at the extreme ends of the line.

• T is not an immediate neighbour of V and A is not an immediate neighbour of F.

• P sits to the left of S.

Q.1. If B and E interchange their places, who amongst the following faces B?

(a) T (b) P (c) Q (d) R (e) None of these

Ans: (c)

Q.2. Which of the following pairs must be at extreme ends but are not facing each other?

(a) R, F (b) T, A (c) D, R (d) C, Q (e) S, A

Ans: (b)

Q.3. If all the 12 persons are members of a family. Females are facing their husbands. All females are sitting at alternate places. If A is the husband of V. Then who is the husband of P?

(a) B (b) A (c) C (d) F (e) D

Ans: (a)

Solutions with Strategies:

1. Take two rows of six places. Consider the upper side of the paper as North and the downside as South.

2. Place the six people above facing downward (South) and six people in other row facing towards the North.

V sits third to the right of S.

S faces F and F does not sit at any of the extreme ends of the line.

D sits third to the right of C. R faces C.

• 
  
  The one facing E sits third to the right of P, and P sits to the left of S. B and P do not sit at the extreme ends of the line, T is not an immediate neighbour of V and A is not an immediate neighbour of F.

Combining all the figures above

CAT Exam: References to Study (Theory):

• Verbal and Non-Verbal Reasoning by R S Agrawal

• Cat preparation-iQuanta (YouTube Channel)

• MBAP CAT EBook (Concept Theory)

CAT Exam: References to Study (Practice):

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

• MBAP CAT EBook (Practice Questions).

• Previous Year's Question Papers

Part-C: CAT Exam VARC

Topic: Strategies to prepare for Conditionals

Conditionals and Rules, Tips to Prepare:
Conditionals are phrases used to explain the result of an event that might happen in the present or future or that could have happened in the past but didn’t.
Conditionals are divided into four parts:

1. Zero Conditionals: sentence formed two present simple verbs (one in the 'if clause' and one in the 'main clause'):
   Rule: If + present simple, .... present simple.
   Trick: Used when the result will always happen.
   Examples:
   If people eat too much, they get fat.
   If you touch a fire, you get burned.
   If babies are hungry, they cry.

2. First Conditional:
   Rule: If + Present Simple, (If Clause) ... will + infinitive (main clause)
   Trick: used to talk about things which might happen in the future.
   Examples:
   If it rains, I won't go to the park.
   If I study today, I'll go to the party tonight.
   If I see her, I'll tell her.

3. The Second Conditional: Uses the past simple after if, then 'would' and the infinitive:
   Rule: if + past simple, ...would + infinitive
   Tricks: We can use 'were' instead of 'was' with 'I' and 'he/she/it. This is mostly done in formal writing. We can use it to talk about things in the future that are probably not going to be true. Maybe I'm imagining some dream for example. We can use it to talk about something in the present which is impossible because it's not true.
   Examples:
   If I won the lottery, I would buy a big house.
   If I met the Queen of England, I would say hello.
   She would pass the exam if she ever studied. (She never studies, so this won't happen)
   If I were you, I wouldn't go out with that man.
   
   How is this different from the first conditional?
   We can understand this with the following example:
   For example (second conditional): If I had enough money, I would buy a house with twenty bedrooms and a swimming pool (I'm probably not going to have this much money, it's just a dream, not very real)
   But (first conditional): If I have enough money, I'll buy some new shoes (It's much more likely that I'll have enough money to buy some shoes)
   
   

4. The Third Conditional: Use past perfect after 'if' and then 'would have' and the past participle in the second part of the sentence:
   Rule: If + past perfect, ...would + have + past participle
   Trick: It represents the incidents in Past. It's used to describe a situation that didn't happen and to imagine the result of this situation.
   Examples:
   If she had studied, she would have passed the exam (but, really, we know she didn't study and so she didn't pass)
   If I hadn't eaten so much, I wouldn't have felt sick (but I did eat a lot, and so I did feel sick).
   If we had taken a taxi, we wouldn't have missed the plane.

Some Additional Tips:

• Differentiate between the use of first (for particular situations) vs zero conditional (for general activities).

• Differentiate between the use of first (for situations likely to happen in future) vs second conditional (for general activities that I don't think will really happen).

Practice Questions for CAT pdf on Conditionals [Based on CAT previous year papers]

Q1. What would you do differently if you _____ to do this again?

1. have

2. had

3. would have

4. None of these

Ans: (1) According to the third conditional.

Q2. The work _____ delayed if payment from the ministry don't credit on time.

1. is

2. will be

3. would be

4. None of these

Ans: (2); According to the first conditional.

Q3. When I watch TV for too long, I _____ a headache.

1. get

2. got

3. would get

4. None of these

Ans: (1); According to Zero Conditional

Prerequisite

• Parts of Speech

• Tenses.

CAT Exam: References to Study (Theory)

• Word Power Made Easy.

• CAT Edurav (YouTube Channel)

CAT Exam: References to Study (Practice)

• Wren and Martin

• Practice Sample Question Papers.

Practice and remain focused; Day 19 will be posted tomorrow.

Free PDFs for CAT 2025 preparation

• CAT 2025 Mastery: Chapter-wise MCQs for Success for VARC, DILR, Quant

• CAT DILR Questions with Solution, Download LRDI Questions for CAT Pdf