Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 510

Number of terms in expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) is \(\ldots \ldots\) (a) 5 (b) 6 (c) 7 (d) 8

Short Answer

Expert verified
The number of terms in the expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) is \(6\).

Step by step solution

01

Understand the Binomial Theorem

The binomial theorem is used to find the expansion of a binomial expression raised to some power. It states that for any positive integer n, and any real numbers a and b: \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\) Where \({n \choose k} = \frac{n!}{k!(n-k)!}\) denotes the binomial coefficient.
02

Expanding \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}\) using Binomial Theorem

Using the binomial theorem, we can expand the first term as follows: \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (\sqrt{\mathrm{y}})^k\)
03

Expanding \((\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) using Binomial Theorem

Similarly, we can expand the second term as follows: \((\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (-\sqrt{\mathrm{y}})^k\)
04

Combine the expansions of Step 2 and Step 3

Now, we need to combine the expansions from Step 2 and Step 3: \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (\sqrt{\mathrm{y}})^k + \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (-\sqrt{\mathrm{y}})^k\) Observe that when k is odd, the terms in the second sum will become negative and cancel out the corresponding terms in the first sum. When k is even, the terms in the second sum will become positive and double the corresponding terms in the first sum. Finally, the combined expression will contain the terms with even k values only. Let's count these terms.
05

Counting the terms with even k values

The possible even k values are 0, 2, 4, 6, 8, and 10. Therefore, there are 6 terms in the expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) with even k values. So, the number of terms in the expansion is 6 which corresponds to option (b).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If coefficients of \(\mathrm{x}^{7}\) and \(\mathrm{x}^{8}\) are equal in expansion of \([2+(\mathrm{x} / 3)]^{\mathrm{n}}\) then \(\mathrm{n}=\) (a) 55 (b) 56 (c) 54 (d) 58

It \(\mathrm{A}\) and \(\mathrm{B}\) are coefficients of \(\mathrm{x}^{\mathrm{r}}\) and \(\mathrm{x}^{\mathrm{n}-\mathrm{r}}\) respectively in expansion of \((1+x)^{n}\) then \(=\) (a) \(\mathrm{A}+\mathrm{B}=\mathrm{n}\) (b) \(\mathrm{A}=\mathrm{B}\) (c) \(\mathrm{A}+\mathrm{B}=2^{\mathrm{n}}\) (d) \(A-B=2^{n}\)

Constant term in expansion of \([1+(2 / \mathrm{x})-(2 / \mathrm{x})]^{4}\) is ........ (a) 5 (b) \(-5\) (c) 4 (d) \(-4\)

\(\left[5^{(1 / 2)}+7^{(1 / 8)}\right]^{1024}\) has number of rational terms \(=\) (a) 0 (b) 129 (c) 229 (d) 178

\((\sqrt{2}+1)^{5}+(\sqrt{2}-1)^{5}=\) (a) 58 (b) \(58 \sqrt{2}\) (c) \(-58\) (d) \(-58 \sqrt{2}\)
See all solutions

Recommended explanations on English Textbooks

Essay Prompts

Read Explanation

Language Acquisition

Read Explanation

Essay Plan

Read Explanation

Language and Social Groups

Read Explanation

Synthesis Essay

Read Explanation

Research and Composition

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free