{"id":93094,"date":"2025-06-01T11:40:53","date_gmt":"2025-06-01T11:40:53","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=93094"},"modified":"2025-06-01T11:40:53","modified_gmt":"2025-06-01T11:40:53","slug":"when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/","title":{"rendered":"When the square of the sum of two numbers are added to the square of t"},"content":{"rendered":"

When the square of the sum of two numbers are added to the square of their difference, we get 416. The difference between the square of the sum and square of the difference is 384. What are the numbers ?<\/p>\n

[amp_mcq option1=”18 and 24″ option2=”12 and 16″ option3=”8 and 12″ option4=”10 and 12″ correct=”option3″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CISF-AC-EXE – 2022<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nLet the two numbers be ‘a’ and ‘b’. The sum of the numbers is (a + b) and the difference is (a – b). The problem gives two conditions based on the squares of the sum and difference.
\nCondition 1: The square of the sum added to the square of the difference is 416.
\n(a + b)\u00b2 + (a – b)\u00b2 = 416.
\nUsing the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 – 2ab + b\u00b2, we get:
\n(a\u00b2 + 2ab + b\u00b2) + (a\u00b2 – 2ab + b\u00b2) = 416.
\n2a\u00b2 + 2b\u00b2 = 416.
\nDividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1)
\n<\/section>\n
\nCondition 2: The difference between the square of the sum and the square of the difference is 384.
\n(a + b)\u00b2 – (a – b)\u00b2 = 384.
\nUsing the identities again:
\n(a\u00b2 + 2ab + b\u00b2) – (a\u00b2 – 2ab + b\u00b2) = 384.
\na\u00b2 + 2ab + b\u00b2 – a\u00b2 + 2ab – b\u00b2 = 384.
\n4ab = 384.
\nDividing by 4, we get: ab = 96. (Equation 2)
\n<\/section>\n
\nWe now have a system of two equations:
\n1) a\u00b2 + b\u00b2 = 208
\n2) ab = 96
\nWe can check the options given to see which pair of numbers satisfies both equations.
\nOption C gives the numbers 8 and 12.
\nLet’s test if a=8 and b=12 satisfy the equations:
\nEquation 1: 8\u00b2 + 12\u00b2 = 64 + 144 = 208. (Satisfied)
\nEquation 2: 8 * 12 = 96. (Satisfied)
\nSince the numbers 8 and 12 satisfy both conditions derived from the problem statement, they are the correct numbers. Alternatively, we can solve the system using the identities (a+b)\u00b2 = a\u00b2+b\u00b2+2ab and (a-b)\u00b2 = a\u00b2+b\u00b2-2ab.
\n(a+b)\u00b2 = 208 + 2(96) = 208 + 192 = 400 => a+b = \u221a400 = 20 (assuming positive numbers).
\n(a-b)\u00b2 = 208 – 2(96) = 208 – 192 = 16 => a-b = \u221a16 = 4 (assuming a>b).
\nSolving a+b=20 and a-b=4: Adding gives 2a=24 => a=12. Substituting gives 12+b=20 => b=8. The numbers are 12 and 8.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

When the square of the sum of two numbers are added to the square of their difference, we get 416. The difference between the square of the sum and square of the difference is 384. What are the numbers ? [amp_mcq option1=”18 and 24″ option2=”12 and 16″ option3=”8 and 12″ option4=”10 and 12″ correct=”option3″] This … <\/p>\n

Detailed SolutionWhen the square of the sum of two numbers are added to the square of t<\/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":[1089],"tags":[1108,1102],"class_list":["post-93094","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1108","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nWhen the square of the sum of two numbers are added to the square of t<\/title>\n<meta name=\"description\" content=\"Let the two numbers be 'a' and 'b'. The sum of the numbers is (a + b) and the difference is (a - b). The problem gives two conditions based on the squares of the sum and difference. Condition 1: The square of the sum added to the square of the difference is 416. (a + b)\u00b2 + (a - b)\u00b2 = 416. Using the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 - 2ab + b\u00b2, we get: (a\u00b2 + 2ab + b\u00b2) + (a\u00b2 - 2ab + b\u00b2) = 416. 2a\u00b2 + 2b\u00b2 = 416. Dividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1) Condition 2: The difference between the square of the sum and the square of the difference is 384. (a + b)\u00b2 - (a - b)\u00b2 = 384. Using the identities again: (a\u00b2 + 2ab + b\u00b2) - (a\u00b2 - 2ab + b\u00b2) = 384. a\u00b2 + 2ab + b\u00b2 - a\u00b2 + 2ab - b\u00b2 = 384. 4ab = 384. Dividing by 4, we get: ab = 96. (Equation 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\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"When the square of the sum of two numbers are added to the square of t\" \/>\n<meta property=\"og:description\" content=\"Let the two numbers be 'a' and 'b'. The sum of the numbers is (a + b) and the difference is (a - b). The problem gives two conditions based on the squares of the sum and difference. Condition 1: The square of the sum added to the square of the difference is 416. (a + b)\u00b2 + (a - b)\u00b2 = 416. Using the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 - 2ab + b\u00b2, we get: (a\u00b2 + 2ab + b\u00b2) + (a\u00b2 - 2ab + b\u00b2) = 416. 2a\u00b2 + 2b\u00b2 = 416. Dividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1) Condition 2: The difference between the square of the sum and the square of the difference is 384. (a + b)\u00b2 - (a - b)\u00b2 = 384. Using the identities again: (a\u00b2 + 2ab + b\u00b2) - (a\u00b2 - 2ab + b\u00b2) = 384. a\u00b2 + 2ab + b\u00b2 - a\u00b2 + 2ab - b\u00b2 = 384. 4ab = 384. Dividing by 4, we get: ab = 96. (Equation 2)\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:40:53+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=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"When the square of the sum of two numbers are added to the square of t","description":"Let the two numbers be 'a' and 'b'. The sum of the numbers is (a + b) and the difference is (a - b). The problem gives two conditions based on the squares of the sum and difference. Condition 1: The square of the sum added to the square of the difference is 416. (a + b)\u00b2 + (a - b)\u00b2 = 416. Using the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 - 2ab + b\u00b2, we get: (a\u00b2 + 2ab + b\u00b2) + (a\u00b2 - 2ab + b\u00b2) = 416. 2a\u00b2 + 2b\u00b2 = 416. Dividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1) Condition 2: The difference between the square of the sum and the square of the difference is 384. (a + b)\u00b2 - (a - b)\u00b2 = 384. Using the identities again: (a\u00b2 + 2ab + b\u00b2) - (a\u00b2 - 2ab + b\u00b2) = 384. a\u00b2 + 2ab + b\u00b2 - a\u00b2 + 2ab - b\u00b2 = 384. 4ab = 384. Dividing by 4, we get: ab = 96. (Equation 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\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/","og_locale":"en_US","og_type":"article","og_title":"When the square of the sum of two numbers are added to the square of t","og_description":"Let the two numbers be 'a' and 'b'. The sum of the numbers is (a + b) and the difference is (a - b). The problem gives two conditions based on the squares of the sum and difference. Condition 1: The square of the sum added to the square of the difference is 416. (a + b)\u00b2 + (a - b)\u00b2 = 416. Using the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 - 2ab + b\u00b2, we get: (a\u00b2 + 2ab + b\u00b2) + (a\u00b2 - 2ab + b\u00b2) = 416. 2a\u00b2 + 2b\u00b2 = 416. Dividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1) Condition 2: The difference between the square of the sum and the square of the difference is 384. (a + b)\u00b2 - (a - b)\u00b2 = 384. Using the identities again: (a\u00b2 + 2ab + b\u00b2) - (a\u00b2 - 2ab + b\u00b2) = 384. a\u00b2 + 2ab + b\u00b2 - a\u00b2 + 2ab - b\u00b2 = 384. 4ab = 384. Dividing by 4, we get: ab = 96. (Equation 2)","og_url":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:40:53+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/","url":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/","name":"When the square of the sum of two numbers are added to the square of t","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:40:53+00:00","dateModified":"2025-06-01T11:40:53+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"Let the two numbers be 'a' and 'b'. The sum of the numbers is (a + b) and the difference is (a - b). The problem gives two conditions based on the squares of the sum and difference. Condition 1: The square of the sum added to the square of the difference is 416. (a + b)\u00b2 + (a - b)\u00b2 = 416. Using the identities (a+b)\u00b2 = a\u00b2 + 2ab + b\u00b2 and (a-b)\u00b2 = a\u00b2 - 2ab + b\u00b2, we get: (a\u00b2 + 2ab + b\u00b2) + (a\u00b2 - 2ab + b\u00b2) = 416. 2a\u00b2 + 2b\u00b2 = 416. Dividing by 2, we get: a\u00b2 + b\u00b2 = 208. (Equation 1) Condition 2: The difference between the square of the sum and the square of the difference is 384. (a + b)\u00b2 - (a - b)\u00b2 = 384. Using the identities again: (a\u00b2 + 2ab + b\u00b2) - (a\u00b2 - 2ab + b\u00b2) = 384. a\u00b2 + 2ab + b\u00b2 - a\u00b2 + 2ab - b\u00b2 = 384. 4ab = 384. Dividing by 4, we get: ab = 96. (Equation 2)","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/when-the-square-of-the-sum-of-two-numbers-are-added-to-the-square-of-t\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CISF-AC-EXE","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cisf-ac-exe\/"},{"@type":"ListItem","position":3,"name":"When the square of the sum of two numbers are added to the square of t"}]},{"@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\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?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\/93094","targetHints":{"allow":["GET"]}}],"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=93094"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/93094\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=93094"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=93094"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=93094"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}