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CAT Aspirant's Triad: Mastering Quadratic Equations, Introduction to Cubes, and Tenses All in One - Day 14
Edited By Team Careers360 | Updated on Sep 19, 2024 03:22 PM IST | #CAT
With the CAT 2024 exam date approaching, you may be confused about how to study effectively in just 2 months, what strategy to use, and the best sources for theory and practice. Questions about whether you can crack the CAT in 2 months, how much CAT preparation time is needed, and the best study plans are common. You might also be considering a 2-month crash course.
CAT 2025: VARC, DILR, and Quant: MCQs & Weightages | Chapter-wise MCQs
CAT 2025 Study Material: VARC | Quantitative Aptitude | 60 Days Study Material by Experts
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This Story also Contains
- Part-1: Quantitative Aptitude for CAT 2024
- Part 2: CAT Logical Reasoning and Data Interpretation:
- Part-3: CAT Exam VARC
- Free PDFs for CAT 2024 preparation
For Day 14 of our CAT preparation, we’re focusing on Quadratic Equations questions. This article will help you practice and understand different types of Quadratic Equations that frequently appear on the exam.
Following the 60-day CAT exam preparation schedule we’ve provided will keep you on track with a structured plan. We’ve already covered strategies for Algebra (Linear Equations ), Direction ion (Logical Reasoning), and Future Tenses on Day 13. Today, we’ll dive into quadratic equations to further enhance your preparation.
Part-1: Quantitative Aptitude for CAT 2024
Topic: Quadratic Equation (Algebra) And Quadratic Equation Questions For CAT
Quadratic Equations are the equations with degree 2. Quadratic Equations are an important part of Algebra as well as for the CAT exam. You can expect 1 or 2 algebra questions for the CAT Exam (based on analysis of previous year's question papers).
Quadratic Equations are of the form of ax2 + bx + c =0.
Here ‘x’ denotes variables (unknown) and a, b, and c are constants and a≠ 0.
To prepare CAT Quadratic Equations questions, you should invest 4-5 hours. Here we will discuss the quadratic equation CAT questions, quadratic equation formula, quadratic equation solutions, roots of quadratic equations, Quadratic Equations examples etc.
Quadratic equation problems for CAT can be categorised into the following types:
Questions that involve calculations of some unknown variable.
Questions that involve calculations of the nature of roots.
Formation of the quadratic equation.
CAT 2025: VARC, DILR, and Quant MCQs & Weightages
Comprehensive CAT prep guide with focused practice on Verbal Ability, Data Interpretation & Logical Reasoning, and Quantitative Aptitude.
Download NowHow to solve quadratic Equations problems for CAT and quadratic equation questions for CAT.
There are three methods which are generally used to find the roots (zeros) of a quadratic equation.
Method of factorization
Example: 2x2 + 5x + 3 = 0
⇒ 2x2 + 2x + 3x + 3 = 0
⇒ 2x(x + 1) + 3(x + 1) = 0
⇒ (x + 1) (2x + 3) = 0
⇒ x = -1 or x = -3/2.
This method is suitable only if factors of the quadratic polynomial are possible.
Quadratic Formula (Sridharacharya Formula)
For equation ax2 + bx + c =0.
x = [-b ± √D] /2a
where D is discriminant which is equal to (b2 - 4ac).
Example: 2x2 + 5x + 3 = 0
D = 52 - 4 x 2 x 3 = 1
x = [-5 ± √1] /4 = [-5 ± 1] /4
⇒ x = -1 or -3/2.
Method of completing the square:
Convert the given equation in the form of either (A+B)2 or (A-B)2
Example:
2x2+ 5x + 3 = 0
⇒ x2 + 5x/2 + 3/2 = 0
⇒ x2 + 2 x 5x/4 + 25/16 – 25/16 + 3/2 = 0
⇒ (x + 5/4)2 – 1/16 = 0
⇒ (x + 5/4)2 = 1/16
⇒ (x + 5/4) = ± ¼
⇒ x = ± ¼ - 5/4
⇒ x = -1 or -3/2
CAT quadratic equation questions with solutions:
The two Previously Asked Questions from Quadratic Equations in the CAT exam are given below, aspirants can also download the CAT Previous Year Algebra Questions pdf for more practice and references.
Q.1. Suppose one of the roots of the equation ax2 - bx + c = 0 is 2 + √3, where a, b, and c are rational numbers and a ≠ 0. If b = c3 then |a| equals [CAT question paper - 2021]
2
3
4
1
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Solution: (a)
For a quadratic equation, ax2 + bx + c = 0.
Since the roots, a, b, and c are all rational and if one of the roots is irrational, the other root of the equation will also be irrational and a conjugate of the first root.
So, the other root is 2 - √ 3.
Sum of roots = −(-b)/a = b/a = (2 + √ 3) + (2 - √ 3) = 4
And Product of roots = c/a = (2 + √ 3) (2 - √ 3) = 4 - 3 = 1
But here the quadratic equation is ax2 - bx + c = 0.
Therefore, b = 4a and c = a
It is given that b = c3
4a = a3
So, a = 2 or -2.
So, |a| = 2.
Q.2. Let m and n be positive integers, If x2 + mx + 2n = 0 and x2 + 2nx + m = 0 have real roots, then the smallest possible value of m + n is [CAT Question Paper 2020]
8
6
5
7
Ans: (b)
CAT 2024 quadratic equations questions with answers
For real roots b2 ≥ 4ac
From first equation: m2 ≥ 4 (2n)
⇒ m2 ≥ 8n
From second equation: (2n)2 ≥ 4m
⇒ 4n2 ≥ 4m
⇒ n2 ≥ m
If n = 1; m2≥ 8; m ≥ 3
If m ≥ 3 at n = 1, n2 ≥ m is not true.
If n = 2; m2 ≥ 16; m ≥ 4
If m ≥ 4 at n = 2, n2 ≥ m which is a valid result, if m = 4 and n = 2
So, the minimum sum of m and n is 4+2 = 6.
Q.3. The sum of twice a number and thrice its reciprocal is 25/2. What is the number?
(a) 7 (b) 6 (c) 5 (d) 4
Ans: (b)
Given: The sum of twice a number and thrice its reciprocal is 25/2.
Let the number be x.
According to the question,
⇒ 2x + 3/x = 25/2
⇒ 4x2 + 6 = 25x
⇒ 4x2 – 25x + 6 = 0
⇒ 4x2 – 24x – x + 6 = 0
⇒ (4x – 1)(x – 6) = 0
⇒ (4x–1) = 0 or (x–6) = 0
⇒ x = 6 or ¼
After checking the options, we get, x=6
Hence, the correct answer is option (b).
Q.4. If (x – 2)(x – p) = x2 – ax + 6, then the value of (a – p) is:
(a) 0 (b) 1 (c) 2 (d) 3
Ans: (c)
Given: (x – 2)(x – p) = x2 – ax + 6
⇒ x2– px – 2x + 2p=x2 – ax + 6
⇒ x2 – (p + 2)x + 2p=x2 –ax + 6
By comparing both sides of the equation,
p + 2 = a
⇒ a – p = 2
Hence, the correct answer is option (c).
Q.5. If x2 + y2 + 2x + 1 = 0, then the value of x31 + y35 is:
(a) –1 (b) 1 (c) 2 (d) 0
Ans: (a)
Given:
x2 + y2 + 2x + 1 = 0
⇒ x2 + 2x + 1 + y2 = 0
⇒ (x+1)2 + y2 = 0
Since the addition of the squares of the two terms is zero,
So, both the terms are individually zero.
Therefore, (x + 1)2 = 0 and y2 = 0
⇒ x = –1 and y = 0
Now, x31 + y35
= (–1)31 + (0)35
= –1
Hence, the correct answer is option (a).
Prerequisite to Solve Quadratic Equation Questions For CAT
For this topic, students must have knowledge of
Degree of Polynomials and zeros of a polynomial.
Tips to Prepare CAT previous year quadratic equations questions:
Learn all the methods of solving Quadratic Equations and use those methods according to the situation.
If the sum of zeros of a quadratic equation is ‘s’ and the product of zeros is ‘p’. Then the quadratic equation will be x2 - sx + p = 0
The nature of zeros is calculated by just finding the value of D.
If D = 0, zeros/ roots are real and equal.
If D > 0, zeros/ roots are real and unequal.
If D < 0, zeros/ roots are imaginary.The sum of the roots = -b/a; Product of roots = c/a.
Always check for the feasibility of solutions.
References to Study (Theory):
How to Prepare Quantitative Aptitude by Arun Sharma (6th edition)
2iim (You-Tube)
References to Study (Practice) Quadratic Equation Problems For CAT:
NCERT Class 9-10
https://iim-cat-questions-answers.2iim.com
Part 2: CAT Logical Reasoning and Data Interpretation:
Questions based on cubes play an important role in CAT. Here we will discuss cube CAT questions, Cube Logical Reasoning questions, cube logical reasoning tricks etc.
Topic: Cube Part-1 (LR) And CAT Questions On Cubes
A cube is a 3-D figure which consists of 6 faces, 12 edges, and 8 vertices.
CAT questions on Cube are based on the following concepts:
Painted surface
Opposite faces
Consecutive faces
Number of cuts to make smaller cubes of equal dimension
Approaches and Tips to Solve Cube Questions Logical Reasoning:
If all the faces of a cube (n x n x n) painted with the same colour are cut into (n x n x n) equal parts.
Number of cubes with 3 faces painted = 8 cubes
Number of cubes with exactly 2 faces painted = (n-2) x 12
Number of cubes with exactly 1 face painted = (n – 2)2 x 6
Number of cubes with 0 faces painted = (n-2)3
You can assume your room is a cube to analyze the situation.
You can use the approach of the open cube to solve the question.
Six faces can be seen like this
(1,6), (2,4), and (3,5) form opposite pairs.
Adjacent faces to face 3 are 1,2,4 and 6. Similarly, we can find the others.
Number of cuts to small cubes of equal size (n) from a larger cuboid of dimension (a x b x c) = (a/2 -1) + (b/2 -1) + (c/2 -1)
Questions that can help you to build the concept of Cubes (Based on CAT EXAM):
An Iron cube is coloured black on four adjoining surfaces while the opposite sides are coloured red. The cube is then cut two times vertically and four times horizontally at right angles. Based on the same answer to the next question.
Q.1. How many cubes (cuboids) have exactly one face coloured black?
1
2
3
4
Ans: D
One cube/cuboid at each black face has exactly one face black colour.
Q.2. How many cubes/cuboids have only one face painted that too in red?
2
3
4
5
Ans: A
The cube/cuboid with each red face has exactly one face red coloured.
Q.3. How many cubes/cuboids have more than 2 sides painted?
2
4
6
8
Ans: D
More than two faces painted means 3 faces painted i.e. a number of cubes formed at each corner.
Prerequisite to solve Cube And Cuboid Questions:
To Prepare the questions based on Cube, students must have the knowledge of
This topic requires basic calculations
Imagination
Analytical Approach
References to 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
Part-3: CAT Exam VARC
Topic: How to solve problems of sentence correction based on Tenses.
In previous articles, we have made strategies to prepare tenses. Today we are going to discuss how to solve problems of sentence correction based on Tenses. Also, we will explain error correction based on tenses, grammatical error correction, error correction examples etc.
Errors in sentences can be due to any of the following reasons:
Confusion in present tenses
Confusion in past tenses
Confusion in future tenses
Use of the wrong verb
You have to identify which type of error is there in the sentence and prepare them accordingly.
Practice Questions on Sentence correction that can boost your preparation: [Based on PYQs]
CAT Verbal Ability Questions
Here are some CAT verbal ability questions curated for aspirants to practice:
Find the correct phrase for underlining.
Q.1. The glass of the car broke while the players were playing football.
Broke
Was broken
No correction is required
Was broke
Ans: a
The action was done in the past but the time is not known. So, Past indefinite tense is used.
Q.2. I will be meeting you in the morning.
Meet
Meeting
No correction is required
Met
Ans: (a)
Future indefinite tense should be used.
Q.3. Preeti attended dance class every Thursday.
Attends
No correction required
Is attending
Will attending
Ans: (a)
Action is in the present time at a regular interval of time. Present indefinite should be used.
Q.4. Rani was playing when her father entered the room.
Played
No correction required
Plays
Is playing
Ans: (b)
Past continuous tense is to be used as the other sentence is in Past indefinite tense.
Q.5. It had stopped raining when I was leaving the office.
Leave
Was leaving
Left
Leaves
Ans: (c)
Past perfect tense should be used as the first phrase in the past perfect tense.
Prerequisite to solve problems of sentence correction based on tenses:
Basic Knowledge of Nouns, Pronouns and Verbs.
Basic knowledge of Tenses.
Knowledge of the use of helping verbs.
Tips to Prepare Error Correction in Tenses:
Practice a lot to improve your grammar.
Write rules and revise them on a regular basis.
During the examination, check on every possibility of error.
Attempt Mock-Test papers and solutions after the test to check your mistakes.
References to Study (Theory):
Advanced English grammar by Wren and Martin
GMATPrepNow (Youtube Channel)
References to Study (Practice):
How to Prepare for Verbal Ability & Reading Comprehension for CAT by Meenakshi Upadhyay and Arun Sharma (9th Edition)
Practice CAT Sample Question Papers.
Free PDFs for CAT 2024 preparation
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Questions related to CAT
Have a question related to CAT ?
Greetings! Here's a detailed response to your query:-
Determination of the best time to start the journey of CAT examination is based on individual factors like your current skill level, knowledge ,potential, learning pace and availability. To give yourself the best chance of success on the CAT exam, which usually takes place in November, it's advisable to begin studying between January and March of that same year. This 8 to 12-month preparation window is widely recommended by experts. But if you're managing your college along with the CAT preparations make sure you start from 2nd year onwards. It will hep you manage stress and you'll have ample amount of time to get along with the syllabus , pattern and various practice and mocks .
You'll be able to build a strong foundation in each sections and be able to know your strengths and weaknesses in different topics. The CAT syllabus is extensive, and early preparation allows for thorough coverage. Assess your learning speed and adjust your preparation timeline accordingly.
Essentially, while the ideal start time is subjective, beginning your CAT preparation well in advance is generally considered the most effective strategy for achieving optimal results.
SUKRITI GUPTA 01 April,2025
The CUSAT CAT previous year question papers are available on the Careers360 website. You can find question papers from 2012 to 2023, which can be downloaded in PDF format HERE
Additionally, the official website of Cochin University of Science and Technology (CUSAT) also provides previous year question papers. To access them, you can follow these steps:
- Visit the CUSAT official website - admissions.cusat.ac.in.
- Click on the resources tab from the homepage.
- Select the previous year's question papers from the drop-down menu.
- Select the Common Admission Test (CAT) question papers link.
- Now click on the B.Tech course.
By practicing with these previous year question papers, you can get familiar with the exam pattern, identify important topics, and improve your time management skills.
sadafreen356 28 March,2025
Hey Shathvika,
Let's dive deep into MBA Entrance Exams (Other than CAT)
1. XAT (Xavier Aptitude Test)
Conducted by: XLRI, Jamshedpur
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Decision Making, Verbal & Logical Ability, Quantitative Ability & Data Interpretation, and General Knowledge.
- Duration: 3 hours.
- Marks: No negative marking for General Knowledge.
Key Highlights:
- The Decision-Making section tests analytical and situational judgment skills.
- General Knowledge is evaluated during interviews, not for percentile calculation.
Official Website: https://xatonline.in
---
2. SNAP (Symbiosis National Aptitude Test)
Conducted by: Symbiosis International (Deemed) University
Eligibility: Bachelor’s degree with 50% marks (45% for reserved categories).
Exam Pattern:
- Sections: General English, Quantitative, Data Interpretation & Data Sufficiency, and Analytical & Logical Reasoning.
- Duration: 1 hour.
- Marks: Total 60 marks (MCQs).
Key Highlights:
- Specifically for admission to Symbiosis institutes like SIBM Pune and SCMHRD.
- Easy-to-moderate difficulty level.
Official Website: https://snaptest.org
---
3. CMAT (Common Management Admission Test)
Conducted by: National Testing Agency (NTA)
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Quantitative Techniques & Data Interpretation, Logical Reasoning, Language Comprehension, General Awareness, and Innovation & Entrepreneurship.
- Duration: 3 hours.
- Marks: 400 marks.
Key Highlights:
- Accepted by over 1,000 AICTE-approved institutes.
- Includes an optional Innovation & Entrepreneurship section.
Official Website: https://cmat.nta.nic.in
---
4. MAT (Management Aptitude Test)
Conducted by: All India Management Association (AIMA)
Eligibility: Bachelor’s degree in any discipline.
Exam Pattern:
- Sections: Language Comprehension, Mathematical Skills, Data Analysis & Sufficiency, Intelligence & Critical Reasoning, and Indian & Global Environment.
- Duration: 2.5 hours.
- Marks: 200 marks.
Key Highlights:
- Held multiple times a year in PBT, CBT, and IBT modes.
- Accepted by more than 600 MBA colleges in India.
Official Website: https://mat.aima.in
---
5. NMAT (NMIMS Management Aptitude Test)
Conducted by: Graduate Management Admission Council (GMAC)
Eligibility: Bachelor’s degree with 50% marks.
Exam Pattern:
- Sections: Language Skills, Quantitative Skills, and Logical Reasoning.
- Duration: 2 hours.
- Marks: 360 marks.
Key Highlights:
- Allows three attempts, with the best score considered.
- Accepted by NMIMS and partner universities worldwide.
Official Website: https://www.mba.com/exams/nmat
---
6. IIFT (Indian Institute of Foreign Trade)
Conducted by: National Testing Agency (NTA)
Eligibility: Bachelor’s degree (final-year students can also apply).
Exam Pattern:
- Sections: Quantitative Analysis, Reading Comprehension & Verbal Ability, Logical Reasoning & Data Interpretation, and General Knowledge.
- Duration: 2 hours.
- Marks: 300 marks.
Key Highlights:
- Specifically for IIFT’s flagship MBA in International Business.
- General Knowledge plays a significant role in shortlisting.
Official Website: https://iift.nta.nic.in
---
7. TISSNET (Tata Institute of Social Sciences National Entrance Test)
Conducted by: Tata Institute of Social Sciences
Eligibility: Bachelor’s degree.
Exam Pattern:
- Sections: English Proficiency, Mathematics & Logical Reasoning, and General Awareness.
- Duration: 100 minutes.
- Marks: 100 marks.
Key Highlights:
- Focused on social sciences, HRM, and public policy programs.
- General Awareness is crucial (with emphasis on current affairs).
Official Website: https://admissions.tiss.edu
---
8. MICAT (MICA Aptitude Test)
Conducted by: MICA, Ahmedabad
Eligibility: Bachelor’s degree and a valid CAT/XAT/GMAT score.
Exam Pattern:
- Sections: Psychometric Test, Verbal Ability, Quantitative Ability, General Awareness, Divergent-Convergent Thinking, and Descriptive Test.
- Duration: 2.5 hours.
Key Highlights:
- Psychometric Test is a qualifying section.
- Specifically for MICA’s flagship PGDM-C program in Strategic Marketing and Communications.
Official Website: https://www.mica.ac.in
---
9. GMAT (Graduate Management Admission Test)
Conducted by: Graduate Management Admission Council (GMAC)
Eligibility: No specific eligibility; bachelor’s degree recommended.
Exam Pattern:
- Sections: Analytical Writing, Integrated Reasoning, Quantitative Reasoning, and Verbal Reasoning.
- Duration: 3 hours 7 minutes.
- Marks: 800 (scaled score).
Key Highlights:
- Globally recognized for MBA programs.
- Popular for Indian B-schools like ISB, SPJIMR, and Great Lakes.
Official Website: https://www.mba.com
---
10. MAH-CET (Maharashtra Common Entrance Test)
Conducted by: Maharashtra State CET Cell
Eligibility: Bachelor’s degree with a minimum of 50% marks (45% for reserved categories).
Exam Pattern:
- Sections: Logical Reasoning, Abstract Reasoning, Quantitative Aptitude, and Verbal Ability & Reading Comprehension.
- Duration: 2.5 hours.
- Marks: 200 marks (MCQs).
Key Highlights:
- Accepted by top Maharashtra B-schools like JBIMS, SIMSREE, and PUMBA.
- Known for its speed-intensive nature with 200 questions in 150 minutes.
Official Website: https://cetcell.mahacet.org
Let me know if you have any doubts regarding a specific exam or any other query.
Have a nice day !!
RAJIV TIWARI 27 March,2025
Hello Ayush,
K.R. Mangalam University utilizes a blend of MCQs, descriptive formats, and practical assessments to evaluate student performance. The continuous evaluation system ensures that students are assessed regularly throughout the semester, contributing to a more comprehensive understanding of their academic progress.
Narendra Ranawat 26 March,2025
Hello,
Here are some simple tips to prepare for the CUSAT CAT exam:
1. Focus on Physics, Chemistry, and Maths (PCM).
2. Give equal time to all subjects.
3. Use NCERT Books as they are best for basics.
You can refer to following study materials for better practice :-
Hope it helps !
Prachi Kumari 26 March,2025
Popular CAT Questions
Directions for question :
M/s Deloitte Touche Tohmatsu Limited, one of the top four audit and accounting firms in the world with headquarters at London, UK, and with an operational presence in 153 countries, hires Management Trainees (MT) from all the premier management institutes of India thrice every year, in the months of January, May and September.
Each new group of Management Trainees (MT) have to go through a four month rigorous training schedule, after which they have to pass through a test consisting of a written assessment and a case-analysis. The top hundred ranked Management Trainees (MT) based on the performance in the test are confirmed as Management Executives (ME). The rest are given the opportunity of undergoing the training for four months one more time along with the next batch of Management Trainees (MT) and then passing through the subsequent test consisting of the written assessment and case-analysis. The Management Trainee (MT) who fails to get confirmed as a Management Executive (ME) the second time is fired.
The scatter-graph below depicts the number of Management Trainees (MT) at Deloitte taking the tests from January 2020 till May 2022, and the vis-à-vis hired Management Trainees (MT) at Deloitte who were fired :
It is also known that for the month of September 2019 at Deloitte, 96 hired Management Trainees (MT) failed to be confirmed as a Management Executive (ME) the first time, and that 36 hired Management Trainees (MT) were fired.
Question :
In which test did the minimum number of Management Trainees (MT) get confirmed as a Management Executive (ME) in the second attempt ?
Option: 1
September 2020
Option: 2
May 2021
Option: 3
January 2021
Option: 4
January 2022
Directions for question:
The bar-graph given below shows the foreign exchange reserves of Nepal (in million Rupees) from 2014 to 2021. Answer the following questions based on the graph :
Question:
What was the percentage increase (rounded to the nearest integer, if deemed necessary) in the foreign exchange reserves in 2020 over 2016 ?
Option: 1 None
Option: 2 None
Option: 3 None
Option: 4 None
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Marketing Manager
A marketing manager is a person who oversees a company or product marketing. He or she can be in charge of multiple programmes or goods or can be in charge of one product. He or she is enthusiastic, organised, and very diligent in meeting financial constraints. He or she works with other team members to produce advertising campaigns and decides if a new product or service is marketable.
A Marketing manager plans and executes marketing initiatives to create demand for goods and services and increase consumer awareness of them. A marketing manager prevents unauthorised statements and informs the public that the business is doing everything to investigate and fix the line of products. Students can pursue an MBA in Marketing Management courses to become marketing managers.
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SEO Analyst
An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns.
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Digital Marketing Executive
Digital marketing is growing, diverse, and is covering a wide variety of career paths. Each job function aids in the development of effective digital marketing strategies and techniques. The aims and objectives of the individuals who opt for a career as a digital marketing executive are similar to those of a marketing professional: to build brand awareness, promote company services or products, and increase conversions. Individuals who opt for a career as Digital Marketing Executives, unlike traditional marketing companies, communicate effectively through suitable technology platforms.
2 Jobs Available
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