Practice Assessment Task SET 3

PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t, y = t as an equation in Cartesian form 4 Show that AF < CD given AC and FD are straight lines Find the equation of the locus of a point whose distance from the line 3x - 4y + = 0 is 3 units AB, AC and CB are tangents with CZ = 3 cm, ZB = 7 cm and AY = cm Find the perimeter of T ABC 5 Find the centre and radius of the circle with equation x + 6x + y - 0y - 5 = 0 6 If a and b are the roots of the quadratic equation 3x - x - = 0, find the value of (a) a + b (b) ab (c) a + b 7 Find the coordinates of the focus and the equation of the directrix of the parabola x =- 8y 8 Solve ] x + 3g + 5( x + 3) + 6 = 0 Find the coordinates of the vertex and focus of the parabola y = x + 8x - 3 Solve x x - 9 + 8 = 0 4 Find the value of k correct to decimal place 5 Find the equation of the tangent to the parabola x = 6y at the point ^-4, h 6 For what values of b does the equation x + 4x - b = 0 have real roots? 9 Find the value of k in the equation x -] k - 4g x + 3k = 0 if the sum of the roots is - 5

656 Maths In Focus Mathematics Extension Preliminary Course 7 5 i 0 3c Find i O is the centre of the circle 8 A and B are the points ^-4, 0h and ^40, h respectively Point P^x, yh moves so that PA + PB = 64 Find the equation of the locus of P and describe it geometrically 9 Find the equation of the circle with centre ^-, -3h and radius 5 units 0 The lines PA and PB are perpendicular, where A is ^-, 7h, B is ^5, -h and P is ^x, yh Find the equation of the locus of P Find x and y O is the centre of the circle 6 Differentiate 9 - x 7 The point P_ ap, ap i lies on the parabola x = 4ay (a) Find the equation of the tangent to the curve at P (b) Find the point R where this tangent meets the directrix (c) Find the equation for FR where F is the focus 8 Find the locus of the point that is equidistant from the point ^5, h and the line y =- 3 9 Show that D ABC is similar to D CDE and hence find y, correct to decimal place O is the centre of the circle Show + DAE = 90c - + BDC Find the gradient of the normal to the curve x =- 6y at the point where x =- 4 3 Find the locus of a point moving so that the ratio of PA to PB is :3 where A is ^3, h and B is ^07, h 4 If x - 3x + / ax ( - ) + bx ( - ) + c, find the values of a, b and c 30 Find the equation of the tangent to the curve ] x - g = 8y at the point where x = 6 3 Find the equation of the locus of point P^x, yh that moves so that it is always equidistant from the point ^-, 3h and the line y =- 5 3 Solve x x - 5 + 4 = 0

PRACTICE ASSESSMENT TASK 3 657 33 Show that - x + x - 9 0 for all x 34 Differentiate ^3x - h^x + 5h 4 35 Simplify cot x + tan x 36 Prove that the opposite angles are supplementary in any cyclic quadrilateral 37 Find the centre and radius of the circle whose equation is x + 0x + y - 6y + 30 = 0 45 (a) Change the set of parametric equations x = t, y = 4t - into Cartesian form (b) Find the coordinates of the point where t =- (c) Find the equation of the normal to the curve at the point where t =- 46 Find the value of i in degrees and minutes 38 47 AC is a tangent and AC < DE Prove FGED is a cyclic quadrilateral 39 Show that x - x + 3 0 for all x 40 Find the value of k in the quadratic equation x - 3x + k + = 0 if the roots are consecutive numbers 4 Find the equation of the locus of the point that is equidistant from ^-, h and ^45, h 4 A ship sails from port due east for 50 km, then turns and sails on a bearing of 95c for 00 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? 43 Find the values of a, b and c if 3x - 7 / a] x + 3g + bx + c 44 Solve x - 7 O is the centre of the circle Find x 48 Show that the quadratic equation 6x + x - 5 = 0 has real, rational roots 49 Find the equation of the normal to the 4 curve y = x - 5x - at the point ^-, -4h 50 Find values of k for which the quadratic equation x - x + k - = 0 has real roots x 5 Sketch y = x + 5 Find the equation of the straight line through ^5, -4h, that is parallel to the line through ^74, h and ^3, -h 53 Divide the interval AB where A = ^, -4h and B = ^70, h in the ratio :3 54 Find the exact value of tan 75c y 55 Solve $ 5 y +

658 Maths In Focus Mathematics Extension Preliminary Course 56 Rationalise the denominator of + 3 3 + 5 57 Find the values of x and y correct to decimal place 58 Given f] xg = 8x - 3, find the value of x for which f] xg = 5 59 Find the distance between ^07, h and ^-, -h correct to 3 significant figures 60 Find the value of p correct to decimal place 3 4 a ^ b h 6 Simplify if a a - ^ h = b 7 3 and 4 b = 9 6 Solve cos x =- for 0c# x # 360c 63 Find the equation of the straight line through ^3, -h perpendicular to the line 3x - y - 7 = 0 64 Solve 5y - 3 = 5 - y 65 Find the size of each internal angle in a regular 0-sided polygon 66 AB is a diameter of the larger circle and DB is a straight line Show AD is a diameter of the smaller circle 67 Solve cos x = for 0c # x # 360c 68 Solve equations x + xy + = 0 and 3x - y + 5 = 0 simultaneously 69 Factorise a 3-8b 3 x + x + 70 Solve - = 7 3 7 Find the gradient of the normal to the 3 curve y = x + 7x + at the point where x =- 7 Find the perpendicular distance from ^3, -h to the line 4x - 3y - 9 = 0 73 Simplify ] sec i + g] sec i -g 4 74 Differentiate ] x + 5g ( x -) x - 75 Find lim x " x - 4 76 Find the equation of the locus of point P ( x, y ) if PA is perpendicular to PB, given A = ^3, -h and B = ^-55, h 77 Find the coordinates of the focus and the equation of the directrix of the parabola x - 4x + 8y - 0 = 0 78 Find the equation of the tangent to the parabola x = 36y at the point P (8 p, 9 p ) 79 Find the equation of the normal to the parabola x =- y at the point where x = 80 If points P ( ap, ap ) and Q ( aq, aq ) lie on the parabola x = 4ay, find (a) the equation of chord PQ (b) the equation of the locus of the midpoint of PQ if PQ passes through (0, a ) (c) Describe the shape of this locus

PRACTICE ASSESSMENT TASK 3 659 8 The equation of the locus of point P ( x, y ) that moves so that it is always 4 units from ^-, 3h is (a) ^ x - h + ^ y + 3h = 4 (b) ^ x + h + ^ y - 3h = 4 (c) ] x + g + ^ y - 3h = 6 (d) ^ x - h + ^ y + 3h = 6 8 If a and b are the roots of the quadratic equation x - 5x + = 0, a b evaluate + b a (a) (b) (c) (d) 0 83 The equation of the locus of point P ( x, y ) moving so that it is equidistant from (3, ) and the line x =- is given by (a) x - x + 8y - 5 = 0 (b) y - 4y - 8x + = 0 (c) x - x - 8y + 7 = 0 (d) y - 4y + 8x - 4 = 0 84 The quadratic equation x + ] k - 3g x + k = 0 has real roots Evaluate k (a) k #, k $ 9 (b) k = 9, (c) # k # 9 (d) k, k 9 85 Find the centre and radius of the circle x + x + y - 8y + 3 = 0 (a) Centre ^-, 4h, radius 4 (b) Centre ^, -4h, radius (c) Centre ^-, 4h, radius (d) Centre ^, -4h, radius 4 86 In the circle, O is the centre Evaluate x (a) x = 4c (b) x = 68c (c) x = 84c (d) x = 96c O 84 87 Find the Cartesian equation for x = 8t, y = 4t (a) x = 3y (b) x = 4y (c) x = 6y (d) x 8y = 88 The equation of the normal to the curve x = 0y at the point (0 p, 5 p ) is (a) x + py = 5p 3 + 0p (b) x - py = 5p 3-0p (c) px + y = 5p (d) px - y + 5p = 0 x

660 Maths In Focus Mathematics Extension Preliminary Course 89 AB is a tangent to the circle Which statement is true (there may be more than one answer)? A D 90 For the quadratic function y = ax + bx + c to be positive definite (a) a 0, 0 (b) a 0, 0 (c) a 0, 0 (d) a 0, 0 C B AB BD (a) = BC AB AB CD (b) = BC AB (c) AB = BC $ CD (d) AB = BC $ BD