{"id":46235,"date":"2024-04-15T22:09:26","date_gmt":"2024-04-15T22:09:26","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=46235"},"modified":"2024-04-15T22:09:26","modified_gmt":"2024-04-15T22:09:26","slug":"the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/","title":{"rendered":"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?"},"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_6a971da3001f5\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    P and S                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    P and R                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    Q and S                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    Q and R                <\/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: <strong>A. P and S<\/strong><\/p>\n<p>A real-valued periodic function $f(x)$ can be represented as a Fourier series, which is a sum of sine and cosine functions with frequencies that are integer multiples of the fundamental frequency. The Fourier series of a real-valued even function $f(x)$ has only cosine terms, while the Fourier series of a real-valued odd function $f(x)$ has only sine terms.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<ul>\n<li><strong>P. Cosine terms if it is even<\/strong><\/li>\n<\/ul>\n<p>A real-valued even function $f(x)$ is a function that satisfies $f(-x) = f(x)$. For example, the function $f(x) = x^2$ is an even function. The Fourier series of an even function has only cosine terms. This is because <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 <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>                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> the cosine function is an even function, and the Fourier series is a sum of sine and cosine functions.<\/p>\n<ul>\n<li><strong>Q. Sine terms if it is even<\/strong><\/li>\n<\/ul>\n<p>A real-valued odd function $f(x)$ is a function that satisfies $f(-x) = -f(x)$. For example, the function $f(x) = x$ is an odd function. The Fourier series of an odd function has only sine terms. This is because the sine function is an odd function, and the Fourier series is a sum of sine and cosine functions.<\/p>\n<ul>\n<li><strong>R. Cosine terms if it is odd<\/strong><\/li>\n<\/ul>\n<p>This statement is incorrect. A real-valued odd function $f(x)$ has only sine terms, not cosine terms.<\/p>\n<ul>\n<li><strong>S. Sine terms if it is odd<\/strong><\/li>\n<\/ul>\n<p>This statement is correct. A real-valued odd function $f(x)$ has only sine terms.<\/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":[959],"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>The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?<\/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\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T22:09:26+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":"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?","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\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/","og_locale":"en_US","og_type":"article","og_title":"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T22:09:26+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\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/","name":"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T22:09:26+00:00","dateModified":"2024-04-15T22:09:26+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-of-a-real-periodic-function-has-only-p-cosine-terms-if-it-is-even-q-sine-terms-if-it-is-even-r-cosine-terms-if-it-is-odd-s-sine-terms-if-it-is-odd-which-of-the-above-statements\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"Signal processing","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/signal-processing\/"},{"@type":"ListItem","position":3,"name":"The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?"}]},{"@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\/46235"}],"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=46235"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/46235\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=46235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=46235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=46235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}