Master Logarithm and Modulus, PYQs for Circular Arrangement Problems for Top Scores- Day 21

Part 1: CAT 2024 QA Preparation Tips (Quantitative Aptitude)

Topic: Algebra (CAT Logarithm questions and Modulus)

Logarithm is an important part of algebra and is consistently asked in the CAT entrance exam. We will also cover a few Logarithm CAT questions and modulus. Students can also refer to the logarithm questions for the CAT pdf as well for more practice.

In the simplest terms, what a logarithm is:

If a^x = N, then x = log_a N.

Here ‘a’ is the base of the logarithm.

For Example:

10² = 100; So, 2 = log₁₀100

A modulus is a function which always returns a positive value. The graph of the modulus function is symmetric about the y-axis.

For Example:

If |x| =2, then x can be 2 or -2.

Tips to solve CAT questions on Logarithm and Modulus

(a part of CAT Arithmetic questions) Learn all the properties related to Logarithm.

The following are some of the key characteristics of logarithms:

1. log_a xy=log_a x + log_a y

2. loga x/y=log_a x - log_a y

3. log_axⁿ = nlog_a x

4. log_aⁿ xm =m/n log_ax

5. log_ax = 1/ log_xa (Base Change Operation)

6. log_aa= 1

7. log_a1 = 0

8. a^(logax) = x

Some CAT common mistakes to avoid:

• log_a (x+ y) ≠ log_a x + log_a y.

• loga xy ≠ (log_ax) (log_ay)

• The logarithm to base 1 is not defined.

• Questions of modulus can be solved by squaring both sides

For example:

If x =y, then x² = y² and we can solve accordingly.

• Practice inequality problems based on logarithms.

• Candidates preparing for the CAT should dedicate time to practice a wide range of inequality questions to ensure they're well-prepared for any variation that may appear on the exam.

• The characteristic of a number is defined as

• If log N = 4.73, then the characteristic of N is 4 and Mantissa is .73.

• This property is used to find the number of digits in a given number.

• i.e. Number of digits = characteristics + 1.

For example: find the number of digits in 2¹²

log 2¹² = 12 log 2 = 12 x 0.3010 = 3.612

Therefore, the number of digits in 212 will be (3+1 = 4)

Importance of Logarithm and Modulus

Logarithms are used to solve exponential equations and hence can be used for analysis while designing software or hardware.

• The modulus Function gives magnitude only irrespective of the direction of the variable.

• In the CAT exam, 1-2 questions are expected from this function.

CAT Previous Year logarithm questions and questions on modulus for CAT:

Q1. For a real number a, if (log15a+log32a)/[(log15a) (log32a)] = 4 then a must lie in the range [CAT 2021, slot-3]

1. 4 < a < 5

2. 3 < a < 4

3. a > 5

4. 2 < a < 3

Solution: (1)

(log15a + log32a)/[(log15a) (log32a)] = 4
⇒ 1/ log32a + 1/log15a = 4
⇒ log_a 32 + log_a15 = 4 (Using base change property)
⇒ loga(32x15) = 4
⇒ a⁴ = 32 x 15 = 480
Doing hit and trial;
2⁴ = 16

3⁴ = 81
4⁴ = 256
5⁴ = 625; clearly a⁴ lies between 256 and 625.
So, a must lie between 4 and 5.

Q.2

equals ____________ [TITA, CAT 2020]

Solution:

Find the value of the denominator first as follows:

(log₂ 4)² = 4

Using properties of logarithm

log₄ 8 = log₂8/ log₂4 = 3/2

So, (Log₄8)³ = (3/2)³ = 27/8

Similarly,

log₈16 = log₂16/ log₂ 8 = 4/3

So, (Log₈16)⁴ = 256/81

Therefore, denominator becomes 4 x (27/8) x (256/81) = 128/3

So,

= (2x 4 x 8 x 16)/(128/3) = 24

Hence, the correct answer is 24.

Prerequisite:

• Knowledge of Exponents

• Knowledge of some basic functions

• Knowledge of Linear and Quadratic Equations

References for Study (Theory):

• For Properties go through NCERT class 11.

• How to Prepare Quantitative Aptitude by Arun Sharma (7th edition)

• Mission CAT (You-Tube)

References to Study (Practice):

• Pearson (How to prepare quantitative aptitude for CAT)

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

• Previous Year's Question Papers

Part 2: CAT 2024 Preparation Tips: CAT DILR tips (Logical Reasoning and Data Interpretation)

Topic: Circular Arrangement questions for CAT (LR)

Type 1: All Persons facing the same direction (Simple)

Directions: Study the following information carefully and answer the questions given below:

Bhuvan, Mohan, Kartik, Hina, Tina, Ramu, David, and Aastha are sitting around a circle facing the centre. Ramu is third to the right of Bhuvan. Hina is second to the right of Aastha who is second to the right of Ramu. Karthik is third to the right of Tina who is not an immediate neighbour of Hina. David is second to the left of Tina.

Q1: Who is to the immediate left of Mohan?

1. Ramu

2. Tina

3. Bhuvan

4. Kartik

Q2: Who is the third to the left of Bhuvan?

1. Bhuvan

2. Aastha

3. Kartik

4. Tina

Q3: Who is the third to the left of Hina?

1. Astha

2. Tina

3. Kartik

4. Ramu

Solution: From the given information, we can make the following figure.

Tricks: Use the Initials of the name while drawing diagrams.

Draw all possibility diagrams.

Ans 1: (d)

Ans 2: (b)

Ans 3: (b)

Type 2: Half are Males and Half are females

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

Asha, Binay, Chintu, Daman, Esha, Fatima, Gurmeet and Haresh are sitting around a circular table facing away from the centre where half of them are males and half are females. No two males or two females are immediate neighbours of each other.

Asha is the wife of Haresh. Asha sits third to the left of Esha. Binay sits second to the right of Daman. Daman is not an immediate neighbour of Asha or Esha. Haresh and Chintu are immediate neighbours of each other. Binay is not an immediate neighbour of his wife Fatima.

Q4: Which of the following is true about Gurmeet?

1. Gurmeet is a male

2. Gurmeet sits exactly between Fatima and Haresh

3. Gurmeet sits third to the left of Esha

4. None is true

Q5: Who sits third to the left of Binay?

1. Fatima

2. Haresh

3. Daman

4. Asha

Q6: How many people sit between Binay and Fatima when counted in an anti-clockwise direction from Binay?

1. One

2. Two

3. Three

4. Four

5. More than four

Solution:

Trick: Notice the important points first

Half are males and Half are females. No two males and no two females are together.

Do not consider a person male or female by their name. Haresh is male (Husband of Asha). Chintu should be female.

All are facing away from the centre:

Left: Anticlockwise

Right: Clockwise

Since Binay sits second to the right of Daman, Daman must be male as Binay is male.

So, male members are Haresh, Binay, Daman, and Esha

And female members are Asha, Fatima, Gurmeet, and Chintu

Since Daman cannot be a neighbour of Esha or Asha, Daman is at position 7, and Binay must be at 5. Haresh will be at 3. So, Chintu will be at 2 and Fatima should be at 8.

Gurmeet will be at 6.

The arrangement will look like

Ans 4: (d)

Ans 5: (a)

Ans 6: (b)

Type 3: Half facing Centre and half facing away from the centre

Direction: Study the following information carefully and answer the questions given below:

Eight persons A, B, C, D, E, F, G and H are sitting around a rectangular table in such a way that two persons sit on each of the four sides of the table. Half of them are facing the centre and the remaining half are facing away from the centre. Persons sitting on opposite sides are exactly opposite to each other.

D faces North (Towards the centre) and sits exactly opposite to H. E is to the immediate left of H. A and G sit on the same side. G is exactly opposite of B who is to the immediate right of C. A is sitting to the left of D.

Q7: From the following, who is facing South?

1. G

2. D

3. H

4. A

Ans: (C)

Q8: If G is facing centre, who is right of B?

1. G

2. C

3. F

4. A or F

Ans: (B)

Q9: Which of the following pairs of persons has both the persons sitting on the same side with the first person sitting to the right of the second person?

1. DF

2. CB

3. FC

4. Cannot be determined

Ans: (D)

Solution:

References for Study (Theory):

• Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)

• CAT entrance test 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 Years CAT Question Papers

Part-3: CAT Exam Preparation 2024: CAT exam tips for VARC (Verbal Ability & Reading Comprehension)

Topic: To solve cloze tests based on articles, tenses, prepositions etc.

The Cloze test is a combination of fill-in-the-blank and reading comprehension. In this exercise, incomplete words must be filled in using the available options.

A cloze test can be based on the following:

• Parts of Speech

• Tenses

• Articles

• Prepositions etc

CAT Tips and Tricks to Solve Cloze Tests:

• Read the passage in the CAT question paper thoroughly to get an idea about the flow of the passage.

• Write down the key points as you read slowly.

• Connect (link) the next sentence with the previous sentence. Reading articles, novels, or newspapers can help with this. Thus, cultivate a reading habit.

• Look at the options to get the best word to fit in and eliminate the most unfit word first.

• Check for grammatical mistakes, missing articles or prepositions, and other errors.

• Attempt CAT mock tests.

CAT Exam Practice Questions on Cloze test

VARC CAT questions

You can better understand cloze tests by using the example below.

Direction (Q1-Q5)

Before the giant waves slammed into the coast in India and Sri Lanka, wild and domestic animals....(1)....to know what was about to happen.....(2)....fled to safety. According to eyewitnesses....(3)...., elephants screamed and ran for....(4)....grounds, dogs refused to go outdoors, flamingoes abandoned....(5)....low-lying breeding areas and zoo animals rushed into their shelters.

1. A. Seeming B. Were Seemed C. Seemed D. Seem

2. A. We B. Them C. Us D. They

3. A. Statement B. Accounts C. Explanations D. Narrations

4. A. High B. Higher C. Highest D. More high

5. A. our B. their C. theirs D. her

Ans:

(B) Were Seemed

(B) them

(D) Narrations is a suitable word.

(B) Comparative degree of high should be taken.

(B), since Flamingoes is plural.

References for 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 CAT Sample Papers and CAT Previous Year Question Papers.

Free PDFs for CAT 2024 preparation

• CAT preparation strategy by experts

• CAT 2020 Topper Interviews