Mathematics Exam > Mathematics Questions > The sum of the binomial coefficients in the e...
Start Learning for Free
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a is
- a)1
- b)2
- c)1/2
- d)for no value of a
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/...
Most Upvoted Answer
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/...
Free Test
FREE
| Start Free Test |
Community Answer
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/...
Given:
The expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ has the following conditions:
1) The sum of the binomial coefficients is between 200 and 400.
2) The term independent of $x$ is 448.
To Find:
The value of $a$.
Solution:
1. Finding the Sum of Binomial Coefficients:
The sum of binomial coefficients in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ can be calculated using the binomial theorem.
The binomial theorem states that the $k^{th}$ term in the expansion of $(a+b)^n$ is given by:
$T_k = \binom{n}{k}a^{n-k}b^k$
In our case, $a = x$ and $b = -\frac{3}{4}ax^{\frac{5}{4}}$, and we need to find the sum of the binomial coefficients. Let's denote this sum as $S$.
$S = \binom{n}{0}a^n(-\frac{3}{4}ax^{\frac{5}{4}})^0 + \binom{n}{1}a^{n-1}(-\frac{3}{4}ax^{\frac{5}{4}})^1 + \binom{n}{2}a^{n-2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$
Since we are interested in the sum of binomial coefficients, we can ignore all terms with powers of $a$. This leaves us with:
$S = \binom{n}{0} + \binom{n}{1}(-\frac{3}{4}ax^{\frac{5}{4}}) + \binom{n}{2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$
Notice that this is the expansion of $(1 - \frac{3}{4}ax^{\frac{5}{4}})^n$.
So, $S = (1 - \frac{3}{4}ax^{\frac{5}{4}})^n$
Given that the sum of binomial coefficients is between 200 and 400, we have:
$200 < (1="" -="" \frac{3}{4}ax^{\frac{5}{4}})^n="" />< />
2. Finding the Term Independent of x:
The term independent of $x$ is obtained when all the powers of $x$ cancel out in the expansion. This occurs when the exponent of $x$ in each term is zero. In other words, the exponent of $x$ in each term is a multiple of $\frac{5}{4}$.
The term independent of $x$ in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^
The expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ has the following conditions:
1) The sum of the binomial coefficients is between 200 and 400.
2) The term independent of $x$ is 448.
To Find:
The value of $a$.
Solution:
1. Finding the Sum of Binomial Coefficients:
The sum of binomial coefficients in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ can be calculated using the binomial theorem.
The binomial theorem states that the $k^{th}$ term in the expansion of $(a+b)^n$ is given by:
$T_k = \binom{n}{k}a^{n-k}b^k$
In our case, $a = x$ and $b = -\frac{3}{4}ax^{\frac{5}{4}}$, and we need to find the sum of the binomial coefficients. Let's denote this sum as $S$.
$S = \binom{n}{0}a^n(-\frac{3}{4}ax^{\frac{5}{4}})^0 + \binom{n}{1}a^{n-1}(-\frac{3}{4}ax^{\frac{5}{4}})^1 + \binom{n}{2}a^{n-2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$
Since we are interested in the sum of binomial coefficients, we can ignore all terms with powers of $a$. This leaves us with:
$S = \binom{n}{0} + \binom{n}{1}(-\frac{3}{4}ax^{\frac{5}{4}}) + \binom{n}{2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$
Notice that this is the expansion of $(1 - \frac{3}{4}ax^{\frac{5}{4}})^n$.
So, $S = (1 - \frac{3}{4}ax^{\frac{5}{4}})^n$
Given that the sum of binomial coefficients is between 200 and 400, we have:
$200 < (1="" -="" \frac{3}{4}ax^{\frac{5}{4}})^n="" />< />
2. Finding the Term Independent of x:
The term independent of $x$ is obtained when all the powers of $x$ cancel out in the expansion. This occurs when the exponent of $x$ in each term is zero. In other words, the exponent of $x$ in each term is a multiple of $\frac{5}{4}$.
The term independent of $x$ in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^
|
Explore Courses for Mathematics exam
|
|
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer?
Question Description
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer?.
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer?.
Solutions for The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics 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