{"id":20132,"date":"2024-04-15T05:48:49","date_gmt":"2024-04-15T05:48:49","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20132"},"modified":"2024-04-15T05:48:49","modified_gmt":"2024-04-15T05:48:49","slug":"given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/","title":{"rendered":"Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} &#8211; 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} &#8211; 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ &#8211; 5}}{{12}}$$ D. -1"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a98d6af27776\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    $$\frac{3}{{16}}$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    0                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    $$\frac{{ - 5}}{{12}}$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    -1                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{3}{16}}$.<\/p>\n<p>The line integral $\\int_c \\mathbf{F} \\cdot d\\mathbf{l}$ is the work done by the force $\\mathbf{F}$ in moving a particle along the curve $c$. In this case, the force $\\mathbf{F}$ is given by<\/p>\n<p>$$\\mathbf{F} = (x^2 &#8211; 2y) \\hat{\\imath} &#8211; 4yz \\hat{\\jmath} + 4xz^2 \\hat{k}$$<\/p>\n<p>and the curve $c$ is the straight line from $(0, 0, 0)$ to $(1, 1, 1)$.<\/p>\n<p>To evaluate the line integral, we can use the following formula:<\/p>\n<p>$$\\int_c \\mathbf{F} \\cdot d\\mathbf{l} = \\int_a^b F_x dx + F_y dy + F_z dz$$<\/p>\n<p>where $a$ and $b$ are the endpoints of the curve $c$. In this case, $a = (0, 0, 0)$ and $b = (1, 1, 1)$. Substituting in the expression for $\\mathbf{F}$ and evaluating the integral, we get<\/p>\n<p>$$\\begin{align<em>}<br \/>\n\\int_c \\mathbf{F} \\cdot d\\mathbf{l} <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> &amp;= \\int_0^1 (x^2 &#8211; 2y) dx &#8211; 4yz dy + 4xz^2 dz \\<br \/>\n&amp;= \\left[ \\frac{x^3}{3} &#8211; 2xy \\right]_0^1 &#8211; 4y \\left[ \\frac{z^2}{2} \\right]_0^1 + 4x \\left[ \\frac{z^3}{3} \\right]_0^1 \\<br \/>\n&amp;= \\frac{1}{3} &#8211; 2(0) &#8211; 4(0) <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> + \\frac{4}{3} \\<br \/>\n&amp;= \\frac{3}{16}<br \/>\n\\end{align<\/em>}$$<\/p>\n<p>Therefore, the value of the line integral is $\\boxed{\\frac{3}{16}}$.<\/p>\n<p>The other options are incorrect because they do not give the correct value of the line integral.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Subscribe on YouTube<\/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":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} - 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} - 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ - 5}}{{12}}$$ D. -1<\/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\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} - 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} - 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ - 5}}{{12}}$$ D. -1\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:48:49+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":"Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} - 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} - 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ - 5}}{{12}}$$ D. -1","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\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/","og_locale":"en_US","og_type":"article","og_title":"Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} - 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} - 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ - 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4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ - 5}}{{12}}$$ D. -1","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:48:49+00:00","dateModified":"2024-04-15T05:48:49+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/given-overrightarrow-textf-left-textx2-2texty-rightoverrightarrow-texti-4textyzoverrightarrow-textj-4textxtextz2overrigh\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Engineering maths","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/"},{"@type":"ListItem","position":4,"name":"Calculus","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/calculus\/"},{"@type":"ListItem","position":5,"name":"Given $$\\overrightarrow {\\text{F}} = \\left( {{{\\text{x}}^2} &#8211; 2{\\text{y}}} \\right)\\overrightarrow {\\text{i}} &#8211; 4{\\text{yz}}\\overrightarrow {\\text{j}} + 4{\\text{x}}{{\\text{z}}^2}\\overrightarrow {\\text{k}} ,$$ the value of the line integral $$\\int\\limits_{\\text{c}} {\\overrightarrow {\\text{F}} \\cdot d\\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is A. $$\\frac{3}{{16}}$$ B. 0 C. $$\\frac{{ &#8211; 5}}{{12}}$$ D. -1"}]},{"@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\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?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\/20132"}],"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=20132"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20132\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}