{"id":90019,"date":"2025-06-01T10:19:14","date_gmt":"2025-06-01T10:19:14","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90019"},"modified":"2025-06-01T10:19:14","modified_gmt":"2025-06-01T10:19:14","slug":"suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/","title":{"rendered":"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2"},"content":{"rendered":"

Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2x^2 – 1$ as shown in the following diagram:
\n[Diagram shows a region R between two parabolas]
\nTwo distinct lines are drawn such that each of these lines partitions the region R into at least two parts. If ‘n’ is the total number of regions generated by these lines, then :<\/p>\n

[amp_mcq option1=”‘n’ can be 4 but not 3″ option2=”‘n’ can be 4 but not 5″ option3=”‘n’ can be 5 but not 6″ option4=”‘n’ can be 6″ correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2016<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions ‘n’ can be 3 or 4, but not more with simple lines.
\n<\/section>\n
\nThe region R is bounded by the parabolas Y = x^2 and Y = 2x^2 – 1 between their intersection points at (-1, 1) and (1, 1).
\nTwo distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R).
\nLet’s consider the number of regions created by 2 lines inside a bounded region R:
\n1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R.
\n2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R.
\n3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions.<\/p>\n

The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves.<\/p>\n

Thus, ‘n’ can be 3 or 4.
\nEvaluating the options:
\nA) ‘n’ can be 4 but not 3 (False, n can be 3)
\nB) ‘n’ can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6)
\nC) ‘n’ can be 5 but not 6 (False, n cannot be 5)
\nD) ‘n’ can be 6 (False, n cannot be 6)
\n<\/section>\n

\nThe formula for the maximum number of regions created by n lines in a plane is $\\frac{n(n+1)}{2} + 1$. For n=2 lines, this is 4 regions. This applies to the whole plane. For a bounded region, the number of regions is typically limited by the number of intersection points within the region plus the number of times lines cross the boundary. For two lines acting as simple chords, at most one intersection point can be inside R, leading to a maximum of 4 regions.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2x^2 – 1$ as shown in the following diagram: [Diagram shows a region R between two parabolas] Two distinct lines are drawn such that each of these lines partitions the region R into at least two parts. If ‘n’ … <\/p>\n

Detailed SolutionSuppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2<\/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":[1098,1102],"class_list":["post-90019","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1098","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nSuppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2<\/title>\n<meta name=\"description\" content=\"The correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions 'n' can be 3 or 4, but not more with simple lines. The region R is bounded by the parabolas Y = x^2 and Y = 2x^2 - 1 between their intersection points at (-1, 1) and (1, 1). Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R). Let's consider the number of regions created by 2 lines inside a bounded region R: 1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R. 2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R. 3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions. The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves. Thus, 'n' can be 3 or 4. Evaluating the options: A) 'n' can be 4 but not 3 (False, n can be 3) B) 'n' can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6) C) 'n' can be 5 but not 6 (False, n cannot be 5) D) 'n' can be 6 (False, n cannot be 6)\" \/>\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\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2\" \/>\n<meta property=\"og:description\" content=\"The correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions 'n' can be 3 or 4, but not more with simple lines. The region R is bounded by the parabolas Y = x^2 and Y = 2x^2 - 1 between their intersection points at (-1, 1) and (1, 1). Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R). Let's consider the number of regions created by 2 lines inside a bounded region R: 1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R. 2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R. 3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions. The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves. Thus, 'n' can be 3 or 4. Evaluating the options: A) 'n' can be 4 but not 3 (False, n can be 3) B) 'n' can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6) C) 'n' can be 5 but not 6 (False, n cannot be 5) D) 'n' can be 6 (False, n cannot be 6)\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:19:14+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":"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2","description":"The correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions 'n' can be 3 or 4, but not more with simple lines. The region R is bounded by the parabolas Y = x^2 and Y = 2x^2 - 1 between their intersection points at (-1, 1) and (1, 1). Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R). Let's consider the number of regions created by 2 lines inside a bounded region R: 1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R. 2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R. 3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions. The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves. Thus, 'n' can be 3 or 4. Evaluating the options: A) 'n' can be 4 but not 3 (False, n can be 3) B) 'n' can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6) C) 'n' can be 5 but not 6 (False, n cannot be 5) D) 'n' can be 6 (False, n cannot be 6)","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\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/","og_locale":"en_US","og_type":"article","og_title":"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2","og_description":"The correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions 'n' can be 3 or 4, but not more with simple lines. The region R is bounded by the parabolas Y = x^2 and Y = 2x^2 - 1 between their intersection points at (-1, 1) and (1, 1). Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R). Let's consider the number of regions created by 2 lines inside a bounded region R: 1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R. 2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R. 3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions. The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves. Thus, 'n' can be 3 or 4. Evaluating the options: A) 'n' can be 4 but not 3 (False, n can be 3) B) 'n' can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6) C) 'n' can be 5 but not 6 (False, n cannot be 5) D) 'n' can be 6 (False, n cannot be 6)","og_url":"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:19:14+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\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/","url":"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/","name":"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:19:14+00:00","dateModified":"2025-06-01T10:19:14+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct option is B. When two distinct lines partition a bounded region R, the number of resulting regions 'n' can be 3 or 4, but not more with simple lines. The region R is bounded by the parabolas Y = x^2 and Y = 2x^2 - 1 between their intersection points at (-1, 1) and (1, 1). Two distinct lines are drawn such that each partitions the region R into at least two parts (meaning each line must cross R). Let's consider the number of regions created by 2 lines inside a bounded region R: 1. If the two lines are parallel and both cross R, they divide R into 3 regions. Imagine two parallel horizontal lines within R. 2. If the two lines intersect *inside* R, they divide R into 4 regions. Imagine two horizontal lines intersecting inside R, or a horizontal and a vertical line intersecting inside R. 3. If the two lines intersect *outside* R but both cross R, they function like two non-parallel chords that do not intersect inside R. This configuration still divides R into 3 regions. The maximum number of regions created by 2 lines partitioning a bounded region (where each line enters and exits the region at most twice) is 4. More regions require lines to intersect multiple times within the region or interact more complexly with the boundary, which is not possible with just two straight lines partitioning a simple bounded region formed by smooth curves. Thus, 'n' can be 3 or 4. Evaluating the options: A) 'n' can be 4 but not 3 (False, n can be 3) B) 'n' can be 4 but not 5 (True, n can be 4, and based on analysis, cannot be 5 or 6) C) 'n' can be 5 but not 6 (False, n cannot be 5) D) 'n' can be 6 (False, n cannot be 6)","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/suppose-r-is-the-region-bounded-by-the-two-curves-y-x2-and-y-2\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"Suppose R is the region bounded by the two curves $Y = x^2$ and $Y = 2"}]},{"@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\/90019","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=90019"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90019\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}