446
Maths In Focus Mathematics Preliminary Course
Practice Assessment Task SET 2 1.
Evaluate sin 309c 41l to 2 decimal places.
2.
Simplify sin 2 38c + sin 2 52c.
3.
Find the equation of the straight line through ^ -1, 3 h that is perpendicular to the line 2x - 5y = 9.
4.
Calculate the point of intersection of the lines 2x - 3y - 10 = 0 and 5x + 4y - 2 = 0.
angle must a shot be made in order to score a goal? 13. Find the domain and range of y =
3 . 2x - 1
14. (a) On a number plane, draw the line 2x - y = 4. (b) On your diagram, shade the region given by y $ 0, 2x - y $ 4. 15. Sketch y = x 2 - 5x + 4.
5.
Evaluate cot 107c 9l to 3 decimal places.
16. Simplify sin 2 i - sin 2 i cos 2 i.
6.
ABCD is a parallelogram in which BC = 2 cm, +ABC = 150c and AB = 3 cm. Find (a) the exact area of the parallelogram (b) the exact lengths of both diagonals.
17. Find the perpendicular distance from ^ -2, 3 h to the line 5x - 12y + 20 = 0.
7.
The lines AB and AC have equations 3x - 4y + 9 = 0 and 8x + 6y - 1 = 0 respectively. Show that the lines are perpendicular and find the coordinates of A.
8.
If sin ^ x + 5 hc = cos 70c, find a possible value of x.
9.
Simplify as a fraction the expression sin 2 60c - cos 2 45c + tan 2 30c without using a calculator.
10. Sketch 5x - 2y - 10 = 0. 11. On the number plane, shade in the region given by x 2 + y 2 # 9 and x - y $ 2. 12. A soccer goal is 8 m wide. A man shoots for goal when he is 9 m from one post and 11 m from the other. Within what
18. Find the equation of the straight line through ^ 1, 2 h parallel to the line through ^ -3, 4 h and ^ 5, 5 h . 19. Solve cos i = -
1 for 0c # i # 360c. 2
20. Find the gradient of the line through the origin and ^ -3, 5 h . 21. Simplify
tan 20c + cot 70c . tan 20c
22. Find a in degrees and minutes.
23. Find the value of y correct to 3 significant figures.
PRACTICE ASSESSMENT TASK 2
24. Find the equation of the line passing through ^ 1, 2 h that passes through the intersection of lines x - 2y - 11 = 0 and 5x - y - 19 = 0.
36. Simplify
25. Solve 3x - 7 1 2.
(b) Line l cuts the x-axis at P and the y-axis at Q. Find the coordinates of P and Q. (c) Find the area of TOPQ where O is the origin. 38. Two points A and B are 100 m apart on the same side of a tower. The angle of elevation of A to the top of the tower is 20c and the angle of elevation from B is 27c. Find the height of the tower, to the nearest metre.
26. If f ] x g = 9 - 2x 2, find the value of f ] -1 g . 27. Find the value of x if f ] x g = 7 where f ] x g = 2 x - 1. 28. Find the exact value of tan 300c. 29. Show that 3x - 4y + 10 = 0 is a tangent to the circle x 2 + y 2 = 4. 3-x if x 2 1 30. If g ] x g = ) 2x if x # 1 (a) find g ] 2 g and g ] -3 g (b) sketch the graph of g ] x g. 3x 2 h - 4xh + 2h 2 . h "0 h
31. Find lim
32. Find the exact value of cos 135c. 33. A bird at the top of a 10 m tree sees a mouse on the ground. If the angle of depression is 34c51l, how far, to 1 decimal place, does the bird need to fly to reach the mouse? 34. If the point ^ 3, -7 h is the midpoint of ^ x, 3 h and ^ 8, y h, find the values of x and y. 35.
(a) Find AB, correct to 1 decimal place. (b) Find the area of TABC, to 3 significant figures.
9 - 9 sin 2 i .
37. (a) Find the equation of the straight line l through ^ -1, 2 h that is perpendicular to the line 3x + 6y - 7 = 0.
39. The lines 3x - y = 6, 2x + y = 14 and y = 0 enclose a triangle. Find the area of the triangle. 40. Show that f ] x g = x 6 - x 2 - 3 is an even function. 4 41. Differentiate x ^ 2x 2 + 1 h .
42. Solve 3y - 7 # 20. 3 43. Differentiate x . 44. (a) Find the equation of the tangent to the curve y = x 3 - 2 at the point P ^ 1, -1 h . (b) The curve y = x 3 - 2 meets the y-axis at Q. Find the equation of PQ. (c) Find the equation of the normal to y = x 3 - 2 at the point ^ -1, - 3 h . (d) Find the point R where this normal cuts the x-axis. 45. Find the distance between the parallel lines 5x - 12y - 3 = 0 and 5x - 12y + 5 = 0. 46. What is the domain and range of the 3 function f ] x g = ? x+4
447
448
Maths In Focus Mathematics Preliminary Course
47. Differentiate
2x - 7 .
48. Justin walks for 3.1 km due west, then turns and walks for 2.7 km on a bearing of 205c. How far is he from his starting point? Give your answer correct to 1 decimal place. 49. Find the derivative of 4x 2 - 7x + 3 + 5x - 2 . 5x 50. Find the derivative of . x+1
60. For the interval AB where A = ^ 3, -2 h and B = ^ -1, 4 h find (a) the midpoint (b) the exact distance (c) the gradient (d) the equation of line AB. 61. Differentiate ] 3x - 2 g ] 2x + 1 g6 (a) 12 ] 3x - 2 g ] 2x + 1 g5 + 3 ] 2x + 1 g6 (b) 36 ] 2x + 1 g5 (c) 6 ] 3x - 2 g ] 2x + 1 g5 + 3 ] 2x + 1 g6 (d) 18 ] 2x + 1 g5.
51. Differentiate from first principles f ] x g = x 2 - 3x + 2. 52. Write as a single fraction 3 4 2 - + . x 2 + 5x x x + 5
62. Find an expression involving i for the triangle below (there may be more than one answer).
53. Find the equation of the normal to the curve y = 3x 2 - 6x + 9 at the point where x = 2. 54. If f ] x g = 2x 3 - 5x 2 + 4x - 1, find f ] -2 g and f l ] -2 g . 55. At the point ^ 2, -3 h on the curve y = ax 2 + bx + 7, the tangent is inclined at 135c to the x-axis. Find the values of a and b. 56. Find the equation of the straight line passing through ^ 3, 6 h that also passes through the intersection of the lines x - 2y = 0 and 3x + y + 7 = 0. 57. Find the equation of the tangent to the curve f ] x g = 2x 3 - 7x + 1 at the point where x = -3. 58. Solve for x: 4 2x - 1 =
1 . 8
59. Find the domain and range of (a) y = 2x - 1 (b) y =
5 x+7
(c) y = - 4 - x 2
4
5
i
a 7
52 + 42 - 72 2#5#4 sin i sin a = (b) 4 5 sin i sin a = (c) 5 4 52 + 72 - 42 (d) cos i = 2#5#7 (a) cos i =
8x 3 if x 2 3 2 ] g 63. If f x = * 3x - 2 if 0 # x # 3 9 if x 1 0 evaluate f ] 3 g + f ] 1 g + f ] -1 g (a) 35 (b) 226 (c) 233 (d) 53
PRACTICE ASSESSMENT TASK 2
64. The equation of the normal to the parabola y = 3x 2 - 5x + 1 at the point ^ 2, 3 h is (a) 7x - y - 11 = 0 (b) 7x - y - 17 = 0 (c) x + 7y - 23 = 0 (d) x + 7y - 19 = 0
65. The linear function with equation 4x - 2y + 3 = 0 has 1 (a) gradient -2, y-intercept -1 2 3 1 (b) gradient , y-intercept 4 2 1 (c) gradient 2, y-intercept 1 2 (d) gradient 4, y-intercept 3.
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