{"id":1291,"date":"2020-03-19T13:43:35","date_gmt":"2020-03-19T13:43:35","guid":{"rendered":"http:\/\/ibalmaths.com\/?page_id=1291"},"modified":"2022-01-30T08:53:55","modified_gmt":"2022-01-30T08:53:55","slug":"paper-3-examples-hl","status":"publish","type":"page","link":"https:\/\/ibalmaths.com\/index.php\/paper-3-examples-hl\/","title":{"rendered":"Paper 3 examples (HL)"},"content":{"rendered":"

Q1.\u00a0 [Maximum mark: 24]<\/strong><\/p>\r\n

(a) (i)<\/strong> Use trigonometric ratios to prove that in a triangle ABC<\/p>\r\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\na<\/mi>sin<\/mi>A<\/mi><\/mrow><\/mfrac>=<\/mo>b<\/mi>sin<\/mi>B<\/mi><\/mrow><\/mfrac>=<\/mo>c<\/mi>sin<\/mi>C<\/mi><\/mrow><\/mfrac><\/math>\r\n .\"\"<\/p>\r\n

(ii)\u00a0 \u00a0<\/strong>Prove that the area of the triangle is\u00a0 \r\n1<\/mn>2<\/mn><\/mfrac>a<\/mi>b<\/mi>sin<\/mi>C<\/mi><\/math>\r\n .<\/p>\r\n

(iii)<\/strong>\u00a0 Given that R<\/em> denotes the radius of the circumscribed circle prove that\u00a0 \u00a0 \u00a0\u00a0 \r\na<\/mi>sin<\/mi>A<\/mi><\/mrow><\/mfrac>=<\/mo>b<\/mi>sin<\/mi>B<\/mi><\/mrow><\/mfrac>=<\/mo>c<\/mi>sin<\/mi>C<\/mi><\/mrow><\/mfrac>=<\/mo>2<\/mn>R<\/mi><\/math>\r\n .<\/p>\r\n

(iv)\u00a0 \u00a0<\/strong>Hence show that the area of the triangle ABC is \r\na<\/mi>b<\/mi>c<\/mi><\/mrow>4<\/mn>R<\/mi><\/mrow><\/mfrac><\/math>\r\n .<\/p>\r\n

(b)<\/strong>\u00a0 \u00a0A new triangle DEF is positioned within a circle radius R such that DF is a diameter as shown in the following diagram.\"\"<\/p>\r\n

(i)<\/strong>\u00a0 Find in terms of R, the two values of (DE)2<\/sup> such that the area of the shaded region is twice the area of the triangle DEF.<\/p>\r\n

(ii)\u00a0 \u00a0<\/strong>Using two diagrams, explain why there are two values of (DE)2<\/sup> .<\/p>\r\n

(c)<\/strong>\u00a0 \u00a0A parallelogram is positioned inside a circle such that all four vertices lie on the circle. Prove that it is a rectangle.<\/p>\r\n

Q2.\u00a0 \u00a0[Maximum mark: 14]<\/strong>\"\"<\/p>\r\n

(a)\u00a0 \u00a0<\/strong>Figure 1 shows a tangent [PQ] at the point Q of a circle and a line [PS] meeting the circle at the points R , S and passing through the centre O of the circle. Show that PQ2<\/sup> = PR \u00d7 PS .<\/p>\r\n

\"\"<\/p>\r\n

(b)<\/strong>\u00a0 \u00a0Figure 2 shows a triangle ABC inscribed in a circle. The tangents at the points A , B , C meet the opposite sides of the triangle externally at the points D , E , F respectively.<\/p>\r\n

(i)<\/strong>\u00a0 \u00a0 Show that \r\nA<\/mi>D<\/mi>2<\/mn><\/msup><\/mrow>B<\/mi>D<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>=<\/mo>C<\/mi>D<\/mi><\/mrow>B<\/mi>D<\/mi><\/mrow><\/mfrac><\/math>\r\n .
\r\n(ii)<\/strong>\u00a0 \u00a0By considering a pair of similar triangles, show that \r\nA<\/mi>D<\/mi><\/mrow>B<\/mi>D<\/mi><\/mrow><\/mfrac>=<\/mo>A<\/mi>C<\/mi><\/mrow>A<\/mi>B<\/mi><\/mrow><\/mfrac><\/math>\r\n and hence that \r\nC<\/mi>D<\/mi><\/mrow>B<\/mi>D<\/mi><\/mrow><\/mfrac>=<\/mo>A<\/mi>C<\/mi>2<\/mn><\/msup><\/mrow>A<\/mi>B<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac><\/math>\r\n .
\r\n(iii)<\/strong>\u00a0 \u00a0By writing down and using two further similar expressions, show that the points D , E , F are collinear.<\/p>\r\n

Q3.\u00a0 [Maximum mark: 15]<\/strong><\/p>\r\n

\"\"<\/p>\r\n