2010HIGHER SCHOOL CERTIFIC ATEEXAMINATIONMathematics Extension 1General Instructions Reading time 5 minutes Working time 2 hours Write using black or blue pen Board-approved calculators may be used A table of standard integrals is provided at the back of this paper All necessary working should be shown in every question Total marks 84 Attempt Questions 17 All questions are of equal value 3370 Total marks 84 Attempt Questions 17 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Question 1 (12 marks) Use a SEPARATE writing booklet. (a) Use the table of standard integrals to find 1dx . 4 2 x 1(d) Solve30 and .2 2 3 cos cos + + 0 2 0.3 1 (c) In the diagram, ST is tangent to both the circles at A. The points B and C are on the larger circle, and the line BC is tangent to the smaller circle at D. The line AB intersects the smaller circle at X. S C A D X B T Copy or trace the diagram into your writing booklet.(i) Explain why AXD = ABD + XDB.(ii) Explain why AXD = TAC + CAD.(iii) Hence show that AD bisects BAC. End of Question 5 9 Question 6 (12 marks) Use a SEPARATE writing booklet. (a) (i) Show that cos(A B) = cos A cos B(1 + tan A tan B).1 1 (ii) Suppose that and . Deduce that if , then 0 2 0 x 1ax ax edx = e , a 0 a 1 cosax dx = sinax, a a 0 a 1 sin ax dx = cosax, a 0 a 2 sec ax dx = 1 tantax , a 0 a 1 secax tanax dx = secax , a 0 a 11 x dx = tan1, a 0 22 aa a a + x 1 x dx = sin1, a a 0, a a 0 x2 a2 1 22 dx = ln(x + x + a ) 22 x + a NOTE : ln x = log x x , x 0e 16 Board of Studies NSW 2010