{"id":90364,"date":"2025-06-01T10:27:06","date_gmt":"2025-06-01T10:27:06","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90364"},"modified":"2025-06-01T10:27:06","modified_gmt":"2025-06-01T10:27:06","slug":"let-x%c2%b2-y%c2%b2-1-u%c2%b2-v%c2%b2-1-and-xu-yv-0-thenwhich-of-the-above","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-x%c2%b2-y%c2%b2-1-u%c2%b2-v%c2%b2-1-and-xu-yv-0-thenwhich-of-the-above\/","title":{"rendered":"Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then\nWhich of the above"},"content":{"rendered":"<p>Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then<br \/>\nWhich of the above is\/are true ?<\/p>\n<ul>\n<li>1. x\u00b2 + u\u00b2 = 1<\/li>\n<li>2. y\u00b2 + v\u00b2 = 1<\/li>\n<li>3. xy + uv = 0<\/li>\n<\/ul>\n<p>[amp_mcq option1=&#8221;3 only&#8221; option2=&#8221;1 and 2 only&#8221; option3=&#8221;1, 2 and 3&#8243; option4=&#8221;2 and 3 only&#8221; correct=&#8221;option3&#8243;]<\/p>\n<div class=\"psc-box-pyq-exam-year-detail\">\n<div class=\"pyq-exam\">\n<div class=\"psc-heading\">This question was previously asked in<\/div>\n<div class=\"psc-title line-ellipsis\">UPSC CAPF &#8211; 2019<\/div>\n<\/div>\n<div class=\"pyq-exam-psc-buttons\"><a href=\"\/pyq\/pyq-upsc-capf-2019.pdf\" target=\"_blank\" class=\"psc-pdf-button\" rel=\"noopener\">Download PDF<\/a><a href=\"\/pyq-upsc-capf-2019\" target=\"_blank\" class=\"psc-attempt-button\" rel=\"noopener\">Attempt Online<\/a><\/div>\n<\/div>\n<section id=\"pyq-correct-answer\">\nAll three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$.<br \/>\n<\/section>\n<section id=\"pyq-key-points\">\nThe conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. The condition $xu + yv = 0$ means the dot product of the vectors $(x, y)$ and $(u, v)$ is zero, which implies these vectors are orthogonal (perpendicular) to each other.<br \/>\n<\/section>\n<section id=\"pyq-additional-information\">\nIf $(x, y)$ is a unit vector, the unit vectors orthogonal to it are $(-y, x)$ and $(y, -x)$. So, $(u, v)$ must be either $(-y, x)$ or $(y, -x)$.<br \/>\nCase 1: $u = -y, v = x$.<br \/>\nStatement 1: $x^2 + u^2 = x^2 + (-y)^2 = x^2 + y^2 = 1$. (True, since $x^2+y^2=1$)<br \/>\nStatement 2: $y^2 + v^2 = y^2 + x^2 = x^2 + y^2 = 1$. (True, since $x^2+y^2=1$)<br \/>\nStatement 3: $xy + uv = xy + (-y)(x) = xy &#8211; xy = 0$. (True)<br \/>\nCase 2: $u = y, v = -x$.<br \/>\nStatement 1: $x^2 + u^2 = x^2 + y^2 = 1$. (True, since $x^2+y^2=1$)<br \/>\nStatement 2: $y^2 + v^2 = y^2 + (-x)^2 = y^2 + x^2 = 1$. (True, since $x^2+y^2=1$)<br \/>\nStatement 3: $xy + uv = xy + (y)(-x) = xy &#8211; xy = 0$. (True)<br \/>\nIn both possible scenarios derived from the given conditions, all three statements hold true.<br \/>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above is\/are true ? 1. x\u00b2 + u\u00b2 = 1 2. y\u00b2 + v\u00b2 = 1 3. xy + uv = 0 [amp_mcq option1=&#8221;3 only&#8221; option2=&#8221;1 and 2 only&#8221; option3=&#8221;1, 2 and 3&#8243; option4=&#8221;2 &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then\nWhich of the above\" class=\"read-more button\" href=\"https:\/\/exam.pscnotes.com\/mcq\/let-x%c2%b2-y%c2%b2-1-u%c2%b2-v%c2%b2-1-and-xu-yv-0-thenwhich-of-the-above\/#more-90364\">Detailed Solution<span class=\"screen-reader-text\">Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then<br \/>\nWhich of the above<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1119,1102],"class_list":["post-90364","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1119","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"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>Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above<\/title>\n<meta name=\"description\" content=\"All three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$. The conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. The condition $xu + yv = 0$ means the dot product of the vectors $(x, y)$ and $(u, v)$ is zero, which implies these vectors are orthogonal (perpendicular) to each other.\" \/>\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-x\u00b2-y\u00b2-1-u\u00b2-v\u00b2-1-and-xu-yv-0-thenwhich-of-the-above\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above\" \/>\n<meta property=\"og:description\" content=\"All three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$. The conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. The condition $xu + yv = 0$ means the dot product of the vectors $(x, y)$ and $(u, v)$ is zero, which implies these vectors are orthogonal (perpendicular) to each other.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/let-x\u00b2-y\u00b2-1-u\u00b2-v\u00b2-1-and-xu-yv-0-thenwhich-of-the-above\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:27:06+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 x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above","description":"All three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$. The conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. The condition $xu + yv = 0$ means the dot product of the vectors $(x, y)$ and $(u, v)$ is zero, which implies these vectors are orthogonal (perpendicular) to each other.","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-x\u00b2-y\u00b2-1-u\u00b2-v\u00b2-1-and-xu-yv-0-thenwhich-of-the-above\/","og_locale":"en_US","og_type":"article","og_title":"Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above","og_description":"All three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$. The conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. The condition $xu + yv = 0$ means the dot product of the vectors $(x, y)$ and $(u, v)$ is zero, which implies these vectors are orthogonal (perpendicular) to each other.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/let-x\u00b2-y\u00b2-1-u\u00b2-v\u00b2-1-and-xu-yv-0-thenwhich-of-the-above\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:27:06+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-x%c2%b2-y%c2%b2-1-u%c2%b2-v%c2%b2-1-and-xu-yv-0-thenwhich-of-the-above\/","url":"https:\/\/exam.pscnotes.com\/mcq\/let-x%c2%b2-y%c2%b2-1-u%c2%b2-v%c2%b2-1-and-xu-yv-0-thenwhich-of-the-above\/","name":"Let x\u00b2 + y\u00b2 = 1; u\u00b2 + v\u00b2 = 1 and xu + yv = 0, then Which of the above","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:27:06+00:00","dateModified":"2025-06-01T10:27:06+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"All three statements are true given the conditions $x^2 + y^2 = 1$, $u^2 + v^2 = 1$, and $xu + yv = 0$. The conditions $x^2 + y^2 = 1$ and $u^2 + v^2 = 1$ imply that $(x, y)$ and $(u, v)$ are unit vectors in a 2-dimensional space. 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