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Unformatted text preview: Student Number: .. 2002 HIGHER SCHOOL CERTIFICATE Sample Examination Paper MATHEMATICS Extension 1 General Instructions Reading time  5 minutes Working time  2 hours Attempt ALL questions Show all necessary working, marks may be deducted for careless or untidy work Standard integrals are printed on the last page Boardapproved calculators may be used Additional Answer Booklets are available Directions to School or College To ensure maximum confidentiality and security, examination papers must NOT be removed from the examination room. Examination papers may not be returned to students or used for revision purposes till September 2002. These examination papers are supplied Copyright Free, as such the purchaser may photocopy and/or cut and paste same for educational purposes within the confines of the school or college. All care has been taken to ensure that this examination paper is error free and that it follows the style, format and material content of previous Higher School Certificate Examination. Candidates are advised that authors of this examination paper cannot in any way guarantee that the 2002 HSC Examination will have a similar content or format. C Mathematics Extension 1, HSC  2002 2 Question 1 Marks (a) Prove: tan cos cot sec 1 = + + ec 2 (b) For what value of x will the geometric series ........ ) 1 2 ( ) 1 2 ( 1 2 3 2 + + + x x x x x x have a sum to infinity? 2 (c) Solve for x and y if 3 4 4 2 = + + y x x 2 (d) Show that 1 x h x h x h x + + = + and use this result to find from first principles the derivative of y= x 3 (e) Sketch, stating the domain and range: ) 1 ( sin 3 1 = x y 3 Mathematics Extension 1, HSC  2002 3 Question 2 Marks (a) Show that form. surd simplest in 2 1 67 tan express hence and tan 2 sin 2 cos 1 = 3 (b) Find dx x x l 3 n using the substitution u = ln 3 x 2 (c) The two functions f ( x ) = ln ( x +1) and g( x )= x 2 meet in 2 places, one of which is clearly x = 0 (i) Show that they also meet between x = 0.7 and x = 0.8 1 (ii) By taking 0.8 as a first approximation, use Newtons method once to find a better approximation, correct to 3 decimal places. 3 (d) Prove by mathematical induction that n n x ) 1 ( ) sin( = + sin x where n is a positive integer. 3 Mathematics Extension 1, HSC  2002 4 Question 3 (a) Show that cos1 2 ) 4 3 ( tan 5 3 1 = 3 (b) MN is a tangent at R and PQRS is a cyclic quadrilateral. Prove (i) QR bisects PQT 2 (ii) MN R 37 2 [ Hint: Draw RT.] (c) Find the cubic equation whose roots are each one less than the roots of the cubic equation 1 5 2 24 2 3 = + x x x 5 R N S P T Q M Mathematics Extension 1, HSC  2002 5 Question 4 Marks (a) (i) Express x x cos 5 sin 3 + in the form ) sin( + x R , where is in radians....
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