Mathematics Extension 1

Transcription

1 013 HIGHER SCHL CERTIFICATE EXAMINATIN Mathematics Etension 1 General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be used A table of standard integrals is provided at the back of this paper In Questions 11 14, show relevant mathematical reasoning and/or calculations Total marks 70 Section I Pages 7 10 marks Attempt Questions 1 10 Allow about 15 minutes for this section Section II Pages marks Attempt Questions Allow about 1 hour and 45 minutes for this section 1100

2 Section I 10 marks Attempt Questions 1 10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions The polnomial P () = k has a factor. What is the value of k? (A) (B) 1 (C) 0 (D) 36 The diagram shows the graph = ƒ ( ). Which diagram shows the graph = ƒ 1 ( )? (A) (B) (C) (D)

3 3 The points A, B and C lie on a circle with centre, as shown in the diagram. The size of AC is 3π radians. 5 A 3p 5 NT T SCALE B C What is the size of ABC in radians? (A) (B) (C) (D) 3π 10 π 5 7π 10 4π 5 3

4 4 Which diagram best represents the graph = (1 ) 3 (3 )? (A) 1 3 (B) 1 3 (C) 1 3 (D) 1 3 4

5 5 Which integral is obtained when the substitution u = 1 + is applied to 1 + d? (A ) (B ) 1 u 1 4 ( ) 1 u 1 ( ) u du u du (C ) ( u 1) u du (D) u 1 ( ) u du 6 Let a 1. What is the general solution of sin = a? n sin a (A) = nπ + ( 1 ) 1, n is an integer (B) n ( ) = nπ + 1 sin 1 a, n is an integer 1 sin a (C) = nπ ±, n is an integer (D) 1 n π ± sin a =, n is an integer 5

6 7 A famil of eight is seated randoml around a circular table. What is the probabilit that the two oungest members of the famil sit together? (A) (B) (C) (D) 6!! 7! 6! 7!! 6!! 8! 6! 8!! 5 8 The angle θ satisfies sinθ = and π < θ < π. 13 What is the value of sin θ? (A) (B) (C) (D) 169 6

7 9 The diagram shows the graph of a function. p p 1 1 Which function does the graph represent? (A) = cos 1 π (B) = + sin 1 (C) (D) = cos 1 π = s in 1 10 Which inequalit has the same solution as = 5? (A) (B) (C) 6 0 (D) 1 5 7

8 Section II 60 marks Attempt Questions Allow about 1 hour and 45 minutes for this section Answer each question in a SEPARATE writing booklet. Etra writing booklets are available. In Questions 11 14, our responses should include relevant mathematical reasoning and/or calculations. Question 11 (15 marks) Use a SEPARATE writing booklet. (a) The polnomial equation = 0 has roots α, β and γ. 1 Find αβγ. (b) Find d. (c) An eamination has 10 multiple-choice questions, each with 4 options. In each question, onl one option is correct. For each question a student chooses one option at random. Write an epression for the probabilit that the student chooses the correct option for eactl 7 questions. (d) Consider the function ƒ ( ) =. 4 (i) Show that ƒ ( ) > 0 for all in the domain of ƒ ( ). (ii) Sketch the graph = ƒ ( ), showing all asmptotes. Question 11 continues on page 9 8

9 Question 11 (continued) (e) Find sin lim (f) Use the substitution 3 u = e to evaluate 3 e 3 d. 6 0 e (g) Differentiate sin 1 5. End of Question 11 9

10 Question 1 (15 marks) Use a SEPARATE writing booklet. π (a) (i) Write 3cos sin in the form cos ( + α), where 0 < α <. 1 (ii) Hence, or otherwise, solve 3 cos = 1 + sin, where 0 < < π. (b) The region bounded b the graph = 3 sin and the -ais between = 0 3 and = 3 π is rotated about the -ais to form a solid. = 3 sin Find the eact volume of the solid. 3p (c) A cup of coffee with an initial temperature of 80 C is placed in a room with a constant temperature of C. The temperature, T C, of the coffee after t minutes is given b T = A + Be kt, where A, B and k are positive constants. The temperature of the coffee drops to 60 C after 10 minutes. How long does it take for the temperature of the coffee to drop to 40 C? Give our answer to the nearest minute. 3 Question 1 continues on page 11 10

11 Question 1 (continued) (d) The point P(t, t + 3) lies on the curve = + 3. The line has equation = 1. The perpendicular distance from P to the line is D( t). = + 3 P(t, t + 3) NT T SCALE D(t) = 1 t t + 4 (i) Show that Dt ()=. 5 (ii) Find the value of t when P is closest to. 1 (iii) Show that, when P is closest to, the tangent to the curve at P is parallel to. 1 (e) A particle moves along a straight line. The displacement of the particle from the origin is, and its velocit is v. The particle is moving so that v + 9 = k, where k is a constant. π Show that the particle moves in simple harmonic motion with period. 3 End of Question 1 11

12 Question 13 (15 marks) Use a SEPARATE writing booklet. (a) A spherical raindrop of radius r metres loses water through evaporation at a rate that depends on its surface area. The rate of change of the volume V of the raindrop is given b dv = 10 4 A, dt where t is time in seconds and A is the surface area of the raindrop. The surface area and the volume of the raindrop are given b respectivel. A = 4πr and V = 4 πr 3 3 (i) (ii) Show that dr dt is constant. 1 How long does it take for a raindrop of volume 10 6 m 3 to completel evaporate? (b) The point P(ap, ap ) lies on the parabola = 4a. The tangent to the parabola at P meets the -ais at T( ap, 0). The normal to the tangent at P meets the -ais at N(0, a + ap ). N(0, a + ap ) = 4a P(ap, ap ) T(ap, 0) The point G divides NT eternall in the ratio :1. (i) Show that the coordinates of G are (ap, a ap ). (ii) Show that G lies on a parabola with the same directri and focal length as the original parabola. Question 13 continues on page 13 1

13 Question 13 (continued) (c) Points A and B are located d metres apart on a horizontal plane. A projectile is fired from A towards B with initial velocit u ms 1 at angle α to the horizontal. At the same time, another projectile is fired from B towards A with initial velocit w ms 1 at angle β to the horizontal, as shown on the diagram. The projectiles collide when the both reach their maimum height. u a A d B The equations of motion of a projectile fired from the origin with initial velocit V ms 1 at angle θ to the horizontal are b w g = Vt cosθ and = Vt sinθ t. (Do NT prove this.) (i) How long does the projectile fired from A take to reach its maimum height? (ii) Show that u sinα = w sinβ. 1 (iii) uw Show that d = +. g sin( α β) Question 13 continues on page 14 13

14 Question 13 (continued) (d) The circles C 1 and C touch at the point T. The points A and P are on C 1. The line AT intersects C at B. The point Q on C is chosen so that BQ is parallel to PA. 3 Q A C T C 1 B Cop or trace the diagram into our writing booklet. Prove that the points Q, T and P are collinear. P End of Question 13 14

15 Question 14 (15 marks) Use a SEPARATE writing booklet. (a) (i) Show that for k > 0, < 0. 1 k + 1 k k + 1 ( ) (ii) Use mathematical induction to prove that for all integers n, <. 1 3 n n 3 (b) (i) Write down the coefficient of n in the binomial epansion of (1 + ) 4n. 1 (ii) Show that ( + ) = n k=0 k n n n k n k ( ). (iii) It is known that n k n k n k n k 0 1 n k ( + ) n k = n k n k n k 0 4n k + +. n k (Do NT prove this.) Show that n 4n n k n n k =. n k k k =0 (c) The equation t 1 e = t has an approimate solution t 0 = 0.5. (i) Use one application of Newton s method to show that t 1 = 0.56 is t 1 another approimate solution of e =. t (ii) Hence, or otherwise, find an approimation to the value of r for which r the graphs = e and = loge have a common tangent at their point of intersection. 3 End of paper 15

16 STANDARD INTEGRALS n 1 n+1 d =, n 1; 0, if n < 0 n d = ln, > 0 a 1 a e d = a e, a 0 1 cosa d = sina, a 0 a 1 sin a d = cosa, a 0 a 1 sec a d = tana, a 0 a 1 seca tana d = seca, a 0 a 1 1 d = tan 1, a 0 a + a a 1 1 d = sin, a > 0, a < < a a a 1 d = ln( ) + a, > a > 0 a 1 d = ln( ) + + a + a NTE : ln = log, > 0 e 16 Board of Studies NSW 013