{"id":90971,"date":"2025-06-01T10:42:15","date_gmt":"2025-06-01T10:42:15","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90971"},"modified":"2025-06-01T10:42:15","modified_gmt":"2025-06-01T10:42:15","slug":"if-xfrac1x2-then-which-one-of-the-following-is-the-value-of","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/","title":{"rendered":"If $x+\\frac{1}{x}=2$, then which one of the following is the value of"},"content":{"rendered":"

If $x+\\frac{1}{x}=2$, then which one of the following is the value of $x^{32}+\\frac{1}{x^{32}}$ ?<\/p>\n

[amp_mcq option1=”-1″ option2=”0″ option3=”1″ option4=”2″ correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2024<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is 2.
\n<\/section>\n
\nGiven the equation $x + \\frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x \\neq 0$, which must be true for $1\/x$ to be defined).
\n$x(x + \\frac{1}{x}) = 2x$
\n$x^2 + 1 = 2x$
\nRearranging the terms gives a quadratic equation:
\n$x^2 – 2x + 1 = 0$
\nThis equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$.
\nSolving for $x$, we get $x-1 = 0$, which means $x = 1$.
\nNow, we need to find the value of the expression $x^{32} + \\frac{1}{x^{32}}$.
\nSubstitute $x = 1$ into the expression:
\n$1^{32} + \\frac{1}{1^{32}} = 1 + \\frac{1}{1} = 1 + 1 = 2$.
\n<\/section>\n
\nThis problem utilizes a common algebraic trick where if $x + \\frac{1}{x} = 2$, then $x$ must be 1 (for real values of x). If $x + \\frac{1}{x} = -2$, then $x$ must be -1. For other values like $x + \\frac{1}{x} = 1$ or $-1$, $x$ would be complex numbers. The property $1^n = 1$ for any integer $n$ is used in the final calculation.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

If $x+\\frac{1}{x}=2$, then which one of the following is the value of $x^{32}+\\frac{1}{x^{32}}$ ? [amp_mcq option1=”-1″ option2=”0″ option3=”1″ option4=”2″ correct=”option4″] This question was previously asked in UPSC CAPF – 2024 Download PDFAttempt Online The correct answer is 2. Given the equation $x + \\frac{1}{x} = 2$. To find the value of $x$, we can multiply … <\/p>\n

Detailed SolutionIf $x+\\frac{1}{x}=2$, then which one of the following is the value of<\/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":[1103,1102],"class_list":["post-90971","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1103","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIf $x+\\frac{1}{x}=2$, then which one of the following is the value of<\/title>\n<meta name=\"description\" content=\"The correct answer is 2. Given the equation $x + frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x neq 0$, which must be true for $1\/x$ to be defined). $x(x + frac{1}{x}) = 2x$ $x^2 + 1 = 2x$ Rearranging the terms gives a quadratic equation: $x^2 - 2x + 1 = 0$ This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$. Solving for $x$, we get $x-1 = 0$, which means $x = 1$. Now, we need to find the value of the expression $x^{32} + frac{1}{x^{32}}$. Substitute $x = 1$ into the expression: $1^{32} + frac{1}{1^{32}} = 1 + frac{1}{1} = 1 + 1 = 2$.\" \/>\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\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If $x+\\frac{1}{x}=2$, then which one of the following is the value of\" \/>\n<meta property=\"og:description\" content=\"The correct answer is 2. Given the equation $x + frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x neq 0$, which must be true for $1\/x$ to be defined). $x(x + frac{1}{x}) = 2x$ $x^2 + 1 = 2x$ Rearranging the terms gives a quadratic equation: $x^2 - 2x + 1 = 0$ This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$. Solving for $x$, we get $x-1 = 0$, which means $x = 1$. Now, we need to find the value of the expression $x^{32} + frac{1}{x^{32}}$. Substitute $x = 1$ into the expression: $1^{32} + frac{1}{1^{32}} = 1 + frac{1}{1} = 1 + 1 = 2$.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:42:15+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":"If $x+\\frac{1}{x}=2$, then which one of the following is the value of","description":"The correct answer is 2. Given the equation $x + frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x neq 0$, which must be true for $1\/x$ to be defined). $x(x + frac{1}{x}) = 2x$ $x^2 + 1 = 2x$ Rearranging the terms gives a quadratic equation: $x^2 - 2x + 1 = 0$ This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$. Solving for $x$, we get $x-1 = 0$, which means $x = 1$. Now, we need to find the value of the expression $x^{32} + frac{1}{x^{32}}$. Substitute $x = 1$ into the expression: $1^{32} + frac{1}{1^{32}} = 1 + frac{1}{1} = 1 + 1 = 2$.","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\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/","og_locale":"en_US","og_type":"article","og_title":"If $x+\\frac{1}{x}=2$, then which one of the following is the value of","og_description":"The correct answer is 2. Given the equation $x + frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x neq 0$, which must be true for $1\/x$ to be defined). $x(x + frac{1}{x}) = 2x$ $x^2 + 1 = 2x$ Rearranging the terms gives a quadratic equation: $x^2 - 2x + 1 = 0$ This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$. Solving for $x$, we get $x-1 = 0$, which means $x = 1$. Now, we need to find the value of the expression $x^{32} + frac{1}{x^{32}}$. Substitute $x = 1$ into the expression: $1^{32} + frac{1}{1^{32}} = 1 + frac{1}{1} = 1 + 1 = 2$.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:42:15+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\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/","url":"https:\/\/exam.pscnotes.com\/mcq\/if-xfrac1x2-then-which-one-of-the-following-is-the-value-of\/","name":"If $x+\\frac{1}{x}=2$, then which one of the following is the value of","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:42:15+00:00","dateModified":"2025-06-01T10:42:15+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is 2. Given the equation $x + \\frac{1}{x} = 2$. To find the value of $x$, we can multiply the entire equation by $x$ (assuming $x \\neq 0$, which must be true for $1\/x$ to be defined). $x(x + \\frac{1}{x}) = 2x$ $x^2 + 1 = 2x$ Rearranging the terms gives a quadratic equation: $x^2 - 2x + 1 = 0$ This equation is a perfect square trinomial, which can be factored as $(x-1)^2 = 0$. Solving for $x$, we get $x-1 = 0$, which means $x = 1$. Now, we need to find the value of the expression $x^{32} + \\frac{1}{x^{32}}$. 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