00 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Approved scientific calculators and templates may be used Attempt all questions Start a new booklet for each question A standard integral sheet is included on the back of this paper Total marks 0 All questions should be attempted All questions are of equal value Examiner MN June 00
HSC Mathematics Extension Trial Examination June 00 Total Marks 0 Attempt Questions -8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Question One (5 marks). Use a SEPARATE writing booklet. Marks (a) θ θ θ. 5 sin cos d (b) (i) Use partial fractions to find the values of A, B and C if x x Ax+ B C = +. x + 4 x x + 4 x ( )( ) (ii) x x Hence find dx. ( x + 4)( x ) (c) π 4 Use integration by parts to evaluate xsec x dx. 0 θ (d) By using the substitution t tan, π = show that. secθ dθ = ln ( + ) 0 5
HSC Mathematics Extension Trial Examination June 00 Question Two (5 marks). Use a SEPARATE writing booklet. (a) Let z = 4 + 8i, and w= + 5. i (i) Find z + w. ( ) (ii) Find arg z+ w. Give your answer in radians correct to significant figures. (iii) Express z w in the form a + ib. (b) Express i in modulus-argument form. (c) (i) Sketch on an Argand diagram the locus defined by arg ( z i) π + = 4 (ii) Let z = + i andz = i. Sketch on an Argand diagram the locus defined by z z z z arg. π = π π (d) In an Argand diagram the point A represents the complex number z = cos + isin. 6 6 (i) Let C represent the number w, where w = iz. Find w in the form a + ib. (ii) The point B completes the rectangle COAB, where O is the origin Let u be the number represented by B. Find u in the form a + ib. (iii) Find the value of w u z u.
Question Three (5 marks).use a SEPARATE writing booklet. HSC Mathematics Extension Trial Examination June 00 (a) Let α, β and γ be the roots of x + x = 0. (i) Find the value of α + β + γ. (ii) Form the equation whose roots are α, β andγ. (b) Let Px ( ) be a polynomial. (i) Prove that if α is a double zero of Px ( ), then P '( α ) = 0. (ii) Hence find the roots of the equation x + 44x 5x 00 = 0, given that two of the roots are equal. (c) Let and z be complex numbers. z (i) Prove that = z z z. (ii) By using the fact that z z z z, = prove that ( ) (iii) Hence prove that z z z z Re( zz) z z = z z. + = + + (iv) Hence prove that z+ z z + z
Question Four (5 marks). Use a SEPARATE writing booklet. HSC Mathematics Extension Trial Examination June 00 The graph below is y x y = e sin x D not to scale A π π A x π A x (i) Find the coordinates of D, the absolute maximum of y = e sin x. (ii) (iii) Prove that the shaded area A is equal to Prove that the shaded area A is equal to 0 e + e π. 4 e π + e π. (iv) Write down the value of A. A (v) Show that A π = e. (vi) Given that the shaded areas form a geometric progression, find the limiting sum of such areas as x. 4
Question Five (5 marks). Use a SEPARATE writing booklet. HSC Mathematics Extension Trial Examination June 00 The point P ct, c t lies on the hyperbola xy = c. (i) Sketch the hyperbola and mark on it the point P where t. (ii) Derive the equation of the tangent at P. (iii) Prove that the equation of the normal at P is given by y = t x+ ct. t c (iv) The tangent at P meets the line y = x at T. Find the coordinates of T. (v) The normal at P meets the line y = x at N. Find the coordinates of N. (vi) Prove that OT ON = 4c 4 Question Six (5 marks). The ellipse E has equation Use a SEPARATE writing booklet. x y + =. P is a point on E. 4 (i) Calculate the eccentricity (ii) Write down the coordinates of the foci S and S. (iii) Write down the equation of each directrix. (iv) Sketch E showing all important features. (v) Prove that the sum of the distances SP + S P is independent of P. (vi) Derive the equation of the normal at P. (vii) Prove that the normal at P bisects SPS. 4 5
Question Seven (5 marks). Use a SEPARATE writing booklet. HSC Mathematics Extension Trial Examination June 00 (a) The curve f ( x) x 6x. = + is defined over the domain x (i) Show that this function has no turning points. (ii) Show that f ( x) x 6x = + is an odd function. (iii) Draw a neat sketch of the function, about one third of a page in size. (iv) On the same diagram, sketch the solid formed when y = f( x) is rotated about the y-axis. (v) Use the method of cylindrical shells to find the exact volume of this solid. 4 (b) A solid is constructed on a circular base of radius 6cm. Parallel cross-sections 5 are right-angled isosceles triangles with the hypotenuse in the base of the solid. Find the volume of the solid. 6
Question Eight (5 marks). Use a SEPARATE writing booklet. HSC Mathematics Extension Trial Examination June 00 α (a) (i) Prove that cot cotα = cosecα (b) n r (ii) Hence find a simplified expression for cosec( α ). y r= 4 5 n n x The graph above is of the curve y = ln x. (i) Find n ln x dx (ii) Prove that the sum of the areas of the rectangle is given by ln ( n )!. (iii) What can you say about your answer in (ii) compared to your answer in (i)? (iv) Prove that for any integer n >, ln n n n. ( n! ) > 7