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Practice Exam Semester 1 2012 2 The next three questions refer to the diagram below. B Solutions Name: ___________________________ 1. Vectors shown: a ~ and b ~ C are represented as An expression for b ~ is: ab ~ ~ A a b ~ ~ B ab ~ ~ C ab ~ ~ D a ~ E Their sum is best represented by: A B 3 C D 4 E The expression represented by: A (10, 0) B (5, 1) C (1, 3) D (1, 3) E (1, 3) c in terms of ~ ab ~ ~ a ~ may be c A vector parallel to ~ but half its magnitude may be represented by: A (1, 2) B (0.5, 2) C (2, .05) D (1, 4) E (1, 2) and A D 5 C Simplify OA AC BC OB A 0 9 B OB C 2OB AB D CA E 6 A vector represented by a displacement of (5, 12) has a magnitude of: 119 A B 13 C 13 D 119 B 10 C 10 D 4 2 E 2 a 5 i 12 j b 5 i 12 j ~ ~ ~ and ~ ~ , then If ~ which of the following is true? ab ~ ~ A C D 7 A vector has a displacement of (5, 2). A The direction it makes with the positive x-axis is: A 21.8 B 68.2 C 68.2 D 111.8 E 158.2 a 2i 4 j 8 ~ If 3a b ~ A B C D E ~ ~ is: 5i 9 j b i 3 j and ~ ~ ~ ~ ~ 7 i 15 j ~ ~ 5i j ~ ~ 7i9 j ~ ~ 5 i 15 j ~ ~ B a ~ a ~ ~ // > b ~ b ~ a E 11 ~ b > ~ The length x is: B then ~ ~ B ab B 2 17 E c 3i j 2c ~ ~ ~ then the magnitude of If ~ is: A 4 B 2 10 A B C D E 5.14 6.13 9.53 10.44 12.44 B 12 2The angle to the nearest degree, is: A B C D E 13 14 D 16 32° 38° 51° 58° 60° cos 39 is equal to: A sin 39 B cos 39 C tan 39 D tan 51 E sin 51 A B C D E 17 3From a vertical fire tower 70 m high, an D observer records the angle of depression to a small fire as 5°. The distance of the fire from the foot of the tower to the nearest metre is: A 6m B 70 m C 350 m D 800 m E 803 m 4A bearing of 290°T is the same as: A N20°W B S20°W C S70°W D N70°W E N20°E 31° 44° 57° 63° 78° E 6The length of the side marked x is: A B C D E 18 15 5The smallest angle to the nearest degree B is: 6.8 11.5 12.5 21.3 22.1 7The area of the triangle is: D A B C D E D 5.9 square units 8.8 square units 11.5 square units 13.2 square units 17.6 square units B 19 8The length of the arc which subtends an C angle of 135° with radius 8 cm is: A B C D E 20 5.2 cm 16.8 cm 18.8 cm 33.5 cm 75.4 cm 9The sector has an area of 150 cm2 and subtends an angle of 70°. What is the radius of the circle correct to 1 decimal place? A B C D E 21 6.1 cm2 11.1 cm2 15.7 cm2 19.1 cm2 20.2 cm2 1Which is the area of the shaded 0segment, to the nearest whole number? A B C D E C A 9 cm2 18 cm2 35 cm2 51.5 cm2 61.4 cm2 22 11.56 radians is closest to: 1A 45° B 89° C 140° D 179° E 280° B 23 1A clock has an hour hand 40 cm long. 2The area swept after 30 minutes is nearest to: A 5 cm2 B 10 cm2 C 2513 cm2 D 5026 cm2 E 1838 cm2 C 24 Which of the following is NOT a polynomial? A 3x 4 3 2x 6 5 B C 8x 7 D E 1 3x x 2 3x 2 x 1 4 D 25 3 2 If 3x x 5 x 6 2a b x3 c a x2 c a x 6, then: A a 1, b 2, c 3 B a 3, b 2, c 1 C C D E 26 27 28 3x 2 5 x 4 x 2 2 x 3 in partial fractions is: 3x 4 x 3 x 1 A 4 3x x 3 x 1 B 4 3x x 3 x 1 C 4x 3 x 3 x 1 D 4x 3 x 3 x 1 E B 30 2 x 2 3x 4 x3 2 x 2 x = 4 2 3 x x 1 ( x 1) 2 A E a 2, b 1, c 3 a 2, b 1, c 3 a 2, b 1, c 3 3x 5 If x 4 is expressed in the form b A x 4 then: A A 3 and b 15 B A 3 and b 12 E C A 3 and b 12 D A 3 and b 15 E A 3 and b 17 3 2 If 3x 12 x 36 x ax( x b)( x c), then a, b and c respectively are: A −3, 12, 36 B −3, 12, 3 C −3, −12, 36 D 3, 12, 36 E −3, -−2, −36 C If A 2 3 2 ( x 4)(ax bx c) 2 x 7 x x 20 then a, b and c respectively are: A 2, −1, 5 B 2, 1, −5 C −2, −1, −5 D 1, 2, 5 E −1, 2, 5 29 31 B 4 2 3 x x 1 ( x 1) 2 C 2 3 4 x x 1 ( x 1) 2 D 3 4 2 x x 1 ( x 1) 2 E 4 2 3 x x 1 ( x 1) 2 The point(s) of intersection between the A 2 parabola y x 2 x 5 and the straight line y 3x 12 is/are: A (3.65, 1.05), (4.65, 25.95) B (3.65, 1.05) C (4.65, 25.96) D (2.32, 5.03) E (2.32, 5.03), (7.32, 33.97) 32 The distance between (5, 1) and (4, 3) is found by calculating: A B C E 33 36 (4 5) 2 ( 3 1) 2 A (4 5) 2 ( 3 1) 2 ( 4 5) ( 3 1) 2 The gradient of the line perpendicular to B 2 x 5 y 1 0 is: (4 5) 2 (3 1) 2 D E B 2 C (4 5) 2 ( 3 1) 2 The midpoint of the line segment between the point (2, 8) and (4, 6) is: A (2, 2) B (1, 1) C (3, 7) D (1, 7) E (1, 1) The equation of the line that contains the points (2, 5) and (5, 10) is: 5x 3 y 5 0 A B C D E 35 5 2 2 5 D E D 37 2 5 3 The equation of the line parallel to A 5 x 2 y 9 0 and passing through the point (1, 2) is: A 5x 2 y 1 0 B C 34 5 2 E D E 2x 5 y 9 0 5x 2 y 9 0 5x 2 y 9 0 2x 5 y 8 0 5x 3 y 5 0 3x 5 y 5 0 38 Josh walks in a line from A(4, 1) to A B(1,7), then directly from B to C(2, 7). The total distance travelled is: A 9.54 B 8.54 C 8.94 D 9.94 E 15.94 39 The points (1, 3), (2, 5) and (5, y) are collinear. The value of y is: A 11 B 25 C 20 D 29 E 35 3x 5 y 5 0 5x 3 y 5 0 M divides AB externally in the ratio 3:2. Given A(5, 2) and B(5, 8), M is the point: A (25, 28) B (0, 3) C (1, 2) D (12, 18) E (25, 28) E D 40 Joining the points A(3, 4), B(0, 1) and C(8, 7) forms: A a straight line B an isosceles triangle C an equilateral triangle D a right-angled triangle at B E a right-angled triangle at C A 41 PQRS is a parallelogram with P(2, 3), Q(1, 7) and R(6, 3). The coordinates of S are: A (3, 1) B (5, 7) C (9, 2) D (1, 3) E (2, 1) A 42 The equation of the line perpendicular to 3 y 4 x 13 0 and has and x- A intercept of -3. A 4 y 3x 9 0 B C D E 43 4 y 3x 9 0 3 y 4 x 12 0 3 y 3x 6 0 4 y 3x 3 0 The equation of the perpendicular bisector of the line joining the points (−10, −2) and (2, 0). A y 6x 3 0 B C D E y 6x 3 0 y 6 x 23 0 6y x 2 0 6 y 6 x 23 0 C Name: ___________________________ Section B Short/Extended answer 1 The angle of depression from a lighthouse 200 m high to a cargo ship in the sea is 3°. What is the distance of the ship from the foot of the lighthouse? 2 opp adj 200 tan 3 x 200 x tan 3 x = 3816.2 m (correct to 1 decimal place) tan 2 The angle of elevation from a Melbourne Airport runway to a plane at an altitude of 300 m is 7°. How far is the plane from the runway, correct to 1 decimal place? 3 opp adj 300 tan 7 d 300 d tan 7 d = 2443.3 m (correct to 1 decimal place) tan 3 A cross-country runner starts at checkpoint A and runs for 8 km on a bearing of 50°T to reach checkpoint B, then heads directly east for 10 km to checkpoint C. How far is checkpoint C from the starting point A? 3 b2 = a2 c2 (2ac cos B) d2 = 82 + 102 – (2 8 10 cos 140°) d2 = 64 + 100 – (–122.6) d2 = 286.6 d = 286.6 d = 16.93 km (correct to 2 decimal places) 4 A boat sails 14 km west, then 12 km south. Find the bearing from its starting point. 2 14 12 14 tan 1 12 = 49.4° Bearing is (180° + 49.4°)T = 229.4°T or S49.4 W tan 5 A helicopter flies 25 km in the direction of N52°W. How far west of the starting point is it, correct to 1 decimal place? 2 adj hyp d cos 38° = 25 d = 25 × cos 38° d = 19.7 km cos 6 In triangle ABC, a = 12, c = 8 and C = 35°. Find two possible values of A. Case 1 a c sin A sin C sin A sin C a c a sin C sin A c 12 sin 35 sin A 8 sin A = 0.8604 A = 59.4° or (180° – 59.4°)T = 120.6° Case 2 3 7 Find the largest angle in the triangle with sides 4 cm, 5 cm and 6 cm. 2 a2 b2 c2 2ab 2 4 52 6 2 cos C = 2 45 5 cos C = 40 cos C = 0.125 C = cos1 0.125 C = 82.8° cos C = 8 Convert the following angles to radian measure. (a) 135 (b) 280 (c) 328 35 (a) 135 = 135 135 180 3 135 = 4 3 180 135 = (b) 280 = 280 280 = 180 14 = 9 180 (c) 328 35 = 328 35 = 5.7349 c 180 9 Convert the following radian measures to degrees and minutes (where appropriate). (a) 10 8 3 3 (a) 180 8 8 = 3 3 1440 = 3 = 480 (b) 2.573c (b) 2.573c = 2.573 180 (c) 0.625 c (c) 0.625c = 0.625 180 = 147.4220407 = 14725 = 35.8098622 = 3549 A goat farm is in the shape of a sector with a radius 90 m and a subtended angle of 78°. Find the length of fencing needed to enclose the farm. 4 78 c = 1.36 radians 78 180 l = r l = 90 1.36 l = 122.4 m Length of fence = (90 122.4 90) = 302.4 m 11 A playing field is designed in the shape of a sector with an area of 8000 m2, and subtends an angle of 80°. Find the radius of the sector. 5 80 c = 1.40 radians 80 180 1 A r 2 2 1 8000 r 2 1.40 2 8000 2 r2 1.40 2 r 11 428.57 r 11 428.57 r 106.9 m (correct to 1 decimal place) 12 From a point A on the level ground, the angle of elevation of the top of a tree is 30°. From a point B on the ground 20 m away from point A and in line with A, and the foot of the tree, the angle of elevation to the top of the tree is 60°. How tall is the tree, to the nearest metre? 6 tan 60° = h x [1] tan 30° = h 20 x [2] h tan 60 Substituting into [2] gives: h tan 30° = h 20 tan 60 h tan 30 20 h tan 60 h tan 30 20 tan 30 h tan 60 20 tan 30° tan 60° h tan 30° = h tan 60° h tan 60° h tan 30° = 20 tan 30° tan 60° h (tan 60° tan 30°) = 20 tan 30° tan 60° 20 tan 30 tan 60 h tan 60 tan 30 h = 17 m (to the nearest metre) From [1], x = 13 In a cross-country run, the competitors run 200 m along a track from the starting point A, directly north to a checkpoint B. From checkpoint B they make their way across to checkpoint C, then back to starting point A. The distance between C and A is 250 m. The bearing of C from starting point A has been recorded as 068°T. 8 (a) A = 68° (b) a2 = b2 c2 (2bc cos A) a2 = 2002 2502 (2 200 250 cos 68°) = 40000 62500 – 37460.7 = 65039.3 a = 65 039.3 = 255 m (to the nearest metre) Distance BC = 212 m (a) Write the value of angle A. (b) Find the distance between the checkpoints B and C. (c) Bearing required is 180 B. (c) Determine the bearing of C from B to the nearest degree. b a sin B sin A sin B sin A b a b sin A sin B a 250 sin 68 sin B 255 sin B = 0.9090 B = sin1 (0.9090) B = 65° The bearing of C from B = (180° 65°) = 115°T or S 65 E The next three questions refer to b ~ (3, 2). 14 15 a ~ (4, 2) and On the same set of axes, draw the vectors a b ~ and ~ . Calculate 16 3a 2b ~ . ~ a b Calculate ~ ~ 2 3(4, 2) 2(3, 2) (12, 6) (6, 4) (6, 10) 2 ab 2 ~ . a b ~ 17 (4, 2) (3, 2) (7, 0) ~ Show that AB BC AD DC . ~ 7 0 49 7 2 2 LHS AB BC AD AC AD AC DA DA AC DC RHS. 3 18 Find the horizontal and vertical components of Horizontal component: a vector of magnitude 6, which makes an angle 6 cos (120) of 120 with the positive x-axis. 1 6 3 2 Vertical component: 6 sin (120) 3 6 3 3 2 19 If d 2 i 7 j e 2 i j f i 2 j ~ ~ ~ ~ ~ , ~ ~ and ~ ~ , calculate d e f 5 i 10 j ~ ~ d e f ~ ~ ~ ~ . ~ ~ d e f ~ 3 ~ 3 ~ (5) 2 10 2 125 5 5 3i 3 j 20 ~ What angle does the vector the positive x-axis? 21 ~ ~ make with 3 1 3 , 4th quadrant 45 tan OA 7 i 3 j If ~ OB i 5 j and ~ ~ , find AB . AB OB OA 8 i 8 j ~ ~ 3 2 22 7 A hiker is at a position (5, 15) where the coordinates represent the distances in kilometres east and north of O, respectively. If the campsite is at a position given by (17, 25), find the distance and the direction the hiker must take to reach the campsite. Let OH (5, 15) OC (17, 25) HC (17, 25) (5, 15) (22, 10) HC 222 102 24.17 10 22 24.44 2427 24.17 km to the campsite, on a course E24 27 N (or N 6533 E) tan 23 Analysis A triathlon course is in the shape of a triangle. O is the start and finish point of the event. The swimming leg goes from O to A. The cycling leg is from A to B. The final leg, running from B to O. The coordinates of A and B are (3, 1) and (1, 14) respectively. The coordinates represent the distance in kilometres east and j i north of O. Take ~ and ~ as the unit vectors 9 along OX and OY . (a) Express the vectors OA and OB in terms (a) j i of ~ and ~ . ~ ~ (c) 3i j ~ ~ OB i 14 j ~ ~ AB OB OA 2 i 15 j AB 2 i 15 j (b) Hence show that OA (b) . Calculate the magnitudes of OA , OB and (c) ~ ~ OA 3 2 12 10 3.16 OB 12 14 2 197 14.04 AB . AB 2 2 15 2 229 15.13 Total distance 3.16 14.04 15.13 32.33 km 1 tan 1 18.4 (e) Find the angle that OA makes with the x- (e) 3 axis. 15 tan 1 82.41 2 (f) (3rd quadrant) AB (f) Calculate the angle that makes with (that is, 262.41) the x-axis and hence show that the bearing of B Bearing from A (90 82.41) from A is S7.59W. (d) Find the distance of the course. (d) S7.59W 24 2 In the regular hexagon shown, AB a and ~ FA b . ~ Express the following in terms of a and b . ~ (a) OB ~ (a) (b) OC OB b ~ (b) OC a ~ (c) FC (d) OA (e) AD (c) FD ~ (d) (f) FC 2 a OA b a ~ ~ (e) AD 2 a 2 b ~ ~ (f) 25. Find the values of a, b and c if ( x 2)(ax 2 bx c) 2 x3 5x 2 6 x 8 FD 2 a b ~ ~ LHS ax 3 bx 2 cx 2ax 2 2bx 2c ax 3 b 2a x 2 c 2b x 2c hence a 2, c 4 b 2a 5 so b 1 2 26. Find the values of a, b and c if ax x b x c 7 x3 5x2 6x 7 LHS ax x 2 cx bx bc 7 3 ax 3 acx 2 abx 2 abcx 7 ax 3 ac ab x 2 abcx 7 hence a 1 ac ab 5 b c 5 b 5 c abc 6 bc 6 5c c 2 6 5c c 6 c 2 5c 6 0 (c 2)(c 3) 0 c 2, b 3 or c 3, b 2 27 7 x 14 Express x 3 x 10 as partial fractions. 2 7 x 14 7 x 14 x 3 x 10 ( x 5)( x 2) A B x5 x 2 7 x 14 A( x 2) B ( x 5) 3 2 7 x 14 ( A B ) x 2 A 5 B A B 7 [1] 2 A 5 B 14 [2] 2 [1] [2] 7 B 28 B4 A3 28 Solve the following equations simultaneously: y 4x 5 2 y 4x 2 2(4 x 5) 4 x 2 2 8 x 10 4 x 2 4 x 12 x3 y7 29 Find the point(s) of intersection between the 2 x x 2 9 10 0 parabola x2 2 x 1 0 2 y x 9 and the straight line with equation 2 x y 10 0 ( x 1) 2 0 x 1 point of intersection is ( 1, 8) 2 30 (2, 5) is the midpoint of the segment AB. If A x1 x2 y1 y2 , has coordinates (w, v) and B has coordinates 2 Midpoint 2 (3, 6) find w and v. w3 2 2 w=1 v 6 5 2 v = 4 w = 1 and v = 4 3 31 If the distance between A(3, 4) and B(3, p) is 13 units, find the value of p. 5 (3 3)2 ( 4 p)2 169 ( 4 p)2 169 4 p 13 p 4 13 p 17 or 9 32 Find the equation of the line passing through P(4, 1) and parallel to the line joining A(2, 5) and B(4, 9). 33 Find the equation of the perpendicular bisector of the segment AB, where A has coordinates (2, 3) and B has coordinates (4, 7) 34 Show that triangle ABC is isosceles given A(1, 5), B(2, 1) and C(4, 1). 9 5 14 mAB 7 42 2 y 1 7( x 4) y 1 7 x 28 7 x y 29 0 4 2 4 37 , 3, 2 2 2 5 Midpoint 7 3 10 mAB 5 42 2 1 m 5 1 y 2 ( x 3) 5 5 y 10 x 3 x 5 y 13 0 AB (1 2) 2 (5 1) 2 25 5 AC (4 1) 2 (1 5) 2 25 5 BC (4 2) 2 (1 1) 2 36 6 Since AB AC, then triangle is isosceles. 3 35 Find the equation of the line perpendicular to 3x 2 y 8 0 and having the same y-intercept. 36 Find the coordinates of the point C that divides the line joining A(7, 4) and B(1, 2) internally in the ratio 2:5. y-intercept: (0, 4) 2 y 3x 8 3 y x4 2 3 m 2 2 m 3 2 y4 x 3 2 y x4 3 or 2x + 3y 12 = 0 5 2 1 5 7 37 2 5 25 7 7 2 2 5 4 16 2 y 2 25 7 7 4 x 2 2 5 , 2 7. Coordinates of point C are 7 37 If the points A(7, 2), B(5, 4) and C(4, z) are collinear, find the value of z. mAB mAC 4 4 2 z 2 57 47 1 z2 2 3 2z + 4 = 3 2z = 1 1 z=2 38 A(7, 2), B(2, 14), C(10, 9) and D(5, 3) form the quadrilateral ABCD. Find: (a) the gradient of AB (a) (b) the gradient of DC (c) (b) (e) the length of AB (f) 9 14 10 2 5 12 mBC the length of BC. Describe the quadrilateral ABCD. 39 5 10 12 5 mDC the gradient of BC (d) the gradient of AD 14 2 2 7 12 5 mAB (c) (d) (e) 3 2 5 7 5 12 mAD d AB ( 2 7) 2 (14 2) 2 13 d BC (10 2) 2 (9 14) 2 13 (f) mAB mDC AB//DC mBC mAD BC//AD 12 5 mAB mBC 1 AB BC 5 12 d AB d BC 13 ABCD is a square. 8