{"id":4575,"date":"2016-05-17T09:49:00","date_gmt":"2016-05-17T04:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous\/"},"modified":"2016-05-17T10:16:50","modified_gmt":"2016-05-17T04:46:50","slug":"ncert-solutions-class-11-maths-miscellaneous","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous\/","title":{"rendered":"NCERT Solutions class-11 Maths Miscellaneous"},"content":{"rendered":"<div>\n<p style=\"text-align: justify;\"><strong>Miscellaneous Exercise<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>1. The relation <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> is defined by <\/strong><img decoding=\"async\" style=\"height: 51px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image002.png\" \/><strong>. The relation <\/strong><img decoding=\"async\" style=\"height: 16px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image003.png\" \/><strong> is defined by <\/strong><img decoding=\"async\" style=\"height: 51px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image004.png\" \/><strong>. Show that <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> is a function and <\/strong><img decoding=\"async\" style=\"height: 16px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image003.png\" \/><strong> is not a function.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image005.png\" \/>and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image007.png\" \/><img decoding=\"async\" style=\"height: 29px; width: 102px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image008.png\" \/>and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image009.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is observed that <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> takes unique value at each point in its domain [0, 10]. Therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> is a function.<\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image011.png\" \/>and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image012.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image013.png\" \/><img decoding=\"async\" style=\"height: 29px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image014.png\" \/>and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image015.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image016.png\" \/> does not have unique value at <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image017.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image016.png\" \/> is not a function.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>2. If <\/strong><img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image018.png\" \/><strong> find <\/strong><img decoding=\"async\" style=\"height: 49px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image019.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image020.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image021.png\" \/> <img decoding=\"async\" style=\"height: 29px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image022.png\" \/><\/p>\n<p style=\"text-align: justify;\">and <img decoding=\"async\" style=\"height: 29px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image023.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/><img decoding=\"async\" style=\"height: 49px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image025.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>3. Find the domain of the function <\/strong><img decoding=\"async\" style=\"height: 44px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 44px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image027.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> is a rational function of <img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image028.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> assumes real values of all <img decoding=\"async\" style=\"height: 15px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image029.png\" \/> except for those values of <img decoding=\"async\" style=\"height: 15px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image029.png\" \/> for which<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image030.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image033.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Domain of function = R \u2013 {2, 6}<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>4. Find the domain and range of the real function <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> defined by <\/strong><img decoding=\"async\" style=\"height: 28px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 28px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image035.png\" \/> assumes real values if <img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image036.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 19px; width: 34px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image037.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image038.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Domain of <img decoding=\"async\" style=\"height: 27px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image039.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Range of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> = all real numbers <img decoding=\"async\" style=\"height: 16px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image041.png\" \/> 0 = <img decoding=\"async\" style=\"height: 27px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image042.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>5. Find the domain and range of the real function <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> defined by <\/strong><img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image043.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image044.png\" \/><\/p>\n<p style=\"text-align: justify;\">The function <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> is defined for all values of <img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image028.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Domain of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> = R<\/p>\n<p style=\"text-align: justify;\">When <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image045.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image046.png\" \/><\/p>\n<p style=\"text-align: justify;\">When <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image047.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image048.png\" \/><\/p>\n<p style=\"text-align: justify;\">When <img decoding=\"async\" style=\"height: 19px; width: 34px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image049.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Range of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image010.png\" \/> = All real numbers <img decoding=\"async\" style=\"height: 16px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image041.png\" \/> 0 = <img decoding=\"async\" style=\"height: 27px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image042.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>6. Let <\/strong><img decoding=\"async\" style=\"height: 53px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image051.png\" \/><strong> be a function from R into R. Determine the range of <\/strong><img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image052.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Here <img decoding=\"async\" style=\"height: 44px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image053.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 44px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image055.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 27px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image056.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 44px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image057.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 49px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 15px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image029.png\" \/> will be real if <img decoding=\"async\" style=\"height: 44px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image059.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 44px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 21px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image061.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/> <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image062.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Range of <img decoding=\"async\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image063.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>7. Let <\/strong><img decoding=\"async\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image064.png\" \/><strong> be defined respectively by <\/strong><img decoding=\"async\" style=\"height: 27px; width: 177px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image065.png\" \/><strong> Find <\/strong><img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image066.png\" \/><strong> and <\/strong><img decoding=\"async\" style=\"height: 44px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image067.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 82px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image068.png\" \/> and <img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image069.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 27px; width: 320px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 323px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image071.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 49px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image072.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>8. Let <\/strong><img decoding=\"async\" style=\"height: 27px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image073.png\" \/><strong> be a function from Z to Z defined by <\/strong><img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image074.png\" \/><strong> for some integers <\/strong><img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image075.png\" \/><strong> Determine <\/strong><img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image075.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image074.png\" \/> and<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image076.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 268px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image077.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image078.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image079.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image080.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image081.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image082.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image083.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Solving eq. (i) and (ii), we get <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image084.png\" \/> and <img decoding=\"async\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image085.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>9. Let R be a relation from N to N defined by R = <\/strong><img decoding=\"async\" style=\"height: 29px; width: 186px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image086.png\" \/><strong> Are the following true:<\/strong><\/p>\n<\/div>\n<div>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image087.png\" \/><strong> R for all <\/strong><img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image088.png\" \/><strong> N<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image089.png\" \/><strong> R implies <\/strong><img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image090.png\" \/><strong> R<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image089.png\" \/><strong> R, <\/strong><img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image091.png\" \/><strong> R implies <\/strong><img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image092.png\" \/><strong> R<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: R = <img decoding=\"async\" style=\"height: 29px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image093.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong>No, (3, 3) <img decoding=\"async\" style=\"height: 16px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image094.png\" \/> R because <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image095.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong>No, (9, 3) <img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> R but (3, 9) <img decoding=\"async\" style=\"height: 16px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image094.png\" \/> R<\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong>No, (81, 9) <img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> R but (81, 3) <img decoding=\"async\" style=\"height: 16px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image094.png\" \/> R<\/p>\n<hr \/>\n<p><strong>10. Let A = {1, 2, 3, 4}, B = {1, 5 9, 11, 15, 16} and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true:<\/strong><\/p>\n<\/div>\n<div>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> is a relation from A to B.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> is a function from A to B.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Justify your answer in each case.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Here A = {1, 2, 3, 4} and B = {1, 5, 9, 11, 15, 16}<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image097.png\" \/>= {(1, 1), (1, 5), (1, 9), (1, 11), (1, 15), (1, 16), (2, 1), (2, 5), (2, 9), (2, 11),<\/p>\n<p style=\"text-align: justify;\">(2, 15), (2, 16), (3, 1), (3, 5), (3, 9), (3, 11), (3, 15), (3, 16), (4, 1), (4, 5),<\/p>\n<p style=\"text-align: justify;\">(4, 9), (4, 11), (4, 15), (4, 16)}<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}<\/p>\n<p style=\"text-align: justify;\">Now, (1, 5), (2, 9), (3, 1), (4, 5), (2, 11) <img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> <img decoding=\"async\" style=\"height: 17px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image097.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> is a relation from A to B.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>11. Let <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> be a subset of <img decoding=\"async\" style=\"height: 17px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image098.png\" \/>defined by <\/strong><img decoding=\"async\" style=\"height: 27px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image099.png\" \/><strong> Is <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><strong> a function from Z to Z? Justify your answer.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>We observed that <img decoding=\"async\" style=\"height: 17px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image100.png\" \/>= 4 and <img decoding=\"async\" style=\"height: 17px; width: 34px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image101.png\" \/> = 4<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/><img decoding=\"async\" style=\"height: 26px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image102.png\" \/><img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> and <img decoding=\"async\" style=\"height: 26px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image103.png\" \/><img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image031.png\" \/>(4, 5) <img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/>and (4, 4) <img decoding=\"async\" style=\"height: 14px; width: 14px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image096.png\" \/> <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/><\/p>\n<p style=\"text-align: justify;\">It shows that <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> is not a function from Z to Z.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>12. Let A = {9, 10, 11, 12 13} and let <\/strong><img decoding=\"async\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image104.png\" \/><strong> be defined by <\/strong><img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image105.png\" \/><strong> the highest prime factor of <\/strong><img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image106.png\" \/><strong> Find the range of <\/strong><img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image052.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Here A = {9, 10, 11, 12, 13}<\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image107.png\" \/><\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 19px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image108.png\" \/>and 3 is highest prime factor of 9]\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 19px; width: 78px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image110.png\" \/>]\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image111.png\" \/><\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 17px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image112.png\" \/>]\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 19px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image114.png\" \/>]\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 27px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 19px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image116.png\" \/>]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 14px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image024.png\" \/>Range of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/11\/maths\/ch02\/Misc\/image001.png\" \/> = {5, 11, 3, 13}<\/p>\n<p style=\"text-align: justify;\">= {3, 5, 11, 13}<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Miscellaneous Exercise 1. The relation is defined by . The relation is defined by . Show that is a function and is not a function. Ans. Given: and At and It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function. Now, and At and Therefore, does &#8230; <a title=\"NCERT Solutions class-11 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous\/\" aria-label=\"More on NCERT Solutions class-11 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[47,281],"tags":[],"class_list":["post-4575","post","type-post","status-publish","format-standard","hentry","category-cbse-class-11","category-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class-11 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions class-11 Maths Miscellaneous | myCBSEguide\" \/>\n<meta property=\"og:description\" content=\"Miscellaneous Exercise 1. 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