{"id":133947,"date":"2023-03-12T16:53:01","date_gmt":"2023-03-12T11:23:01","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=133947"},"modified":"2023-03-12T16:53:01","modified_gmt":"2023-03-12T11:23:01","slug":"show-that-the-determinate-x-sin0-cos0-is-independent-of-0-sin0-x-1-cos0-1-x","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/show-that-the-determinate-x-sin0-cos0-is-independent-of-0-sin0-x-1-cos0-1-x","title":{"rendered":"Show that the determinate \/x sin0 cos0\/ is independent of 0. \/-sin0 -x 1\/ \/cos0 1 x\/"},"content":{"rendered":"<h2>Question: Show that the determinate \/x sin0 cos0\/ is independent of 0. \/-sin0 -x 1\/ \/cos0 1 x\/<\/h2>\n<p>&nbsp;<\/p>\n<h3>The correct answer is &#8211;<\/h3>\n<div class=\"flex-1 overflow-hidden\">\n<div class=\"react-scroll-to-bottom--css-gletb-79elbk h-full dark:bg-gray-800\">\n<div class=\"react-scroll-to-bottom--css-gletb-1n7m0yu\">\n<div class=\"flex flex-col items-center text-sm dark:bg-gray-800\">\n<div class=\"group w-full text-gray-800 dark:text-gray-100 border-b border-black\/10 dark:border-gray-900\/50 bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0 m-auto\">\n<div class=\"relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<h4>To show that the determinant is independent of 0, we need to show that its value remains constant as 0 varies.<\/h4>\n<h4>Expanding the determinant along the first row, we get:<\/h4>\n<h4>|x sin 0 cos 0| = x |-sin 0 1| &#8211; sin 0 |cos 0 1| |cos 0 x| |-sin 0 x|<\/h4>\n<h4>= -xsin0(-sin0) &#8211; cos0(x)(cos0) + sin0(cos0)(x) = x(sin^2 0 + cos^2 0) = x<\/h4>\n<h4>Therefore, we can see that the value of the determinant is x, which is independent of 0.<\/h4>\n<h4>Now, let&#8217;s check whether the given matrix satisfies the determinant equation:<\/h4>\n<h4>|x sin 0 cos 0| |-sin 0 -x 1| |cos 0 1 x|<\/h4>\n<h4>Expanding the determinant along the first row, we get:<\/h4>\n<h4>x |-sin 0 1| &#8211; sin 0 |cos 0 1| |1 x| |0 cos 0|<\/h4>\n<h4>= -xsin0(cos0) &#8211; sin0(1)(x) + (cos0)(1)(0) = -xsin0<\/h4>\n<h4>Therefore, we can see that the determinant of the given matrix is -xsin0, which is not equal to x. Hence, the given matrix does not satisfy the determinant equation.<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"flex justify-between\">\n<h4 class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-3 md:gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/h4>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"w-full h-32 md:h-48 flex-shrink-0\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"absolute bottom-0 left-0 w-full border-t md:border-t-0 dark:border-white\/20 md:border-transparent md:dark:border-transparent md:bg-vert-light-gradient bg-white dark:bg-gray-800 md:!bg-transparent dark:md:bg-vert-dark-gradient\">\n<form class=\"stretch mx-2 flex flex-row gap-3 last:mb-2 md:last:mb-6 lg:mx-auto lg:max-w-3xl\">\n<div class=\"relative flex h-full flex-1 md:flex-col\">\n<div class=\"flex ml-1 mt-1.5 md:w-full md:m-auto md:mb-2 gap-0 md:gap-2 justify-center\"><\/div>\n<\/div>\n<\/form>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question: Show that the determinate \/x sin0 cos0\/ is independent of 0. \/-sin0 -x 1\/ \/cos0 1 x\/ &nbsp; The correct answer is &#8211; To show that the determinant is independent of 0, we need to show that its value remains constant as 0 varies. Expanding the determinant along the first row, we get: |x [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-133947","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133947","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=133947"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133947\/revisions"}],"predecessor-version":[{"id":133948,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133947\/revisions\/133948"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=133947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=133947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=133947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}