CAT with Confidence: Diving into Complex Numbers, Calendar Puzzles, and PYQs on Critical Reasoning- Day 54

Part-1: Quantitative Aptitude for CAT exam

Practise additional complex-numbers problems to strengthen your understanding of modulus, conjugates and algebraic operations.

Important Concepts and Formulas Used in CAT Complex Numbers Questions

The following concepts and formulas are useful for solving CAT Complex Numbers questions.

1. Complex Number: A complex number consists of a real part and an imaginary part.

   Example: $z=2+3i$

   • Real part: $\operatorname{Re}(z)=2$

   • Imaginary part: $\operatorname{Im}(z)=3$

2. Imaginary Unit ((i)): The imaginary unit is defined as $i=\sqrt{-1}$, which means $i^2=-1$.

3. Complex Conjugate: The complex conjugate of $z=a+ib$ is $\overline{z}=a-ib$.

   Example: If $z=2+3i$, then $\overline{z}=2-3i$.

CAT Previous Years Questions On Complex Numbers

CAT Previous Years Questions on complex numbers with solutions are given below:

Q1. CAT 2023 Complex Numbers Question and Answer

If $z$ is a complex number, then the minimum value of $|z-1|+|z+1|$ is ______.

Solution:

Let $z=a+ib$, where $a,b\in\mathbb{R}$.

Then,

$|z-1|=\sqrt{(a-1)^2+b^2}$

and

$|z+1|=\sqrt{(a+1)^2+b^2}$

Therefore,

$|z-1|+|z+1|=\sqrt{(a-1)^2+b^2}+\sqrt{(a+1)^2+b^2}$

By the triangle inequality,

$|z-1|+|z+1|\geq |(z-1)-(z+1)|$

$|z-1|+|z+1|\geq |-2|=2$

Equality holds when $z$ lies on the real line segment joining $-1$ and $1$, that is, when $b=0$ and $-1\leq a\leq1$.

Therefore, the minimum value is $2$.

Q2. Complex Numbers Question with Solution

If the cube roots of unity are $1$, $\omega$, and $\omega^2$, where $\omega=e^{2\pi i/3}$, find the roots of the equation $(x-1)^3+64=0$.

[Practice Question]

Solution:

Given,

$(x-1)^3+64=0$

Therefore,

$(x-1)^3=-64$

Taking the three cube roots of $-64$,

$x-1=-4,\quad -4\omega,\quad -4\omega^2$

Hence,

$x=-3,\quad 1-4\omega,\quad 1-4\omega^2$

Since $\omega=-\frac12+\frac{\sqrt3}{2}i$ and $\omega^2=-\frac12-\frac{\sqrt3}{2}i$, the roots can also be written as:

$x=-3,\quad 3-2\sqrt3,i,\quad 3+2\sqrt3,i$

CAT 2026 Complex Numbers: Important Preparation Tips

Use the following concepts and strategies to prepare for complex-number questions in CAT 2026.

1. Simplify Powers of $i$

Use the identity:

$i^{4k}=1$

where $k$ is a non-negative integer. The powers of $i$ repeat in a cycle of four.

2. Learn the Algebra of Complex Numbers

Let $N_1=a_1+ib_1$ and $N_2=a_2+ib_2$.

Addition

$N_1+N_2=(a_1+a_2)+i(b_1+b_2)$

Subtraction

$N_1-N_2=(a_1-a_2)+i(b_1-b_2)$

Multiplication

$N_1N_2=(a_1a_2-b_1b_2)+i(a_1b_2+a_2b_1)$

Division

$\frac{N_1}{N_2}=\frac{a_1+ib_1}{a_2+ib_2},\quad N_2\neq0$

To simplify the division, multiply the numerator and denominator by the conjugate of the denominator.

3. Prioritise Your Preparation

Complex numbers are a specialised Quantitative Aptitude topic. Understand the fundamental concepts and practise representative questions before deciding how much additional time to allocate to them.

4. Practise Previous Years' Questions

Solve previous years' questions to understand how complex-number concepts are tested and to improve your calculation accuracy.

Prerequisites for CAT Complex Numbers Questions

Before studying complex numbers, revise the following topics:

1. Basics of the number system

2. Imaginary numbers and the properties of $i$

CAT 2026 Complex Numbers: How to Analyse Unfamiliar Question Patterns

Complex Numbers questions can become challenging when the problem combines algebraic expressions, equations, and conditions on the value of a complex number. Rather than immediately applying a memorised formula, candidates should first identify the mathematical relationship being tested and determine which conditions the answer must satisfy.

How to Identify the Main Requirement of a Complex Numbers Question

Before solving a problem, identify whether it asks for a numerical value, a set of roots, a minimum or maximum value, or a relationship between complex numbers. This distinction helps candidates decide which mathematical approach is appropriate.

For example, a question asking for the roots of an equation requires all valid solutions to be considered. A question asking for the minimum value of an expression requires both a lower bound and a demonstration that the bound can actually be attained.

Candidates should therefore distinguish between finding a possible value and proving that it is the required value.

How to Verify Solutions to Complex Equations

When solving an equation involving complex numbers, candidates should check the original equation rather than relying only on the intermediate calculations.

Use the following verification process:

1. Determine the mathematical conditions imposed by the original equation.

2. Find the candidate solutions using the relevant algebraic method.

3. Substitute the solutions back into the original equation.

4. Check that each solution satisfies the equation and any additional restrictions.

5. Confirm that no valid solutions have been omitted.

This process is particularly useful when equations involve powers, roots, or multiple possible values.

Part-2: CAT 2026 Preparation Tips: CAT DILR tips (Logical Reasoning and Data Interpretation)

The following calendar reasoning questions demonstrate how to apply the odd-days method and leap-year rules. Practise each question step by step to improve your accuracy in determining weekdays and solving date-based problems.

Important Concepts and Formulas Used in Calendar Questions for CAT

1. Concept of Odd Days

Odd days are the days remaining after dividing the total number of days by 7.

$\text{Odd Days}=\text{Total Days}\bmod 7$

For example, 10 days contain 1 odd day because $10\div7$ leaves a remainder of 3. Therefore, the correct number of odd days is 3, not 1.

2. Concept of Leap Year

A year is a leap year if it is divisible by 4, except for century years, which must be divisible by 400.

Examples:

• 2024 was a leap year.

• 1900 was not a leap year.

• 2000 was a leap year.

A leap year has 366 days, including 29 days in February. An ordinary year has 365 days, including 28 days in February.

3. Finding the Day on a Given Date

To determine the day of the week for a given date, calculate the number of days between the known date and the required date. Divide this number by 7 and use the remainder to determine the weekday, accounting for whether you are moving forward or backward in time.

4. Calendar Calculations Using the Odd-Days Method

Calendar repetition depends on the sequence of leap years and the weekdays on which dates fall. Instead of relying on fixed repetition intervals, use the odd-days method to calculate the difference between dates and determine the corresponding weekday.

CAT Calendar Questions from Previous Years' Papers

Question 1. Given that 27 February 2003 was a Thursday, what day of the week was 27 February 1603? [CAT 2016]

1. Monday

2. Thursday

3. Sunday

4. Tuesday

Answer: Option 2 - Thursday

Solution:

The two dates are exactly 400 years apart. In the Gregorian calendar, the total number of days in a complete 400-year cycle is divisible by 7. Therefore, the same date falls on the same weekday.

Hence, 27 February 1603 was also a Thursday.

Question 2. Today is Monday. What day of the week will it be after 73 days? [SNAP Sample Question]

1. Thursday

2. Friday

3. Saturday

4. Sunday

Answer: Option 1 - Thursday

Solution:

Divide 73 by 7:

$73=7\times10+3$

The remainder is 3, so there are 3 odd days.

Starting from Monday and moving forward by 3 days:

• Tuesday: 1 day

• Wednesday: 2 days

• Thursday: 3 days

Therefore, the correct answer is Option 1: Thursday.

Tips/Tricks for CAT on calendar based CAT questions

1. An ordinary year starts with and end on the same day.

2. A leap year-end with a day after it starts.

3. Odd days in a week is ‘0’.

4. A century can not end on Tuesday, Thursday or Saturday.

5. After every 400 years the calendar repeats.

6. Practice separately each kind of question.

7. Practice previous years questions from various MBA entrance Exams to get an idea of the pattern of questions being asked.

CAT entrance exam: Prerequisite to calendar reasoning questions

1. Knowledge of reading a calendar

2. Basics Calculations

CAT 2026 Calendar Reasoning: Solving Questions Involving Multiple Dates

Calendar-based reasoning questions may require candidates to compare two dates, determine the interval between events, or establish the relationship between dates in different years. These problems become more manageable when candidates clearly identify the starting date, the target date, and the exact information being requested.

How to Interpret Questions with Multiple Date Conditions

Read the complete question before beginning the calculation. Determine whether the problem asks for the day of the week on a particular date, the number of days between two events, or the date on which a recurring event will occur.

These requirements are different and should not be treated as interchangeable.

For instance, a question may provide the weekday for one date and ask for the weekday on another date. Another question may provide two event dates and ask for the interval between them. In the first case, the objective is to determine a weekday; in the second, it is to establish a time difference.

Write down the known date and the required date separately to avoid confusing the starting point with the target.

How to Handle Questions Involving Recurring Events

Some calendar reasoning problems describe events that occur repeatedly, such as meetings, examinations, or scheduled activities. The key is to identify the recurrence condition stated in the question and apply it consistently.

Candidates should pay attention to whether the event repeats after a fixed number of days, occurs on a particular weekday, or follows a condition that changes from one month or year to another.

A systematic approach involves:

• Recording the date of the first event.

• Identifying the exact recurrence condition.

• Determining how that condition applies to subsequent events.

• Checking whether the final date satisfies all the stated restrictions.

Do not assume that an event repeats under the same calendar conditions unless the question provides enough information to establish that relationship.

Part-3: CAT Exam Preparation 2026: CAT exam tips for VARC (Verbal Ability & Reading Comprehension)

CAT Exam Important Topic: Strategy to Prepare “Statement and Conclusions” (Critical Reasoning) for CAT-2026

Important Concept of CAT Statement and Conclusions questions

The concepts used in Statement and Conclusions CAT questions are given below:

The question has three parts

Statement + Question+ Options (Conclusions)

Statement: A statement is a formal account of certain facts, views or situations in words.

Conclusions:

• No degree of assumptions

• It is derived

• It is not an opinion

• It should not be forced

CAT Practice Questions on Statements and Conclusions (critical reasoning questions for CAT)

Question 1: Statements and Conclusions (critical reasoning questions for CAT) are given below:

Read the passage and answer the questions based on it. (CAT 2017)

Typewriters are the epitome of a technology that has been comprehensively rendered obsolete by the digital age. The ink comes off the ribbon, they weigh a ton, and second thoughts are a disaster. But they are also personal, portable and, above all, private. Type a document and lock it away and more or less the only way anyone else can get it is if you give it to them. That is why the Russians have decided to go back to typewriters in some government offices, and why in the US, some departments have never abandoned them. Yet it is not just their resistance to algorithms and secret surveillance that keeps typewriter production lines - well one, at least - in business (the last British one closed a year ago). Nor is it only the nostalgic appeal of the metal body and the stout well-defined keys that make them popular on eBay. A typewriter demands something particular: attentiveness. By the time the paper is loaded, the ribbon tightened, the carriage returned, the spacing and the margins set, there's a big premium on hitting the right key. That means sorting out ideas, pulling together a kind of order and organising details before actually striking off. There can be no thinking on screen with a typewriter. Nor are there any easy distractions. No online shopping. No urgent emails. No Twitter. No need even for electricity - perfect for writing in a remote hideaway. The thinking process is accompanied by the encouraging clack of keys, and the ratchet of the carriage return. Ping!

Question. Which one of the following best describes what the passage is trying to do?

1. It describes why people continue to use typewriters even in the digital age.

2. It argues that typewriters will continue to be used even though they are an obsolete technology.

3. It highlights the personal benefits of using typewriters.

4. It shows that computers offer fewer options than typewriters.

Ans: [1]

Explanation:

The author is trying to explain:

why the typewriter continues to be used even in today's digital age. For this, he has given reasons that typewriters are personal and private and cannot be hacked. These can be used anywhere without the requirement of electricity.

Q2. CAT exam 2023 statements and Conclusions questions with solutions are given below:

Comprehension: [CAT 2018]

[The] Indian government [has] announced an international competition to design a National War Memorial in New Delhi, to honour all of the Indian soldiers who served in the various wars and counter-insurgency campaigns from 1947 onwards. The terms of the competition also specified that the new structure would be built adjacent to the India Gate – a memorial to the Indian soldiers who died in the First World War. Between the old imperialist memorial and the proposed nationalist one, India’s contribution to the Second World War is airbrushed out of existence.

The Indian government’s conception of the war memorial was not merely absent-minded. Rather, it accurately reflected the fact that both academic history and popular memory have yet to come to terms with India’s Second World War, which continues to be seen as little more than mood music in the drama of India’s advance towards independence and partition in 1947. Further, the political trajectory of the postwar subcontinent has militated against popular remembrance of the war. With partition and the onset of the India-Pakistan rivalry, both of the new nations needed fresh stories for self-legitimisation rather than focusing on shared wartime experiences.

However, the Second World War played a crucial role in both the independence and partition of India. . . . The Indian army recruited, trained and deployed some 2.5 million men, almost 90,000 of which were killed and many more injured. Even at the time, it was recognised as the largest volunteer force in the war. . . .

India’s material and financial contribution to the war was equally significant. India emerged as a major military-industrial and logistical base for Allied operations in Southeast Asia and the Middle East. This led the United States to take considerable interest in the country’s future and ensured that this was no longer the preserve of the British government.

Other wartime developments pointed in the direction of India’s independence. In a stunning reversal of its long-standing financial relationship with Britain, India finished the war as one of the largest creditors to the imperial power.

Such extraordinary mobilization for war was achieved at a great human cost, with the Bengal famine the most extreme manifestation of widespread wartime deprivation. The costs on India’s home front must be counted in millions of lives.

Indians signed up to serve on the war and home fronts for a variety of reasons. . . . [M]any were convinced that their contribution would open the doors to India’s freedom. . . . The political and social churn triggered by the war was evident in the massive waves of popular protest and unrest that washed over rural and urban India in the aftermath of the conflict. This turmoil was crucial in persuading the Attlee government to rid itself of the incubus of ruling India. . . .

Seventy years on, it is time that India engaged with the complex legacies of the Second World War. Bringing the war into the ambit of the new national memorial would be a fitting – if not overdue – recognition that this was India’s War.

Question: In the first paragraph, the author laments the fact that

1. the new war memorial will be built right next to India Gate.

2. there is no recognition of the Indian soldiers who served in the Second World War.

3. funds will be wasted on another war memorial when we already have the India Gate memorial.

4. India lost thousands of human lives during the Second World War

Ans: [2]

In the first paragraph, the author says that ‘India’s contribution to the second world war is airbrushed out of existence.’ This means, there is no recognition of the Indian soldiers’ contribution to the second world war.

Tips and Tricks for Statement and Conclusions questions for CAT exam 2026

1. Conclusions are not generalised.

2. A valid conclusion must follow logically from the statement. It should not introduce unsupported facts, make unjustified generalisations or assume that a possibility is certain.

3. Understand the difference between Inference and Conclusion

4. Rephrase the question in your words.

5. Generally, indirect questions are asked in CAT Exam (generally in the RCs). So, Practice different RCs appeared in CAT Exam.

6. Practice questions daily (Fewer questions but daily)

CAT 2026: Prerequisite to Statement and Conclusions

1. Knowledge of basic components of statements and arguments.

2. Reading Skills.

CAT 2026 Critical Reasoning: Common Traps in Statements and Conclusions Questions

Statements and Conclusions questions require candidates to determine whether a conclusion follows from the information provided. The challenge is to evaluate the logical relationship between the statement and the proposed conclusion without introducing assumptions that the question does not support.

Even when an option appears reasonable in everyday life, it may not follow logically from the given information. Candidates should therefore assess the evidence in the statement rather than rely on personal opinions or outside knowledge.

How to Identify Conclusions That Go Beyond the Given Information

A conclusion must remain within the scope of the statement. If the statement establishes a limited fact, the conclusion should not extend that fact to every situation or every member of a group without sufficient evidence.

For example, suppose a statement says that several organisations have introduced flexible working arrangements. A conclusion that all organisations have adopted flexible working would go beyond the information provided.

When evaluating an option, ask:

• Does the statement directly support the conclusion?

• Does the conclusion introduce a new fact?

• Does it make a claim that is broader than the available evidence?

• Does it confuse a possibility with a certainty?

These questions help identify conclusions that sound convincing but are not logically established.

How to Eliminate Incorrect Conclusion Options

When several answer choices appear plausible, evaluate each option independently against the statement.

Do not eliminate an option merely because it uses strong wording; check whether the statement actually supports that wording. Likewise, do not accept an option simply because it repeats a phrase from the passage.

For CAT 2026 preparation, candidates should review incorrect answers by identifying the exact logical gap in the rejected option. This develops a more consistent method for evaluating statements and conclusions.

Short Tricks to Solve Complex Number Questions in CAT

Complex number questions in the CAT exam often appear in the Quantitative Aptitude section and can be solved quickly with the right approach. Here are six effective tricks to simplify your preparation:

1. Use $i^2=-1$ Efficiently

Always reduce powers of $i$ using the following cycle:

• $i^1=i$

• $i^2=-1$

• $i^3=-i$

• $i^4=1$

Repeat this cycle to simplify powers of i instantly.

2. Convert to Polar Form When Multiplying

For multiplication or division of complex numbers, convert them to polar form:

$z=r(\cos\theta+i\sin\theta)$

Use De Moivre’s Theorem to handle powers and roots easily.

3. Use Conjugates to Simplify Divisions

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

4. Apply Modulus for Distance-Based Questions

Use the modulus formula $|z|=x^2+y^2$ for any complex number $z=x+iy$. This is helpful in geometry-based or coordinate-format questions.

5. Visualise on the Argand Plane

Plotting complex numbers on the Argand plane (real vs. imaginary axes) can simplify addition, subtraction, and understanding geometrical relationships like collinearity or symmetry.

6. Use Properties of Roots of Quadratic Equations

If a quadratic equation has complex roots, use the identity: if z is a root, its conjugate z is also a root, provided the equation has real coefficients.

Short Tricks to Solve Calendar Puzzle Questions in CAT

Calendar-based puzzles in CAT test logical reasoning and date-based calculations. With smart shortcuts, you can solve them quickly and accurately. Here are six effective tricks to tackle calendar questions with ease:

1. Memorise Odd Days of Each Month

Know how many odd days each month contributes:

January: 3, February: 0 or 1 (non-leap year/leap year), March: 3, April: 2, and so on.

Use this to calculate the day of the week efficiently.

2. Use the “Odd Days” Concept for Day Calculations

$\text{Odd Days}=\text{Total Days}\bmod 7$

The remainder gives the odd days, which determine the day of the week. For example, 3 odd days after Sunday means Wednesday.

3. Learn Century Code Tricks

Each century has a fixed code in the Gregorian calendar method:

• 1600s: 6

• 1700s: 4

• 1800s: 2

• 1900s: 0

• 2000s: 6

This helps in quickly determining the base weekday for a given date when used with the appropriate calendar formula.

4. Zeller’s Rule Shortcut

Use Zeller’s Congruence formula for exact day calculation, or its simplified version if you're comfortable with modular arithmetic.

5. Check the Leap Year Pattern

A year is a leap year if it is divisible by 4, except for century years, which must be divisible by 400. Leap years have 366 days, with February having 29 days instead of 28. This rule must be considered when calculating the day of the week for a given date.

6. Use Date Reversal Logic

Some puzzles ask questions like, “What day is it 100 days from now?” Quickly find the remainder when 100 is divided by 7:

$100\bmod 7=2$

Then count forward by 2 days using the odd-days method.

CAT 2026 Preparation Resources

Explore CAT 2026 preparation resources, including study materials, section-wise preparation tips, practice questions, and question papers. Strengthen your VARC, DILR, and Quantitative Aptitude skills with useful resources to boost your exam preparation.

Practice, stay focused, and we'll be back tomorrow with Day 55 CAT Preparation Tips.