{"id":20182,"date":"2024-04-15T05:49:28","date_gmt":"2024-04-15T05:49:28","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20182"},"modified":"2024-04-15T05:49:28","modified_gmt":"2024-04-15T05:49:28","slug":"let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/","title":{"rendered":"Let the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f’}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. Which of the following statements is\/are TRUE? I. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f’}}\\left( \\theta \\right) = 0.\\] II. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f’}}\\left( \\theta \\right) \\ne 0\\] A. l only B. ll only C. Both l and ll D. Neither l nor ll"},"content":{"rendered":"

[amp_mcq option1=”l only” option2=”ll only” option3=”Both l and ll” option4=”Neither l nor ll” correct=”option2″]<\/p>\n

The correct answer is $\\boxed{\\text{C}}$.<\/p>\n

We can find the derivative of $f$ as follows:<\/p>\n

\\begin{align}
\n{\\text{f’}}\\left( \\theta \\right) &= \\det \\left( {\\begin{array}{<\/em>{20}{c}} {\\cos \\theta }&{-\\sin \\theta }&{1} \\ {-\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2}}&0 \\ {\\frac{\\sqrt{3}}{2}}&-\\frac{1}{2}&1 \\end{array}} \\right) \\\\[6pt] &= \\left( -\\cos \\theta – \\frac{\\sqrt{3}}{2} \\right) \\left( -\\frac{\\sqrt{3}}{2} – \\frac{1}{2} \\right) – \\left( -\\sin \\theta \\right) \\left( \\frac{\\sqrt{3}}{2} \\right) \\\\[6pt] &= \\sin \\theta \\cos \\theta + \\frac{\\sqrt{3}}{2} \\sin \\theta – \\frac{\\sqrt{3}}{2} \\cos \\theta \\\\[6pt] &= \\frac{\\sqrt{3}}{2} \\left( \\sin \\theta – \\cos \\theta \\right)
\n\\end{align*}<\/p>\n

We can see that ${\\text{f’}}\\left( \\theta \\right) = 0$ when $\\sin \\theta = \\cos \\theta$. This occurs when $\\theta = \\frac{\\pi}{4}$ or $\\theta = \\frac{5\\pi}{4}$. Therefore, there exists $\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)$ such that ${\\text{f’}}\\left( \\theta \\right) = 0$.<\/p>\n

However, we can also see that ${\\text{f’}}\\left( \\theta \\right)$ is not continuous on $\\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]$. This is because ${\\text{f’}}\\left( \\frac{\\pi}{4} \\right)$ and ${\\text{f’}}\\left( \\frac{5\\pi}{4} \\right)$ have different signs. Therefore, there exists $\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)$ such that ${\\text{f’}}\\left( \\theta \\right) \\ne 0$.<\/p>\n

In conclusion, both statements I and II are true.<\/p>\n","protected":false},"excerpt":{"rendered":"

[amp_mcq option1=”l only” option2=”ll only” option3=”Both l and ll” option4=”Neither l nor ll” correct=”option2″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20182","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\nLet the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f'}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. Which of the following statements is\/are TRUE? I. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) = 0.\\] II. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) \\ne 0\\] A. l only B. ll only C. Both l and ll D. Neither l nor ll<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f'}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. Which of the following statements is\/are TRUE? I. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) = 0.\\] II. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) \\ne 0\\] A. l only B. ll only C. Both l and ll D. Neither l nor ll\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=”l only” option2=”ll only” option3=”Both l and ll” option4=”Neither l nor ll” correct=”option2″]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:49:28+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Let the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f'}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. Which of the following statements is\/are TRUE? I. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) = 0.\\] II. There exists \\[\\theta \\in \\left( {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right)\\] such that \\[{\\text{f'}}\\left( \\theta \\right) \\ne 0\\] A. l only B. ll only C. Both l and ll D. Neither l nor ll","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/","og_locale":"en_US","og_type":"article","og_title":"Let the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f'}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. 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Neither l nor ll","og_description":"[amp_mcq option1=”l only” option2=”ll only” option3=”Both l and ll” option4=”Neither l nor ll” correct=”option2″]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:49:28+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/","url":"https:\/\/exam.pscnotes.com\/mcq\/let-the-function-textfleft-theta-right-left-beginarray20c-sin-theta-cos-theta-tan-theta-sin-left-fracpi-6-rightcos\/","name":"Let the function \\[{\\text{f}}\\left( \\theta \\right) = \\left| {\\begin{array}{*{20}{c}} {\\sin \\theta }&{\\cos \\theta }&{\\tan \\theta } \\\\ {\\sin \\left( {\\frac{\\pi }{6}} \\right)}&{\\cos \\left( {\\frac{\\pi }{6}} \\right)}&{\\tan \\left( {\\frac{\\pi }{6}} \\right)} \\\\ {\\sin \\left( {\\frac{\\pi }{3}} \\right)}&{\\cos \\left( {\\frac{\\pi }{3}} \\right)}&{\\tan \\left( {\\frac{\\pi }{3}} \\right)} \\end{array}} \\right|\\] where \\[\\theta \\in \\left[ {\\frac{\\pi }{6},\\,\\frac{\\pi }{3}} \\right]\\] and \\[{\\text{f'}}\\left( \\theta \\right)\\] denote the derivative of f with respect to \\[\\theta \\]. 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Neither l nor ll"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20182"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20182\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}