Ascham 2003T

# Ascham 2003T - \‘l-ET-ANIMI ago a so ooa Atnamames...

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Unformatted text preview: \‘l-ET-ANIMI) ago a so ooa Atnamames emanatu i; L EXAMENA’EEN Z@@3 time allowed: 2 hours plus 5 minutes reading time All questions may be attempted All questions are of equal value. All necessary working should be shown in every question. Approved calculators may be used. Standard integrals are printed at the end of the exam paper. Start each question in a new booklet. Form 6 Mathematics Extension 1 Trial Exam 2003 QEES'HON l a) Find ill—sin”l 2x dx . 5 b) Find j———2dx 2 + 3x 2 c) Solve for X: ~2x—l x (1) Find the acute angle between the lines y = —x and \/§ y = x u . _] 1 e) Find the exact value of cos(s1n (—1)) 2 l - x t) Use the substitution u = l + x to ﬁnd —————3dx 1 (1 + x) QEE§TE®N 2 a) Find sin2 2xdx b) Find the co—ordinates of the point P which divides AB externally in the ratio 2:3 where A is (1,4) and B(6,9). c) Solve [3x—2l >x+l d) on and [3 are acute angles such that cos CL = 2 and sin [3 =-1—. 5 «5 Without ﬁnding the size of either angle, show that 0!. = 2B. e) The roots of the equation x3 —— 6x2 + 3x + k = 0 are consecutive terms of an Arithmetic Sequence. Find k. (3) (2) (2) t2) (2) t3} t2) (2) t3) (2) t3) Form 6 Mathematics Extension 1 Trial Exam 2003 3 3 it QUESTEQN 3 é a) A spherical bubble is expanding so that its volume is increasing at the constané rate of 12mm3 per second. What is the rate of increase of the radius when the "" surface area is 500mm2 ? (4) b) Find 9 if sinB—cos6=l I (4-) c) . Prove by mathematical induction that 1x20 +2>< 21+3>< 22 +'.....+ 22 x2“ =1+(n —1)2” for all integers n 21 (4) QUESTEON 4 a) A particle moves on a line so that its distance from the origin at time t is x and its velocity is v. dzx d l 7 i Prove —— = —— ~12“ 2 ) dtz dx (A2 ) i i _, d 2x 2 . , 11) If d 2 = n (3 — x) where n IS a constant 1‘ and ifthe particle is released from rest at x = 0, show iv2 —nz(3x—lx2)=O (2) 2 2 , iii) Hence show that the particle never moves outside a certain interval. (2) . x + 4 . . b) 1) Draw a large sketch of y = , shQng all essential features. x(x + 8) (4) -- - x + 4 , 11) Find the area bounded by the curve )2 = and the x was x(x + 8) between x = l and x = 2. (2) , 25/3 M v f: w 1 a , w» “53’ r ’ . a . a \ a “M 3 x L a,” \j w” v i 13» 2» ., ' ' g i 2M. > 1/ as 1;; g9}. gitéwlingxwié f3” a") ' “s 45 (9,) PM. i 2’ t .3 {if/grids Jag/r W” Form 6 Mathematics Extension 1 Trial Exam 2003 i 4 QUESTEON 5 a) For what values ofx will l—- tan2 x+ tan4 x — tan6 x + A have a limiting sum for 0 S x S 27x ? (4) r b) A particle is oscillating in simple harmonic motion such that its displacement x metres from a given origin 0 satisﬁes the equation dzx . . . dtz = ~4x where t is .the tune in seconds. i) Show that x = a cos(2t + ,6) is a possible equation of motion for the particle when a and B are constants. (i) ii) The particle is observed at time t = O to have a velocity of 2m/s and a displacement from the origin of 4m. Find the amplitude of oscillation. (3} iii) Determine the maximum velocity of the particle. (14) b) A T Two circles touch externally at C. The circles are touched by the common tangent at A and B respectively. The common tangent at C meets AB in T. Show that AACB = 90°. (3) Form 6 Mathematics Extension 1 Trial Exam 2003 anagrams 6 a) b) A particle is ﬁred ﬁom a point 0 on the ﬂoor of a horizontal tunnel of height 15 metres at a speed of 30m/s and at an angle 6 above the horizontal where O < 6 < ~75. 2 Assume that the horizontal displacement x metres and the vertical displacement y metres of the particle from O at time t seconds after ﬁring are given by x: 30100s6 and y : 3Otsinl9—Sz‘2. Find the maximum horizontal range of the particle along the tunnel. A function is deﬁned as f(x) : l — 0035— Where 0 S x S a 2 i) Find the largest value of a for which the inverse function f“(x) exists. ii) Find f“ (x) iii) Sketch the graph ofy = f‘1(x) iv) Find the area enclosed between the curve y = f '1 (x) , the x axis and x = 2. (5) (2) (l) (2) (2) maxim» Form 6 Mathematics Extension 1 Trial Exam 2003 QUES’HON '7 a) b) The following question appears in a textbook: Use Newton ’s method to ﬁnd an approximation to the solution of f(x)= —5. Take )60 = as a ﬁrst approximation. Christiana writes down: i) What is x0 ? ii) It is known that f(X) = asinx +bx, where a and b are integers. Write down 2 equations involving a and 1). Do not solve the equations. The straight line y = mx + b meets the parabola x2 = 4ay at the points P( 2ap,ap2) and Q( 2aq, aq2 ). i) Find the equation of the chord PQ and hence or otherwise Show that pq : :2 a .. 2 2 2 2b 11) Prove that p + q = 4m + —— a iii) Given that the equation of the normal to the parabola at P is x + py = Zap + ap3 and that N, the point of intersection of the normals at P and Q , has co-ordinates [—apq(p + q),a(2 + p2 + 1%] + 612)], express these co—ordinates in terms of a, m and b. iv) Suppose that the chord PQ is free to move while maintaining a ﬁxed gradient. Find the locus of N and show that this locus is a straight line. Verify that this line is a normal to the parabola. V END GE? EXAM (1) t3} t2) (2) (3i ...
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## This note was uploaded on 11/01/2011 for the course ECON 101 taught by Professor Adam during the Two '11 term at University of Technology, Sydney.

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Ascham 2003T - \‘l-ET-ANIMI ago a so ooa Atnamames...

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