MATHEMATICS 4 UNIT (ADDITIONAL) HIGHER SCHOOL CERTIFICATE EXAMINATION. Time allowed Three hours (Plus 5 minutes reading time)

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1 N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 997 MATHEMATICS UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value. All necessar working should be shown in ever question. ma be deducted for careless or badl arranged work. Standard integrals are printed on page. Board-approved calculators ma be used. Answer each question in a separate Writing Booklet. You ma ask for etra Writing Booklets if ou need them.

2 QUESTION. Use a separate Writing Booklet. (a) Evaluate 5 d. + (b) Evaluate π sinθ d. cos θ θ (c) Find d. + + (d) t 6 Find dt. ( t + )( t + ) π (e) Evaluate sec d.

3 QUESTION. Use a separate Writing Booklet. (a) (i) Epress i in modulus argument form. ( ) (ii) Hence evaluate i. 6 (b) (i) Simplif ( i ). (ii) Hence find all comple numbers z such that z = 8i. Epress our answers in the form + i. (c) Sketch the region where the inequalities z + i 5 and z+ z both hold. (d) i Let w = + 5 i and z = 5+, so that w = z =. (i) Find wz and wz in the form + i. (ii) Hence find two distinct was of writing 65 as the sum a + b, where a and b are integers and < a< b.

4 QUESTION. Use a separate Writing Booklet. (a) C(, ) (, ) O In the diagram, the shaded region is bounded b the ais, the line =, and the circle with centre C, ( ) and radius. Find the volume of the solid formed when the region is rotated about the ais. 5 (b) Let f( )= (i) Show that f ( ) for all. (ii) For what values of is f ( ) positive? Sketch the graph of = f( ), indicating an turning points and points of inflection. (c) In a game, two plaers take turns at drawing, and immediatel replacing, a marble from a bag containing two green and three red marbles. The game is won b plaer A drawing a green marble, or plaer B drawing a red marble. Plaer A draws first. Find the probabilit that: 5 (i) (ii) (iv) A wins on her first draw; B wins on her first draw; A wins in less than four of her turns; A wins eventuall.

5 5 QUESTION. Use a separate Writing Booklet. (a) 7 M M P (, ) S O T S d d The point P (, ) lies on the hperbola =. The tangent to the 9 7 hperbola at P cuts the ais at T, and has equation =. 9 7 The two foci of the hperbola are S and S, and the two directrices are d and d. The points M and M are the closest points to P on the directrices d and d. (i) (ii) Find the coordinates of the foci. Find the equations of the directrices. Show that T has coordinates 9,. (iv) Using the focus directri definition, or otherwise, show that PS TS PS = TS. (b) (i) Find an epression for cot A in terms of tan A. 8 (ii) Show that tan A and cot A satisf the equation + cot A =. Hence, or otherwise, find the eact value of tan π 8. (iv) π π Hence find the eact value of tan cot. 6 6

6 6 QUESTION 5. Use a separate Writing Booklet. (a) (i) Sketch the graph of = for. 5 (ii) Hence, without using calculus, sketch the graph of = e for. The region between the curve e =, the ais, the ais, and the line = is rotated around the ais to form a solid. Using the method of clindrical shells, find the volume of the solid. (b) N F θ mg A particle of mass m is ling on an inclined plane and does not move. The plane is at an angle θ to the horizontal. The particle is subject to a gravitational force mg, a normal reaction force N, and a frictional force F parallel to the plane, as shown in the diagram. Resolve the forces acting on the particle, and hence find an epression for F N in terms of θ. (c) Suppose that b and d are real numbers and d. Consider the polnomial P()= z z + bz + d. 7 The polnomial has a double root α. (i) Prove that P ( z ) is an odd function. (ii) (iv) (v) Prove that α is also a double root of Pz (). b Prove that d =. For what values of b does P() z have a double root equal to i? For what values of b does P() z have real roots?

7 7 QUESTION 6. Use a separate Writing Booklet. (a) The series + + n has n + terms. 7 (i) Eplain wh n + + = + + n+. (ii) Hence show that + n n+. Hence show that, if, then tan 5 n+ + + tan n n +. (iv) Deduce that < π <. (b) A ball of mass kilograms is thrown verticall upward from the origin with an initial speed of 8 metres per second. The ball is subject to a downward gravitational force of newtons and air resistance of v 5 ( ) newtons in the opposite direction to the velocit, v metres per second. Hence, until the ball reaches its highest point, the equation of motion is 8 where metres is its height. v ẏ =, (i) Using the fact that ẏ = v dv, show that, while the ball is rising, d v e 5 = 6. (ii) Hence find the maimum height reached. Using the fact that ẏ maimum height. dv =, find how long the ball takes to reach this dt (iv) How fast is the ball travelling when it returns to the origin?

8 8 QUESTION 7. Use a separate Writing Booklet. (a) S O U T R The points R and S lie on a circle with centre O and radius. The tangents to the circle at R and S meet at T. The lines OT and RS meet at U, and are perpendicular. Show that OU OT =. (b) S T 7 O U P θ Q R ( ) and radius r, lies inside the The circle ( r) + = r, with centre Qr, circle + =, with centre O and radius. The point Pr ( + rcos θ, rsinθ) lies on the inner circle, and P and O do not coincide. The tangent to the inner circle at P meets the outer circle at R and S, and the tangents to the outer circle at R and S meet at T. The lines OT and RS meet at U, and are perpendicular. (i) Show that OT is parallel to QP. ( ) (ii) Show that the equation of RS is cosθ + sinθ = r + cosθ. (iv) Find the length of OU. B using the result of part (a), show that T lies on the curve r + r =.

9 9 QUESTION 7. (Continued) (c) 6 L J N M K O The parabola = a touches the circle + + g + f + c = at J, and cuts it at K and L. The midpoint of KL is M, and the line JM cuts the ais at N, as shown on the diagram. (i) Find a quartic equation whose roots are the coordinates of J, K, and L. (ii) Show that JN = NM. Hence show that the area of JKN is one-quarter of the area of JKL.

10 QUESTION 8. Use a separate Writing Booklet. (a) D B F X C A E Triangle ABC is scalene. Eternal equilateral triangles ABF, BCD, and CAE are constructed on the sides of triangle ABC as shown. Lines AD and BE meet at X. Cop or trace this diagram into our Writing Booklet. (i) Show that BCE = DCA. (ii) Show that BCE DCA. Show that BDCX is a cclic quadrilateral. (iv) Show that BXD = DXC = CXE = EXA = π. (v) Show that CF passes through X. (vi) Show that AD = BE = CF.

11 QUESTION 8. (Continued) (b) U T 5 O S R The diagram shows points O, R, S, T, and U in the comple plane. These points correspond to the comple numbers, r, s, t, and u respectivel. The π π triangles ORS and OTU are equilateral. Let ω = cos + i sin. (i) Eplain wh u = ω t. (ii) Find the comple number r in terms of s. Using comple numbers, show that the lengths of RT and SU are equal.

12 STANDARD INTEGRALS n d n+ =, n ;, if n< n + d = ln, > e a d a e a =, a cosa d = sin a, a a sin a d = cos a, a a sec a d = tan a, a a sec a tan a d = sec a, a a a d = a tan, + a a a d = sin, a>, a< < a a ( ) > > d = ln + a, a a ( ) d = ln + + a + a NOTE : ln = log, > e Board of Studies NSW 997