{"id":1157,"date":"2020-03-08T08:31:30","date_gmt":"2020-03-08T08:31:30","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1157"},"modified":"2020-07-05T08:50:43","modified_gmt":"2020-07-05T08:50:43","slug":"circular-measure-practice-questions","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/circular-measure\/circular-measure-practice-questions\/","title":{"rendered":"IBDP Past year Exam Questions &#8211; Circular measure"},"content":{"rendered":"<p><strong>Q1.\u00a0 \u00a0[M09.P1.TZ1]<\/strong><\/p>\r\n<p>The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius <em>R <\/em>and the inner circle has radius <em>r <\/em>.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1158\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-5.jpg\" alt=\"\" width=\"445\" height=\"272\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-5.jpg 445w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-5-300x183.jpg 300w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><\/p>\r\n<p>Consider the shaded regions with areas <em>A <\/em>and <em>B <\/em>. Given that <em>A<\/em>: <em>B <\/em>= 2 :1, find the <strong>exact <\/strong>value of the ratio <em>R <\/em>: <em>r <\/em>.\u00a0 \u00a0 \u00a0 \u00a0 [5 marks]\u00a0 \u00a0<\/p>\r\n<div id=\"link1-link-1157\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link1-content-1157\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1184\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-6.jpg\" alt=\"\" width=\"955\" height=\"268\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-6.jpg 955w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-6-300x84.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-6-768x216.jpg 768w\" sizes=\"auto, (max-width: 955px) 100vw, 955px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q2.\u00a0 [N09.P2]<\/strong><\/p>\r\n<p>The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm. The points P and Q lie on the larger circle and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>P<\/mi><mover><mi>O<\/mi><mo>^<\/mo><\/mover><mi>Q<\/mi><mo>=<\/mo><mi>x<\/mi><\/math>\r\n , where \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#60;<\/mo><mi>x<\/mi><mo>&#60;<\/mo><mfrac><mi mathvariant=\"normal\">&#960;<\/mi><mn>2<\/mn><\/mfrac><\/math>\r\n .\u00a0 \u00a0 \u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1159\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-3.jpg\" alt=\"\" width=\"423\" height=\"281\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-3.jpg 423w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-3-300x199.jpg 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\r\n<p><strong>(a)<\/strong> Show that the area of the shaded region is \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>8<\/mn><mo>&#160;<\/mo><mi>sin<\/mi><mi>x<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><mi>x<\/mi><\/math>\r\n .\u00a0 \u00a0 [3 marks]\u00a0 \u00a0<\/p>\r\n<p><strong>(b)<\/strong> Find the maximum area of the shaded region.\u00a0 \u00a0 [4 marks]<\/p>\r\n<div id=\"link2-link-1157\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link2-content-1157\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1186\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-4.jpg\" alt=\"\" width=\"955\" height=\"568\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-4.jpg 955w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-4-300x178.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-4-768x457.jpg 768w\" sizes=\"auto, (max-width: 955px) 100vw, 955px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q3.\u00a0 \u00a0[M11.P2.TZ1]<\/strong><\/p>\r\n<p>The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB.\u00a0 [7 marks]<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1162\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-2.jpg\" alt=\"\" width=\"586\" height=\"339\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-2.jpg 586w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-2-300x174.jpg 300w\" sizes=\"auto, (max-width: 586px) 100vw, 586px\" \/><\/p>\r\n<div id=\"link3-link-1157\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link3-content-1157\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1187\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-3.jpg\" alt=\"\" width=\"955\" height=\"943\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-3.jpg 955w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-3-300x296.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-3-768x758.jpg 768w\" sizes=\"auto, (max-width: 955px) 100vw, 955px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q4.\u00a0 \u00a0[M12\/P2\/TZ2]<\/strong><\/p>\r\n<p>Two discs, one of radius 8 cm and one of radius 5 cm, are placed such that they touch each other. A piece of string is wrapped around the discs. This is shown in the diagram below.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1164\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-2.jpg\" alt=\"\" width=\"527\" height=\"338\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-2.jpg 527w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-2-300x192.jpg 300w\" sizes=\"auto, (max-width: 527px) 100vw, 527px\" \/><\/p>\r\n<p>Calculate the length of string needed to go around the discs.\u00a0 [8 marks]<\/p>\r\n<div id=\"link4-link-1157\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link4-content-1157\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1189\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-3.jpg\" alt=\"\" width=\"967\" height=\"862\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-3.jpg 967w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-3-300x267.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-3-768x685.jpg 768w\" sizes=\"auto, (max-width: 967px) 100vw, 967px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q5.\u00a0 \u00a0[N13\/P2]<\/strong><\/p>\r\n<p>The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mover><mi>O<\/mi><mo>^<\/mo><\/mover><mi>B<\/mi><mo>=<\/mo><mi>&#952;<\/mi><\/math>\r\n , where \u03b8 is in radians.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1166\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-1.jpg\" alt=\"\" width=\"608\" height=\"299\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-1.jpg 608w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-1-300x148.jpg 300w\" sizes=\"auto, (max-width: 608px) 100vw, 608px\" \/><\/p>\r\n<p><strong>(a)<\/strong> Show that the shaded area can be expressed as\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>50<\/mn><mi>&#952;<\/mi><mo>&#8211;<\/mo><mn>50<\/mn><mi>sin<\/mi><mi>&#952;<\/mi><\/math>\r\n .\u00a0 \u00a0 \u00a0 [2marks]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/p>\r\n<p><strong>(b)<\/strong> Find the value of \u03b8 for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures.\u00a0 \u00a0 [3 marks]<\/p>\r\n<div id=\"link5-link-1157\" class=\"sh-link link5-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link5', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link5-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link5-content-1157\" class=\"sh-content link5-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1190\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-2.jpg\" alt=\"\" width=\"976\" height=\"813\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-2.jpg 976w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-2-300x250.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-2-768x640.jpg 768w\" sizes=\"auto, (max-width: 976px) 100vw, 976px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q6.\u00a0 \u00a0[M13\/P2\/TZ2]<\/strong><\/p>\r\n<p>A circle of radius 4 cm , centre O , is cut by a chord [AB] of length 6 cm.\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1167\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-5.jpg\" alt=\"\" width=\"468\" height=\"295\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-5.jpg 468w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-5-300x189.jpg 300w\" sizes=\"auto, (max-width: 468px) 100vw, 468px\" \/><\/p>\r\n<p><strong>(a)<\/strong> Find , expressing your answer in radians correct to four significant figures. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [2 marks]<\/p>\r\n<p><strong>(b)<\/strong> Determine the area of the shaded region.\u00a0 \u00a0 [3 marks]<\/p>\r\n<div id=\"link6-link-1157\" class=\"sh-link link6-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link6', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link6-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link6-content-1157\" class=\"sh-content link6-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1192\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-6.jpg\" alt=\"\" width=\"991\" height=\"528\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-6.jpg 991w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-6-300x160.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-6-768x409.jpg 768w\" sizes=\"auto, (max-width: 991px) 100vw, 991px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q7.\u00a0 \u00a0[M13\/P2\/TZ1]<\/strong><\/p>\r\n<p>A rectangle is drawn around a sector of a circle as shown. If the angle of the sector is 1 radian and the area of the sector is , find the dimensions of the rectangle, giving your answers to the nearest millimetre.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1169\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-3.jpg\" alt=\"\" width=\"421\" height=\"245\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-3.jpg 421w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-3-300x175.jpg 300w\" sizes=\"auto, (max-width: 421px) 100vw, 421px\" \/><\/p>\r\n<div id=\"link7-link-1157\" class=\"sh-link link7-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link7', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link7-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link7-content-1157\" class=\"sh-content link7-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1194\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-4.jpg\" alt=\"\" width=\"927\" height=\"258\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-4.jpg 927w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-4-300x83.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-4-768x214.jpg 768w\" sizes=\"auto, (max-width: 927px) 100vw, 927px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q8.\u00a0 \u00a0[M11\/P2\/TZ2]<\/strong><\/p>\r\n<p>The points P and Q lie on a circle, with centre O and radius 8 cm, such that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>P<\/mi><mover><mi>O<\/mi><mo>^<\/mo><\/mover><mi>Q<\/mi><mo>=<\/mo><msup><mn>59<\/mn><mo>&#8728;<\/mo><\/msup><\/math>\r\n .<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1171\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-3.jpg\" alt=\"\" width=\"447\" height=\"259\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-3.jpg 447w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-3-300x174.jpg 300w\" sizes=\"auto, (max-width: 447px) 100vw, 447px\" \/><\/p>\r\n<p>Find the area of the shaded segment of the circle contained between the arc PQ and the chord [PQ].<\/p>\r\n<div id=\"link8-link-1157\" class=\"sh-link link8-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link8', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link8-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link8-content-1157\" class=\"sh-content link8-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1195\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-4.jpg\" alt=\"\" width=\"932\" height=\"257\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-4.jpg 932w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-4-300x83.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-4-768x212.jpg 768w\" sizes=\"auto, (max-width: 932px) 100vw, 932px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q9.\u00a0 \u00a0[M14\/P2\/TZ2]<\/strong><\/p>\r\n<p>The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1172\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-6.jpg\" alt=\"\" width=\"447\" height=\"312\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-6.jpg 447w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-6-300x209.jpg 300w\" sizes=\"auto, (max-width: 447px) 100vw, 447px\" \/><\/p>\r\n<p>Find:\u00a0 \u00a0 \u00a0 \u00a0<strong>(a)\u00a0\u00a0<\/strong> \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><mover><mi>O<\/mi><mo>^<\/mo><\/mover><mi>C<\/mi><\/math>\r\n ;\u00a0 \u00a0 \u00a0 [2 marks]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\r\n<p><strong>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(b)<\/strong> the area of the shaded region.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [4 marks]\u00a0 \u00a0 \u00a0<\/p>\r\n<div id=\"link9-link-1157\" class=\"sh-link link9-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link9', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link9-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link9-content-1157\" class=\"sh-content link9-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1197\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-7.jpg\" alt=\"\" width=\"972\" height=\"683\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-7.jpg 972w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-7-300x211.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-7-768x540.jpg 768w\" sizes=\"auto, (max-width: 972px) 100vw, 972px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q10.\u00a0 \u00a0[M11\/P1\/TZ2]<\/strong><\/p>\r\n<p>The diagram shows a tangent, (TP) , to the circle with centre O and radius <em>r <\/em>. The size of is \u03b8 radians.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1173\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-5.jpg\" alt=\"\" width=\"463\" height=\"276\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-5.jpg 463w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-5-300x179.jpg 300w\" sizes=\"auto, (max-width: 463px) 100vw, 463px\" \/><\/p>\r\n<p><strong>(a)<\/strong> Find the area of triangle AOP in terms of <em>r <\/em>and \u03b8 .\u00a0 \u00a0 \u00a0 \u00a0\u00a0 [1 mark]\u00a0 \u00a0 \u00a0\u00a0<\/p>\r\n<p><strong>(b)<\/strong> Find the area of triangle POT in terms of <em>r <\/em>and \u03b8 .\u00a0 \u00a0 \u00a0 \u00a0 [2 marks]\u00a0 \u00a0 \u00a0<\/p>\r\n<p><strong>(c)<\/strong> Using your results from part (a) and part (b), show that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>sin<\/mi><mi>&#952;<\/mi><mo>&#60;<\/mo><mi>&#952;<\/mi><mo>&#60;<\/mo><mi>tan<\/mi><mi>&#952;<\/mi><\/math>\r\n .\u00a0[2 marks]\u00a0<\/p>\r\n<div id=\"link10-link-1157\" class=\"sh-link link10-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link10', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link10-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link10-content-1157\" class=\"sh-content link10-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1199\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-6.jpg\" alt=\"\" width=\"946\" height=\"504\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-6.jpg 946w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-6-300x160.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-6-768x409.jpg 768w\" sizes=\"auto, (max-width: 946px) 100vw, 946px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q11.\u00a0 \u00a0[N11\/P1]<\/strong><\/p>\r\n<p>From a vertex of an equilateral triangle of side 2<em>x <\/em>, a circular arc is drawn to divide the triangle into two regions, as shown in the diagram below.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1174\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-4.jpg\" alt=\"\" width=\"447\" height=\"259\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-4.jpg 447w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-4-300x174.jpg 300w\" sizes=\"auto, (max-width: 447px) 100vw, 447px\" \/><\/p>\r\n<p>Given that the areas of the two regions are equal, find the radius of the arc in terms of <em>x <\/em>.\u00a0 \u00a0 \u00a0 \u00a0[6 marks]<\/p>\r\n<div id=\"link11-link-1157\" class=\"sh-link link11-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link11', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link11-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link11-content-1157\" class=\"sh-content link11-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1200\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-5.jpg\" alt=\"\" width=\"958\" height=\"417\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-5.jpg 958w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-5-300x131.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-5-768x334.jpg 768w\" sizes=\"auto, (max-width: 958px) 100vw, 958px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>Q12.\u00a0 \u00a0[N15\/P1]<\/strong><\/p>\r\n<p>The following diagram shows a sector of a circle where A\u00d4B = <em> \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math>\r\n <\/em>radians and the length of the arc AB = \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><mi>x<\/mi><\/math>\r\n cm .<\/p>\r\n<p>Given that the area of the sector is \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>16<\/mn><mi>c<\/mi><msup><mi>m<\/mi><mn>2<\/mn><\/msup><\/math>\r\n , find the length of the arc AB.\u00a0 \u00a0[4 marks]<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1175\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-2.jpg\" alt=\"\" width=\"278\" height=\"234\" \/><\/p>\r\n<div id=\"link12-link-1157\" class=\"sh-link link12-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link12', 1157, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link12-toggle-1157\">Solution<\/span><\/a><\/div><div id=\"link12-content-1157\" class=\"sh-content link12-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1201\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-3.jpg\" alt=\"\" width=\"1019\" height=\"352\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-3.jpg 1019w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-3-300x104.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-3-768x265.jpg 768w\" sizes=\"auto, (max-width: 1019px) 100vw, 1019px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>&nbsp;<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>&nbsp;<\/p><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"Q1.\u00a0 \u00a0[M09.P1.TZ1] The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius R and the inner circle has radius r . Consider the shaded regions with areas A and B . Given that A: B = 2 :1, find the exact value of [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1157","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-circular-measure","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1157"}],"version-history":[{"count":19,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1157\/revisions"}],"predecessor-version":[{"id":1202,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1157\/revisions\/1202"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}