N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value. All necessary working should be shown in every question. may be deducted for careless or badly arranged work. Standard integrals are printed on page. Board-approved calculators may be used. Answer each question in a separate Writing Booklet. You may ask for etra Writing Booklets if you need them.
QUESTION. Use a separate Writing Booklet. (a) Evaluate 4 ( + ) d. (b) Find sec θtan θdθ. (c) Find 5t + dt. tt + ( ) (d) Using integration by parts, or otherwise, find tan d. (e) Using the substitution = sinθ, or otherwise, calculate 5 4 d.
QUESTION. Use a separate Writing Booklet. (a) Suppose that c is a real number, and that z = c i. Epress the following in the form + iy, where and y are real numbers: ( iz); z. (b) On an Argand diagram, shade the region specified by both the conditions Re() 4 z and z 4+ 5i. n ( ) = ( )+ ( ) (c) Prove by induction that cosθ + isinθ cos nθ isin nθ integers n. Epress w = i in modulus argument form. for all 7 Hence epress w 5 in the form + iy, where and y are real numbers. (d) θ z 4 O The diagram shows the locus of points z in the comple plane such that π arg( z ) arg( z+ )=. This locus is part of a circle. The angle between the lines from to z and from to z is θ, as shown. Copy this diagram into your Writing Booklet. π Eplain why θ =. Find the centre of the circle.
4 QUESTION. Use a separate Writing Booklet. (a) y 5 S O 4 + y = 0 The shaded region is bounded by the lines =, y =, and y = and by the curve + y = 0. The region is rotated through 60 about the line = 4 to form a solid. When the region is rotated, the line segment S at height y sweeps out an annulus. Show that the area of the annulus at height y is equal to 4 π( y + 8 y + 7 ). Hence find the volume of the solid. π k (b) Show that ( sin ) cos d =, k + 0 ( ) n ( ) = By writing cos sin, show that n where k is a positive integer. 6 π 0 n+ n cos d. k + k ( ) = n k= 0 ( ) k Hence, or otherwise, evaluate π 0 5 cos d.
5 QUESTION. (Continued) (c) v 4 O 4 5 6 7 8 9 t A particle moves along the ais. At time t = 0, the particle is at = 0. Its velocity v at time t is shown on the graph. Trace or copy this graph into your Writing Booklet. At what time is the acceleration greatest? Eplain your answer. At what time does the particle first return to = 0? Eplain your answer. Sketch the displacement graph for the particle from t = 0 to t = 9.
6 QUESTION 4. Use a separate Writing Booklet. (a) By differentiating both sides of the formula find an epression for + + + + + = n n+, + + 4+ 4 8+ + n n. (b) On the same set of aes, sketch and label clearly the graphs of the functions 6 y = and y = e. Hence, on a different set of aes, without using calculus, sketch and label clearly the graph of the function y = e. Use your sketch to determine for which values of m the equation e = m+ has eactly one solution. (c) Consider a lotto-style game with a barrel containing twenty similar balls numbered to 0. In each game, four balls are drawn, without replacement, from the twenty balls in the barrel. The probability that any particular number is drawn in any game is 0. 6 Find the probability that the number 0 is drawn in eactly two of the net five games played. Find the probability that the number 0 is drawn in at least two of the net five games played. Let j be an integer, with 4 j 0. (iv) Write down the probability that, in any one game, all four selected numbers are less than or equal to j. Show that the probability that, in any one game, j is the largest of the four numbers drawn is j. 0 4
7 QUESTION 5. Use a separate Writing Booklet. (a) For any non-zero real number t, the point t, t lies on the graph of y =. 8 Show that y = 4 is the equation of the locus of the midpoint of the straight line joining t, t and t,, as t varies. t Show that the line joining t, t and t, t part. is tangent to the locus in Show that the equation of the normal to y = at the point t, t written in the form 4 t t + ty = 0. may be (iv) R(0, h) is a point on the y ais. Show that there are eactly two points on the hyperbola y = with normals that pass through R. (b) Consider the polynomial equation 4 + a + b + c+ d = 0, where a, b, c, and d are all integers. Suppose the equation has a root of the form ki, where k is real, and k 0. 7 State why the conjugate ki is also a root. Show that c = k a. Show that c + a d = abc. (iv) If is also a root of the equation, and b = 0, show that c is even.
8 QUESTION 6. Use a separate Writing Booklet. (a) Solve. (b) A circular drum is rotating with uniform angular velocity round a horizontal ais. A particle P is rotating in a vertical circle, without slipping, on the inside of the drum. The radius of the drum is r metres and its angular velocity is ω radians/second. Acceleration due to gravity is g metres/second, and the mass of P is m kilograms. The centre of the drum is O, and OP makes an angle θ to the horizontal. The drum eerts a normal force N on P, as well as a frictional force F, acting tangentially to the drum, as shown in the diagram. F P O N θ mg By resolving forces perpendicular to and parallel to OP, find an epression for F in terms of the data. N
9 QUESTION 6. (Continued) (c) y 9 P θ L φ γ T O Q y = 4a A mirror is described by the parabola y = 4a, where a is a constant. Light travels parallel to the ais from a point, L, towards the mirror. The light travels along the line y = at, and meets the mirror at Pat (, at). The reflected light meets the ais at Q. The tangent to the parabola at P meets the ais at T. The light is reflected at P by the mirror in such a way that θ, the angle of incidence between the light ray and the tangent, is equal to φ, the angle of reflection between the tangent and the reflected ray, i.e., θ = φ. The reflected ray makes an angle γ with the ais. Show that γ = θ. By considering the gradient of PT, show that tanθ = t. Hence show that the equation of the line PQ is ( ) +. t = t y at (iv) (v) Show that Q is the point (a, 0), the focus of the parabola. Use the focus directri definition of the parabola to show that the path LPQ is the shortest path from L to Q via the parabola.
0 QUESTION 7. Use a separate Writing Booklet. (a) A particle is moving along the ais. Its acceleration is given by d 5 = dt and the particle starts from rest at the point =. 6 Show that the particle starts moving in the positive direction. Let v be the velocity of the particle. Show that v = + 4 5 for. Describe the behaviour of the velocity of the particle after the particle passes = 5. (b) Let f( )= ln a+ b, for > 0, where a and b are real numbers and a > 0. Show that y = f( ) has a single turning point which is a maimum. 9 The graphs of y = ln and y = a b intersect at points A and B. Using the result of part, or otherwise, show that the chord AB lies below the curve y = ln. Using integration by parts, or otherwise, show that k ln d= kln k k+. (iv) Use the trapezoidal rule on the intervals with integer endpoints,,,..., k to show that k ln d [( )] is approimately equal to ln k + ln k!. (v) Hence deduce that k k! < e k k. e
QUESTION 8. Use a separate Writing Booklet. π π (a) Let w = cos + isin. 9 9 7 (b) Show that w k is a solution of z 9 = 0, where k is an integer. 4 5 6 7 8 Prove that w+ w + w + w + w + w + w + w =. Hence show that cos π π π cos 4 cos. 9 9 9 = 8 T P Q R The points P, Q, R lie on a straight line, in that order, and T is any point not on the line. Using the fact that PR PQ = QR, show that QT QP > RT RP. (c) B 7 K A L N D M C ABCD is a quadrilateral, and the sides of ABCD are tangent to a circle at points K, L, M, and N, as in the diagram. Show that AB + CD = AD + BC. ABCD is a quadrilateral, with all angles less than 80. Let X be the point of intersection of the angle bisectors of ABC and of BCD. Prove that X is the centre of a circle to which AB, BC, and CD are tangent. ABCD is a quadrilateral, with all angles less than 80. Given that AB + CD = AD + BC, show that there eists a circle to which all sides of ABCD are tangent. You may use the result of part (b).
n d STANDARD INTEGRALS n+ =, n ; 0, if n< 0 n + d = ln, > 0 e a d a e a =, a 0 cosa d = sin a, a 0 a sin a d = cos a, a 0 a sec a d = tan a, a 0 a sec a tan a d = sec a, a a 0 a d = a tan, 0 + a a a d = sin, a> 0, a< < a a ( ) > > d = ln + a, a a ( ) d = ln + + a + a NOTE : ln = log, >0 e 0 Board of Studies NSW 996