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Objective Response Bank YEAR 12 Mathematics Extension 2 Topics 1. Graphs 2. Complex numbers 3. Conics 4. Integration 5. Volumes 6. Mechanics 7. Polynomials 8. Harder Extension 1 topics Year 12 Mathematics Extension 2 Graphs 1 Solutions The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y f ( x) ? (B) (A) (D) (C) 2 Main Menu Year 12 Mathematics Extension 2 2 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y f ( x)2 ? (B) (A) (D) (C) 3 Year 12 Mathematics Extension 2 3 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y f ( x) ? (B) (A) (D) (C) 4 Year 12 Mathematics Extension 2 4 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y 1 ? f x (A) (B) (C) (D) 5 Year 12 Mathematics Extension 2 5 The diagram below shows the graph of the function y f ( x) . Which diagram represents the graph of y 2 f ( x) ? (A) (B) (C) (D) 6 Year 12 Mathematics Extension 2 6 What is the graph of y f x given that f ( x) 4 x 2 ? (A) (B) (C) (D) 7 Which of the following is the sketch of y log 2 x (A) (B) (C) (D) 7 1 ? x Year 12 Mathematics Extension 2 8 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y f ( x) ? (B) (A) (D) (C) 8 Year 12 Mathematics Extension 2 9 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y f x ? (B) (A) (D) (C) 9 Year 12 Mathematics Extension 2 10 The diagram shows the graph of the function y f ( x) . Which of the following is the graph of y 2 f x ? (B) (A) (D) (C) 10 Year 12 Mathematics Extension 2 Complex numbers Solutions 11 Let z 1 i and w 1 2i . What is the value of zw ? (A) 1 i 1 i (C) 3 i (D) 3 i (B) 12 Let z 3 4i and w 3 i . What is the value of (A) 3 3 4 (4 3 3)i 4 4 (B) 3 3 4 (4 3 3)i 4 4 (C) 3 3 4 (4 3 3)i 2 2 (D) 3 3 4 (4 3 3)i 2 2 13 Let z 1 2i and w 2 i . What is the value of (A) z ? w 5 ? iw 1 2i 1 2i (C) 1 2i (D) 1 2i (B) 14 Let z 3 i . What is the value of iz ? (A) 1 3i 1 3i (C) 1 3i (D) 1 3i (B) 15 Let z 2 i and w 1 i . What is the value of 3z iw ? (A) 5 4i 5 4i (C) 7 4i (D) 7 4i (B) 11 Main Menu Year 12 Mathematics Extension 2 16 What is the value of arg z given the complex number z 1 i 3 ? 3 2 (B) 3 (C) 3 (D) 3 (A) 17 What is the value of z1 given the complex numbers z1 2 2i and z2 1 i 3 ? z2 (A) 3 1 3 1 i 2 2 (B) 1 3 3 1 i 2 2 (C) 3 1 3 1 i 4 4 (D) 1 3 3 1 i 4 4 18 It is given that 3 i is a root of P( z ) z 3 az 2 bz 10 where a and b are real numbers. Which expression factorises P(z) over the real numbers? (A) ( z 1)( z 2 6 z 10) (B) ( z 1)( z 2 6 z 10) (C) ( z 1)( z 2 6 z 10) (D) ( z 1)( z 2 6 z 10) 19 What is the solution to the equation z 2 i z ? (A) (0, 0) and (0,1) (B) (0, 0) and (0, 1) (C) (0, 0) , (0, 1) , ( (D) (0, 0) , (0,1) , ( 3 1 3 1 , ) and ( , ) 2 2 2 2 3 1 3 1 , ) and ( , ) 2 2 2 2 12 Year 12 Mathematics Extension 2 20 Consider the Argand diagram below. Which inequality could define the shaded area? (A) 0 z 2 (B) 1 z 2 (C) 0 z 1 2 (D) 1 z 1 2 21 Consider the Argand diagram below. y 3 2 (1,1) 1 -2 -1 1 -1 -2 Which inequality could define the shaded area? (A) | z | 1 and | z (1 i ) | 1 (B) | z | 1 and | z (1 i) | 1 (C) | z | 1 and | z (1 i ) | 1 (D) | z | 1 and | z (1 i) | 1 13 2 3 x Year 12 Mathematics Extension 2 22 Consider the Argand diagram below. Which inequality could define the shaded area? (A) z 1 2 and 0 arg( z i ) (B) z 1 2 and 0 arg( z i) (C) z 1 1 and 0 arg( z i ) (D) z 1 1 and 0 arg( z i) 4 4 4 4 23 Consider the Argand diagram below. y 4 3 2 1 -2 -1 1 2 x -1 -2 Which inequality could define the shaded area? (A) | z i | 2 and 0 arg( z 1) (C) | z i | 2 and 0 arg( z 1) 3 4 (B) | z i | 2 and 0 arg( z 1) (D) | z i | 2 and 0 arg( z 1) 4 14 3 4 4 Year 12 Mathematics Extension 2 24 What is 1 i expressed in modulus-argument form? (A) (cos (B) (C) (D) i sin ) 4 4 i sin ) 4 4 3 3 (cos i sin ) 4 4 2(cos 2(cos 3 3 i sin ) 4 4 25 What is 3 i expressed in modulus-argument form? (A) (B) (C) (D) 2(cos 2(cos i sin ) 6 6 i sin ) 6 6 2(cos 2(cos 5 5 i sin ) 6 6 5 5 i sin ) 6 6 26 What is 2 2 3i expressed in modulus-argument form? (A) (B) 2 2 i sin ) 3 3 2 2 4(cos i sin ) 3 3 2(cos (C) 2(cos (D) 4(cos i sin ) 3 3 i sin ) 3 3 27 What is (1 3i ) 1 expressed in modulus-argument form? (A) 1 (cos i sin ) 4 3 3 (B) 1 (cos i sin ) 4 3 3 (C) 1 (cos i sin ) 2 3 3 (D) 1 (cos i sin ) 2 3 3 15 Year 12 Mathematics Extension 2 28 What are the three roots of z 3 1 0 in modulus argument form? 2π 3 2π 2π cis0, cis , cis 3 3 (A) cis0, cis (B) (C) cis0, cis π 3 π π (D) cis0, cis , cis 3 3 29 Which of the following complex numbers equals ( 3 i)4 ? (A) 2 2 (B) 8 8 (C) 2 2 3i (D) 8 8 3i 3 3 i i 30 Let the point R represent the complex number z on an Argand diagram. Which of the following describes the locus of R specified by | z || z 4 | ? (A) Perpendicular bisector of (0,0) and (4,0) (B) Perpendicular bisector of (0,0) and (4,0) (C) Circle with a centre (0,0) and radius of 2 (D) Circle with a centre (0,0) and radius of 4 16 Year 12 Mathematics Extension 2 Conics Solutions 31 For the ellipse with the equation (A) 1 4 (C) 3 4 Main Menu x2 y 2 1 . What is the eccentricity? 4 3 1 (B) 2 (D) 9 16 x2 y2 1 where a b 0 . The tangent at P meets the a 2 b2 tangents at the ends of the major axis at R and T. 32 The point P lies on the ellipse y R P T x What is the equation of the tangent at P? (A) ax by a 2 b2 cos sin (B) ax by a 2 b2 sec tan (C) x y sec tan 1 a b (D) x y cos sin 1 a b 17 Year 12 Mathematics Extension 2 x2 y2 1 and a 2 b2 the chord PQ subtends a right angle at (0,0) . Which of the following is the correct expression? 33 The points P(a cos , b sin ) and Q (a cos , b sin ) lie on the ellipse (A) b2 tan tan 2 a (B) tan tan (C) tan tan b2 a2 (D) tan tan a2 b2 a2 b2 x2 y 2 1 where a b 0 . The points a 2 b2 P (a sec , b tan ) and Q (a sec , b tan ) lie on the hyperbola and the chord PQ subtends a right angle at the origin. 34 The diagram below shows the hyperbola y Q O x P Use the parametric representation of the hyperbola to determine which of the following expressions is correct? a2 (A) sin sin 2 b a2 b2 (B) sin sin (C) tan tan (D) tan tan a2 b2 a2 b2 18 Year 12 Mathematics Extension 2 c c ) and Q (cq, ) lie on the same branch of the hyperbola xy c 2 (p p q q). The tangents at P and Q meet at the point T. What is the equation of the normal to the hyperbola at P? 35 The points P (cp, (A) p 2 x py c cp 4 0 (B) p3 x py c cp 4 0 (C) x p 2 y 2c 0 (D) x p 2 y 2cp 0 36 Consider the hyperbola with the equation x2 y 2 1. 16 9 What is the eccentricity of the hyperbola? (A) 3 4 (B) 5 4 (C) 9 16 (D) 25 16 37 Consider the hyperbola with the equation x2 y 2 1. 144 25 What are the equations of the directrices? (A) x 13 144 (B) x 13 25 (C) x 25 13 (D) x 144 13 19 Year 12 Mathematics Extension 2 x2 y 2 1. 16 9 What are the coordinates of the foci of the hyperbola? (A) ( 4, 0) (B) (0, 4) 38 Consider the hyperbola with the equation (C) (0, 5) (D) (5, 0) x2 y 2 39 Consider the hyperbola with the equation 1. 4 3 What are the coordinates of the vertex of the hyperbola? (A) ( 2, 0) (B) (0, 2) (C) (0, 4) (D) ( 4, 0) c c 40 The points P cp, and Q cq, , p q, lie on the same branch of the hyperbola p q 2 xy c . The tangents at P and Q meet at the point T. Which of the following expressions is the equation of the tangent to the hyperbola at Q? (A) x q 2 y 2cq (B) x q 2 y 2c 2 (C) x p 2 y 2cp (D) x p 2 y 2c 2 20 Year 12 Mathematics Extension 2 Integration Solutions 41 What is the value of 1 0 xe x dx ? 2 (A) 1 e 2e (B) e 1 2e (C) 2e 1 e (D) 1 2e e 42 What is the value of ex 0 1 e x dx ? 1 (A) loge 1 e (C) log e (B) 1 1 e (D) log e 2 43 Which of the following is an expression for x 16 x 2 e 2 2 dx ? (A) 2 16 x2 c (B) 16 x2 c (C) 1 16 x 2 c 2 (D) 44 Which of the following is an expression for Use the substitution u 4 sin x . (A) 4ln | 4 sin x | c (C) Main Menu sin x cos x dx ? 4 sin x (B) sin x 4ln | 4 sin x | c 1 16 x 2 c 2 4ln | 4 sin x | c (D) sin x 4ln | 4 sin x | c 45 Which of the following is an expression for Use the substitution u cos x . cos3 x 3cos5 x 3cos 7 x cos9 x c 3 5 7 9 (A) (B) cos3 x 3cos5 x 3cos 7 x cos9 x c (C) cos3 x 3cos5 x 3cos 7 x cos9 x c 3 5 7 9 (D) cos3 x 3cos5 x 3cos 7 x cos9 x c 21 cos2 x sin 7 xdx ? Year 12 Mathematics Extension 2 46 Which of the following is an expression for 1n x 3 1n x 3 1n x 3 1 x 2 6 x 10 dx ? x 6 x 10 c x 6 x 10 c x 6 x 10 c (A) 1n x 3 x 2 6 x 10 c (B) (C) (D) 2 2 2 47 Which of the following is an expression for x 3 (A) sin 1 c 2 (C) 1 7 6 x x2 (B) x 3 sin 1 c 4 dx ? x 3 sin 1 c 2 x 3 (D) sin 1 c 4 48 Which of the following is an expression for x 2 2 dx ? 4 x 13 (A) 1 1 ( x 2) tan c 3 3 (B) 2 ( x 2) tan 1 c 3 3 (C) 1 1 ( x 2) tan c 9 9 (D) 2 ( x 2) tan 1 c 9 9 49 Which of the following is an expression for (A) ln( x 2 1) ln | x 2 | c (B) ln( x 2 1) 2ln | x 2 | c (C) ln( x 2 1) 3tan 1 x ln | x 2 | c (D) ln( x 2 1) 3tan 1 x 2ln | x 2 | c 22 x 2 7x 4 dx ? 1 ( x 2) Year 12 Mathematics Extension 2 50 Which of the following is an expression for (A) 1 4 ln x 1 ln x 4 c 5 5 (B) 1 4 ln x 1 ln x 4 c 5 5 (C) 1 5 ln x 1 ln x 4 c 4 4 (D) 1 5 ln x 1 ln x 4 c 4 4 51 Which of the following is an expression for x x 1 ( x 4) dx ? 4 x2 5x 1 x 3 x2 1 dx ? (A) 1n x 3 x 2 1 c (B) 1n x 3 x2 1 c 2 (C) 1n x 3 x 2 1 tan -1 x c (D) 1n x 3 x2 1 tan -1 x c 2 52 What is the value of 2 0 (A) 0.322 (C) 1.107 1 d ? Use the substitution t tan 2 . cos 2sin 3 (B) 0.785 (D) 1.570 53 What is the value of sec xdx ? Use the substitution t tan 2x . 1 t | c t 1 t 1 | c (D) ln | t 1 (A) ln | (t 1)(t 1) | c (C) (B) ln | (1 t )(t 1) | c 54 What is the value of ln | x x 2 dx ? Use the substitution u x 2 . 3 5 1 (A) 1 7 (B) 2 7 (C) 1 3 (D) 2 3 23 Year 12 Mathematics Extension 2 55 Which of the following is an expression for (A) x2 x2 log e x c 2 4 (B) x2 x log e x c 2 2 (C) (D) x2 c 4 x x log e x c 2 (A) x 4 log e x x5 c 4 20 (B) x 4 log e x x5 c 4 5 (C) x5 log e x x5 c 5 20 (D) x5 log e x x5 c 5 5 57 What is the value of (C) e x log e x 56 Which of the following is an expression for (A) x log xdx ? Use integration by parts. 12 4 log e xdx ? Use integration by parts. 6 0 x cos xdx ? Use integration by parts. 3 1 12 2 x (B) 3 1 2 x 58 Let I n cosntdt , where 0 x (D) 3 1 12 2 12 . 2 Which of the following is the correct expression for I n ? 0 (A) n 1 In I n 2 with n 2 . n (B) n 1 In I n 2 with n 2 . n (C) I n n n 1 I n 2 with n 2 . (D) I n n n 1 I n 2 with n 2 . 24 3 1 2 Year 12 Mathematics Extension 2 59 Let I n x n sin xdx , where 0 x . 2 Which of the following is the correct expression for I n ? 0 (A) n n(n 1) I n 2 (B) n n(n 1) I n 2 (C) n n(n 2) I n 2 (D) n n(n 2) I n 2 60 Let I n x n e ax dx . Which of the following is the correct expression for I n ? (A) x neax In nI n 1 a (B) In x n eax n I n 1 a a (C) In x neax nI n 1 a (D) In x neax n I n 1 a a 25 Year 12 Mathematics Extension 2 Volumes Solutions Main Menu 61 The parabola y x3 is rotated about the y axis {x : 0 x 2} to form a solid. What is the volume of this solid using the method of slicing? (A) 2 cubic units 5 (B) 3 cubic units 5 (C) (D) 93 cubic units 5 96 cubic units 5 62 The area between the curve y 3x x 2 , the x-axis, x 0 and x 3 , is rotated about the y-axis to form a solid. What is the volume of this solid using the method of slicing? (A) 9 cubic units 4 (B) 9 cubic units 2 (C) 27 cubic units 4 (D) 27 cubic units 2 26 Year 12 Mathematics Extension 2 63 The region enclosed by the ellipse ( x 1) 2 y2 1 is rotated about the y axis to 4 form a solid. What is the correct expression for volume of this solid using the method of slicing? 2 (A) V 1 y 2 dy 2 2 (B) V 2 1 y 2 dy 2 2 (C) V 4 y 2 dy 2 2 (D) V 2 4 y 2 dy 2 64 The region is bounded by the lines x 1 , y 1 , y 1 and by the curve x y 2 . The region is rotated through 360º about the line x 2 to form a solid. What is the correct expression for volume of this solid? (A) V y 4 4 y 2 3dy 1 1 (B) V y 4 4 y 2 3dy 1 1 (C) V y 4 4 y 2 4dy 1 1 (D) V y 4 4 y 2 4dy 1 1 27 Year 12 Mathematics Extension 2 65 What is the volume of the solid formed when the region bounded by the curves y 2 x3 and y 2 x is rotated about the x-axis? Use the method of slicing. (A) 5 cubic units 14 (B) 10 cubic units 14 (C) 5 cubic units 7 (D) 10 cubic units 7 66 What is the volume of the solid formed when the region bounded by the curves y x 2 , y 30 x 2 and the y-axis is rotated about the y-axis? Use the method of slicing. What is the correct expression for volume of this solid using the method of cylindrical shells? 5 (A) V 2 x 2 30 x 2 dx 0 5 (B) V 2 x x 2 30 x 2 dx 0 5 (C) V 2 0 5 (D) V 2 x 0 30 x 2 x 2 dx 30 x 2 x 2 dx 28 Year 12 Mathematics Extension 2 67 The region bounded by y 4 x 2 x 4 and 0 x 2 is rotated about the y axis to form a solid. What is the volume of this solid using the method of cylindrical shells? (A) 16 units3 3 (B) 8 units3 3 (C) 20 units3 3 (D) 32 units3 3 68 A solid is formed by rotating the region enclosed by the parabola y 2 4ax , its vertex (0,0) and the line x a , about the x-axis. y x What is the volume of this solid using the method of cylindrical shells? (A) 7 a 3 units3 4 (B) 7 a 3 units3 8 (C) 7 a 3 units3 16 (D) 2 a 3 units3 29 Year 12 Mathematics Extension 2 69 The region enclosed by y sin x , y 0 and x is rotated around the y-axis to 2 produce a solid. What is the volume of this solid using the method of cylindrical shells? (A) units3 (C) units3 2 3 units3 (D) 2 units3 (B) 2 30 Year 12 Mathematics Extension 2 Mechanics 70 71 72 Solutions Main Menu A particle of mass m falls from rest under gravity and the resistance to its motion is mkv 2 , where v is its speed and k is a positive constant. Which of the following is the correct expression for square of the velocity where x is the distance fallen? (A) v 2 g 1 e2kx k (B) v2 g 1 e 2 kx k (C) v2 g 1 e 2 kx k (D) v 2 g 1 e2kx k A rock is projected vertically upwards from ground level. Assume air resistance is kv, where v is the velocity of the rock and k is a positive constant. The rock falls back to ground level under the influence of g, the acceleration due to gravity. Consider the rock’s motion starting from maximum height. Let y be the displacement and t be the time elapsed after the rock has reached maximum height. Assume the rock has a unit mass. Which of the following is the correct expression for velocity? (A) v g kt (e 1) k (B) v g kt (e 1) k (C) v k kt (e 1) g (D) v k kt (e 1) g A particle of mass m is moving in a straight line under the action of a force. m (6 10 x) x3 What of the following is an expression for its velocity in any position, if the particle starts from rest at x 1 ? F (A) v 1 (3 10 x 7 x 2 ) x (B) v x (3 10 x 7 x 2 ) (C) v 1 2(3 10 x 7 x 2 ) x (D) v x 2(3 10 x 7 x 2 ) 31 Year 12 Mathematics Extension 2 73 A particle of mass m is projected vertically upwards with an initial velocity of u ms-1 in a medium in which the resistance to the motion is proportional to the square of the velocity v ms-1 of the particle or mkv2. Let x be the displacement in metres of the particle above the point of projection, O, so that the equation of motion is x g kv 2 where g ms-2 is the acceleration due to gravity. Assume k = 10 and the acceleration due to gravity is 10 ms-2. Which of the following gives the correct expression for the time taken? 74 (A) t 1 tan -1 u tan -1 v 10 (B) t 1 tan -1 v tan -1 u 10 (C) t 1 tan -1 u tan -1 v 10 (D) t 1 tan -1 v tan -1 u 10 A conical pendulum consists of a body P of mass m kg and a string of length l metres. Point A is fixed and the body P rotates in a horizontal circle of radius r and centre O at a constant angular velocity of radians per second. OA is vertical and has a length of h metres. The angle OAP is radians. The body, P, is subject to a gravitational force of mg newtons. The tension in the string is T newtons. Which of the following gives the correct resolution of forces on P in the horizontal and vertical directions? (A) T sin mg 0 and T cos mr 2 (B) T sin mg 0 and T cos mr 2 (C) T cos mg 0 and T sin mr 2 (D) T cos mg 0 and T sin mr 2 32 Year 12 Mathematics Extension 2 75 Two light inextensible strings are attached to a particle of mass m. The particle describes a horizontal circle with constant angular velocity . Which of the following gives the correct resolution of forces in the horizontal and vertical directions? (A) T sin T sin m 2 r and T cos T cos mg (B) T sin T cos m 2 r and T sin T cos mg (C) T sin T sin m 2 r and T cos T cos mg (D) T sin T cos m 2 r and T sin T cos mg 76 A body of mass m kg is attached by two light rods AB and BC. Both rods are l metres in length. Rod AB is hinged at point A and makes an angle with the vertical shaft. Rod BC is attached to a ring that can slide freely along the vertical shaft. T1 T2 What are the tensions in the rods? 1 1 mg sec ml 2 and T2 ml 2 mg sec 2 2 1 1 T1 mg sin ml 2 and T2 ml 2 mg sin 2 2 (A) T1 (B) (C) T1 1 1 mg sec ml 2 and T2 ml 2 mg sec 2 2 (D) T1 1 1 mg sin ml 2 and T2 ml 2 mg sin 2 2 33 Year 12 Mathematics Extension 2 77 Two light inextensible strings PQ and QR each of length l are attached to a particle of mass m at Q. The other ends P and R are fixed to two points in a vertical line such that P is a distance l above R. The particle describes a horizontal circle with constant angular velocity . P l l r Q l mg R What is the tension in the strings? (A) T1 m 2 m (lw 2 g ) and T2 (lw2 g ) 2 2 T1 m 2 m (lw 2 g ) and T2 (lw2 g ) 2 2 (B) (C) T1 m(lw2 2 g ) and T2 m(lw2 g ) (D) T1 m(lw2 2 g ) and T2 m(lw2 g ) 78 79 What is the angle at which a road must be banked so that a car may round a curve with a radius of 200 metres at 100 km/h without sliding? Assume that the road is smooth. (A) 21.49 (B) 22.49 (C) 23.49 (D) 24.49 A conical pendulum consists of a body P of mass m kg and a string of length l metres. A is fixed and the body P rotates in a horizontal circle of radius r and centre O at a constant angular velocity of radians per second. OA is vertical and OA = h metres. The angle OAP is . The body, P, is subject to a gravitational force of mg newtons. The tension in the string is T newtons. What is the angular velocity? (A) g h (B) h g (C) 2 g h (D) 2 h g 34 Year 12 Mathematics Extension 2 Polynomials 80 81 82 83 Solutions Main Menu Let , and be roots of the equation x3 x 2 2 x 5 0 . Which of the following polynomial equations have roots 2 , 2 and 2 ? (A) x3 7 x 2 14 x 3 0 (B) x3 7 x 2 21x 3 0 (C) x3 x 2 6 x 9 0 (D) x3 2 x 2 6 x 9 0 The polynomial equation x3 5 x 2 6 0 has roots , and . Which of the following polynomial equations have roots 1 , 1 and 1 ? (A) x3 8 x 2 13x 0 (B) x3 8 x 2 7 x 0 (C) x3 3x 2 7 x 2 0 (D) x3 2 x 2 7 x 2 0 The polynomial equation x3 3x 2 x 2 0 has roots , and . Which of the following polynomial equations have roots 2 , 2 and 2 ? (A) x3 6 x 2 44 x 49 0 (B) x3 12 x 2 44 x 49 0 (C) x3 3x 2 36 x 5 0 (D) x3 6 x 2 36 x 5 0 Let , and be roots of the equation x3 3x 2 4 0 . Which of the following polynomial equations have roots 2 , 2 and 2 ? (A) x3 9 x 2 24 x 4 0 (B) x3 9 x 2 12 x 4 0 (C) x3 9 x 2 24 x 16 0 (D) x3 9 x 2 12 x 16 0 35 Year 12 Mathematics Extension 2 84 The polynomial equation x3 3x 2 x 2 0 has roots , and . Which of the 1 1 1 following polynomial equations have roots , and ? 85 (A) x3 x 2 3x 1 0 (B) x3 2 x 2 3x 1 0 (C) 2 x3 x 2 3x 1 0 (D) 2 x3 2 x 2 3x 1 0 The polynomial equation P( x) 2 x 4 3x3 2 x 2 7 x 3 has roots , , and . 1 1 1 1 Which of the following polynomial equations have roots , , and ? (A) 2 x 4 3 x3 x 2 5 x 4 (B) 2 x 4 3 x3 x 2 5 x 4 (C) 3 x 4 7 x3 2 x 2 3 x 2 (D) 3x4 7 x3 2 x2 3x 2 86 87 The polynomial equation x3 x 2 2 x 5 0 has roots , and . Which of the following polynomial equations have roots 2 , 2 and 2 ? (A) x3 5 x 2 6 x 25 0 (B) x3 5 x 2 14 x 25 0 (C) x3 4 x 2 5 x 1 0 (D) x3 4 x 2 5 x 1 0 The polynomial equation x3 5 x 2 6 0 has roots , and . Which of the following polynomial equations have roots 2 , 2 and 2 ? (A) x3 25 x 2 60 x 36 0 (B) x3 25 x 2 60 x 12 0 (C) x3 x 2 12 x 36 0 (D) x3 x 2 12 x 12 0 36 Year 12 Mathematics Extension 2 88 The equation 24 x3 12 x 2 6 x 1 has roots , and . What is the value of if ? 89 90 (A) (C) 1 2 1 2 (B) 1 4 (D) 1 What are the zeros of the equation x 4 x 2 6 x 4 0 over the complex field given that it has a rational zero of multiplicity 2? (A) 1 , 1 5i and 1 5i (B) 1 , 1 3i and 1 3i (C) 1 , 1 5i and 1 5i (D) 1 , 1 3i and 1 3i The polynomial P( x) x 4 ax 2 bx 28 has a double root at x 2 . What are the values of a and b? (A) a 11 and b 12 a 5 and b 12 (C) a 11 and b 12 (D) a 5 and b 12 (B) 91 92 The polynomial P( x) x5 3x4 4 x3 4 x2 3x 1 has x 1 as a root of multiplicity 3 and x i as a root. Which of the following expressions is a factorised form of P ( x ) over the complex numbers? (A) P( x) ( x 1)3 ( x 1)( x 1) (B) P( x) ( x 1)3 ( x 1)( x 1) (C) P( x) ( x 1)3 ( x i)( x i) (D) P( x) ( x 1)3 ( x i)( x i) What are the values of real numbers p and q such that 1 i is a root of the equation z 3 pz q 0 ? (A) p 2 and q 4 (B) p 2 and q 4 (C) p 2 and q 4 (D) p 2 and q 4 37 Year 12 Mathematics Extension 2 Harder Extension 1 topics 93 Solutions Main Menu Two equal circles touch externally at B. XB is a diameter of one circle. XZ is the tangent from X to the other circle and cuts the first circle at Y. Z Y X B A Which is the correct expression that relates XZ to XY? (A) 3XZ 4 XY (B) XZ 2 XY 94 (C) 2 XZ 3XY (D) 2 XZ 5 XY What is the derivative of sin 1 x 1 x2 ? (A) (B) (C) (D) 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 38 C Year 12 Mathematics Extension 2 n 95 Using the binomial theorem (1 x) n nC0 nC1 x1 nC2 x 2 ... nCn x n nCk x k k 0 which of the following expressions is correct? n 1 n(n 1)(n 2)...(n k ) 1 (A) (1 ) n n nk k! k 0 (B) n 1 n(n 1)(n 2)...(n k ) 1 (1 )n k n n (k 1)! k 0 (C) n 1 n(n 1)(n 2)...(n k 1) 1 (1 ) n n nk k! k 0 n 1 n(n 1)(n 2)...(n k 1) 1 (D) (1 ) n k n (k 1)! n k 0 96 The labor party conducted a survey for the 2010 election. The ratio of the votes in three seats X, Y and Z was 4:3:2 respectively. The percentage of votes for Ms Gillard in these seats was 60%, 30% and 90% respectively. Ten voters were chosen at random, what is the probability that Ms Gillard gained at least eight votes? (A) 0.1672897536 (B) 0.2509346304 (C) 0.3345795072 (D) 0.418224384 97 A coin is tossed 20 times. What is the probability of obtaining at most 3 heads? (A) 0.0000029 (B) 0.0002 (C) 0.0013 (D) 0.0059 98 1 What is the solution to the equation tan -1 4 x tan -1 3x tan -1 ? 7 (A) x 1 2 or x 7 7 (B) x 1 2 or x 3 3 (C) x 1 1 or x 3 4 (D) x 3 or x 4 39 Year 12 Mathematics Extension 2 99 What is the solution to the inequation 2sin3x 1 if 0 x 2 ? Hint: Use a sketch. (A) (B) (C) (D) 6 6 18 18 x 5 13 17 25 29 x x , , 6 6 6 6 6 x 7 13 20 25 31 x x , , 6 6 6 6 6 x 5 13 17 25 29 x x , , 18 18 18 18 18 x 7 13 20 25 31 x x , , 18 18 18 18 18 100 What is the solution to the inequation x(5 x) 3 ? x4 2 x 4 or x 6 (B) 1 x 4 or x 5 (A) 4 x 6 or x 2 (D) 4 x 5 or x 1 (C) 101 A rock is projected to just clear two poles of height h metres at distances of b and c metres from the point of projection. If v is the velocity of the projection at an angle to the horizontal. Which of the following is the correct expression for square of the velocity? (A) v 2 2(b c) tan g sec2 (B) v2 2(b c) tan g sec2 (C) (b c) g sec2 v 2 tan 2 (D) v 2 (b c) g sec2 2 tan 40

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