UPSC Exam > UPSC Questions > The interval in which x (> 0) must lie so tha...
Start Learning for Free
The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient is
a.
b.
c.
d. None of these?
a.
b.
c.
d. None of these?
Most Upvoted Answer
The interval in which x (> 0) must lie so that the greatest term in th...
Understanding the Problem
In the expansion of (1 + x)^2n, the coefficient of a term represents the number of ways that term can be formed. To find the greatest coefficient, we must find the interval in which x must lie so that the greatest term also has the greatest coefficient.
Analysis of Terms
- The general term in the expansion of (1 + x)^2n is given by C(2n, k) * (1)^(2n-k) * x^k, where k is the power of x.
- The coefficient of this term is C(2n, k), which represents the number of ways this term can be formed.
Finding the Greatest Coefficient
- The greatest coefficient occurs when k is maximum, which is when k = n.
- Therefore, the greatest term in the expansion is C(2n, n) * x^n.
Interval for x
- To ensure that the greatest term also has the greatest coefficient, we need to find the interval for x.
- Since the coefficient C(2n, n) is maximum when x^n is maximum, x must be greater than 0 for x^n to be maximum.
- Therefore, the interval for x (> 0) is the set of all positive real numbers.
Conclusion
- In conclusion, the interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)^2n has the greatest coefficient is all positive real numbers.
- By ensuring that x is positive, we guarantee that x^n is maximized, leading to the greatest coefficient in the expansion.
In the expansion of (1 + x)^2n, the coefficient of a term represents the number of ways that term can be formed. To find the greatest coefficient, we must find the interval in which x must lie so that the greatest term also has the greatest coefficient.
Analysis of Terms
- The general term in the expansion of (1 + x)^2n is given by C(2n, k) * (1)^(2n-k) * x^k, where k is the power of x.
- The coefficient of this term is C(2n, k), which represents the number of ways this term can be formed.
Finding the Greatest Coefficient
- The greatest coefficient occurs when k is maximum, which is when k = n.
- Therefore, the greatest term in the expansion is C(2n, n) * x^n.
Interval for x
- To ensure that the greatest term also has the greatest coefficient, we need to find the interval for x.
- Since the coefficient C(2n, n) is maximum when x^n is maximum, x must be greater than 0 for x^n to be maximum.
- Therefore, the interval for x (> 0) is the set of all positive real numbers.
Conclusion
- In conclusion, the interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)^2n has the greatest coefficient is all positive real numbers.
- By ensuring that x is positive, we guarantee that x^n is maximized, leading to the greatest coefficient in the expansion.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Top Courses for UPSCView all
The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these?
Question Description
The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these?.
The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these?.
Solutions for The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? defined & explained in the simplest way possible. Besides giving the explanation of
The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these?, a detailed solution for The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? has been provided alongside types of The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? theory, EduRev gives you an
ample number of questions to practice The interval in which x (> 0) must lie so that the greatest term in the expansion of (1 + x)2n has the greatest coefficient isa. b. c. d. None of these? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup