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536

Maths In Focus Mathematics Preliminary Course

Practice Assessment Task SET 1 3 1.

Solve m 2 - 5m + 6 $ 0.

2.

Find the locus of point P that moves so that it is equidistant from the points A^ -3, 1 h and B ^ 5, 7 h .

3.

Find the centre and radius of the circle with equation x 2 + 6x + y 2 - 10y - 15 = 0.

4.

If a and b are the roots of the quadratic equation 3x 2 - 2x - 1 = 0, ﬁnd the value of (a) a + b (b) ab (c) a 2 + b2

5.

Find the coordinates of the focus and the equation of the directrix of the parabola x 2 = - 8 y.

6.

Solve ] x + 3 g + 5(x + 3) + 6 = 0.

7.

Find the value of k in the equation x 2 - ] k - 4 g x + 3k = 0 if the sum of the roots is -5.

8.

Find the equation of the locus of a point whose distance from the line 3x - 4y + 1 = 0 is 3 units.

9.

2

Find the coordinates of the vertex and focus of the parabola y = x 2 + 8x - 1. 2x

x

10. Solve 2 - 9.2 + 8 = 0. 11. Find the equation of the tangent to the parabola x 2 = 16y at the point ^ -4, 1h . 12. For what values of b does the equation x 2 + 4x - 2b = 0 have real roots?

13. A and B are the points ^ -4, 0 h and ^ 4, 0 h respectively. Point P ^ x, y h moves so that PA 2 + PB 2 = 64. Find the equation of the locus of P and describe it geometrically. 14. Find the equation of the circle with centre ^ -2, -3 h and radius 5 units. 15. The lines PA and PB are perpendicular, where A is ^ -2, 7 h, B is ^ 5, -1 h and P is ^ x, y h . Find the equation of the locus of P. 16. Find the gradient of the normal to the curve x 2 = - 6y at the point where x = - 4. 17. Find the locus of a point moving so that the ratio of PA to PB is 2:3 where A is ^ 3, 2 h and B is ^ 0, 7 h . 18. If 2x 2 - 3x + 1 / a(x - 1) 2 + b(x - 1) + c, ﬁnd the values of a, b and c. 19. Differentiate

9 - x2 .

20. Find the locus of the point that is equidistant from the point ^ 2, 5 h and the line y = -3. 21. Show that D ABC is similar to DCDE and hence ﬁnd y, correct to 1 decimal place.

PRACTICE ASSESSMENT TASK 3

22. Find the equation of the tangent to the curve ] x - 2 g2 = 8y at the point where x = 6. 23. Find the equation of the locus of point P ^ x, y h that moves so that it is always equidistant from the point ^ -1, 3 h and the line y = - 5. 24. Solve 2 2x - 5.2 x + 4 = 0. 25. Show that - x 2 + x - 9 1 0 for all x. 26. Differentiate ^ 3x - 1 h ^ 2x + 5 h . 4

27. Simplify cot x + tan x. 28. Find the centre and radius of the circle whose equation is x 2 + 10x + y 2 - 6y + 30 = 0. 29. Show that x 2 - x + 3 2 0 for all x.

36. Show that the quadratic equation 6x 2 + x - 15 = 0 has 2 real, rational roots. 37. Find the equation of the normal to the curve y = 2x 4 - 5x 2 - 1 at the point ^ -1, -4 h. 38. Find values of k for which the quadratic equation x 2 - 2x + k - 2 = 0 has real roots. 39. Find the equation of the straight line through ^ 5, -4 h , that is parallel to the line through ^ 7, 4 h and ^ 3, -1 h. 40. Rationalise the denominator of 2+1 3 3+ 5

.

41. Find the values of x and y correct to 1 decimal place.

30. Find the value of k in the quadratic equation x 2 - 3x + k + 1 = 0 if the roots are consecutive numbers. 31. Find the equation of the locus of the point that is equidistant from ^ -2, 1 h and ^ 4, 5 h . 32. A ship sails from port due east for 150 km, then turns and sails on a bearing of 195c for 200 km. (a) How far from port is the ship, to the nearest kilometre? (b) On what bearing, to the nearest degree, is the ship from port?

42. Given f ] x g = 8x - 3, ﬁnd the value of x for which f ] x g = 5. 43. Find the distance between ^ 0, 7 h and ^ -2, -1 h correct to 3 signiﬁcant ﬁgures. 44. Find the value of p correct to 1 decimal place.

33. Find the values of a, b and c if 3x 2 - 7 / a ] x + 3 g2 + bx + c. 34. Solve 2x - 7 2 1. 35. Find the value of i in degrees and minutes.

a3 ^ b2 h 2 4 if a = and b = . 2 7 1 3 9 ^a h b 4

45. Simplify

46. Solve cos 2x = -

1 for 0c # x # 360c . 2

47. Find the equation of the straight line through ^ 3, -1 h perpendicular to the line 3 x - 2 y - 7 = 0.

537

538

Maths In Focus Mathematics Preliminary Course

48. Solve 5y - 3 = 5 - y.

64. Solve x 2 - 4 2 0.

49. Find the size of each internal angle in a regular 20-sided polygon. 50. Solve 2 cos 2 x = 1 for 0c # x # 360c .

65. A circle with centre at the origin O passes through the point (2, 5 ). Find the radius of the circle, and hence its equation.

51. Solve equations x 2 + xy + 1 = 0 and 3x - y + 5 = 0 simultaneously.

66. Find values of a, b and c for which 3x 2 - 2x - 7 / a (x + 2) 2 + b (x + 2) + c.

52. Factorise a 3 - 8b 3 .

67. What is the domain and range of y = x 2 - 3?

x+1 x+2 53. Solve = 7. 2 3 54. Find the gradient of the normal to the curve y = 2x 3 + 7x + 1 at the point where x = - 2.

68. Prove that TABC is congruent to TCDE.

55. Find the perpendicular distance from ^ 3, -2 h to the line 4x - 3y - 9 = 0. 56. Simplify

] sec i + 1 g ] sec i - 1 g .

69. Find the area of the ﬁgure below.

57. Differentiate ] 2x + 5 g (x - 1) . 2

4

x-2 . x2 - 4 59. Find the equation of the locus of point P(x, y) if PA is perpendicular to PB, given A = ^ 3, -2 h and B = ^ -5, 5 h .

58. Find lim x "2

60. Find the coordinates of the focus and the equation of the directrix of the parabola x 2 - 4x + 8y - 20 = 0. 61. Find the equation of the normal to the parabola x 2 = -12y at the point where x = 12.

70. Find the equation of the straight line through the midpoint of (-5, 7) and (1, 3) and making an angle of 135c with the x-axis.

62. Prove that the line 6x - 8y + 40 = 0 is a tangent to the circle with centre the origin and radius 4 units.

71. Complete the square on x 2 - 12x.

63. In the quadratic equation (k -1) x 2 - 5x + 3k + 4 = 0, the roots are reciprocals of each other. Find the value of k.

72. Solve 2y + 4 $ 9. 73. (a) Find the equation of the tangent to the curve y = x 3 - 3x at the point P (-2, -2).

PRACTICE ASSESSMENT TASK 3

(b) Find the equation of the normal to y = x 3 - 3x at P. (c) Find the point Q where this normal cuts the x-axis. 74. (a) Find the equation of the normal to the curve y = x 2 - 6x + 9 at the point where x = -1. (b) This normal cuts the curve again at point R. Find the coordinates of R. 75. The function f (x) = ax 2 + bx + c has a tangent at (1, -3) with a gradient of -1. It also passes through (4, 3) . Find the values of a, b and c. 76. The equation of the locus of point P(x, y) that moves so that it is always 4 units from ^ -1, 3 h is (a) ^ x - 1 h2 + ^ y + 3 h2 = 4 (b) ^ x + 1 h2 + ^ y - 3 h2 = 4 (c) ] x + 1 g2 + ^ y - 3 h2 = 16 (d) ^ x - 1 h + ^ y + 3 h = 16 2

2

77. If a and b are the roots of the quadratic equation x 2 - 5x + 2 = 0, a b evaluate + b a 1 2 1 (b) 12 2 1 (c) 2 2 1 (d) 10 2 (a) 11

78. The equation of the locus of point P(x, y) moving so that it is equidistant from (3, 2) and the line x = -1 is given by (a) x 2 - 2x + 8y - 15 = 0 (b) y 2 - 4y - 8x + 12 = 0

(c) x 2 - 2x - 8y + 17 = 0 (d) y 2 - 4y + 8x - 4 = 0 79. The quadratic equation x 2 + ] k - 3 g x + k = 0 has real roots. Evaluate k (a) k # 1, k $ 9 (b) k = 1, 9 (c) 1 # k # 9 (d) k 1 1, k 2 9 80. Find the centre and radius of the circle x 2 + 2x + y 2 - 8y + 13 = 0. (a) Centre ^ -1, 4 h, radius 4 (b) Centre ^ 1, -4 h, radius 2 (c) Centre ^ -1, 4 h, radius 2 (d) Centre ^ 1, -4 h, radius 4 81. For the quadratic function y = ax 2 + bx + c to be positive deﬁnite (a) a 2 0, b 2 - 4ac 2 0 (b) a 1 0, b 2 - 4ac 2 0 (c) a 2 0, b 2 - 4ac 1 0 (d) a 1 0, b 2 - 4ac 1 0

539

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Maths In Focus Mathematics Preliminary Course

Practice Assessment Task SET 1 3 1.

Solve m 2 - 5m + 6 $ 0.

2.

Find the locus of point P that moves so that it is equidistant from the points A^ -3, 1 h and B ^ 5, 7 h .

3.

Find the centre and radius of the circle with equation x 2 + 6x + y 2 - 10y - 15 = 0.

4.

If a and b are the roots of the quadratic equation 3x 2 - 2x - 1 = 0, ﬁnd the value of (a) a + b (b) ab (c) a 2 + b2

5.

Find the coordinates of the focus and the equation of the directrix of the parabola x 2 = - 8 y.

6.

Solve ] x + 3 g + 5(x + 3) + 6 = 0.

7.

Find the value of k in the equation x 2 - ] k - 4 g x + 3k = 0 if the sum of the roots is -5.

8.

Find the equation of the locus of a point whose distance from the line 3x - 4y + 1 = 0 is 3 units.

9.

2

Find the coordinates of the vertex and focus of the parabola y = x 2 + 8x - 1. 2x

x

10. Solve 2 - 9.2 + 8 = 0. 11. Find the equation of the tangent to the parabola x 2 = 16y at the point ^ -4, 1h . 12. For what values of b does the equation x 2 + 4x - 2b = 0 have real roots?

13. A and B are the points ^ -4, 0 h and ^ 4, 0 h respectively. Point P ^ x, y h moves so that PA 2 + PB 2 = 64. Find the equation of the locus of P and describe it geometrically. 14. Find the equation of the circle with centre ^ -2, -3 h and radius 5 units. 15. The lines PA and PB are perpendicular, where A is ^ -2, 7 h, B is ^ 5, -1 h and P is ^ x, y h . Find the equation of the locus of P. 16. Find the gradient of the normal to the curve x 2 = - 6y at the point where x = - 4. 17. Find the locus of a point moving so that the ratio of PA to PB is 2:3 where A is ^ 3, 2 h and B is ^ 0, 7 h . 18. If 2x 2 - 3x + 1 / a(x - 1) 2 + b(x - 1) + c, ﬁnd the values of a, b and c. 19. Differentiate

9 - x2 .

20. Find the locus of the point that is equidistant from the point ^ 2, 5 h and the line y = -3. 21. Show that D ABC is similar to DCDE and hence ﬁnd y, correct to 1 decimal place.

PRACTICE ASSESSMENT TASK 3

22. Find the equation of the tangent to the curve ] x - 2 g2 = 8y at the point where x = 6. 23. Find the equation of the locus of point P ^ x, y h that moves so that it is always equidistant from the point ^ -1, 3 h and the line y = - 5. 24. Solve 2 2x - 5.2 x + 4 = 0. 25. Show that - x 2 + x - 9 1 0 for all x. 26. Differentiate ^ 3x - 1 h ^ 2x + 5 h . 4

27. Simplify cot x + tan x. 28. Find the centre and radius of the circle whose equation is x 2 + 10x + y 2 - 6y + 30 = 0. 29. Show that x 2 - x + 3 2 0 for all x.

36. Show that the quadratic equation 6x 2 + x - 15 = 0 has 2 real, rational roots. 37. Find the equation of the normal to the curve y = 2x 4 - 5x 2 - 1 at the point ^ -1, -4 h. 38. Find values of k for which the quadratic equation x 2 - 2x + k - 2 = 0 has real roots. 39. Find the equation of the straight line through ^ 5, -4 h , that is parallel to the line through ^ 7, 4 h and ^ 3, -1 h. 40. Rationalise the denominator of 2+1 3 3+ 5

.

41. Find the values of x and y correct to 1 decimal place.

30. Find the value of k in the quadratic equation x 2 - 3x + k + 1 = 0 if the roots are consecutive numbers. 31. Find the equation of the locus of the point that is equidistant from ^ -2, 1 h and ^ 4, 5 h . 32. A ship sails from port due east for 150 km, then turns and sails on a bearing of 195c for 200 km. (a) How far from port is the ship, to the nearest kilometre? (b) On what bearing, to the nearest degree, is the ship from port?

42. Given f ] x g = 8x - 3, ﬁnd the value of x for which f ] x g = 5. 43. Find the distance between ^ 0, 7 h and ^ -2, -1 h correct to 3 signiﬁcant ﬁgures. 44. Find the value of p correct to 1 decimal place.

33. Find the values of a, b and c if 3x 2 - 7 / a ] x + 3 g2 + bx + c. 34. Solve 2x - 7 2 1. 35. Find the value of i in degrees and minutes.

a3 ^ b2 h 2 4 if a = and b = . 2 7 1 3 9 ^a h b 4

45. Simplify

46. Solve cos 2x = -

1 for 0c # x # 360c . 2

47. Find the equation of the straight line through ^ 3, -1 h perpendicular to the line 3 x - 2 y - 7 = 0.

537

538

Maths In Focus Mathematics Preliminary Course

48. Solve 5y - 3 = 5 - y.

64. Solve x 2 - 4 2 0.

49. Find the size of each internal angle in a regular 20-sided polygon. 50. Solve 2 cos 2 x = 1 for 0c # x # 360c .

65. A circle with centre at the origin O passes through the point (2, 5 ). Find the radius of the circle, and hence its equation.

51. Solve equations x 2 + xy + 1 = 0 and 3x - y + 5 = 0 simultaneously.

66. Find values of a, b and c for which 3x 2 - 2x - 7 / a (x + 2) 2 + b (x + 2) + c.

52. Factorise a 3 - 8b 3 .

67. What is the domain and range of y = x 2 - 3?

x+1 x+2 53. Solve = 7. 2 3 54. Find the gradient of the normal to the curve y = 2x 3 + 7x + 1 at the point where x = - 2.

68. Prove that TABC is congruent to TCDE.

55. Find the perpendicular distance from ^ 3, -2 h to the line 4x - 3y - 9 = 0. 56. Simplify

] sec i + 1 g ] sec i - 1 g .

69. Find the area of the ﬁgure below.

57. Differentiate ] 2x + 5 g (x - 1) . 2

4

x-2 . x2 - 4 59. Find the equation of the locus of point P(x, y) if PA is perpendicular to PB, given A = ^ 3, -2 h and B = ^ -5, 5 h .

58. Find lim x "2

60. Find the coordinates of the focus and the equation of the directrix of the parabola x 2 - 4x + 8y - 20 = 0. 61. Find the equation of the normal to the parabola x 2 = -12y at the point where x = 12.

70. Find the equation of the straight line through the midpoint of (-5, 7) and (1, 3) and making an angle of 135c with the x-axis.

62. Prove that the line 6x - 8y + 40 = 0 is a tangent to the circle with centre the origin and radius 4 units.

71. Complete the square on x 2 - 12x.

63. In the quadratic equation (k -1) x 2 - 5x + 3k + 4 = 0, the roots are reciprocals of each other. Find the value of k.

72. Solve 2y + 4 $ 9. 73. (a) Find the equation of the tangent to the curve y = x 3 - 3x at the point P (-2, -2).

PRACTICE ASSESSMENT TASK 3

(b) Find the equation of the normal to y = x 3 - 3x at P. (c) Find the point Q where this normal cuts the x-axis. 74. (a) Find the equation of the normal to the curve y = x 2 - 6x + 9 at the point where x = -1. (b) This normal cuts the curve again at point R. Find the coordinates of R. 75. The function f (x) = ax 2 + bx + c has a tangent at (1, -3) with a gradient of -1. It also passes through (4, 3) . Find the values of a, b and c. 76. The equation of the locus of point P(x, y) that moves so that it is always 4 units from ^ -1, 3 h is (a) ^ x - 1 h2 + ^ y + 3 h2 = 4 (b) ^ x + 1 h2 + ^ y - 3 h2 = 4 (c) ] x + 1 g2 + ^ y - 3 h2 = 16 (d) ^ x - 1 h + ^ y + 3 h = 16 2

2

77. If a and b are the roots of the quadratic equation x 2 - 5x + 2 = 0, a b evaluate + b a 1 2 1 (b) 12 2 1 (c) 2 2 1 (d) 10 2 (a) 11

78. The equation of the locus of point P(x, y) moving so that it is equidistant from (3, 2) and the line x = -1 is given by (a) x 2 - 2x + 8y - 15 = 0 (b) y 2 - 4y - 8x + 12 = 0

(c) x 2 - 2x - 8y + 17 = 0 (d) y 2 - 4y + 8x - 4 = 0 79. The quadratic equation x 2 + ] k - 3 g x + k = 0 has real roots. Evaluate k (a) k # 1, k $ 9 (b) k = 1, 9 (c) 1 # k # 9 (d) k 1 1, k 2 9 80. Find the centre and radius of the circle x 2 + 2x + y 2 - 8y + 13 = 0. (a) Centre ^ -1, 4 h, radius 4 (b) Centre ^ 1, -4 h, radius 2 (c) Centre ^ -1, 4 h, radius 2 (d) Centre ^ 1, -4 h, radius 4 81. For the quadratic function y = ax 2 + bx + c to be positive deﬁnite (a) a 2 0, b 2 - 4ac 2 0 (b) a 1 0, b 2 - 4ac 2 0 (c) a 2 0, b 2 - 4ac 1 0 (d) a 1 0, b 2 - 4ac 1 0

539

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