PRACTICE ASSESSMENT TASK 3 655

Practice Assessment Task

SET 3
1. Solve m 2 - 5m + 6 $ 0. 10.

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. Write x = 4t, y = 2t 2 as an equation in

Cartesian form.

4. Show that AF < CD given AC and FD are

straight lines.

11. Find the equation of the locus of a

point whose distance from the line
3x - 4y + 1 = 0 is 3 units.

12. Find the coordinates of the vertex and

AB, AC and CB are tangents with focus of the parabola y = x 2 + 8x - 1.
CZ = 3 cm, ZB = 7 cm and AY = 2 cm.
13. Solve 2 2x - 9.2 x + 8 = 0.
Find the perimeter of TABC.

5. Find the centre and radius of the 14.

circle with equation
x 2 + 6x + y 2 - 10y - 15 = 0.

6. If a and b are the roots of the quadratic

equation 3x 2 - 2x - 1 = 0, find the
value of
(a) a + b
Find the value of k correct to 1 decimal
(b) ab
place.
(c) a 2 + b2
7. Find the coordinates of the focus and the 15. Find the equation of the tangent to
equation of the directrix of the parabola the parabola x 2 = 16y at the point ^ -4, 1h .
x 2 = - 8 y.
16. For what values of b does the equation
8. Solve ] x + 3 g + 5(x + 3) + 6 = 0.
2
x 2 + 4x - 2b = 0 have real roots?

9. Find the value of k in the equation

x 2 - ] k - 4 g x + 3k = 0 if the sum of the
roots is -5.
656 Maths In Focus Mathematics Extension 1 Preliminary Course

17. 25.

i
32c
0

Find i. O is the centre of the circle.

Find x and y. O is the centre of the circle.
18. A and B are the points ^ -4, 0 h and
26. Differentiate 9 - x2 .
^ 4, 0 h respectively. Point P ^ x, y h moves
so that PA 2 + PB 2 = 64. Find the
27. The point P _ 2ap, ap 2 i lies on the
equation of the locus of P and describe
parabola x 2 = 4ay.
it geometrically.
(a) Find the equation of the tangent to
19. Find the equation of the circle the curve at P.
with centre ^ -2, -3 h and radius (b) Find the point R where this tangent
5 units. meets the directrix.
(c) Find the equation for FR where F is
20. The lines PA and PB are perpendicular,
the focus.
where A is ^ -2, 7 h, B is ^ 5, -1 h and
P is ^ x, y h . Find the equation of the 28. Find the locus of the point that is
locus of P. equidistant from the point ^ 2, 5 h and the
line y = -3.
21.
29. Show that D ABC is similar to DCDE and
hence find y, correct to 1 decimal place.

O is the centre of the circle. Show

+DAE = 90c - +BDC.
30. Find the equation of the tangent to
22. Find the gradient of the normal to the curve ] x - 2 g2 = 8y at the point
the curve x 2 = - 6y at the point where where x = 6.
x = - 4.
31. Find the equation of the locus of point
23. Find the locus of a point moving so that P ^ x, y h that moves so that it is always
the ratio of PA to PB is 2:3 where A is equidistant from the point ^ -1, 3 h and
^ 3, 2 h and B is ^ 0, 7 h . the line y = - 5.

24. If 2x 2 - 3x + 1 / a(x - 1) 2 + b(x - 1) + c, 32. Solve 2 2x - 5.2 x + 4 = 0.

find the values of a, b and c.
 PRACTICE ASSESSMENT TASK 3 657

33. Show that - x 2 + x - 9 1 0 for all x. 45. (a) Change the set of parametric
equations x = 2t, y = 4t 2 - 1 into
34. Differentiate ^ 3x - 1 h ^ 2x + 5 h4.
Cartesian form.
35. Simplify cot x + tan x. (b) Find the coordinates of the point
where t = -2.
36. Prove that the opposite angles (c) Find the equation of the normal to
are supplementary in any cyclic the curve at the point where t = -2.
quadrilateral.
46. Find the value of i in degrees and
37. Find the centre and radius of minutes.
the circle whose equation is
x 2 + 10x + y 2 - 6y + 30 = 0.

38.

47.

AC is a tangent and AC < DE. Prove O is the centre of the circle. Find x.
FGED is a cyclic quadrilateral.
48. Show that the quadratic equation
39. Show that x 2 - x + 3 2 0 for all x. 6x 2 + x - 15 = 0 has 2 real, rational roots.
40. Find the value of k in the quadratic 49. Find the equation of the normal to the
equation x 2 - 3x + k + 1 = 0 if the roots curve y = 2x 4 - 5x 2 - 1 at the point
are consecutive numbers. ^ -1, -4 h.
41. Find the equation of the locus of the 50. Find values of k for which the
point that is equidistant from ^ -2, 1 h quadratic equation
and ^ 4, 5 h . x 2 - 2x + k - 2 = 0 has real roots.
42. A ship sails from port due east for x
51. Sketch y = .
150 km, then turns and sails on a 2x + 1
bearing of 195c for 200 km. 52. Find the equation of the straight line
(a) How far from port is the ship, to the through ^ 5, -4 h , that is parallel to the
nearest kilometre? line through ^ 7, 4 h and ^ 3, -1 h.
(b) On what bearing, to the nearest
53. Divide the interval AB where A = ^ 1, -4 h
degree, is the ship from port?
and B = ^ 7, 0 h in the ratio 2:3.
43. Find the values of a, b and c if 54. Find the exact value of tan 75c.
3x 2 - 7 / a ] x + 3 g2 + bx + c.
y
55. Solve $ 5.
44. Solve 2x - 7 2 1. y+1
658 Maths In Focus Mathematics Extension 1 Preliminary Course

56. Rationalise the denominator of AB is a diameter of the larger circle

2+1 and DB is a straight line. Show AD is a
.
3 3+ 5 diameter of the smaller circle.

57. Find the values of x and y correct to 67. Solve 2 cos 2 x = 1 for 0c # x # 360c .
1 decimal place.
68. Solve equations x 2 + xy + 1 = 0 and
3x - y + 5 = 0 simultaneously.

69. Factorise a 3 - 8b 3 .
x+1 x+2
70. Solve - = 7.
2 3
71. Find the gradient of the normal to the
58. Given f ] x g = 8x - 3, find the value of x curve y = 2x 3 + 7x + 1 at the point
for which f ] x g = 5. where x = - 2.

59. Find the distance between ^ 0, 7 h and 72. Find the perpendicular distance from
^ -2, -1 h correct to 3 significant figures. ^ 3, -2 h to the line 4x - 3y - 9 = 0.

60. Find the value of p correct to 1 decimal 73. Simplify ] sec i + 1 g ] sec i - 1 g .
place.
74. Differentiate ] 2x + 5 g (x 2 - 1) 4.
x-2
75. Find lim .
x "2
x2 - 4
76. Find the equation of the locus of point
P(x, y) if PA is perpendicular to PB, given
a3 ^ b2 h
4
2 4 A = ^ 3, -2 h and B = ^ -5, 5 h .
61. Simplify if a = and b = .
^ a - 1 h2 b 7 3 9
77. Find the coordinates of the focus and the
1 equation of the directrix of the parabola
62. Solve cos 2x = - for 0c # x # 360c .
2 x 2 - 4x + 8y - 20 = 0.
63. Find the equation of the straight line
through ^ 3, -1 h perpendicular to the line 78. Find the equation of the tangent to the
3 x - 2 y - 7 = 0. parabola x 2 = 36y at the point P(18p, 9p2).

64. Solve 5y - 3 = 5 - y. 79. Find the equation of the normal to

the parabola x 2 = -12y at the point
65. Find the size of each internal angle in a where x = 12.
regular 20-sided polygon.
80. If points P(2ap, ap2) and Q(2aq, aq2) lie
66. on the parabola x 2 = 4ay, find
(a) the equation of chord PQ
(b) the equation of the locus of the
midpoint of PQ if PQ passes through (0, 2a)
(c) Describe the shape of this locus.
 PRACTICE ASSESSMENT TASK 3 659

81. The equation of the locus of point P(x, y) 85. Find the centre and radius of the circle
that moves so that it is always 4 units x 2 + 2x + y 2 - 8y + 13 = 0.
from ^ -1, 3 h is (a) Centre ^ -1, 4 h, radius 4
(a) ^ x - 1 h2 + ^ y + 3 h2 = 4 (b) Centre ^ 1, -4 h, radius 2
(b) ^ x + 1 h + ^ y - 3 h = 4
2 2
(c) Centre ^ -1, 4 h, radius 2
(c) ] x + 1 g2 + ^ y - 3 h2 = 16 (d) Centre ^ 1, -4 h, radius 4
(d) ^ x - 1 h + ^ y + 3 h = 16
2 2
86. In the circle, O is the centre. Evaluate x.
82. If a and b are the roots of the
quadratic equation x 2 - 5x + 2 = 0,
a b
evaluate +
b a O
x
1 84⬚
(a) 11
2
1
(b) 12
2
1
(c) 2
2
1 (a) x = 42c
(d) 10
2 (b) x = 168c
83. The equation of the locus of (c) x = 84c
point P(x, y) moving so that it is (d) x = 96c
equidistant from (3, 2) and the line
87. Find the Cartesian equation
x = -1 is given by
for x = 8t, y = 4t 2 .
(a) x 2 - 2x + 8y - 15 = 0
(a) x 2 = 32y
(b) y 2 - 4y - 8x + 12 = 0
(b) x 2 = 4y
(c) x - 2x - 8y + 17 = 0
2
(c) x 2 = 16y
(d) y - 4y + 8x - 4 = 0
2
(d) x 2 = 8y

84. The quadratic equation 88. The equation of the normal to

x 2 + ] k - 3 g x + k = 0 has real roots. the curve x 2 = 20y at the point
Evaluate k (10p, 5p2) is
(a) k # 1, k $ 9 (a) x + py = 5p 3 + 10p
(b) k = 1, 9 (b) x - py = 5p 3 - 10p
(c) 1 # k # 9 (c) px + y = 15p 2
(d) k 1 1, k 2 9 (d) px - y + 15p 2 = 0
660 Maths In Focus Mathematics Extension 1 Preliminary Course

89. AB is a tangent to the circle. Which 90. For the quadratic function
statement is true (there may be more y = ax 2 + bx + c to be positive definite
than one answer)? (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
A
D (d) a 1 0, b 2 - 4ac 1 0

AB BD
(a) =
BC AB
AB CD
(b) =
BC AB
(c) AB 2 = BC $ CD
(d) AB 2 = BC $ BD