• # question_answer The coefficient of ${{\mathbf{x}}^{\mathbf{3}}}$in ${{\left( \sqrt{{{\mathbf{x}}^{\mathbf{5}}}}+\frac{3}{\sqrt{{{x}^{3}}}} \right)}^{6}}$  is A)  120                              B)  540           C)  150                              D)  250

[b] Let ${{(r+1)}^{th}}$ term be the coefficient of x3 ${{t}_{r+1}}{{=}^{6}}{{C}_{r}}{{\left( \sqrt{{{x}^{5}}} \right)}^{6-r}}{{\left( \frac{3}{\sqrt{{{x}^{3}}}} \right)}^{r}}{{=}^{6}}{{C}_{r.}}{{(x)}^{15-\frac{5r}{2}-\frac{3r}{2}}}.{{(3)}^{r}}$${{=}^{6}}{{C}_{r}}.{{x}^{15-4r}}.{{(3)}^{r}}$ $\because 15-4r=3\Rightarrow 4r=12$ $\therefore r=3$ $\therefore$Coefficient of ${{x}^{3}}{{=}^{6}}{{C}_{3}}.{{(3)}^{3}}$ $=\frac{6}{3.3}\times 27=\frac{6\times 5\times 4}{3\times 2}\times 27=5\times 4\times 27=540$. Hence, option [b] is correct.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://studyadda.com/sample-papers/mathematics-olympiad-sample-paper-3_q5/818/377159", "fetch_time": 1643220336000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-05/segments/1642320304959.80/warc/CC-MAIN-20220126162115-20220126192115-00030.warc.gz", "warc_record_offset": 614742655, "warc_record_length": 21322, "token_count": 358, "char_count": 801, "score": 3.890625, "int_score": 4, "crawl": "CC-MAIN-2022-05", "snapshot_type": "latest", "language": "en", "language_score": 0.33876505494117737, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00048-of-00064.parquet"}